Introduction to Symplectic Geometry [Hardcover ed.] 9811339864, 9789811339868

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Jean-Louis Koszul Yi Ming Zou

Introduction to Symplectic Geometry

Introduction to Symplectic Geometry

Jean-Louis Koszul Yi Ming Zou •

Introduction to Symplectic Geometry

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Jean-Louis Koszul Institut Fourier Université Grenoble Alpes Gières, Grenoble, Isère, France

Yi Ming Zou Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI, USA

ISBN 978-981-13-3986-8 ISBN 978-981-13-3987-5 https://doi.org/10.1007/978-981-13-3987-5

(eBook)

Jointly published with Science Press, Beijing, China The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press Library of Congress Control Number: 2018965909 Mathematics Subject Classification (2010): 53Dxx © Springer Nature Singapore Pte Ltd. and Science Press 2019 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword 1. About This Book

En effet, Barbaresco m’a dit qu’il était curieux de voir le texte qui a été publié d’après un cours sur la géométrie symplectique que j’ai donné à Tianjin, il y a bien longtemps. Je n’ai rien pu lui procurer d’autre que le petit fascicule rédigé en chinois par un assistant de Nankai. Je ne sais pas ce qu’il vaut. De toutes façons, il n’y avait pas beaucoup de choses originales dans ce que j’ai raconté. Indeed, Barbaresco told me that he was curious to see the text that was published according to a course on symplectic geometry that I gave in Tianjin, long time ago. I could not give anything else but the little notebook written in Chinese by a Nankai assistant. I do not know whether it was worthful. In any case, there were no too many original things in what I was speaking about. J.-L. Koszul, 02/02/2017. Above is an excerpt from a message sent to me by Jean-Louis Koszul in February 2017. Earlier, in January 2017, I had written to Koszul about the lectures on Symplectic Geometry he delivered in China. I informed him that the notes of those lectures were missing in my private documentation. I also informed him that Fréderic Barbaresco was interested in these notes. This book is more than an elementary introduction to symplectic structures and their geometry. It highlights the unifying nature of symplectic structures. Often, readings of Koszul works are walks through Algebra, Homological Algebra, Geometry, Differential Geometry, Topology, and Differential Topology. This adage is highlighted by Foreword 3. The lectures in this book were delivered while new developments in Symplectic Geometry were occurring, with the works of A. Weinstein, B. Kostant, V. Guillemin, S. Stenberg, M. Atiyah, R. Bott, and many others. Koszul also emphasizes fruitful exchanges of results and techniques between Symplectic Geometry and other areas. For example, in Chap. 4, the notion of momentum map leads to rich relationships between symplectic structures, homological algebra, affine representations of Lie groups and Lie algebras, and homogeneous spaces. The author introduces many additional structures in manifolds endowed with symplectic structures. Major instances are Lagrangian

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submanifolds, complex symplectic structures, and Kaehler forms. Symplectic structures appear as a unifying framework for several notions worth studying. Koszul’s Geometry and its applications to bounded domains strongly impact many areas which are currently the subject of active and exciting research, such as the Geometric Science of Information and the Topology of Information. According to the Stefan–Sussmann theorem, Poisson geometry is the Differential Topological side of symplectic geometry, while Lagrangian foliations are the differential topological side of locally flat geometry. The starting point of this book is the algebraic counterpart of those subjects. One could claim that quantum data are those data which are Z2 -graded. Under such a simplification, Chap. 6 is an introduction to quantum symplectic structures. I have often talked with Koszul about his stays in China and in India. He and the editors had planned to include a video with this book. However, on November 28, 2017, Koszul wrote to me : Depuis que nous sommes installés dans cette maison de retraite, je suis très mal en point et très affaibli. Dans l’état où je suis, il n’est évidemment pas question que je donne un interview, je regrette bien de vous décevoir. J.-L. Koszul, 28/11/2017. Since we have settled in this retirement house, I feel very badly and very weakened. In the situation that I am now, there is obviously no question that I give an interview, I regret to disappoint you. What is new? As I just mentioned, some of Koszul’s work deeply impacts current research in both the Topology of Information and the Geometric Science of Information as well as their applications in Physics. This relates to his work on the geometry of convex cones, on the affine representations of Lie groups and Lie algebras, and on deformations of locally flat manifolds. Koszul knew of the connection through Barbaresco’s paper Koszul information geometry and Souriau Lie group thermodynamics. He wrote to me : Pour ce qui est des représentations affines, je ne suis pas le premier à les avoir manipulé. Si j’ai bon souvenir, elles interviennent dans le travail des russes sur les domaines bornés. A part cela, cet article de Barbaresco contient bien de choses que j’aimerais comprendre. Je vais essayer de m’y mettre. As for the affine representations, I am not the first to have manipulated them. If I remember correctly, they intervene in the work of the Russians on the bounded domains. Apart from that, this Barbaresco article contains many things that I would like to understand. I will try myself to enter into the matter of the article. J.-L. Koszul. Thence, I undertook to convince him that the mathematical foundation of the Geometric Science of Information is the algebraic topology of Koszul Geometry, viz the cohomology theory of Koszul Geometry. Subsequently, Koszul’s interest in the Geometric Science of Information increased.

Foreword 1. About This Book

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Je suis sensible à l’honneur que me fait le comité d’organisation du congrès 2013 en m’invitant à la conférence de Shima et je vous demande de bien vouloir lui transmettre mes remerciements. J’aimerais aussi pouvoir vous dire de transmettre mon acception. I am sensible to the honor that the organizing committee of the 2013 congress gave me inviting me to the Shima conference and I ask you to forward kindly my thanks. I would also like to tell you to convey my gratitude. J.-L. Koszul, 20/12/2012. On August 29, 2013, H. Shima delivered a keynote conference on Hessian Geometry, whose founder is Jean-Louis Koszul. Je ne regrette pas d’avoir été à Paris le 29 Août, en plus de ces retrouvailles avec Shima, j’ai observé avec intérêt ce colloque GSI dont le contenu et les objectifs étaient pour moi assez mystérieux. J’ai aussi regardé avec curiosité le volume de Lectures Notes publié à l’occasion de cette rencontre et cela m’a bien aidé à comprendre ce que l’on visait. A propos de ce volume, réussir à le sortir dans les délais est une prouesse que j’admire beaucoup. Je crois bien n’avoir jamais vu cela. Encore une fois, merci de m’avoir signalé cette rencontre et de m’avoir encouragé à faire le déplacement. I do not regret having been in Paris on August 29, in addition to this reunion of Shima, I watched with interest this GSI conference whose content and objectives were rather mysterious for me. I also looked with curiosity at the volume of Lecture Notes published on the occasion of this meeting and it helped me to understand what we sought. Concerning this volume, succeeding to release on time is a feat that I admire a lot. I think I have never seen things like that. Once again, thank you for informing me about this meeting and to have encouraged me to make the trip. J.-L. Koszul. Montpellier, France

Michel Nguiffo Boyom Emeritus Professor IMAG: Alexander Grothendieck Research Institute University of Montpellier

Foreword 2. Koszul Contemporaneous Lecture: Elementary Structures of Information Geometry and Geometric Heat Theory

La Physique mathématique, en incorporant à sa base la notion de groupe, marque la suprématie rationnelle… Chaque géométrie—et sans doute plus généralement chaque organisation mathématique de l’expérience—est caractérisée par un groupe spécial de transformations…. Le groupe apporte la preuve d’une mathématique fermée sur elle-même. Sa découverte clôt l’ère des conventions, plus ou moins indépendantes, plus ou moins cohérentes Gaston Bachelard, Le nouvel esprit scientifique, 1934

Jean-Louis Koszul’s Life: The Spirit of Geometry and the Spirit of Finesse of an “Esprit raffiné” Jean-Louis André Stanislas Koszul, born in Strasbourg in 1921, was the fourth child of a family of four (with three older sisters, Marie Andrée, Antoinette, and Jeanne) of André Koszul (born in Roubaix on November 19, 1878, Professor at Strasbourg University), and Marie Fontaine (born in Lyon on June 19, 1887), who was a friend of Henri Cartan’s mother. Henri Cartan writes on this friendship “My mother, in her youth, had been a close friend of the one who was to become Jean-Louis Koszul’s mother” [4]. His paternal grandparents were Julien Stanislas Koszul and Hélène Ludivine Rosalie Marie Salomé. He attended high school in Fustel-de-Coulanges in Strasbourg and the Faculty of Science in Strasbourg and in Paris. He entered ENS Ulm in the class of 1940 and defended his thesis with Henri Cartan. Henri Cartan noted “This promotion included other mathematicians like Belgodère and Godement, and also physicists and some chemists, like Marc Julia and Raimond Castaing” [4] (Fig. 1). On July 17, 1948, Jean-Louis Koszul married Denise Reyss-Brion, who became a student at ENS Sèvre, in 1941. They had three children, Michel (married to Christine Duchemin), Anne (wife of Stanislas Crouzier), and Bertrand. Koszul then taught in Strasbourg and was appointed as Associate Professor at the University of Strasbourg in 1949, and his colleagues including René Thom, Marcel Berger, and Bernard Malgrange. He was promoted to Professor in 1956 and became a member of second generation of Bourbaki, with Jacques Dixmier, Roger Godement, Samuel

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Fig. 1 ENS Ulm students 1940

Eilenberg, Pierre Samuel, Jean-Pierre Serre, and Laurent Schwartz. Henri Cartan remarked in [4] “In the vehement discussions within Bourbaki, Koszul was not one of those who spoke loudly; but we learned to listen to him because we knew that if he opened his mouth he had something to say” (Fig. 2). About Koszul’s period at Strasbourg University, Pierre Cartier [43] said “When I arrived in Strasbourg, Koszul was returning from a year spent at the Institute for

Fig. 2 Jean-Louis Koszul at the Bourbaki seminar 1951

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Fig. 3 Jean-Louis Koszul at the differential topology colloquium, Strasbourg 1953; second row from the bottom, Koszul is the second person from the left before André Weil. We can also see in the picture Chern, de Rham, Eckmann, Ehresmann, Godeaux, Hopf, Lichnerowicz, Malgrange, Milnor, Reeb, Schwartz, Süss, Thom, and Libermann

Advanced Studies in Princeton, and he was, after the departure of Ehresman and Lichnerowicz to Paris, the paternal figure of the Department of Mathematics (despite his young age). I am not sure of his intimate convictions, but he represented for me a typical figure of this Alsatian Protestantism, which I frequented at the time. He shared the seriousness, the honesty, the common sense and the balance. In particular, he knew how to resist the academic attraction of Paris. He left us after two years to go to Grenoble, in a maneuver uncommon at the time, exchanging of positions with Georges Reeb.” In Strasbourg, he supervised Edith Kosmanek Ph.D. [44], a graduate from Louis Pasteur University (Fig. 3). Koszul became Senior Lecturer at the University of Grenoble in 1963, and then an Honorary Professor at Joseph Fourier University [6] and integrated into the Fourier Institute led by Claude Chabauty. During this period, as recalled by Bernard Malgrange [42], Koszul held a seminar on “algebra and geometry” with his three students Jacques Vey [45–46], Domingo Dominique Luna [47], and Jacques Helmstetter [48–50]. In Grenoble, Koszul practiced mountaineering and was a member of the French Alpine Club. He was awarded the Jaffré Prize in 1975 and was elected correspondent at the Academy of Sciences on January 28, 1980. The following year, he was elected to the Academy of São Paulo. Koszul was one of the CIRM conference center founders at Luminy. Jean-Louis Koszul died on January 12, 2018, at the age of 97.

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As early as 1947, Koszul published three articles in the Comptes Rendus of the Academy of Sciences, on the Betti number of a simple compact Lie group, on cohomology rings, generalizing ideas of Jean Leray, and finally on the homology of homogeneous spaces. Koszul’s thesis, defended on June 10, 1949 under the direction of Henri Cartan, dealt with the homology and cohomology of Lie algebras. The jury was composed of Professors Arnaud Denjoy (president), Henri Cartan, Paul Dubreil, and Jean Leray. Under the title “Works of Koszul I, II and III”, Henri Cartan reported Koszul’s Ph.D. results to the Bourbaki seminar (Fig. 4).

Fig. 4 Cover page of Koszul’s Ph.D. report, defended on June 10, 1949 with a Jury composed of Profs. Arnaud Denjoy, Henri Cartan, Paul Dubreil, and Jean Leray

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In 1987, an International Symposium on Geometry was held in Grenoble in honor of Koszul, whose proceedings were published in “les Annales de l’Institut Fourier”, Volume 37, No. 4. This conference began with a presentation by Henri Cartan, who remembered the mention given to Koszul for his aggregation [4]: “Distinguished Spirit; he is successful in his problems. Should beware, orally, of overly systematic trends. A little less subtle complications, baroque ideas, a little more common sense and balance would be desirable.” About his supervision of Koszul’s Ph.D., Henri Cartan wrote “Why did he turn to me as his ‘guide’ (so to speak)? Is it because he found inspiration in Elie Cartan’s work on the topology of Lie groups? Perhaps he was surprised to note that mathematical knowledge is not necessarily transmitted by descent. In any case, he helped me to better know what my father had brought to the theory” [4]. On the work of Koszul algebraic work, Henri Cartan notes “Koszul was the first to give a precise algebraic formalization of the situation studied by Leray in his 1946 publication, which became the theory of spectral sequences. It took a good deal of insight to unravel what lay behind Leray’s study. In this respect, Koszul’s Note in the July 1947 CRAS is of historical significance” [4]. From June 26 to July 2, 1947, CNRS hosted an International conference in Paris, on “Algebraic Topology”. This was the first postwar international diffusion of Leray’s ideas. Koszul writes about this lecture “I can still see Leray putting down his chalk at the end of his talk by saying (modestly?) that he definitely did not understand anything about Algebraic Topology.” In writing his lectures at the Collège de France, Leray adopted the algebraic presentation of the spectral sequence elaborated by Koszul. As early as 1950, Jean-Pierre Serre used the term “Leray–Koszul sequence”. Speaking of Leray, Koszul wrote “around 1955, I remember asking him what had put him on the path of what he called the homology ring of a representation in his Notes to the CRAS of 1946. His answer was Künneth’s theorem; I could not find out more.” Sheaf theory, introduced by Jean Leray, followed in 1947, at the same time as spectral sequences. In 1950, Koszul published an important 62-page book entitled Homology and Cohomology of Lie Algebras in which he studied the links between the homology and cohomology (with real coefficients) of a compact connected Lie group and purely algebraic problems of Lie algebra. Koszul then gave a lecture in São Paulo on the topic “sheaves and cohomology”. The superb lecture notes were published in 1957 and dealt with Čech cohomology with coefficients in a sheaf. In the autumn of 1958, he organized a second series of seminars in São Paulo, on symmetric spaces [8]. R. Bott commented on these seminars “very pleasant. The pace is fast, and the considerable material is covered elegantly. In addition to the more or less standard theorems on symmetric spaces, the author discusses the geometry of geodesics, Bergmann’s metrics, and finally studies the bounded domains with many details.” In the mid-1960s, Koszul taught at the Tata Institute in Bombay on transformation groups [12] and on fiber bundles and differential geometry. The second lecture dealt with the theory of connections and the lecture notes were published in 1965.

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Fig. 5 Henri Cartan lecture on homogeneous domains, Freiburg, March 13, 1987

In 1994, in [3], a comment by Koszul explains the problems he was preoccupied with when he invented what is now called the “Koszul complex”. This was introduced to define a cohomology theory for Lie algebras and proved to be useful in general homological algebra.

Fig. 6 (On the left) Yann Ollivier and Jean-Louis Koszul in the GSI’13 conference group photo at Hôtel de Vendôme, Ecole des Mines de Paris, (on the right) Jean-Louis Koszul at the CIRM anniversary in Luminy

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The Genesis of this Translation of Koszul’s Book “Introduction to Symplectic Geometry” The genesis of the translation of this book dates back to 2013. We got in contact with Professor Koszul in connection with his work on homogeneous bounded domains and their links with Information Geometry. Professor Michel Boyom successfully convinced Professor Koszul to accept our invitation to attend the first GSI “Geometric Science of Information” conference in August 2013 at Ecole des Mines ParisTech in Paris, and more especially to attend Hirohiko Shima talk, given in his honor, on the topic “Geometry of Hessian Structures” (Fig. 7). We were more particularly motivated by Koszul’s work developed in his paper Domaines bornées homogènes et orbites de groupes de transformations affines [9] of 1961, written by Koszul at the Institute for Advanced Studies in Princeton during a stay funded by the National Science Foundation. Koszul proved in this paper that on a complex homogeneous space, an invariant volume defines with the complex structure the canonical invariant Hermitian form introduced in [7]. It is in this article that Koszul uses the affine representation of Lie groups and Lie algebras. By studying the open orbits of the affine representations, he introduced an affine representation of G, writtenðf; qÞ, and the following equation setting f of the

Fig. 7 Jean-Louis Koszul and Hirihiko Shima at the GSI’13 conference in Ecole des Mines ParisTech in Paris, October 2013

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linear representation of the Lie algebra g of G, defined by f, and q the restriction to g of the differential of q (f and q are, respectively, differential of f and q): f ðXÞqðYÞ  f ðYÞqðXÞ ¼ qð½X; Y Þ 8X; Y 2 g with f : g ! glðEÞ and q : g 7! E

ð1Þ

If the homogeneous space is holomorphically isomorphic to a bounded domain of a space Cn, this Hermitian form is positive definite because it coincides with the Bergmann metric of the domain. Koszul demonstrates in this article the converse of this proposition for a class of complex homogeneous spaces. This class consists of some open orbits of complex affine transformation groups and contains all homogeneous bounded domains. Koszul again addresses the problem of knowing if a complex homogeneous space, with a canonical Hermitian form, that is positive definite, is isomorphic to a bounded domain, but via the study of the invariant bilinear form defined on a real homogeneous space by an invariant volume and an invariant flat connection. Koszul demonstrates that if this bilinear form is positive definite, then the homogeneous space with its flat connection is isomorphic to a convex open domain containing no straight line in a real vector space and extends it to the initial problem for the complex homogeneous spaces obtained in defining a complex structure in the variety of vectors of a real homogeneous space provided with an invariant flat connection. Koszul’s use of the affine representation of Lie groups and Lie algebras drew our attention, especially the similarities of his approach with that used by Jean-Marie Souriau in geometric mechanics in the framework of homogeneous symplectic manifolds. We then initiated explorations to make the bridge between Koszul and Souriau’s works. We finally discovered that, in 1986, Koszul published this book “Introduction to symplectic geometry” following a Koszul lectured in English course in China. We also observed that this book analyzes in detail and develops Souriau’s works on homogeneous symplectic manifolds in Geometric Mechanics by the means of the affine representation of Lie algebras and Lie groups. Chuan Yu Ma writes in a review of this book in Chinese that “This work coincided with developments in the field of analytical mechanics. Many new ideas have also been derived using a wide variety of notions of modern algebra, differential geometry, Lie groups, functional analysis, differentiable manifolds, and representation theory. [Koszul’s book] emphasizes the differential-geometric and topological properties of symplectic manifolds. It gives a modern treatment of the subject that is useful for beginners as well as for experts.” We then started an epistolary correspondence with Professor Koszul on Souriau’s works and on the genesis of this book. In May 2015, questioning Koszul on Souriau’s work on Geometric Mechanics and on Lie Group Thermodynamics, Koszul answered me “[A l’époque où Souriau développait sa théorie, l’establishment avait tendance à ne pas y voir des avancées importantes. Je l’ai entendu exposer ses idées sur la thermodynamique mais je n’ai pas du tout réalisé à l’époque que la géométrie hessienne était en jeu.] At the time when Souriau was

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developing his theory, the establishment tended not to see significant progress. I heard him explaining his ideas on thermodynamics but I did not realize at the time that Hessian geometry was at stake.” In September 2016, I asked him about the origins of Lie Group and Lie Algebra Affine representations. Koszul informed me about the seminal work of Elie Cartan, who gave him lectures at ENS Ulm on homogeneous bounded domains, the germinal root of his Ph.D.: “[Il y a là bien des choses que je voudrais comprendre (trop peut-être !), ne serait-ce que la relation entre ce que j’ai fait et les travaux de Souriau. Détecter l’origine d’une notion ou la première apparition d’un résultat est souvent difficile. Je ne suis certainement pas le premier à avoir utilisé des représentations affines de groupes ou d’algèbres de Lie. On peut effectivement imaginer que cela se trouve chez Elie Cartan, mais je ne puis rien dire de précis. A propos d’Elie Cartan: je n’ai pas été son élève. C’est Henri Cartan qui a été mon maître pendant mes années de thèse. En 1941 ou 42 j’ai entendu une brève série de conférences données par Elie à l’Ecole Normale et ce sont des travaux d’Elie qui ont été le point de départ de mon travail de thèse.] There are many things that I would like to understand (too much perhaps!), if only the relationship between what I did and the work of Souriau. Detecting the origin of a notion or the first appearance of a result is often difficult. I am certainly not the first to have used affine representations of groups or Lie algebras. We can imagine that it is Elie Cartan, but I cannot say anything specific. About Elie Cartan: I was not his student. It was Henri Cartan who was my master during my thesis years. In 1941 or 42, I heard a brief series of lectures given by Elie at the Ecole Normale and it was Elie’s work that was the starting point of my thesis work.” After discovering the existence of Koszul’s book, written in Chinese, based on a course “Introduction to Symplectic Geometry”, given at Nakai, in which he made reference to Souriau’s book and developed his main tools, we started to discuss its content. In January 2017, Koszul wrote me with his usual humility “[Ce petit fascicule d’introduction à la géométrie symplectique a été rédigé par un assistant de Nankai qui avait suivi mon cours. Il n’y a pas eu de version initiale en français.] This small introductory booklet on symplectic geometry was written by a Nankai assistant who had taken my course. There was no initial version in French.” I asked him if he had a personal archive of this course. He answered “[Je n’ai pas conservé de notes préparatoires à ce cours. Dites-moi à quelle adresse je puis vous envoyer un exemplaire du texte chinois.] I have not kept any preparatory notes for this course. Tell me where I can send you a copy of the Chinese text.” Professor Koszul then sent me his last copy of this book in Chinese (Fig. 8). I was not able to read the Chinese text, but I have observed in Chap. 4 “Symplectic G-spaces”, and in Chap. 5 “Poisson Manifolds”, that their equations include new original developments of Souriau’s work on moment maps and affine representation of Lie Groups and Lie Algebra. More particularly, Koszul considered in great detail “non-equivariance” case of co-adjoint action on moment map, where I recovered Souriau’s theorem. Koszul shows that when (M; x) is a connected Hamiltonian G-space and l a moment of the action of G, there exists an affine action of G on g (dual Lie algebra), whose linear part is the coadjoint action, for which the moment l is equivariant. Koszul developed Souriau’s idea that this

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Fig. 8 Koszul’s original “little green” book “Introduction to Symplectic Geometry” in Chinese from his lecture at Nankai

affine action is obtained by modifying the coadjoint action by means of a closed cochain (called a cocycle by Souriau), and that (M; x) is a G-Poisson space, making reference to Souriau’s book for more details. About collaboration between Koszul and Souriau and another potential lecture on Symplectic Geometry in Toulouse, Koszul informed me in February 2017 that: “[J’ai plus d’une fois rencontré Souriau lors de colloques, mais nous n’avons jamais collaboré. Pour ce qui est de cette allusion à un “cours” donné à Toulouse, il y erreur. J’y ai peut être fait un exposé en 81, mais rien d’autre.] I have met Souriau more than once at conferences, but we have never collaborated. As for this allusion to a “course” given in Toulouse, there is an error. I could have made a presentation in 81, but nothing else.” Koszul admitted that he had no direct collaboration with Souriau: “[Je ne crois pas avoir jamais parlé de ses travaux avec Souriau. Du reste j’avoue ne pas en avoir bien mesuré l’importance à l’époque] I do not think I ever talked to Souriau about his work. For the rest, I admit that I did not have a good idea of its importance at the time.”

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Considering the importance of this book for different communities, I tried to find an editor for its translation into English. By chance, I met Catriona Byrne from SPRINGER, when I gave a talk at IHES, invited by Pierre Cartier, on applications of Koszul and Souriau’s work to Radar (concluded by beautiful pieces of music written by Julien Koszul, Jean-Louis’ grandfather, performed by Bertrand Maury). With the perseverance of Michel Boyom, we convinced Professor Koszul to translate this book, proposing to contextualize it with regard to contemporary research trends in Geometric Mechanics, Lie Group Thermodynamics, and the Geometric Science of Information. Professors Marle and Boyom agreed to check the translation and help me to write the forewords.

Koszul’s Book: A Joint Source of Geometric Heat Theory and Information Geometry In the Foreword of this book, Koszul writes “The development of analytical mechanics provided the basic concepts of symplectic structures. The term symplectic structure is due largely to analytical mechanics. But in this book, the applications of symplectic structure theory to mechanics is not discussed in any detail.” Koszul considers purely algebraic and geometric developments of Geometric/Analytic Mechanics developed during the 60s, in particular, Jean-Marie Souriau’s works detailed in Chaps. 4 and 5. The originality of this book lies in the fact that Koszul develops new points of view, and demonstrations not initially considered by Souriau and later developed by the Geometric Mechanics community. Jean-Marie Souriau was the Creator of a new discipline called “Mécanique Géométrique (Geometric Mechanics)”. Souriau observed that the collection of motions of a dynamical system is a manifold with an antisymmetric flat tensor that is a symplectic form where the structure contains all the pertinent information on the state of the system (positions, velocities, forces, etc.). Souriau said: “[Ce que Lagrange a vu, que n’a pas vu Laplace, c’était la structure symplectique] What Lagrange saw, that Laplace didn’t see, was the symplectic structure.” Using the symmetries of a symplectic manifold, Souriau introduced a mapping which he called the “moment map”, which takes its values in a space attached to the group of symmetries (in the dual space of its Lie algebra). The moment map allows one to build conserved quantities for the group action, generalizing the classical notions of linear and angular momentum. Souriau associated to this moment maps the notion of symplectic cohomology, linked to the fact that such a moment is defined up to an additive constant that brings into play an algebraic mechanism (called cohomology). Souriau proved that the moment map is a constant of the motion and provided a geometric generalization of Emmy Noether’s invariant theorem (invariants of E. Noether’s theorem are the components of the moment map; but where Noether’s approach is purely algebraic, Souriau’s approach gives geometric roots

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and meanings to these invariants). Souriau has defined in a geometrical way the Noetherian symmetries using the Lagrange–Souriau 2-form with the moment map. Influenced by François Gallissot (Souriau and Galissot both attended ICM’54 in Moscow. Did they discuss this point?), Souriau introduced in Mechanics the Lagrange 2-form, recovering Lagrange’s seminal ideas. Motivated by the need to give a coordinate-independent formulation of the variational principles, inspired by Henri Poincaré and Elie Cartan who introduced a differential 1-form instead of the Lagrangian, Souriau introduced the Lagrange 2-form as the exterior differential of the Poincaré–Cartan 1-form, and obtained the phase space as a symplectic manifold. Souriau proposed to consider this Lagrange 2-form as the fundamental structure for Lagrangian system and not the classical Lagrangian function or the Poincaré–Cartan 1-form. This 2-form is called the Lagrange-Souriau 2-form and is the exterior derivative of the Lepage form (the Poincaré–Cartan form is a first-order Lepage form). This structure is developed in Koszul’s book, where the authors show that when (M; x) is an exact symplectic manifold (when there exists a 1-form a on M such that x ¼ da), and that a symplectic action leaves not only x, but a invariant, this action is strongly Hamiltonian ((M; x) is a g-Poisson space). Koszul shows that a symplectic action of a Lie algebra g on an exact symplectic manifold (M; x ¼ da) that leaves invariant not only x, but also a, is strongly Hamiltonian. In this book, in Chap. 4, Koszul defines symplectic G-space as a symplectic manifold (M; x) on which a Lie group G acts by a symplectic action (an action which leaves unchanged the symplectic form x). Koszul then introduces and develops properties of the moment map l (Souriau’s invention) of a Hamiltonian action of the Lie algebra g . He also defines the Souriau 2-cocycle, considering that the difference of two moments of the same Hamiltonian action is a locally constant function on M, showing that when l is a moment map, for every pair (a, b) of elements of g , the function cl ða; bÞ ¼ fhl; ai; hl; big  hl; fa; bgi is locally constant on M, defining an antisymmetric bilinear map of g  g in H0(M; R) which satisfies Jacobi’s identity. This is the 2-cocycle introduced by Souriau in Geometric Mechanics, which plays a fundamental role in Souriau’s Lie Groups Thermodynamics where it is used to define an extension of the Fisher Metric from Information Geometry (what we will call the Souriau–Fisher metric in the following). To highlight the importance of Koszul’s book, we will illustrate its links of the detailed tools, including demonstrations or original Koszul extensions, with Souriau’s Lie Group Thermodynamics, whose applications range from statistical physics to machine learning in Artificial Intelligence. Koszul originally developed Souriau’s model, in the case of non-equivariance, of the action of the group G on the moment map. As explained in [51] by Thomas Delzant at the 2010 CIRM conference “Action Hamiltoniennes: invariants et classification”, organized with Michel Brion: “The definition of the moment map is due to Jean-Marie Souriau…. In the book of Souriau we find a proof of the proposition: the map J is equivariant for an affine action of G on g whose linear part is Ad  …. In Souriau’s book, we can also find a study of the non-equivariant case and its applications to classical and quantum mechanics. In the case of the Galileo group operating in the phase

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxi

space of space-time, obstruction to equivariance (a class of cohomology) is interpreted as the inert mass of the object under study. We can uniquely define the moment map up to an additive constant of integration that can always be chosen to make the moment map equivariant (a moment map is G-equivariant when G acts on g via the coadjoint action) if the group is compact or semi-simple. In 1969, Souriau considered the non-equivariant case where the coadjoint action must be modified to make the map equivariant under a 1-cocycle on the group with values in the dual Lie algebra g .” The concept and seminal idea of the moment map appeared in the second volume of Sophus Lie’s book, published in 1890, developed for homogeneous canonical transformations. Professor Marsden summarized the development of this concept by Jean-Marie Souriau and Bertram Kostant based on their two testimonials: “In Kostant’s 1965 Phillips lectures at Haverford, and in the 1965 U.S.– Japan Seminar, Kostant introduced the momentum map to generalize a theorem of Wang and thereby classified all homogeneous symplectic manifolds; this is called today ‘Kostant’s coadjoint orbit covering theorem’… . Souriau introduced the momentum map in his 1965 Marseille lecture notes and put it in print in 1966. The momentum map finally got its formal definition and its name, based on its physical interpretation, by Souriau in 1967. Souriau also studied its properties of equivariance, and formulated the coadjoint orbit theorem. The momentum map appeared as a key tool in Kostant’s quantization lectures in 1970 [52], and Souriau discussed it at length in 1970 in his book [31]. Kostant and Souriau realized its importance for linear representations, a fact apparently not foreseen by Lie.” Souriau’s book is referred to be published by Dunod in 1970, but Souriau manuscript and book was available as soon as 1969. Incidentally, Jean-Louis Koszul knew Souriau and Kostant’s work very well, and as early as 1958, Koszul made a survey of Kostant’s first work at the Bourbaki seminar [53]. In 1969, Souriau also introduced the concept of a coadjoint action of a group on its moment space in the framework of Thermodynamics, based on the orbit method, which allows one to define physical observables like energy, heat, and momentum or moment as pure geometrical objects. In the first step to establishing a new foundations of thermodynamics, Souriau defined a Gibbs canonical ensemble on a symplectic manifold M for a Lie group action on M. In classical statistical mechanics, a state is given by the solution of Liouville’s equation on the phase space, the partition function. As symplectic manifolds have a completely continuous measure, invariant under diffeomorphisms (the Liouville measure k), Souriau proved that when statistical states are Gibbs states (as generalized by Souriau), they are the product of the Liouville measure by the scalar function given by the generalized partition function eUðbÞhb;UðnÞi defined by the energy U (defined in the dual of the Lie algebra of this dynamical group) and the geometric temperature b, where U is a normalizing constant such that the mass of probability is equal to 1, R UðbÞ ¼  log ehb;UðnÞi dk. Souriau then generalizes the Gibbs equilibrium state M

to all symplectic manifolds that have a dynamical group. Souriau has observed that theory to dynamical groups in Physics (Galileo or Poincaré groups), the symmetry

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Foreword 2. Koszul Contemporaneous Lecture: Elementary …

will be broken. For each temperature b, element of the Lie algebra g, Souriau ~ b , equal to the sum of the cocycle H ~ and the heat coboundary introduced a tensor H (with [.,.] Lie bracket): ~ b ðZ1 ; Z2 Þ ¼ H ~ ðZ1 ; Z2 Þ þ hQ; adZ ðZ2 Þ i H 1

ð2Þ

~ b has the following properties: HðX; ~ This tensor H YÞ ¼ hHðXÞ; Y i, where the map H is the symplectic one-cocycle of the Lie algebra g with values in g , with ~ ðX; Y Þ is HðXÞ ¼ Te hðXðeÞÞ, where h is the one-cocycle of the Lie group G. H ~ constant on M and the map HðX; Y Þ : g  g ! < is a skew-symmetric bilinear form, and is called the symplectic two-cocycle of the Lie algebra g associated to the moment map J, with the following properties: ~ HðX; YÞ ¼ J½X;Y   fJX ; JY g with J the moment map

ð3Þ

~ ð½X; Y ; Z Þ þ H ~ ð½Y; Z ; X Þ þ H ~ ð½Z; X ; Y Þ ¼ 0 H

ð4Þ

where JX linear map from g to a differential function on M : g ! C1 ðM; RÞ; X ! JX and the associated differentiable map J, called the moment(um) map, is defined by J : M ! g ; x 7! JðxÞ such that JX ðxÞ ¼ hJðxÞ; X i; X 2 g

ð5Þ

The geometric temperature, an element of the algebra g, is in the kernel of the ~ b : b 2 Ker H ~ b such that tensor H ~ b ðb; bÞ ¼ 0; 8b 2 g H

ð6Þ

~ b ðZ1 ; ½b; Z2 Þ, defined on The following symmetric tensor gb ð½b; Z1 ; ½b; Z2 Þ ¼ H all values of adb ð:Þ ¼ ½b; :, is positive definite and defines an extension of the classical Fisher metric in Information Geometry (as the Hessian of the logarithm of the partition function):   ~ b ðZ1 ; Z2 Þ; 8Z1 2 g; 8Z2 2 Im adb ð:Þ gb ð½b; Z1 ; Z2 Þ ¼ H

ð7Þ

  gb ðZ1 ; Z2 Þ  0; 8Z1 ; Z2 2 Im adb ð:Þ

ð8Þ

where

These equations are universal, because they are not dependent on the symplectic manifold but only on the dynamical group G, the symplectic two-cocycle H, the temperature b and the heat Q. Souriau called this “Lie group thermodynamics”.

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxiii

This antisymmetric bilinear map (7) is precisely the mathematical object introduced in Chap.4 of Koszul’s book by: cl ða; bÞ ¼ fhl; ai; hl; big  hl; fa; bgi. For any moment map l, Koszul defines the skew-symmetric bilinear form cl ða,bÞ on a Lie algebra by:   cl ða,bÞ ¼ dhl ðaÞ; b ; a; b 2 g

ð9Þ

Koszul observes that if we use: hl ðstÞ ¼ lðstxÞ  Adst lðxÞ ¼ hl ðsÞ þ Ads lðtxÞ  Ads Adt lðxÞ ¼ hl ðsÞ þ Ads hl ðtÞ

ð10Þ

by developing dlðaxÞ ¼ t ada lðxÞ þ dhl ðaÞ; x 2 M; a 2 g, he obtains:   hdlðaxÞ; bi ¼ hlðxÞ; ½a; bi þ dhl ðaÞ; b ¼ fhl; ai; hl; bigðxÞ; x 2 M; a; b 2 g

ð11Þ

  cl ða,bÞ ¼ fhl; ai; hl; big  hl; ½a; bi ¼ dhl ðaÞ; b ; a; b 2 g

ð12Þ

We then have:

and the property: cl ð½a,b; cÞ þ cl ð½b; c; aÞ þ cl ð½c; a; bÞ ¼ 0; a; b; c 2 g

ð13Þ

Koszul concludes by observing that if the moment map is transformed as l0 ¼ l þ / then we have: cl0 ða; bÞ ¼ cl ða; bÞ  h/; ½a; bi

ð14Þ

  Finally, using cl ða,bÞ ¼ fhl; ai; hl; big  hl; ½a; bi ¼ dhl ðaÞ; b ; a; b 2 g, Koszul highlights the property that:   fl ðaÞ; l ðbÞg ¼ fhl; ai; hl; big ¼ l ½a; b þ cl ða; bÞ ¼ l fa; bgcl

ð15Þ

In Chap. 4, Koszul introduces the equivariance of the moment map l. Based on the definitions of the adjoint and coadjoint representations of a Lie group or a Lie algebra, Koszul proves that when (M; x) is a connected Hamiltonian G-space and l : M ! g a moment of the action of G, there exists an affine action of G on g , whose linear part is the coadjoint action, for which the moment l is equivariant. This affine action is obtained by modifying the coadjoint action by means of a cocycle. This notion is also developed in Chap. 5 for Poisson manifolds.

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxiv

Defining the classical operation Ads a ¼ sas1 ; s 2 G; a 2 g, ada b ¼ ½a; b; a 2 g; b 2 g and Ads ¼ t Ads1 ; s 2 G with classical properties:  t Adexp a ¼ expðada Þ; a 2 g or Adexp a ¼ exp ðada Þ; a 2 g

ð16Þ

Koszul considers: x 7! sx; x 2 M; l : M ! g

ð17Þ

hdlðvÞ; ai ¼ xðax; vÞ

ð18Þ

from which he obtains:

Koszul then studies l  sM  Ads  l : M ! g and develops:     d Ads  l; a ¼ Ads dl; a ¼ hdl; Ads1 ai

ð19Þ

  hdlðvÞ; Ads1 ai ¼ x s1 asx; v ¼ xðasx; svÞ ¼ hdlðsvÞ; ai ¼ ðd hl  sM ; aiÞðvÞ

ð20Þ

  So d Ads  l; a ¼ d hl  sM ; ai then proving that   d l  sM  Ads  l; a ¼ 0

ð21Þ

Koszul considers the cocycle given by hl ðsÞ ¼ lðsxÞ  Ads lðxÞ; s 2 G and observes that: hl ðstÞ ¼ hl ðsÞ  Ads hl ðt) , s; t 2 G

ð22Þ

From this action of the group on the dual Lie algebra: G  g ! g ; ðs; nÞ 7! sn ¼ Ads n þ hl ðsÞ

ð23Þ

Koszul introduces the following properties: lðsxÞ ¼ slðxÞ ¼ Ads lðxÞ þ hl ðsÞ; 8s 2 G; x 2 M

ð24Þ

G  g ! g ; ðe; nÞ 7! en ¼ Ade n þ hl ðeÞ ¼ n þ lðxÞ  lðxÞ ¼ n

ð25Þ

ðs1 s2 Þn ¼ Ads1 s2 n þ hl ðs1 s2 Þ ¼ Ads1 Ads2 n þ hl ðs1 Þ þ Ads1 hl ðs2 Þ   ðs1 s2 Þn ¼ Ads1 Ads2 n þ hl ðs2 Þ þ hl ðs1 Þ ¼ s1 ðs2 nÞ; 8s1 ; s2 2 G; n 2 g

ð26Þ

Koszul’s study of the moment map l equivariance, and the existence of an affine action of G on g , whose linear part is the coadjoint action, for which the moment l is equivariant, is at the cornerstone of Souriau’s Theory of

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxv

Geometric Mechanics and Lie Group Thermodynamics. We illustrate its importance by giving Souriau’s theorem of Lie Group Thermodynamics: Theorem (Souriau’s Theorem of Lie Group Thermodynamics). Let X be the R largest open proper subset of g, the Lie algebra of G, such that ehb;UðnÞi d k and M R n:ehb;UðnÞi d k are convergent integrals, this set X is convex and is invariant M

under every transformation Adg ð:Þ. Then, the fundamental equations of Lie group thermodynamics are given by the action of the group: • • • •

Action of the Lie group on the Lie algebra: b ! Adg ðbÞ (27)   Characteristic function after Lie group action: U ! U  hðg1 Þ; b (28) Invariance of entropy with respect to the action of the Lie group: s ! s (29) Action of the Lie group on geometric heat: Q ! aðg; QÞ ¼ Adg ðQÞ þ hðgÞ (30)

Souriau’s equations of Lie group thermodynamics, related to the moment map l equivariance, and the existence of an affine action of G on g , whose linear part is the coadjoint action, for which the moment l is equivariant, are summarized in the following figures (Figs. 9 and 10). We finally observe that the Koszul antisymmetric bilinear map cl ða; bÞ ¼ fhl; ai; hl; big  hl; fa; bgi is equal to Souriau’s Riemannian metric, introduced by means of a symplectic cocycle. We have observed that this metric is a generalization of the Fisher metric from Information Geometry that we call the Souriau– Fisher metric, defined as a hessian of the partition function logarithm gb ¼  @@bU2 ¼ 2

@ 2 log wX @b2

as in classical information geometry. We can establish the equality of two

terms, between Souriau’s definition based on the Lie group cocycle H and parameterized by “geometric heat” Q (element of dual Lie algebra) and “geometric temperature” b (element of the Lie algebra) and the hessian of the characteristic function UðbÞ ¼  log wX ðbÞ with respect to the variable b:

Fig. 9 Broken symmetry on the geometric heat Q due to the adjoint action of the group on the temperature b as an element of the Lie algebra

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Foreword 2. Koszul Contemporaneous Lecture: Elementary …

Fig. 10 Sourau’s global scheme of Lie group thermodynamics

gb ð½b; Z1 ; ½b; Z2 Þ ¼ hHðZ1 Þ; ½b; Z2 i þ hQ; ½Z1 ; ½b; Z2 i ¼ we differentiate this relation  If  Q Adg ðbÞ ¼ Adg ðQÞ þ hðgÞ, we obtain:

of

@ 2 log wX @b2

Souriau’s

ð31Þ theorem

@Q ~ ðZ1 ; ½b; :Þ þ hQ; Ad:Z ð½b; :Þi ¼ H ~ b ðZ1 ; ½b; :Þ ð½Z1 ; b; :Þ ¼ H 1 @b 

@Q ~ ðZ1 ; ½b; Z2 Þ þ hQ; Ad:Z ð½b; Z2 Þi ¼ H ~ b ðZ1 ; ½b; Z2 Þ ð½Z1 ; b; Z2 Þ ¼ H 1 @b )

@Q ¼ gb ð½b; Z1 ; ½b; Z2 Þ @b

ð32Þ ð33Þ ð34Þ

UðbÞ The Fisher metric IðbÞ ¼  @ @b ¼  @Q 2 @b has been considered by Souriau as a 2

generalization of “heat capacity”. Souriau called it the “geometric capacity”.

Affine Representation of Lie Groups and Lie Algebras: Koszul and Souriau’s Pillars Based on Elie Cartan’s Seminal Work The affine representation of Lie groups/algebras used by Koszul in this book has been intensively studied by Souriau, who called the mechanics deduced from this model “affine mechanics”. We will explain notions such as moment map, equivariance of moment map, and cocycles through this notion of affine representation of Lie Groups and Lie Algebras. Previously, we presented Souriau’s work on the

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxvii

affine representation of a Lie group used to elaborate Lie Group Thermodynamics. Here, we will present Koszul’s original approach to the affine representation of Lie groups and Lie algebras in a finite-dimensional vector space, seen as special examples of actions. Since the work of Henri Poincaré and Elie Cartan, the theory of differential forms has become an essential instrument of modern differential geometry, used by Souriau to identify the space of motions as a symplectic manifold. However, as observed by Paulette Libermann, except for Henri Poincaré, who wrote shortly before his death a report on the work of Elie Cartan during his application to the Sorbonne University, the French mathematicians did not see the importance of Cartan’s breakthroughs. Souriau followed lectures of Elie Cartan in 1945. Elie Cartan’s second student was Jean-Louis Koszul. Koszul studied symmetric homogeneous spaces and defined relations between invariant flat affine connections to affine representations of Lie algebras, and characterized invariant Hessian metrics by affine representations of Lie algebras. Koszul provided a correspondence between symmetric homogeneous spaces with invariant Hessian structures by using affine representations of Lie algebras and proved that a simply connected symmetric homogeneous space with invariant Hessian structure is a direct product of a Euclidean space and a homogeneous self-dual regular convex cone. Let G be a connected Lie group and let G/K be a homogeneous space on which G acts effectively. Koszul gave a bijective correspondence between the set of G-invariant flat connections on G/K, and the set of a certain class of affine representations of the Lie algebra of G. Koszul’s main theorem is: let G/K be a homogeneous space of a connected Lie group G and let g and k be the Lie algebras of G and K, assuming that G/K is endowed with a G-invariant flat connection, then g admits an affine representation (f,q) on the vector space E. Conversely, suppose that G is simply connected and that g is endowed with an affine representation, then G/K admits a G-invariant flat connection. In the foregoing, the basic tool studied by Koszul is the affine representation of Lie algebras and Lie groups. To study these structures, Koszul introduced the following developments. Let X be a convex domain on Rn without any straight lines, and an associated convex cone VðXÞ ¼ fðkx; xÞ 2 Rn  R=x 2 X; k 2 R þ g, then there exists an affine embedding: ‘ : x 2 X 7!

 x 2 VðXÞ 1

ð35Þ

If we consider the group homomorphism g of Aðn; RÞ to GLðn þ 1; RÞ given by: 

fðsÞ qðsÞ s 2 Aðn; RÞ 7! 2 GLðn þ 1; RÞ 0 1

ð36Þ

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxviii

and the affine representation of the Lie algebra: 

f 0

q 0

ð37Þ

where Aðn; RÞ is the group of all affine representations of Rn , we have gðGðXÞÞ  GðVðXÞÞ and the pair ðg; ‘Þ comprising the homomorphism g : GðXÞ ! GðVðXÞÞ and the map ‘ : X ! VðXÞ is equivariant. Observing Koszul’s affine representations of Lie algebras and Lie groups, we have to consider a convex Lie group G and a real or complex vector space E of finite dimension, Koszul introduced an affine representation of G in E such that: E!E a 7! sa

8s 2 G

ð38Þ

is an affine representation. We set AðEÞ the set of all affine transformation of a real vector space E, a Lie group called affine representation group of E. The set GLðEÞ of all regular linear representation of E, a subgroup of AðEÞ. We define a linear representation of G in GLðEÞ: f: G ! GLðEÞ s 7! fðsÞa ¼ sa  so 8a 2 E

ð39Þ

and a map from G to E: q: G ! E s 7! qðsÞ ¼ so 8s 2 G

ð40Þ

then, we have 8s; t 2 G: fðsÞqðtÞ þ qðsÞ ¼ qðstÞ

ð41Þ

Since fðsÞqðtÞ þ qðsÞ ¼ sqðtÞ  so þ so ¼ sqðtÞ ¼ sto ¼ qðstÞ. Conversely, if a map q from G to E and a linear representation f from G to GLðEÞ satisfy the previous equation, then we can define an affine representation from G in E, written ðf; qÞ: Aff ðsÞ : a 7! sa ¼ fðsÞa þ qðsÞ; 8s 2 G; 8a 2 E

ð42Þ

The condition fðsÞqðtÞ þ qðsÞ ¼ qðstÞ is equivalent to the statement that the following mapping is a homomorphism: Aff : s 2 G 7! Aff ðsÞ 2 AðEÞ

ð43Þ

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxix

We denote by f the affine representation of the Lie algebra g of G, defined by f, and q the restriction to g of the differential of q (f and q are the differentials of f and q, respectively). Koszul proved the following equation: f ðXÞqðYÞ  f ðYÞqðXÞ ¼ qð½X; YÞ 8X; Y 2 g and f : g ! glðEÞ and q : g 7! E

ð44Þ

and glðEÞ is the set of all linear endomorphisms of E, the Lie algebra of GLðEÞ. We use the assumption that: qðAds YÞ ¼

dqðs:etY :s1 Þ

1

¼ fðsÞf ðYÞqðs Þ þ fðsÞqðYÞ dt t¼0

ð45Þ

We then obtain:

dqðAdetX YÞ

qð½X; YÞ ¼

¼ f ðXÞqðYÞqðeÞ þ fðeÞf ðYÞðqðXÞÞ þ f ðXÞqðYÞ ð46Þ dt t¼0 where e is neutral element of G. Since fðeÞ is the identity map and qðeÞ ¼ 0, we have the equality: f ðXÞqðYÞ  f ðYÞqðXÞ ¼ qð½X; YÞ

ð47Þ

A pair ðf ; qÞ comprising a linear representation of f of a Lie algebra g on E and a linear map q from g in E is an affine representation of g in E if it satisfies: f ðXÞqðYÞ  f ðYÞqðXÞ ¼ qð½X; YÞ

ð48Þ

Conversely, if we assume that g has an affine representation ðf ; qÞ on E, by using the coordinate systems fx1 ; . . .; xn g on E, we can express the affine map v 7! f ðXÞv þ qðYÞ by a matrix representation of size ðn þ 1Þ  ðn  1Þ: 

f ðXÞ aff ðXÞ ¼ 0

qðXÞ 0

ð49Þ

where f ðXÞ is a matrix of size n  n and qðXÞ a vector of size n. X 7! aff ðXÞ is an injective homomorphism of the Lie algebra g to the Lie algebra of ðn þ 1Þ  ðn þ 1Þ matrices, glðn þ 1; RÞ: g ! glðn þ 1; RÞ X 7! aff ðXÞ

ð50Þ

Writing gaff ¼ aff ðgÞ, we denote by Gaff the linear Lie subgroup of GLðn þ 1; RÞ generated by gaff . An element of s 2 Gaff may be expressed by:

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Foreword 2. Koszul Contemporaneous Lecture: Elementary …



fðsÞ Aff ðsÞ ¼ 0

qðsÞ 1

ð51Þ

Let Maff be the orbit of Gaff from the origin o, then Maff ¼ qðGaff Þ ¼ Gaff =Kaff , where Kaff ¼ fs 2 Gaff =qðsÞ ¼ 0g ¼ KerðqÞ. We can give as an example the following case. Let X be a convex domain in Rn without any straight lines. We define the cone VðXÞ in Rn þ 1 ¼ Rn  R by VðXÞ ¼ fðkx; xÞ 2 Rn  R=x 2 X; k 2 R þ g. Then, there is an affine embedding:  x ‘ : x 2 X 7! 2 VðXÞ 1

ð52Þ

Let g be the group homomorphism of Aðn; RÞ to GLðn þ 1; RÞ given by: 

fðsÞ qðsÞ 2 GLðn þ 1; RÞ s 2 Aðn; RÞ 7! 0 1

ð53Þ

where Aðn; RÞ is the group of all affine transformations in Rn . We have gðGðXÞÞ  GðVðXÞÞ and the pair ðg; ‘Þ comprising the homomorphism g : GðXÞ ! GðVðXÞÞ and the map ‘ : X ! VðXÞ are equivariant: ‘  s ¼ gðsÞ  ‘ and d‘  s ¼ gðsÞ  d‘

ð54Þ

In Table 1, we compare the affine representations of Lie groups and Lie algebras according to each of Souriau and Koszul’s approaches:

Table 1 Table comparing Souriau and Koszul’s affine representations of Lie groups and Lie algebras Souriau’s model of affine representations of Lie groups and algebras (using the notation of Libermann–Marle)

Koszul’s model of affine representations of Lie groups and algebras (using Koszul’s notations)

AðgÞðxÞ ¼ RðgÞðxÞ þ hðgÞ with g 2 G; x 2 E R : G ! GLðEÞ and h : G ! E

Aff ðsÞ : a 7! sa ¼ fðsÞa þ qðsÞ 8s 2 G; 8a 2 E f: G ! GLðEÞ s 7! fðsÞa ¼ sa  so 8a 2 E q: G ! E s 7! qðsÞ ¼ so 8s 2 G

hðghÞ ¼ RðgÞðhðhÞÞ þ hðgÞ with g; h 2 G h : G ! E is a one-cocycle of G with values in E, aðXÞðxÞ ¼ rðXÞðxÞ þ HðXÞ with X 2 g; x 2 E The linear map H : g ! E is a one-cocycle of G with values in E: HðXÞ ¼ Te hðXðeÞÞ; X 2 g

qðstÞ ¼ fðsÞqðtÞ þ qðsÞ v 7! f ðXÞv þ qðYÞ f and q are the differentials of f and q, respectively

(continued)

Foreword 2. Koszul Contemporaneous Lecture: Elementary …

xxxi

Table 1 (continued) Souriau’s model of affine representations of Lie groups and algebras (using the notation of Libermann–Marle)

Koszul’s model of affine representations of Lie groups and algebras (using Koszul’s notations)

Hð½X; Y Þ ¼ rðXÞðHðYÞÞ  rðYÞðHðXÞÞ

qð½X; Y Þ ¼ f ðXÞqðYÞ  f ðYÞqðXÞ 8X; Y 2 g with f : g ! glðEÞ and q : g 7! E  f ðXÞ qðXÞ aff ðXÞ ¼ 0 0  fðsÞ qðsÞ Aff ðsÞ ¼ 0 1

-

Conclusion Koszul’s book, which uses a purely algebraic and geometric approach to reinforce Souriau’s founding work in geometric mechanics, introducing original developments and proofs, is of major interest to various communities. In the first place, the community of physicists can find the mathematical foundations for analytical mechanics, but also for Lie Group Thermodynamics (the covariant theory of classical thermodynamics introduced by Souriau in parallel with geometric mechanics, but ignored by the great majority of the community). Second, this book is of great interest for the emerging field known as the “Geometric Science of Information”, in which the generalization of the Fisher metric is at the heart of the extension of classical tools of Machine Learning and Artificial Intelligence to deal with more abstract objects living in homogeneous manifolds, groups, and structured matrices. Future of Information Geometry and Artificial Intelligence should be based on the pillars developed in Koszul’s book. The “Geometric Science of Information” community (GSI, www.gsi2017.org) has lost a mathematician of great value, who enlightened his views by the depth of his thought. … et sic matheseos demonstrationes cum aleae incertitudine jugendo, et quae contraria videntur conciliando, ab utraque nominationem suam accipiens, stupendum hunc titulum jure sibi arrogat: Aleae Geometria … par l’union ainsi réalisée entre les démonstrations des mathématiques et l’incertitude du hasard, et par la conciliation entre les contraires apparents, elle peut tirer son nom de part et d’autre et s’arroger à bon droit ce titre étonnant: Géométrie du Hasard … by the union thus achieved between the demonstrations of mathematics and the uncertainty of chance, and by the conciliation between apparent opposites, it can take its name from both sides and arrogate to right this amazing title: Geometry of Chance Blaise Pascal—ALEAE GEOMETRIA: De compositione aleae in ludis ipsi subjectis, in Celeberrimae matheseos Academiae Parisiensi, 1654

Limours, France June 2018

Frédéric Barbaresco

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References 1. Cartan, E.: Sur les invariants intégraux de certains espaces homogènes clos et les propriétés topologiques de ces espaces. Ann. Soc. Pol. De Math. 8, 181–225 (1929) 2. Cartan, E.: Sur les domaines bornés de l’espace de n variables complexes. Abh. Math. Semi. Hamburg 1, 116–162 (1935) 3. Selected Papers of Koszul, J.L.: Series in Pure Mathematics, vol. 17. World Scientific Publishing (1994) 4. Cartan, H.: Allocution de Monsieur Henri Cartan, colloques Jean-Louis Koszul. Annales de l’Institut Fourier, tome 37(4), 1–4 (1987) 5. Koszul, J.L.: L’œuvre d’Élie Cartan en géométrie différentielle, in Élie Cartan, 1869–1951. Hommage de l’Académie de la République Socialiste de Roumanie à l’occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Grou-pement des Mathématiciens d’Expression Latine, tenu à Bucarest en 1969 (Editura Academiei Republicii Socialiste Romania, Bucharest), pp. 39–45 (1975) 6. Last interview of J.L. Koszul for the 50th birthday of “laboratoire mathématique de l’Institut Fourier” in 2016: video: https://www.youtube.com/watch?v=AzK5K7Q05sw 7. Koszul, J.L.: Sur la forme hermitienne canonique des espaces homogènes complexes. Can. J. Math. 7, 562–576 (1955) 8. Koszul, J.L.: Exposés sur les Espaces Homogènes Symétriques; Publicação da Sociedade de Matematica de São Paulo: São Paulo, Brazil (1959) 9. Koszul, J.L.: Domaines bornées homogènes et orbites de groupes de transformations affines. Bull. Soc. Math. Fr. 89, 515–533 (1961) 10. Koszul, J.L.: Ouverts convexes homogènes des espaces affines. Math. Z. 79, 254–259 (1962) 11. Koszul, J.L.: Variétés localement plates et convexité. Osaka. J. Math. 2, 285–290 (1965) 12. Koszul, J.L.: Lectures on Groups of Transformations, Tata Institute of Fundamental Research, Bombay (1965) 13. Koszul, J.L.: Déformations des variétés localement plates. Ann. Inst. Fourier 18, 103–114 (1968) 14. Koszul, J.L.: Trajectoires Convexes de Groupes Affines Unimodulaires. In: Essays on Topology and Related Topics, pp. 105–110. Springer, Berlin, Germany (1970) 15. Koszul, J.L.: Introduction to Symplectic Geometry. Science Press, Beijing (1989) (in Chinese); translation in English by Springer (2018) 16. Koszul, J.-L.: Homologie et cohomologie des algèbres de Lie. Bull. Soc. Math. Fr. 78, 65–127 (1950) 17. Koszul, J.-L.: Crochet de Schouten-Nijenhuis et cohomologie. Dans É. Cartan et les mathématiques d’aujourd’hui. Astérisque, numéro hors-série, pp. 257–271 (1985) 18. Barbaresco, F.: Koszul information geometry and Souriau Lie group thermodynamics. AIP Conf. Proc. 1641(74) (2015), MaxEnt’14 Proceedings, Amboise, Septembre 2014 19. Barbaresco, F.: Geometric theory of heat from Souriau Lie Groups thermodynamics and Koszul Hessian geometry: applications in information geometry for exponential families. Entropy 18, 386 (2016) 20. Barbaresco, F.: Poly-symplectic Model of Higher Order Souriau Lie Groups Thermodynamics for Small Data Analytics, GSI’17, Springer LNCS 10589, pp. 432–441 (2017) 21. Barbaresco, F.: Les densités de probabilité « distinguées » et l’équation d’Alexis Clairaut: regards croisés de Maurice Fréchet et de Jean-Louis Koszul, GRETSI’17, Juan-Les-Pins, September 2017 22. Barbaresco, F.: Jean-Louis Koszul and the elementary structures of Information Geometry, submitted to Springer Book “Geometric Structures of Information” (2018)

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23. Barbaresco, F.: Higher Order Geometric Theory of Information and Heat based on Poly-symplectic Geometry of Souriau Lie Groups Thermodynamics and their Contextures, submitted to MDPI Special Issue “Topological and Geometrical Structure of Information”. Selected Papers from CIRM Conferences 2017 24. Libermann, P., Marle, C.-M.: Géométrie symplectique: Bases théorique de la Mécanique classique. Tomes 1, 2, 3, U.E.R. de Mathématiques, L.A. 212 et E.R.A. 944, 1020, 1021 du C.N.R.S. 25. Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987) 26. Marle, C.-M.: On mechanical systems with a Lie group as configuration space. In: Jean Leray ‘99 Conference Proceedings: The Karlskrona Conference in the Honor of Jean Leray (Maurice de Gosson, éditeur), pp. 183–203. Kluwer, Dordrecht (2003) 27. Marle, C.-M.: Symmetries of Hamiltonian systems on symplectic and Poisson manifolds. In: Similarity and Symmetry Methods, Applications in Elasticity and Mechanics of Materials. Lecture Notes in Applied and Computational Mechanics 28. Marle, C.-M.: From tools in symplectic and Poisson geometry to J.-M. Souriau’s theories of statistical mechanics and thermodynamics. Entropy 18, 370 (2016) 29. Marle, C.-M.: Géométrie symplectique et géométrie de Poisson, collection mathématiques en devenir, édition Calvage & Mounet, Paris (2018) 30. de Saxcé, G., Vallée, C.: Galilean Mechanics and Thermodynamics of Continua. Wiley, Hoboken (2016) 31. Souriau, J.-M.: Structure des systèmes dynamiques. Dunod, Paris (1969) 32. Souriau, J.-M.: Géométrie globale du problème à deux corps. Modern developments in analytical mechanics. Accademia delle Scienze di Torino, pp. 369–418. Supplemento al vol. 117. Atti della Accademia della Scienze di Torino (1983) 33. Souriau, J.-M.: La structure symplectique de la mécanique décrite par Lagrange en 1811. Mathématiques et sciences humaines, tome 94, 45–54 (1986) 34. Souriau, J.-M.: Mécanique statistique, groupes de Lie et cosmologie, Colloques int. du CNRS numéro 237, Géométrie symplectique et physique mathématique, pp. 59–113 (1974) 35. Nguiffo Boyom, M.: Varietes symplectiques affines. Manuscripta Mathematica 64(1), 1–33 (1989) 36. Nguiffo Boyom, M.: Structures localement plates dans certaines variétés symplectiques. Math. Scand. 76, 61–84 (1995) 37. Nguiffo Boyom, M.: Métriques kählériennes affinement plates de certaines variétés symplectiques. I., Proc. Lond. Math. Soc. (3), 66(2), 358–380 (1993) 38. Nguiffo Boyom, M.: The cohomology of Koszul-Vinberg algebras. Pac. J. Math. 225, 119–153 (2006) 39. Nguiffo Boyom, M.: Some Lagrangian Invariants of Symplectic Manifolds, Geometry and Topology of Manifolds; Banach Center Institute of Mathematics, Polish Academy of Sciences, Warsaw, vol. 76, pp. 515–525 (2007) 40. Lichnerowicz, A.: Groupes de Lie à structures symplectiques ou Kähleriennes invariantes. In: Albert, C. (ed.) Géométrie Symplectique et Mécanique. Lecture Notes in Mathematics, vol. 1416. Springer (1990) 41. Barbaresco, F.: Jean-Louis Koszul et les structures élémentaires de la géométrie de l’information, revue MATAPLI 2018, SMAI (2018) 42. Malgrange, B.: Quelques souvenirs de Jean-Louis KOSZUL. Gazette des Mathématiciens 156, 63–64 (2018) 43. Cartier, P.: In memoriam Jean-Louis KOSZUL. Gazette des Mathématiciens 156, 64–66 (2018) 44. Kosmanek, E.-E.: Hommage à mon directeur de thèse. Gazette des Mathématiciens 156, 64–66 (2018)

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45. Vey, J.: Sur une notion d’hyperbolicité des variétés localement plates, Thèse de troisième cycle de mathématiques pures, Faculté des sciences de l’université de Grenoble (1969) 46. Vey, J.: Sur les automorphismes affines des ouverts convexes saillants, Annali della Scuola Normale Superiore di Pisa, Classe di Science, 3e série, 24(4), 641–665 (1970) 47. Luna, D.D.: Sur l’intersection générique de deux sous-algèbres d’une algèbre de Lie, Thèse de 3ème cycle de mathématiques pures, Faculté des sciences de l’université de Grenoble (1970) 48. Helmstetter, J.: Algèbres symétriques à gauche. C.R. Acad. Sci. Paris Sér. A-B 272, A1088– A1091 (1971) 49. Helmstetter, J.: Radical et groupe formel d’une algèbre symétrique à gauche, Thèse de 3ème cycle de mathématiques pures, Faculté des sciences de l’université de Grenoble (1975) 50. Helmstetter, J.: Radical d’une algèbre symétrique à gauche. Ann. Inst. Fourier 29, 17–35 (1979) 51. Delzant, T., Wacheux, C.: Actions Hamiltonniennes, vol. 1, no. 1, pp. 23–31, Rencontres du CIRM « Action Hamiltoniennes: invariants et classification », organisé par Michel Brion et Thomas Delzant, CIRM, Luminy (2010) 52. Kostant, B.: Quantization and unitary representations, Lecture Notes in Mathematics, vol. 170. Springer (1970) 53. Koszul, J.L.: Travaux de B. Kostant sur les groupes de Lie semi-simples. Séminaire Bourbaki, Tome 5, Exposé no. 191, pp. 329–337 (1958–1960)

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I was very pleased when Frédéric Barbaresco gave me the opportunity to write a short presentation for this book. Jean-Louis Koszul, recently deceased, was a highly distinguished mathematician whom I respected and admired. It is for me an honor to introduce a book stemming from the lectures he taught in 1983. In this Foreword, I am going to describe the content of this book and highlight its originality. Then I will say a few words about some developments in Symplectic and Poisson geometry, of which I am aware, not discussed in this book, often because they appeared after 1983.

The Content and Originality of this Book The first chapter offers a very nice presentation of the main properties of symplectic vector spaces. The framework used at the beginning, slightly more general than that usually considered, is that of a finite-dimensional vector space V over an arbitrary field k, endowed with a skew-symmetric bilinear form x whose kernel may not be f0g. Orthogonality with respect to x is defined, as well as isotropic, coisotropic, Lagrangian, and symplectic vector subspaces, and their main properties are proven. Then it is assumed that kerx ¼ f0g, which means that ðV; xÞ is a symplectic vector space. Symplectic bases (often called, in other texts, canonical bases or Darboux bases) are defined. A long section is devoted to the canonical representation of the Lie algebra slð2; kÞ in the graded vector space of skew-symmetric multilinear forms on V. The last two sections of the chapter deal with the linear symplectic group and with adapted complex structures on a real finite-dimensional vector space. Symplectic manifolds are introduced in the second chapter, and a symplectic homomorphism u from a symplectic manifold ðM1 ; x1 Þ to another symplectic manifold ðM2 ; x2 Þ is defined. Contrary to the usual practice, it is not assumed that dimM1 ¼ dimM2 . Of course we must have dimM1 dimM2 , and rank u ¼ dimM1 , but there are interesting cases in which dimM1 \dimM2 . Several examples are

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discussed: quotients of R2n by the action of a discrete subgroup of the group of translations, Kähler manifolds, the complex projective space CPn . A long section, which uses results obtained in Chap. 1 about the canonical representation of slð2; kÞ in the graded vector space of skew-symmetric multilinear forms on V, is devoted to operators on the space of differential forms on a symplectic manifold, allowing the construction of a representation of the Lie algebra slð2; RÞ. When there exists a 1-form a such that x ¼ d a, it is proven that this representation can be extended to a representation of the Lie superalgebra ospð2; RÞ. Then symplectic local coordinates (often called, in other texts, canonical coordinates or Darboux coordinates) are defined, and a proof by induction of the Darboux theorem is given. In that proof, the rank of the form x is assumed to be constant, but it may be strictly smaller than the dimension of the considered manifold. Symplectic vector fields on a symplectic manifold are then defined (in other texts, they are often called locally Hamiltonian vector fields), as well as Hamiltonian vector fields. Their properties are discussed. For example, it is proven that the vector space SðM; xÞ of symplectic vector fields is a Lie subalgebra of the Lie algebra of all smooth vector fields, and that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field. After a concise definition of de Rham cohomology groups, it is proven that the sequence f0g ! H 0 ðM; RÞ ! C1 ðM; RÞ ! SðM; xÞ ! H 1 ðM; RÞ ! f0g is exact. Examples are given of vector fields which are symplectic but not Hamiltonian. Next, the Poisson bracket of smooth functions on a symplectic manifold ðM; xÞ is defined. It is proven that C1 ðM; RÞ, endowed with the Poisson bracket, is a Lie algebra, and that the map which associates to each smooth function the corresponding Hamiltonian vector field is a Lie algebra homomorphism whose kernel is the vector subspace H 0 ðMÞ of locally constant functions. Let u : P ! Q be a smooth map between two symplectic manifolds ðP; xP Þ and ðQ; xQ Þ. For each smooth vector field Y defined on Q, there exists a unique vector field on P, denoted by u Y, such that iðu YÞxP ¼ u ðiðYÞxQ Þ. By using this correspondence between vector fields when u is a symplectic homomorphism, several orthogonality properties are established. It is proven that when u is a symplectic homomorphism between symplectic manifolds of the same dimension, for any pair ðf ; gÞ of smooth functions defined on Q, u ðff ; ggQ ¼ fu f ; u ggP . This means that u is a Poisson map. However, this result is not stated in these terms, because Poisson manifolds and Poisson maps (called in this book Poisson homomorphisms) are defined later, in Chap. 4. This result does not hold when u is a symplectic homomorphism between two symplectic manifolds of different dimensions. In other words, symplectic homomorphisms between symplectic manifolds of different dimensions are not Poisson maps. The set of polynomials defined on R2n (endowed with its canonical symplectic structure) has both an associative algebra structure (for the ordinary product) and a Lie algebra structure (for the Poisson bracket). A careful study of its ideals is

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presented. Then the Lie algebra of formal symplectic vector fields defined on a symplectic manifold is discussed, and it is proven that the algebra of jets of symplectic vector fields at all points of a symplectic manifold is a simple algebra. Isotropic, coisotropic, and Lagrangian submanifolds of a symplectic manifold are defined, and their main properties are established. The semi-global generalizations of the Darboux theorem, due to the American mathematician Alan Weinstein, are given. Then, a method for constructing Lagrangian submanifolds, or more generally Lagrangian immersions, called the contraction method, is presented. I believe that this method, called in other texts the use of Morse families, is due to the Swedish mathematician Lars Hörmander [F3 Hor].1 At the end of the second chapter, an exercise is proposed in which it is proven that there exists, on each leaf of a Lagrangian foliation, a flat and torsionless connection. That result was observed as early as 1953 by the French mathematician Paulette Libermann in her thesis [F3 Lib]. The symplectic manifolds considered in Chap. 3 are cotangent bundles endowed with their canonical symplectic form. The Liouville 1-form on the total space of a cotangent bundle is first defined, and its main properties and its behavior under the prolongation of a diffeomorphism to covectors are presented. The canonical symplectic form is then defined as the opposite of the exterior differential of the Liouville form. A detailed discussion of symplectic vector fields on a cotangent bundle is given, in which the Liouville vector field C (the infinitesimal generator of homotheties in the fibers) is used. Functions which are homogeneous of a given degree with respect to the Lie derivation along C, in other words, functions whose restriction to each fiber of the cotangent bundle is a polynomial of a given degree r þ 1, and the associated Hamiltonian vector fields, are considered and a detailed study is made of the cases when r ¼ 1, 0, and 1. A deep study of Lagrangian submanifolds of a cotangent bundle is presented in the next section. It is first proven that the image of a 1-form is a Lagrangian submanifold if and only if that 1-form is closed. When it is the differential of a smooth function, that function is called a generating function for the corresponding Lagrangian submanifold. Then, following the beautiful booklet of Alan Weinstein [29], more general generating functions are introduced, now defined on an open subset of the cotangent bundle, whose associated Lagrangian submanifolds may not be transverse to the fibers. That construction is closely related to the contraction method described in Chap. 2. Chapter 4 begins with the definition of actions of a Lie group and of a Lie algebra on a smooth manifold. A smooth manifold M on which a Lie group G acts smoothly on the left is called a G-space. An action of a finite-dimensional Lie algebra h on a smooth manifold M is a Lie algebra homomorphism of h in the Lie algebra of smooth vector fields on M (equipped with the Lie bracket as composition law). When the smooth manifold M is endowed with a symplectic form x, a smooth 1 Reference tags beginning with F3, such as [F3 Hor], point to the references list at the end of this Foreword, while reference tags such as [29] point to the main references list, at the end of this book.

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action on M of a Lie group G is said to be symplectic, and ðM; xÞ is said to be a symplectic G-space, if the diffeomorphism of M associated to each element in G is a symplectic diffeomorphism. Similarly, an action on M of a finite-dimensional Lie algebra h is said to be symplectic, and ðM; xÞ is said to be a symplectic h-space, when the vector field associated to each element in h is a symplectic vector field. It is shown that the natural prolongation to the cotangent bundle T  N of any smooth action of a Lie group on M is symplectic (later on it will be shown that such an action is Hamiltonian, after the definition of that notion). Other examples are presented. It is shown that each orbit of a symplectic action is an immersed submanifold on which x induces a form of constant rank, and that the set of fixed points of a symplectic action of a compact Lie group on a symplectic manifold is a symplectic submanifold. A symplectic action C of a Lie algebra g on a symplectic manifold ðM; xÞ is said to be Hamiltonian, and ðM; xÞ is said to be a Hamiltonian g-space, when the vector field CðaÞ associated to each a 2 g is a Hamiltonian vector field. A moment map l of the Hamiltonian action C is defined: it is a smooth map l from M to the dual space g of the Lie algebra g such that, for each a 2 g, the smooth function on M: x 7! hlðxÞ; ai is a Hamiltonian for the Hamiltonian vector field CðaÞ. The properties of moment maps are established and it is shown that the difference of two moment maps for the same Hamiltonian action is a locally constant map. Moreover, for any moment map l : M ! g and any pair ða; bÞ of elements in g, the function x 7! cl ða; bÞðxÞ ¼ fhlðxÞ; ai; hlðxÞ; big  hlðxÞ; ½a; bi is locally constant on M. Therefore, cl is a skew-symmetric bilinear map on M which takes its values in H 0 ðMÞ. It satisfies a remarkable identity, a consequence of the Jacobi identity. Called a closed 2-cochain in this book, it is called a 2-cocycle of g in other texts. Its restriction to each connected component of M is a skew-symmetric 2-form on g. When the moment map l is modified by addition of a locally constant map u taking its values in g , cl is modified by addition of the coboundary of u. The cohomology class of cl remains unchanged: it depends only on the considered Hamiltonian g-space ðM; xÞ, not on the choice of a moment map of the g-action. It is denoted by cðM; xÞ. A Hamiltonian g-space ðM; xÞ is said to be strongly Hamiltonian when cðM; xÞ ¼ f0g. For reasons which will become clear in the next chapter, strongly Hamiltonian g-spaces are also called Poisson g-spaces. Then it is proven that when the symplectic manifold ðM; xÞ is exact, i.e., when there exists a 1-form a such that x ¼  d a, and when the considered symplectic action C of the Lie algebra g is such that the Lie derivatives of a with respect to the vector fields in CðgÞ all vanish, that action is strongly Hamiltonian. Using the second Whitehead lemma (quoted without proof), it is shown that any symplectic action of a semi-simple Lie algebra is strongly Hamiltonian. Several examples are given of actions which are symplectic but not Hamiltonian, or Hamiltonian but not strongly Hamiltonian, and of strongly Hamiltonian actions. Some general properties of the moment map l of a Hamiltonian action are proven. For example, it is shown

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that at each point x of a Hamiltonian g-space ðM; xÞ, the kernel of the map d lx (denoted by Tx l in other texts) and fCa ðxÞja 2 gg are two symplectically orthogonal vector subspaces of the symplectic vector space ðTx M; xðxÞÞ. Moreover, the image of Tx l in g is the annihilator of the isotropy subalgebra gx of x (called in this book the orthogonal complement of gx ). When the Hamiltonian action of the Lie algebra g comes from a Hamiltonian action of a Lie group G whose Lie algebra is g, several consequences are deduced from these properties, about the moment map and fixed points of the action. The next section of Chap. 4 is devoted to equivariance properties of the moment map. Definitions of the adjoint and coadjoint actions of a Lie group G and of its Lie algebra g are recalled. Then, it is shown that the moment map l of a Hamiltonian action of a Lie group G on a connected Hamiltonian G-space ðM; xÞ is equivariant with respect to the considered action of G on M, and an affine action of that group on g , obtained by adding to the coadjoint action a closed cochain (in other words, a cocycle). When the considered action of G is strongly Hamiltonian, that cocycle is a coboundary; therefore, the addition of a suitably chosen constant to the moment map yields a moment map equivariant with respect to the coadjoint action. Two special cases are then considered: when the Lie group G is Abelian, and when its action on the manifold M is transitive. Readers are referred to the works of B. Kostant [17] and J.-M. Souriau [23] for more details. Chapter 5 begins with a statement, without proof, of the main properties of the Schouten–Nijenhuis bracket. For each integer p  0, the vector space of smooth p-multivectors on a smooth manifold M is denoted by Dp ðMÞ, with the convention D0 ðMÞ ¼ C1 ðM; RÞ. The direct sum of the vector spaces Dp ðMÞ for all p 2 NÞ is denoted by D ðMÞ. The Schouten–Nijenhuis bracket is a bilinear, graded skew-symmetric composition law on D ðMÞ which maps Dp ðMÞ  Dq ðMÞ into Dp þ q1 ðMÞ. With the use of an element w 2 D2 ðMÞ, one can define a bilinear and skew-symmetric composition law ðf ; gÞ 7! ff ; ggw ¼ wðd f ; d gÞ on C 1 ðM; RÞ and a linear map f 7! Hf from C 1 ðM; RÞ in the vector space D1 ðMÞ of smooth vector fields on M. It is shown that the following four conditions are equivalent: (i) the composition law ðf ; gÞ 7! ff ; ggw satisfies the Jacobi identity; (ii) for any pair ðf ; gÞ of elements in C 1 ðM; RÞ, Hff ;ggw ¼ ½Hf ; Hg ; (iii) for each f 2 C 1 ðM; RÞ, ½Hf ; w ¼ 0 (the bracket in the left-hand side being the Schouten–Nijenhuis bracket which, in this case, is the Lie derivative of w with respect to the vector field Hf ); (iv) ½w; w ¼ 0 (the bracket in the left-hand side being the Schouten–Nijenhuis bracket). A Poisson structure on M is the structure determined on that manifold by a 2-multivector w 2 D2 ðMÞ which satisfies these four equivalent conditions. The manifold M is then called a Poisson manifold and denoted by ðM; wÞ. Poisson homomorphisms (often called in other texts Poisson maps) between two Poisson manifolds are then defined. It is explained in the second section of Chap. 5 that any Poisson manifold is a union of immersed symplectic manifolds, called its symplectic leaves. The proof of that property, rather delicate, is not fully given; it is only observed that the result is true when the rank of the bivector field w which determines the Poisson structure is

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constant, and asserted that the same is true in the general case. It is proven that each integral manifold of the family of vector fields fHf j f 2 C 1 ðM; RÞg is endowed with a symplectic form such that, for each point x 2 M and each pair ðf ; gÞ of smooth functions defined on an open neighborhood of x, the following property is satisfied: ff ; ggw ðxÞ, defined as the value at x of the evaluation wðdf ; dgÞ of w on the two 1-forms d f and dg, is equal to the value at x of the Poisson bracket of the restrictions of f and g to the integral manifold of fHf j f 2 C1 ðM; RRÞg which contains x, calculated with the use of its symplectic form as indicated in Chap. 2. A Poisson manifold which has only one symplectic leaf is a symplectic manifold, the bivector field w being, in a certain sense, the inverse of its symplectic form. A simple example is given at the end of that section. Poisson structures on the dual vector space of a finite-dimensional Lie algebra are studied in detail in the last section of Chap. 5. The bracket of a finite-dimensional Lie algebra g, considered as a composition law of linear functions defined on the dual space g , is first extended to a composition law on C 1 ðg ; RÞ. The Poisson structure so obtained on g was observed by Sophus Lie and rediscovered, much later, independently by A. Kirillov, B. Kostant, and J.-M. Souriau. It is often called the canonical Lie–Poisson structure or the Kirillov– Kostant–Souriau structure on g . By observing that the transpose of a Lie algebra homomorphism is a Poisson homomorphism, it is proven that when g is the Lie algebra of a Lie group G, the canonical Lie–Poisson structure on g is preserved by the coadjoint action of G, and that the symplectic leaves of g are the orbits of the coadjoint action, restricted to the neutral component of G. Restricted to each orbit, the action of G is Hamiltonian and has the canonical injection of that orbit in g as moment map. It is then proven that the Ad -equivariant moment map of a strongly Hamiltonian action is a Poisson homomorphism. The remainder of that section extends these results to Hamiltonian actions, which may not be strongly Hamiltonian, of a Lie group G on a connected symplectic manifold ðM; xÞ. For any moment map l of that action, it was proven in Chap. 4 that there exists an affine action of G on the dual space g of g with respect to which l is equivariant. That action is obtained by adding to the coadjoint action, which is its linear part, a smooth map ul : G ! g which is a 1-cocycle of G with values in g for the coadjoint action. It is now proven that the canonical Lie– Poisson structure on g can be modified, by addition of the 1-cocycle of g with values in g associated to ul , in such a way that for this modified Poisson structure, the moment map l becomes a Poisson homomorphism. This remarkable result was obtained for the first time, I believe, by J.-M. Souriau, and appears in his book [23]. The proof presented here is slightly different, and more algebraic. Moreover, it is proven here that when the Lie group G is simply connected, the 1-cocycle ul of G and the associated 1-cocycle of its Lie algebra for which l is an equivariant Poisson homomorphism can be deduced from the 2-cocycle of g introduced in Chap. 4. Chapter 5 ends with proposed exercises related to Lie groups and Lie algebras endowed with a symplectic structure, in which reference to the works of Vinberg is made [25].

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Finally, Chap. 6, the most original part of this book, is a short introduction to supermanifolds, in particular, symplectic supermanifolds.

Some Developments in Symplectic and Poisson Geometry Since 1983, several textbooks have appeared in the fields of symplectic and Poisson geometry, for example [F3 Cann, F3 Fo, F3 Lib-Ma], in which the readers will find other viewpoints on the subjects dealt with in this book. Readers interested in more specialized and advanced results are referred to the proceedings of conferences, for example [F3 Do, F3 Gra-Urb, F3 Mars-Ra]. Methods in symplectic geometry are used for the study of global geometric properties of completely integrable Hamiltonian systems in the book by M. Audin, A. Cannas da Silva, and E. Lerman [F3 Au-Can-Ler] and in the book by R. Cushman and L. M. Bates [F3 Cush-Ba] for classical mechanical systems. Local properties of Poisson manifolds were thoroughly studied by A. Weinstein [F3 W1]. There exists on a Poisson manifold a cohomology discovered by A. Lichnerowicz, defined by means of the Poisson bivector field, called the Lichnerowicz–Poisson cohomology. When the considered Poisson manifold is in fact a symplectic manifold, its Lichnerowicz–Poisson cohomology coincides with its de Rham cohomology. For general Poisson manifolds, the Lichnerowicz– Poisson cohomology is often more complicated than the de Rham cohomology because it reflects the topological and geometrical properties of the foliation of that manifold in symplectic leaves. It was studied by P. Xu [F3 Xu] and many other authors. Remarkable properties of moment maps, not discussed in this book, have been discovered since 1982. The readers are referred to the important papers of M. Atiyah [F3 Ati], V. Guillemin and S. Sternberg [F3 Gu-Ster1, F3 Gu-Ster2], F. Kirwan [F3 Kir], T. Delzant [F3 Del], and to the beautiful book of M. Audin [F3 Au]. A deep study of moment maps and their use for reduction can be found in the monumental book by J.-P. Ortega and T. Ratiu [F3 Or-Ra]. Following the pioneering work of F. Magri [F3 Mag, F3 Ma-Mo], vector fields defined on a smooth manifold which are Hamiltonian with respect to two different symplectic or Poisson structures, with two different Hamiltonian functions, called bi-Hamiltonian systems, were intensively studied by many authors. Very often these systems are integrable and also occur in the more general setting of evolution partial differential equations. Several important new results were found in the geometry of Poisson manifolds. Jean-Louis Koszul discovered the composition law on the graded vector space of differential forms of all degrees on a Poisson manifold [F3 Kosz]. Its restriction to the vector subspace of differential forms of degree 1 had already been observed a little earlier by F. Magri and C. Morosi [F3 Ma-Mo], and a little later by P. Dazord and D. Sondaz [F3 Da-Son]. It was then found by A. Weinstein, P. Dazord,

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D. Sondaz, and probably many others [F3 W2, F3 W3, F3 Da-W, F3 Da-Son] that the tangent space to a Poisson manifold is endowed with a Lie algebroid structure. J.-P. Dufour and N. T. Zung’s book [F3 Du-Zu] very thoroughly presents normal forms of Poisson structures. The book by C. Laurent-Gengoux, A. Pichereau, and P. Vanhaecke is an invaluable reference work about Poisson structures and their applications. A Poisson Lie group is a Lie group G endowed with a Poisson structure for which the group composition law is a Poisson homomorphism from G  G equipped with the product Poisson structure, to G. Poisson Lie groups were introduced by V. G. Drinfel’d [F3 Dri] as classical analogues of quantum groups. The Lie algebra of a Poisson–Lie group has a remarkable property and is often called a Lie bialgebra: its dual vector space too is endowed with a Lie algebra structure. Poisson–Lie groups have been extensively studied by many authors, in particular, by J.-H. Lu and A. Weinstein [F3 Lu, F3 Lu-W]. Research on the symplectization of Poisson manifolds (the determination of a symplectic manifold of which the considered Poisson manifold is a quotient) and on the integration of Lie algebroids led A. Weinstein, S. Zakzrewski, M. Karasev, and others to the consideration of Lie groupoids in relation with Poisson manifolds and the definition of symplectic groupoids [F3 W2, F3 W3, F3 W4, F3 W5]. The readers are referred to the book by K.C.H. Mackenzie [F3 Mack] for a thorough study of Lie groupoids and to the booklet by A. Cannas da Silva and A. Weinstein [F3 Cann-W] for symplectic groupoids, the links of Lie groupoids with Poisson geometry, and more references. Poisson groupoids, i.e., Lie groupoids whose total space is endowed with a suitable Poisson structure, generalize both Poisson Lie groups and symplectic groupoids. Hamiltonian actions of a Poisson groupoid on a symplectic or Poisson manifold and moment maps for these actions can be defined, which generalize Hamiltonian actions and moment maps for a Lie group action [F3 Mack-Xu, F3 Gra-Urb, F3 Mars-Ra]. Symplectic topology is a development of symplectic geometry which appeared toward the end of the last century, in which symplectic geometric methods are used for the proof of topological properties of symplectic or contact manifolds, and for the definition of topological invariants. That development stemmed from the seminal work of M. Gromov [F3 Grom] and is now a very active domain of research. Readers interested in that subject are referred to the books by D. McDuff and D. Salamon [F3 McD-Sa] and by L. Polterovich and D. Rosen [F3 Pol-Ros] and the proceedings of an international conference [F3 Eli-Tray]. Meudon, France May 2018

Charles-Michel Marle

Foreword 3

xliii

References [F3 Ati] Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982) [F3 Au] Audin, M.: The Topology of Torus Actions on Symplectic Manifolds. Progress in Mathematics, vol. 93. Birkhäuser, Boston, Basel, Berlin (1991) [F3 A-Can-Ler] Audin, M., Cannas da Silva, A., Lerman, E.: Symplectic Geometry of Integrable Hamiltonian Systems. Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Basel, Boston, Berlin (2003) [F3 Cann] Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001). New edition (2004) [F3 Cann-W] Cannas da Silva, A., Weinstein, A.: Geometric Models for Noncommutative Algebras. Berkeley Mathematics Lecture Notes Series, American Mathematical Society (1999) [F3 Cush-Ba] Cushman, R., Bates, L.M.: Global Aspects of Classical Integrable Systems. Birkhäuser, Basel (1997) [F3 Da-Son] Dazord, P., Sondaz, D.: Variétés de Poisson, Algébroïdes de Lie. Publ. Dépt. Math., Univ. Lyon I, nouvelle série 1/B, pp. 1–68 (1988) [F3 Da-W] Dazord, P., Weinstein, A. (eds.): Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol. 20. Springer, New York (1991) [F3 Del] Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. Fr. 116, 315–339 (1988) [F3 Do] Donato, P., Duval, C., Elhadad, J., Tuynman, G.M. (eds.): Symplectic Geometry and Mathematical Physics. Actes du colloque en l’honneur de Jean-Marie Souriau, Progress in Mathematics, vol. 99. Birkhäuser, Boston, Basel, Berlin (1991) [F3 Dri] Drinfel’d, V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Soviet Math. Dokl. 27(1), 68–71 (1983) [F3 Du-Zu] Dufour, J.-P., Zung, N.T.: Poisson structures and their normal forms. Progress in Mathematics, vol. 242. Birkhäuser, Boston, Basel, Berlin (2005) [F3 Eli-Tray] Eliashberg, Y., Traynor, L. (eds.): Symplectic Geometry and Topology. AMS and IAS/Park City Mathematics Institute, Mathematics Series, vol. 7 (1999) [F3 Fo] Fomenko, A.T.: Symplectic Geometry, second edition. Advanced Studies in Contemporary Mathematics, vol. 5. Gordon and Breach, Luxembourg (1995) [F3 Gra-Urb] Grabowski, J., Urbański, P. (eds.): Poisson Geometry, Stanisław Zakrzewski in memoriam, vol. 51. Banach Center Publications, Warszawa (2000) [F3 Grom] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82, 307–347 (1985) [F3 Gu-Ster1] Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982) [F3 Gu-Ster2] Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping, II. Invent. Math. 77, 533–546 (1984) [F3 Hor] Hörmander, L.: Fourier integral operators, I. Acta Mathematica 127, 79–183 (1971) [F3 Kir] Kirwan, F.: Convexity properties of the moment map, III. Invent. Math. 77, 547–552 (1984) [F3 Kosz] Koszul, J.-L.: Crochet de Schouten-Nijenhuis et cohomologie. In É. Cartan et les mathématiques d’aujourd’hui. Astérisque, numéro hors série, pp. 257–271 (1985) [F3 La-Pi-Va] Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson structures. Grundlehren der mathematischen Wissenschaften, vol. 347. Springer, Berlin (2013) [F3 Lib] Libermann, P.: Sur le problème d’équivalence de certaines structures infinitésimales régulières, thèse de doctorat d’État, Strasbourg (1953) and Ann. Mat. Pura Appl. 36, 27–120 (1954)

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[F3 Lib-Ma] Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987) [F3 Lu] Lu, J.-H.: Multiplicative and affine Poisson structures on Lie groups. Thesis, Berkeley (1990) [F3 Lu-W] Lu, J.-H., Weinstein, A.: Poisson Lie groups, dressing transformations and Bruhat decompositions. J. Differ. Geom. 31, 501–526 (1990) [F3 Mack] Mackenzie, K.C.H.: General theory of Lie groupoids and Lie algebroids, London Mathematical Society. Lecture Notes Series, vol. 213. Cambridge University Press, Cambridge (2005) [F3 Mack-Xu] Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73, 415–452 (1994) [F3 Mag] Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19(5), 1156–1162 (1978) [F3 Ma-Mo] Magri, F., Morosi, C.: A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson–Nijenhuis manifolds. Quaderno S. 19, Università di Milano (1984) [F3 Mars-Ra] Marsden, J.E., Ratiu, T.S. (eds.): The Breadth of Symplectic and Poisson Geometry. Festschrift in Honor of Alan Weinstein. Progress in Mathematics, vol. 32. Birkhäuser, Boston, Basel, Berlin (2005) [F3 McD-Sa] McDuff, D., Salamon, D.: Introduction to Symplectic Topology. Oxford Mathematical Monographs, Clarendon Press, Oxford (1998) [F3 Or-Ra] Ortega, J.-P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction. Progress in Mathematics, vol. 222. Birkhäuser, Boston, Basel, Berlin (2004) [F3 Pol-Ros] Polterovich, L., Rosen, D.: Function Theory on Symplectic Manifolds. CRM Monograph Series, vol. 34. American Mathematical Society, Providence (2014) [F3 Vai] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics, vol. 118. Birkhäuser, Basel, Boston, Berlin (1994) [F3 W1] Weinstein, A.: The local structure of Poisson manifolds. J. Differ. Geom. 18, 523–557 (1983) et 22, 255 (1985) [F3 W2] Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. 16, 101–103 (1987) [F3 W3] Weinstein, A.: Some remarks on dressing transformations. J. Fac. Sci. Univ Tokyo, Sect. 1A, Math. 36, 163–167 (1988) [F3 W4] Weinstein, A.: Groupoids: unifying internal and external symmetry, a tour through some examples. Not. Am. Math. Soc. 43, 744–752 (1996) [F3 W5] Weinstein, A.: Lagrangian mechanics and groupoids. Fields Inst. Comm. 7, 207–231 (1996) [F3 Xu] Xu, P.: Poisson cohomology of regular Poisson manifolds. Ann. Inst. Fourier, Grenoble, 42(4), 967–988 (1992)

Preface

I was invited to give lectures at Nankai University in the spring of 1983. This book is based on the lecture notes, translated and written (with minor modifications) by Yi Ming Zou. We hope to introduce symplectic manifold theory to the readers through this introductory book. The development of analytical mechanics provided the basic concepts of symplectic structures. The term symplectic structure is due largely to analytical mechanics. But in this book, the applications of symplectic structure theory to mechanics are not discussed in any detail; and some of the important parts of the theory, especially the application in analysis, are not discussed at all. For those topics, we refer the readers to the references [1, 2, 7, 26]. The emphasis of this book is on the differential properties of manifolds with symplectic structures. The first chapter of the book discusses the symplectic structures of vector spaces. The second chapter discusses symplectic manifolds and introduces the basic concepts and the basic results to the readers. We prove the existence of symplectic coordinates (Darboux Theorem) as early as possible in Chap. 2, so the readers can see the importance of the formulas we give in later discussions. The connection between the differentiable functions and the infinitesimal automorphisms of the symplectic structure on a symplectic manifold is the foundation of the symplectic manifold theory, and it will be discussed in Sects. 2.4 and 2.5. This chapter ends with some results on the submanifolds, especially the Lagrangian submanifolds, of a symplectic manifold. The existence of canonical symplectic structures on cotangent bundles clarifies a lot of questions associated with symplectic structures. Chapter 3 introduces the results on cotangent bundles and symplectic vector fields on cotangent bundles. Chapter 4 discusses symplectic G-spaces, that is, symplectic manifolds with symplectic structures that are invariant under the actions of some Lie group G. For these symplectic manifolds, certain maps we call moment maps provide with us an effective study tool. The study of symplectic G-spaces is a rich topic in symplectic manifold theory, and there are still many problems that deserve further study. The study of the symplectic G-spaces leads us to the study of the dual structures of Lie algebras and the geometry properties of the so-called coadjoint representations. We xlv

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will discuss these subjects in Sect. 4.3, which will last until Chap. 5. In Chap. 5, we first introduce some general properties of the so-called Poisson manifolds. The concept of Poisson structures is a generalization of the concept of symplectic structures, and it allows us to consider the classical contents from a new viewpoint. Poisson structures start from the concept of contravariant skew-symmetric tensors. In Sect. 5.3, we will give the precise results for the Poisson structures in the dual spaces of Lie algebras. Chapter 6, the last chapter, is a special chapter. The purpose of this chapter is to introduce the generalizations of the concepts discussed in Chaps. 2 and 3 to supermanifolds. We only discuss ð0; nÞ-dimensional supermanifolds, that is, we only consider differentiable properties, not geometric properties. In this chapter, we mainly describe the basic properties, omit most the proofs. This is because the materials discussed are rather basic; and, we think that these omitted proofs can serve as exercises for the readers. Finally, we wish to express our sincere thanks to Professor Zhi-da Yan for his help in the process of the translation and writing of this book. Grenoble, France December 1984

Jean-Louis Koszul

Notations

Z Zþ R C Z2 Ap ðVÞ ðV; xÞ P x ¼ ri¼1 fi ^ fr þ i SpðV; xÞ spðV; xÞ OðV; bÞ Oð2nÞ UðnÞ TM T M TðT  PÞ T  ðXÞ C 1 ðMÞ Xp ðMÞ Dp ðMÞ iðXÞ hðXÞ CPn lðaÞ ospð2; 1Þ ðM; xÞ SðM; xÞ Z p ðMÞ H p ðMÞ ¼ H p ðM; RÞ Hf

The ring of integers The set of nonnegative integers Real number field Complex number field The ring of integers modulo 2 Skew-symmetric p-linear forms on V Symplectic vector space Canonical symplectic form on k2r Symplectic group Symplectic Lie algebra Orthogonal transformation group of V w.r.t bilinear form b Orthogonal group Unitary group Tangent bundle of M Cotangent bundle of M Tangent bundle of the cotangent bundle T  P Extension of the vector field X Real C 1 -differentiable functions on M Differential p-forms on M Degree p skew symm. contravariant diff. tensor fields on M The inner product through X The Lie derivation defined by X Complex projective space Left exterior product defined by a 5-dimensional orthosymplectic Lie superalgebra Symplectic manifold Symplectic vector fields on ðM; xÞ Space of all closed p-forms on M p-dimensional de Rham group of M The vector field on M defined by f 2 C1 ðMÞ

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ff ; gg Jx ðMÞ Jx ðXÞ g GðxÞ gx Gx gx Ad ad Ad ½ ; S Kðk n Þ ! 7!

Notations

Poisson bracket of f ; g Jet algebra at the point x 2 M Jet of the vector field X at the point x Lie algebra G-orbit of the point x Tangent space of GðxÞ at x Isotropy subgroup of G at the point x Lie algebra of Gx Adjoint representation of G on g Adjoint representation of g on itself Coadjoint representation of G on g Schouten–Nijenhuis bracket of D ðg Þ Rank n Grassmann algebra over k Map between sets Map between elements

Contents

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1 1 3 6

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8 11 16

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21 21

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26 30 35 44 48

3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Symplectic Vector Fields on a Cotangent Bundle . . . . . . 3.3 Lagrangian Submanifolds of a Cotangent Bundle . . . . . .

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57 61 68

4 Symplectic G-Spaces . . . . . . . . . . . . . . . . . . . 4.1 Definitions and Examples . . . . . . . . . . . . 4.2 Hamiltonian g-Spaces and Moment Maps 4.3 Equivariance of Moment Maps . . . . . . . .

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75 76 79 87

1 Some Algebra Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Skew-Symmetric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Orthogonality Defined by a Skew-Symmetric 2-Form . . . . . . . . 1.3 Symplectic Vector Spaces, Symplectic Bases . . . . . . . . . . . . . . 1.4 The Canonical Linear Representation of sl(2, k) in the Algebra of the Skew-Symmetric Forms on a Symplectic Vector Space . . 1.5 Symplectic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Symplectic Complex Structures . . . . . . . . . . . . . . . . . . . . . . . .

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2 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Symplectic Structures on Manifolds . . . . . . . . . . . . . . . . . . . . . 2.2 Operators of the Algebra of Differential Forms on a Symplectic Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Symplectic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hamiltonian Vector Fields and Symplectic Vector Fields . . . . . 2.5 Poisson Brackets Under Symplectic Coordinates . . . . . . . . . . . 2.6 Submanifolds of Symplectic Manifolds . . . . . . . . . . . . . . . . . .

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Contents

5 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Structure of a Poisson Manifold . . . . . . . . . 5.1.1 The Schouten–Nijenhuis Bracket . . . . . . 5.2 The Leaves of a Poisson Manifold . . . . . . . . . . 5.3 Poisson Structures on the Dual of a Lie Algebra

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91 91 91 95 98

6 A Graded Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 (0, n)-Dimensional Supermanifolds . . . . . . . . . 6.2 (0, n)-Dimensional Symplectic Supermanifolds . 6.3 The Canonical Symplectic Structure on T  P . .

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109 109 114 115

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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1

Some Algebra Basics

1.1 Skew-Symmetric Forms We denote by V a finite-dimensional vector space over a field k of characteristic not equal to 2, and denote by A p (V ) the vector space of skew-symmetric p-linear forms on V with values in k. In particular, A0 (V ) = k and A1 (V ) is the dual vector space V ∗ of V . Definition 1.1.1 The exterior product α ∧ β of a p-form α ∈ A p (V ) and a q-form · · × V is β ∈ Aq (V ) is the ( p + q)-form whose value at (x1 , . . . , x p+q ) ∈ V × · p+q factors

given by (α ∧ β)(x1 , . . . , x p+q ) =



sg(τ )α(xτ (1) , . . . , xτ ( p) )β(xτ ( p+1) , . . . , xτ ( p+q) ),

τ ∈S( p,q)

where S( p, q) is the set of the permutations of the set {1, 2, . . . , p + q} which satisfy τ (1) < τ (2) < · · · < τ ( p) and τ ( p + 1) < τ ( p + 2) < · · · < τ ( p + q), and sg(τ ) is the signature of the permutation τ (its value is 1 when the permutation τ is even and −1 when τ is odd). The exterior  product is associative, therefore it defines a graded algebra structure on A(V ) = p A p (V ). Using the exterior product definition, we can check that the graded algebra A(V ) is Z2 -commutative, that is, when α ∈ A p (V ) and β ∈ Aq (V ), we have α ∧ β = (−1) pq β ∧ α. Let ( f 1 , . . . , f n ) be a basis of V and let λ be a map from {1, 2, . . . , p} into {1, 2, . . . , n}. We set © Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_1

1

2

1 Some Algebra Basics

f λ = f λ(1) ∧ · · · ∧ f λ( p) . When λ runs through all maps from the set {1, 2, . . . , p} into the set {1, 2, . . . , n} such that λ(1) < λ(2) < · · · < λ( p), space A p (V ). If n = dim V , then the resulted collection of f λ forms a basis of the  n for any integer p ≥ 0, we have dim A p (V ) = . p Definition 1.1.2 The inner product of a p-form α ∈ A p (V )with an element x ∈ V is the ( p − 1)-form, denoted by i(x)α, whose value is given by i(x)α(x1 , . . . , x p−1 ) = α(x, x1 , . . . , x p−1 ), ∀ x1 , . . . , x p−1 ∈ V. Using the definition

of i(x),  we see that the map x −→ i(x) is a linear map from V into the space End A(V ) of endomorphisms of A(V ). For arbitrary x, y ∈ V , we have (1.1.1) i(x) ◦ i(y) + i(y) ◦ i(x) = 0. When α ∈ A p (V ) and β ∈ A(V ), we have i(x)(α ∧ β) = (i(x)α) ∧ β + (−1) p α ∧ (i(x)β).

(1.1.2)

This property means that i(x) is a Z2 -derivation of the graded algebra A(V ). Definitions 1.1.3 The kernel of a skew-symmetric p-form α on a vector space V is the vector subspace of V , denoted by ker α, formed by all elements x ∈ V such that i(x)α = 0. The rank of α, denoted by rankα, is the codimension of ker α. Thus rankα = dim V − dim ker α. Definition 1.1.4 Let V , W be two vector spaces over k, and let f be a linear map from V to W . For β ∈ A p (W ) and x1 , . . . , x p ∈ V , we set

 A( f )β (x1 , . . . , x p ) = β( f (x1 ), . . . , f (x p )).

This formula defines a p-form A( f )β ∈ A p (V ) called the pullback of β by f . The map A( f ) : A p (W ) −→ A p (V ), extended by linearity to a linear map A( f ) : A(W ) −→ A(V ), is a graded homomorphism of degree 0 of graded algebras, called the pullback of forms by f . From the definition of A( f ), we see that when f is injective (or onto), then A( f ) is also injective (or onto). The easy proof of the following result is left to the readers as an exercise. Proposition 1.1.5 Let α ∈ A p (V ), let N be a vector subspace of V such that N ⊂ ker α, and let q be the canonical projection from V onto V /N . There exists a unique

1.1 Skew-Symmetric Forms

3

p-form β ∈ A p (V /N ), called the pushforward of α by the projection q, such that α = A(q)β. Its kernel is ker β = q(ker α).

1.2 Orthogonality Defined by a Skew-Symmetric 2-Form Definitions 1.2.1 Let ω be a skew-symmetric 2-form on a finite-dimensional vector space V . Two elements x and y ∈ V are said to be orthogonal (with respect to ω) if ω(x, y) = 0. The orthogonal complement of a vector subspace E of V is the vector subspace E ⊥ of V formed by all elements x ∈ V which satisfy ω(x, y) = 0, ∀ y ∈ E. As an exercise, the reader can try to prove the properties stated without proof in the proposition below. Proposition 1.2.2 With the assumptions and notations of Definition 1.2.1, we have the following properties. (1) For any vector subspace E ⊂ V , E ⊥ ⊃ ker ω, (E ⊥ )⊥ = E + ker ω,

⊥ ⊥ ⊥ (E ) = E ⊥.

(2) If E and F are two vector subspaces of V , then (E + F)⊥ = E ⊥ ∩ F ⊥ . (3) If the vector subspaces E and F of V satisfy E ⊂ F, then E ⊥ ⊃ F ⊥ . (4) If the vector subspaces E and F of V satisfy ker ω ⊂ E ∩ F, then (E ∩ F)⊥ = E ⊥ + F ⊥. Lemma 1.2.3 Let ω ∈ A2 (V ) and let E be a vector subspace of V . Then we have dim E ⊥ = dim V − dim E + dim(E ∩ ker ω). Proof By the definition of the inner product (1.1.2), the map x −→ i(x)ω, x ∈ V , ∗ is a linear map with values

in V , and its kernel is ker ω. The dimension of the image of E is therefore dim i(E)ω = dim E − dim(E ∩ ker ω). By duality, the image i(E)ω is the orthogonal complement of E ⊥ . Its dimension is therefore equal to the  codimension of E ⊥ . Definitions 1.2.4 Let ω be a skew-symmetric 2-form on a vector space V . A vector subspace E of V is said to be • • • •

an isotropic vector subspace of (V, ω) if E ⊂ E ⊥ , a coisotropic vector subspace of (V, ω) if E ⊃ E ⊥ , a Lagrangian vector subspace of (V, ω) if E = E ⊥ , a symplectic vector subspace of (V, ω) if E ∩ E ⊥ = {0}.

4

1 Some Algebra Basics

Using Definitions 1.2.4, we can see that any vector subspace of dimension ≤1 is isotropic, and that any vector subspace of codimension ≤1 which contains ker ω is coisotropic. We can also see that, under the order relation of set inclusion, Lagrangian subspaces are maximal isotropic subspaces as well as minimal coisotropic subspaces. Proposition 1.2.5 Let ω ∈ A2 (V ) and let L be a Lagrangian vector subspace of (V, ω). We have rank ω = 2(dim L − dim ker ω). Proof Since L is a Lagrangian subspace, L ∩ ker ω = L ⊥ ∩ ker ω = ker ω. Thus, by Lemma 1.2.3, dim L = dim V − dim L + dim ker ω. Corollary 1.2.6 The rank of any skew-symmetric 2-form ω on V is an even number, and the dimensions of all Lagrangian vector subspaces of the space (V, ω) are the same. Proposition 1.2.7 Let ω ∈ A2 (V ) and and let W be a coisotropic vector subspace of (V, ω). For any Lagrangian vector subspace L of (V, ω), L ∩ W + W ⊥ is a Lagrangian vector subspace of (W, ω|W ). Proof Since both L and W are coisotropic, ker ω ⊂ L ∩ W . Thus (L ∩ W )⊥ = L ⊥ + W ⊥ = L + W ⊥ . Again, since W is coisotropic, (L ∩ W )⊥ ∩ W = (L + W ⊥ ) ∩ W = L ∩ W + W ⊥ . This shows that the orthogonal complement of L ∩ W + W ⊥ in (W, ω|W ) is itself. Proposition 1.2.8 Let ω ∈ A2 (V ), let N be a vector subspace of V that contained in ker ω, and let q : V −→ V /N be the canonical projection. Let ω ∈ A2 (V /N ) be the unique 2- form which satisfies the relation A(q)ω = ω. Then the map L −→ q(L) is a bijection from the set of Lagrangian vector subspaces of (V, ω) onto the set of Lagrangian subspaces of (V /N , ω ).

 Proof For any x and y ∈ V , we have ω q(x), q(y) = ω(x, y). Thus for any vector subspace E ⊂ V , q(E ⊥ ) is the orthogonal complement of q(E) with respect to ω . Therefore, if L is a Lagrangian vector subspace of (V, ω), then q(L) is a Lagrangian vector subspace of (V /N , ω ). Since N ⊂

ker ω,  any Lagrangian vector subspace of (V, ω) contains N . Therefore L = q −1 q(L) , which proves that the map L −→ q(L) is injective. To prove that this map is also onto, let L  be a Lagrangian subspace of (V /N , ω ), and let L = q −1 (L  ). Then q(L ⊥ ) = q(L)⊥ = L  , thus L ⊥ + N = L + N . Also since N ⊂ ker ω ⊂ L ⊥ and N = ker q ⊂ L, we have L ⊥ = L. Therefore L is a Lagrangian vector subspace of (V, ω) and q(L) = L  .

1.2 Orthogonality Defined by a Skew-Symmetric 2-Form

5

Proposition 1.2.9 Let ω ∈ A2 (V ), let L be a Lagrangian vector subspace of (V, ω), and let J be the set of all isotropic subspaces E of (V, ω) such that E ∩ L = {0}. If F is a maximal element of J for the order relation of set inclusion, then V is the direct sum of L and F. Proof For any x ∈ F ⊥ , F + kx is an isotropic vector subspace of (V, ω). Therefore we have either F + kx ⊂ F or (F + kx) ∩ L = {0}. In both cases, F ⊥ ⊂ F + L. Thus F ⊥ ∩ L = F ⊥ ∩ L ⊥ = (F + L)⊥ ⊂ (F ⊥ )⊥ = F + ker ω. So

and

F ⊥ ∩ L ⊂ (F + ker ω) ∩ L = ker ω, V = (F ⊥ ∩ L)⊥ = F + ker ω + L ⊥ = F + L . 

Definition 1.2.10 Under the assumptions of Proposition 1.2.9, a maximal element of J is called an isotropic complement vector subspace of the Lagrangian vector subspace L. Corollary 1.2.11 (First corollary of Proposition 1.2.9) Let L be a Lagrangian subspace of (V, ω), and let f 1 , . . . , fr be linearly independent 1-forms in the dual space V ∗ of V such that ker f i . L= 1≤i≤r

Then there are r linearly independent 1-forms fr +1 , . . . , f 2r in V ∗ such that f 1 , . . . , fr , fr +1 , . . . , f 2r are linearly independent and ω =

r i

f i ∧ fr +i .

Proof Let F be an isotropic complement subspace of L in V . Let e1 , . . . , er be a set of elements of F satisfying f i (e j ) = δi j , and let fr + j = i(e j )ω, j = 1, . . . , r. Then since F is isotropic, i(e j ) fr +i = ω(ei , e j ) = 0, i, j = 1, . . . , r. Thus, i(e j )(ω −

r 

f i ∧ fr +i ) = fr + j − fr + j = 0

i=1

holds for all 1 ≤ j ≤ r . Therefore the kernel of the form ω − ri=1 f i ∧ fr +i contains F. In addition, by the assumption of the corollary, the restriction of this form to L

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1 Some Algebra Basics

is 0, and by the definition of F, V = F + L . Therefore ω = ri=1 f i ∧ fr +i . To prove that f 1 , . . . , f 2r are linearly independent, note that the kernel of ω contains  1≤i≤2r ker f i , but its kernel has dimension 2 dim L − dim V = dim V − 2r, thus f 1 , . . . , f 2r are linearly independent.  Corollary 1.2.12 (Second corollary of Proposition 1.2.9) Let L be a Lagrangian subspace of (V, ω). Then there exists a Lagrangian subspace L˜ of (V, ω) such that L ∩ L˜ = ker ω, ˜ and thus V = L + L. Proof Keep the notations in Corollary 1.2.11, let L˜ = desired result follows.

 1≤i≤r

ker fr +i , then the 

Corollary 1.2.13 (Third corollary of Proposition 1.2.9) If k has characteristic 0, and the rank of ω is 2r , then ω’s r th exterior power ωr = 0 and its (r + 1)th exterior power ωr +1 = 0.

Proof Keep the notations in Corollary 1.2.11, and let ω = ri=1 f i ∧ fr +i . Then ωr = r ! f 1 ∧ fr +1 ∧ f 2 ∧ fr +2 ∧ · · · ∧ fr ∧ f 2r = (−1)

r (r −1) 2

r ! f 1 ∧ f 2 ∧ · · · ∧ f 2r .

Thus the desired results hold.



1.3 Symplectic Vector Spaces, Symplectic Bases In this section, V is assumed to be a finite-dimensional vector space over a field k of characteristic =2. Definition 1.3.1 A symplectic form on the vector space V is a skew-symmetric 2form whose kernel ker ω = 0. The vector space V endowed with a symplectic form ω is called a symplectic vector space and is denoted by (V, ω). The dimension of a symplectic vector space (V, ω), being equal to the rank of ω, is always an even integer 2n. The dimension of any Lagrangian vector subspace of (V, ω) is n, half the dimension of V . Examples 1.3.2 1. Let W be an r -dimensional vector space over the field k, and let W ∗ be the dual space of W . For any x1 and x2 ∈ W and any f 1 and f 2 ∈ W ∗ , we set ω(( f 1 , x1 ), ( f 2 , x2 )) = f 1 (x2 ) − f 2 (x1 ).

1.3 Symplectic Vector Spaces, Symplectic Bases

7

The 2-form ω so defined is a symplectic form on W ∗ × W . It is easy to see that W ∗ × {0} and {0} × W are both Lagrangian vector subspaces of (W ∗ × W, ω). 2. Let f 1 , . . . , f 2r be the natural coordinates of the vector space k 2r . Then ω=

r 

f i ∧ fr +i

i=1

is a symplectic form on k 2r , called the canonical symplectic form on k 2r . The symplectic vector space (k 2r , ω) is called the 2r -dimensional canonical symplectic k-space. According to Corollary 1.2.12, any Lagrangian vector subspace L of a symplectic space (V, ω) has a Lagrangian complement subspace in V , that is, there exists another ˜ Lagrangian vector subspace L˜ of (V, ω) such that L ∩ L˜ = {0} and V = L + L. Proposition 1.3.3 Let (V, ω) be a 2n-dimensional symplectic vector space, and let L 1 and L 2 two complementary Lagrangian vector subspaces of (V, ω). For any basis (e1 , . . . , en ) of L 1 , there exists a unique basis (en+1 , . . . , e2n ) of L 2 such that ω(ei , en+ j ) = δi j , 1 ≤ i, j ≤ n. Proof The map x −→ (i(x)ω)| L 1 is an isomorphism from L 2 to the dual space L ∗1 of L 1 . Let ( f 1 , . . . , f n ) be the basis of L ∗1 dual to (e1 , . . . , en ). For any 1 ≤ j ≤ n, choose en+ j ∈ L 2 such that (i(en+ j )ω)| L 1 = − f j , then (en+1 , . . . , e2n ) is a basis of L 2 and ω(ei , en+ j ) = f j (ei ) = δi j , 1 ≤ i, j ≤ n.  Definition 1.3.4 A symplectic basis1 of a 2n-dimensional symplectic vector space (V, ω) is a basis (e1 , . . . , e2n ) of V satisfies ω(ei , en+ j ) = δi j and ω(ei , e j ) = ω(en+i , en+ j ) = 0, i, j = 1, 2, . . . , n. According to Corollary 1.2.12 and Proposition 1.3.3, any symplectic vector space (V, ω) has a symplectic basis. Under a symplectic basis, the matrix of ω has the form 0 In , −In 0

 J2n = where In is the n × n identity matrix.

1 Added

by the authors of the Forewords. A symplectic basis of a symplectic vector space is often called, in other texts, a canonical basis or a Darboux basis.

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1 Some Algebra Basics

A basis (e1 , . . . , e2n ) of a 2n-dimensional symplectic vector space (V, ω) is a ∗ symplectic basis,

n if and only if, for the corresponding dual basis ( f 1 , . . . , f 2n ) of V , we have ω = i=1 f i ∧ f n+i . Exercise 1.3.5 Let L 1 and L 2 be two Lagrangian vector subspaces of a symplectic vector space (V, ω). Prove that there exists a vector subspace of V which is a Lagrangian complement to both L 1 and L 2 .

1.4 The Canonical Linear Representation of sl(2, k) in the Algebra of the Skew-Symmetric Forms on a Symplectic Vector Space In this section, k is a field of characteristic 0, and sl(2, k) denotes the 3-dimensional Lie algebra2 over k generated by the basis elements  0 0

 1 0 , 0 −1

 0 1 , 0 0

0 . −1

Let (V, ω) be a 2n-dimensional symplectic space over k. For α ∈ Ar (V ), let μ(α) be the endomorphism of A(V ) defined by β −→ α ∧ β, β ∈ A(V ). Then μ(α) is a degree r endomorphism of A(V ), that is, it maps each subspace A p (V ) into the subspace A p+r (V ). In particular X = μ(ω) is a degree 2 endomorphism. Let (e1 , . . . , e2n ) be a symplectic basis of (V, ω), then the endomorphism Y =

n j=1 i(e j ) i(en+ j ) is a degree −2 endomorphism. Lemma 1.4.1 For every x ∈ V , i(x) = Y ◦ μ( f ) − μ( f ) ◦ Y, where f = i(x)ω. Proof According to the definition, for any e j , 1 ≤ j ≤ 2n, i(e j ) is a degree −1 Z2 derivation, that is, for any α ∈ A p (V ) and any β ∈ Aq (V ), i(e j )(α ∧ β) = (i(e j )α) ∧ β + (−1) p α ∧ (i(e j )β). Thus for any β ∈ A(V ) we have (i(e j ) ◦ μ( f ))(β) = i(e j )((i(x)ω) ∧ β) = (i(e j ) i(x)ω) ∧ β − (i(x)ω) ∧ (i(e j )β), (μ( f ) ◦ i(e j ))(β) = f ∧ (i(e j )β) = (i(x)ω) ∧ (i(e j )β). 2 We assume that the readers have a basic knowledge of Lie algebras, Lie groups, and representation

theory. The readers who need to improve their knowledge of these subjects are referred to [13, 31].

1.4 The Canonical Linear Representation of sl(2, k) in the Algebra …

9

Also μ( f (e j ))(β) = f (e j ) ∧ β = ω(x, e j )β. Thus i(e j ) ◦ μ( f ) + μ( f ) ◦ i(e j ) = μ( f (e j )), 1 ≤ j ≤ 2n. Now let x =

2n j=1

x j e j . Then

μ( f (e j )) = −xn+ j , μ( f (en+ j )) = x j , 1 ≤ j ≤ n. Thus Y ◦ μ( f ) =

n 

i(e j ) i(en+ j )μ( f )

j=1

=

n 

x j i(e j ) −

j=1

μ( f ) ◦ Y =

n 

n 

i(e j )μ( f ) i(en+ j ),

j=1

μ( f ) i(e j ) i(en+ j )

j=1

=−

n 

xn+ j i(en+ j ) −

n 

j=1

i(e j )μ( f ) i(en+ j ).

j=1

Subtracting both sides to get Y ◦ μ( f ) − μ( f ) ◦ Y =

2n 

x j i(e j ) = i(x).

j=1

 Now we prove the endomorphism Y of A(V ) defined just before Lemma 1.4.1 is independent of the choice of the symplectic basis (e1 , . . . , e2n ).  ) be another symplectic basis of (V, ω), and let In fact, let (e1 , . . . , e2n Y =

n 

 i(ej ) i(en+ j ).

j=1

Then by Lemma 1.4.1 (Y  − Y ) ◦ μ( f ) = μ( f ) ◦ (Y  − Y ) holds for any 1-form f = i(x)ω, and thus it holds for any f ∈ V ∗ . Then for any f ∈ V ∗ , the kernel of Y  − Y is an invariant subspace of μ( f ). Thus since the algebra

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1 Some Algebra Basics

A(V ) is generated by V ∗ = A1 (V ), the kernel of Y  − Y is an ideal of A(V ). But it is obvious that the identity of A(V ) is in the kernel of Y  − Y . Therefore Y  = Y . If ( f 1 , . . . , f 2n ) is the basis of V ∗ dual to the basis (e1 , . . . , e2n ), then i(e j )ω = f n+ j , i(en+i )ω = − f j , 1 ≤ j ≤ n. Thus by formulas (1.1.1) and (1.1.2) [Y, X ] = Y ◦ X − X ◦ Y =

n  (− i(en+ j ) ◦ μ( f n+ j ) + μ( f j ) ◦ i(e j )) j=1

=

2n 

μ( f j ) ◦ i(e j ) − n · id,

j=1

where id is the identity map. Since the endomorphism 2n j=1 μ( f j ) ◦ i(e j ) of A(V ) is a degree zero derivation of A(V ), it is easy to see that it is the identity map on A1 (V ), thus its restriction on A p (V ) is equal to p · id. Let H = [Y, X ], then for any α ∈ A p (V ), H (α) = ( p − n)α. Proposition 1.4.2 Define a linear map ρ from sl(2, k) into the endomorphism space of A(V ) by ρ

 0 0

 1 0 = X, ρ 0 −1

 0 1 = Y, ρ 0 0

0 = H, −1

then ρ is a linear representation of the Lie algebra sl(2, k) on the space A(V ). Proof In fact, since [Y, X ] = H , and X and Y are derivations of degree 2 and degree −2, respectively, H is a degree 0 derivation of A(V ). Direct computation shows that [H, X ] = 2X, [H, Y ] = −2Y, thus ρ is a representation of sl(2, k). It is obvious that the eigenvectors of H in A(V ) are homogenous elements of A(V ). In particular, we call the eigenvectors of H that are contained in ker X \{0} the primitive elements of the representation ρ (see Ref. [13]). According to the representation theory of Lie algebras, if ϕ ∈ A(V ) is a primitive element and H (ϕ) = r ϕ, then r is an integer ≥0, and the following elements: ϕ, Y (ϕ), . . . , Y r (ϕ), form a basis of a simple sl(2, k) submodule in A(V ). Since it follows from H (ϕ) = r ϕ that ϕ ∈ An+r (V ), thus all primitive elements have gradings ≥n. Since ker X is a subspace of A(V ) generated by the primitive elements, so ker X is contained in An (V ) + An+1 (V ) + · · · + A2n (V ).

1.4 The Canonical Linear Representation of sl(2, k) in the Algebra …

11

Examples 1.4.3 1. Let ωn be the nth exterior power of ω, then 0 = ωn ∈ A2n (V ) and ωn is a primitive element. Under the representation ρ, it generates an n + 1-dimensional sl(2, k) submodule k + kω + · · · + kωn . 2. Since X is a degree 2 derivation, so any nonzero element in A2n−1 (V ) is a primitive element. Each of these primitive element generates an n-dimensional submodule. Proposition 1.4.4 For r = 0, 1, . . . , n, we have (1) X r maps An−r (V ) isomorphically onto An+r (V ); (2) Y r maps An+r (V ) isomorphically onto An−r (V ). Proof This is a direct consequence of the representation theory of Lie algebras (see Ref. [31] or [27]). Exercise 1.4.5 Show that the dimension of the subspace kerX ∩ An+r (V ) is equal to   2n 2n − . n +r n +r +2

1.5 Symplectic Groups Let (V1 , ω1 ) and (V2 , ω2 ) be two symplectic vector spaces over a field k, and let ϕ be a vector space isomorphism from V1 to V2 . If ϕ satisfies ω1 = A(ϕ)ω2 , then it is called an isomorphism from (V1 , ω1 ) onto (V2 , ω2 ). There exists an isomorphism between (V1 , ω1 ) and (V2 , ω2 ) if and only if dim V1 = dim V2 . In fact, if both V1 and V2 are 2n-dimensional, let (e1 , . . . , e2n )  )) be a symplectic basis of V1 (respectively, V2 ), then the (respectively, (e1 , . . . , e2n map ϕ : V1 −→ V2 defined by ϕ(ei ) = ei , i = 1, . . . , 2n, is an isomorphism from (V1 , ω1 ) to (V2 , ω2 ). The necessary condition is obvious. Let (V, ω) be a symplectic space. An automorphism of (V, ω) is an isomorphism from (V, ω) onto itself. So an automorphism s of (V, ω) is an element of the linear transformation group Gl(V ) that satisfies the following equation: ω(sx, sy) = ω(x, y), ∀ x, y ∈ V. It is easy to see that the automorphisms of (V, ω) form a subgroup of Gl(V ), we denote this subgroup by Sp(V, ω). In particular, the automorphism group of the canonical symplectic space (k 2n , ω) is denoted by Sp(2n, k). If k = R, then Sp(2n, k) is denoted simply as Sp(2n), and is called the 2n-dimensional symplectic group.

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Let (e1 , . . . , e2n ) be a symplectic basis of (V, ω). Let S ∈ Gl(2n, k) be a matrix. Then S is the matrix of an automorphism of (V, ω) with respect to the basis (e1 , . . . , e2n ) if and only if t S J2n S = J2n , where t S is the transpose of S, and J2n is the matrix of ω with respect to the basis (e1 , . . . , e2n ), that is In . 0



J2n

0 = −In

Let S ∈ Gl(2n, k) and write  S=

B , D

A C

where A, B, C, D are all n × n matrices. Then S ∈ Sp(2n, k) if and only if t

C A − t AC = 0,

t

D A − t BC = −In ,

t

C B − t AD = In , t

D B − t B D = 0.

(1.5.1)

Since det J2n = 1, if S ∈ Sp(2n, k), then the identity t S J2n S = J2n implies (det S)2 = 1. More precisely, we have the following proposition. Proposition 1.5.1 Let (V, ω) be a symplectic space, then for any s ∈ Sp(V, ω), we have det(s) = 1. Proof In fact, if dim V = 2n, then since A(s)ω = ω, we have A(s)ωn = ωn . Since ωn ∈ A2n (V ), A(s)ωn = det(s)ωn . If V is a vector space over a field k of characteristic 0, then we have ωn = 0 (Corollary 1.2.13), det(s) = 1. For the general case, use the divided power ω[n] instead of ωn for the discussion, just note that ω[n] is also a basis of A2n (V ). Let T be an indeterminate. We denote by k[T ] the one variable polynomial ring with the indeterminate T over k. Proposition 1.5.2 Let (V, ω) be a 2n-dimensional symplectic space, and let s ∈ Sp(V, ω). If P ∈ k[T ] is the characteristic polynomial of s, then we have T

2n

 1 = P(T ). P T

Proof In fact, denote simply J2n = J , I2n = I , then J 2 = −I . If S is the matrix of s with respect to some symplectic basis of V , then since t S J S = J , t S = −J S −1 J . Thus P(T ) = det(S − T I ) = det(t S − T I ) = det(−J S −1 J − T I ) = det(S −1 − T I ). Moreover, det(S) = 1, therefore

1.5 Symplectic Groups

13

P(T ) = det(S) det(S −1 − T I ) = det(I − T S) = T 2n P

 1 . T



Let L be a Lagrangian vector subspace of a symplectic vector space (V, ω). For any s ∈ Sp(V, ω), s(L) is obviously a Lagrangian subspace of (V, ω). Thus Sp(V, ω) acts on the set of all Lagrangian subspaces of (V, ω). ˆ Proposition 1.5.3 The group Sp(V, ω) acts transitively on the set of all pairs (L , L), where L and Lˆ are arbitrary mutual Lagrangian complementary Lagrangian subspaces of (V, ω). Proof In fact, according to Proposition 1.3.3, for any Lagrangian complementary pair ˆ there exists a symplectic basis (e1 , . . . , e2n ) of (V, ω) such that e1 , . . . , en (L , L), ˆ Therefore since Sp(V, ω) acts form a basis of L and en+1 , . . . , e2n form a basis of L. transitively on the set of all symplectic bases of (V, ω), the proposition follows. Let L be an arbitrary Lagrangian subspace of a symplectic space (V, ω), we denote the stabilizer of L in Sp(V, ω) by S(L). Proposition 1.5.4 Let L and Lˆ be two arbitrary mutual Lagrangian complementary Lagrangian subspaces of (V, ω). Then the map from S(L) into Gl(L) defined by ˆ onto Gl(L). s −→ s| L induces an isomorphism from S(L) ∩ S( L) Proof In fact, choose a symplectic basis of (V, ω) such that e1 , . . . , en form a basis ˆ Then by the relations in (1.5.1), the set of L and en+1 , . . . , e2n form a basis of L. ˆ S(L) ∩ S( L) contains those elements in Gl(V ) such that with respect to the basis (e1 , . . . , e2n ), their matrices have the form 

A 0

0 t −1 , A

where A ∈ Gl(n, k) is the matrix of the restriction s| L of s on L. Thus the proposition holds.  Corollary 1.5.5 Let S(L)0 be the kernel of the homomorphism s −→ s| L from S(L) to Gl(L). Then the group S(L) is the semi-direct product of the normal subgroup ˆ and the group S(L)0 acts simply transitively S(L)0 and the subgroup S(L) ∩ S( L), on the set of all Lagrangian complementary subspaces of L in (V, ω). Next, we determine the structure of S(L)0 . Let q be the canonical map from V onto V /L, and let B(V /L) be the space of the symmetric bilinear forms on V /L. Since the rank of ω is equal to the dimension of V , thus for any b ∈ B(V /L), there exists a unique endomorphism b˜ of V such that ˜ ω(b(x), y) = b(q(x), q(y)) holds for arbitrary x, y ∈ V .

(1.5.2)

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1 Some Algebra Basics

Proposition 1.5.6 The map b −→ idV + b˜ is an isomorphism from the additive group B(V /L) onto the subgroup S(L)0 of Sp(V, ω). ˜ Proof For any b ∈ B(V /L), we have b(L) = {0}, where b˜ is defined by (1.5.2). ˜ ) ⊂ L ⊥ = L, thus b˜1 b˜2 = 0 for all b1 , b2 ∈ B(V /L). Thus the map Also since b(V b −→ idV + b˜ is a group homomorphism from the additive group B(V /L) into the group Gl(V ). This map is injective. This is because q : V −→ V /L is onto, so if ˜ ˜ b˜ = 0, then b = 0. For any b ∈ B(V /L) and any x, y ∈ V , since ω(b(x), b(y)) = 0, thus ˜ ˜ ˜ ˜ ω(x + b(x), y + b(y)) = ω(x, y) + ω(b(x), y) + ω(x, b(y)) = ω(x, y) + b(q(x), q(y)) − b(q(y), q(x)) = ω(x, y). ˜ Thus idV + b˜ ∈ Sp(V, ω). Since b(L) = {0}, idV + b˜ ∈ S(L)0 , ∀ b ∈ B(V /L). Now if s ∈ S(L)0 , we define a bilinear form on V by (x, y) −→ ω(s(x) − x, y) = ω(x, s −1 (y) − y), ∀ x, y ∈ V. If at least one of x, y belongs to L, then obviously (x, y) = 0. Thus for any y ∈ V , we have s(y) − y = s −1 (s(y) − y) = y − s −1 (y). Therefore the bilinear form (x, y) −→ ω(s(x) − x, y) is symmetric. Since its kernel contains L, thus there exists b ∈ B(V /L) such that ω(s(x) − x, y) = b(q(x), q(y)), ∀ x, y ∈ V. Thus s = idV + b˜ and the image of the homomorphism b −→ idV + b˜ is S(L)0 . By Propositions 1.5.4 and 1.5.6, we obtain a canonical exact sequence {0} −→ B(V /L) −→ S(L) −→ Gl(L) −→ (1). From the discussions before, we also know that the group S(L) is isomorphic to the linear group formed by all matrices of the form 

A 0

B t −1 , A

where A ∈ Gl(n, k), and B is an n × n symmetric matrix. This matrix linear group is a linear group of dimension n 2 + n(n+1) . 2 Let (V, ω) be a symplectic space. Then the set of all Lagrangian subspaces of (V, ω) is a submanifold of the Grassmann manifold of the n-dimensional planes of V (assume that dim V = 2n), we denote it by L (V , ω). If L is a Lagrangian

1.5 Symplectic Groups

15

subspace of (V, ω), then the set of all Lagrangian complementary subspaces of L is a Zariski open subset of L (V , ω). By the Corollary of Proposition 1.5.4 and Proposition 1.5.6, there exists an affine space structure on this open subset, its affine transformation group is isomorphic to B(V /L). Since the dimension of B(V, L) is n(n+1) , so L (V , ω) is an n(n+1) dimensional manifold. While the group Sp(V, ω) 2 2 is a dim L (V , ω) + dim S(L) = n(n + 1) + n 2 = 2n 2 + n dimensional linear algebraic group. Let (V, ω) be a symplectic space. We denote by sp(V, ω) the subspace of the endomorphism space gl(V ) of V formed by all α such that ω(α(x), y) + ω(x, α(y)) = 0, ∀ x, y ∈ V. Proposition 1.5.7 The map α −→ (idV − α)(idV + α)−1 is a bijection from the subset of sp(V, ω) formed by all elements α such that idV + α is invertible to the subset of Sp(V, ω) formed by all elements s such that idV + s is invertible. Proof Let idV = I , then for any α ∈ sp(V, ω), the identity ω((I + α)(x), (I + α)(y)) = ω((I − α)(x), (I − α)(y)) holds for all x, y ∈ V . Thus, if I + α is invertible, then the identity ω((I − α)(I + α)−1 (x), (I − α)(I + α)−1 (y)) = ω(x, y) holds for all x, y ∈ V . Hence s = (I − α)(I + α)−1 ∈ Sp(V, ω). Also, since I + s = (I + α)(I + α)−1 + (I − α)(I + α)−1 = 2(I + α)−1 , so I + s is invertible. Conversely, if s ∈ Sp(V, ω) such that I + s is invertible, let α = (I + s)−1 (I − s), then for all x, y ∈ V ,

16

1 Some Algebra Basics

ω(α(x), y) + ω(x, α(y)) = −2ω(x, y) + 2ω((I + s)−1 (x), y) + 2ω(x, (I + s)−1 (y)). Let

x1 = (I + s)−1 x,

y1 = (I + s)−1 y,

then the previous formula implies ω(α(x), y) + ω(x, α(y)) = −2ω((I + s)(x1 ), (I + s)(y1 )) + 2ω(x1 , (I + s)(y1 )) + 2ω((I + s)(x1 ), y1 ) = 0. Thus α = (I + s)−1 (I − s) ∈ sp(V, ω). Also, since I + α = (I + s)−1 (I + s) + (I + s)−1 (I − s) = 2(I + s)−1 , so I + α is invertible, and s = (I − α)(I + α)−1 . Hence the map s −→ (I +  s)−1 (I − s) is the inverse of the map α −→ (I − α)(I + α)−1 . The bijection α −→ (I − α)(I + α)−1 in Proposition 1.5.7 is called the Cayley parametrization (see Ref. [30]). Remark 1.5.8 If α, β ∈ sp(V, ω), then [α, β] = αβ − βα ∈ sp(V, ω). If a bracket operation on sp(V, ω) is defined by the above formula, then sp(V, ω) becomes the Lie algebra of Sp(V, ω). If k = R or k = C, then the exponential map α −→ exp α maps sp(V, ω) into Sp(V, ω) (see Ref. [12]).

1.6 Symplectic Complex Structures In this section, (V, ω) is assumed to be a 2n-dimensional symplectic space over R. Since V is an even dimensional vector space, complex structures exist on V , that is, there exist endomorphisms j of V such that j 2 = −idV . If j is a complex structure of V and j ∈ Sp(V, ω), then we call j a symplectic complex structure. Thus, if j is a symplectic complex structure, then for any x, y ∈ V , ω( j (x), j (y)) = ω(x, y), and ω(x, j (y)) = ω( j (x), −y) = ω(y, j (x)).

1.6 Symplectic Complex Structures

17

Thus (x, y) = ω(x, j (y)), ∀ x, y ∈ V, is a symmetric bilinear form on V . It is obvious that this bilinear form is nondegenerate. For λ + iμ ∈ C, where λ, μ ∈ R, and x ∈ V , if V is defined as a vector space over C by (λ + iμ)x = λx + μj (x), then the complex valued real linear form h(x, y) = ω(x, j (y)) − iω(x, y), x, y ∈ V, is a pseudo-Hermitian form on V . In fact, for all x, y ∈ V , we have the following equalities: h(x, y) = h(y, x), h(i x, y) = h( j (x), y) = ω( j (x), j (y)) − iω( j (x), y) = ω(x, y) + iω(x, j (y)) = i h(x, y). Definition 1.6.1 A complex structure j on a symplectic space (V, ω) is called suitable, if it is symplectic and the symmetric bilinear form (x, y) = ω(x, j (y)) defined by j is positive definite. According to this definition, a complex structure j is a suitable complex structure is equivalent to that the form (x, y) −→ h(x, y) = ω(x, j (y)) − iω(x, y) is a Hermitian form. Proposition 1.6.2 Let (e1 , . . . , e2n ) be a symplectic basis of a symplectic space (V, ω), and let j be a complex structure on V such that for any 1 ≤ i ≤ n, j (ei ) = en+i . Then j is a suitable complex structure. Conversely, if j is a suitable complex structure, then there exists a symplectic basis (e1 , . . . , e2n ) of V such that j (ei ) = en+i , 1 ≤ i ≤ n. Proof In fact, if j (ei ) = en+i , 1 ≤ i ≤ n, then we have j (en+i ) = −ei , therefore the matrix of j with respect to the basis (e1 , . . . , e2n ) is  −J2n =

0 In

−In . 0

Thus by Sect. 1.4, j is symplectic. Also, with respect to the basis (e1 , . . . , e2n ), the matrix of the symmetric bilinear form ω(x, j (y)) is −(J2n )2 = I2n , thus it is positive definite. That is, j is a suitable complex structure. Conversely, let j be a

18

1 Some Algebra Basics

suitable complex structure on (V, ω). Let L be a Lagrangian subspace of (V, ω), let (e1 , . . . , en ) be a basis of L such that it is an orthonormal basis with respect to the bilinear form (x, y) −→ ω(x, j (y)), and let en+i = j (ei ), i = 1, . . . , n. Then (e1 , . . . , e2n ) is a symplectic basis of (V, ω).  Proposition 1.6.3 Let j be a suitable complex structure on (V, ω), then we have: (1) For any Lagrangian subspace L of (V, ω), j (L) is a complementary Lagrangian subspace of L. (2) Any complex subspace of V (that is, a subspace invariant under the action of j) is a symplectic space. Proof Since j ∈ Sp(V, ω), j (L) is a Lagrangian subspace. If x, y ∈ L, then we have ω(x, j ( j (y))) = −ω(x, y) = 0, thus j (L) is orthogonal to L with respect to the bilinear form (x, y) −→ ω(x, j (y)). Since this bilinear form is positive definite, we have L ∩ j (L) = {0}, thus j (L) is a complementary Lagrangian subspace of L, and (1) is proved. Now if E is a complex subspace of V and x ∈ E ∩ E ⊥ , then since j (x) ∈ E, ω(x, j (x)) = 0, and therefore  x = 0. Thus E ∩ E ⊥ = {0}. That is, E is a symplectic subspace. Lemma 1.6.4 Let j be a symplectic complex structure on a symplectic space (V, ω) and let s ∈ Gl(V ). The following conditions are equivalent: (1) s ◦ j = j ◦ s and ω(s(x), s(y)) = ω(x, y) for all x, y ∈ V , (2) s ◦ j = j ◦ s and ω(x, j (y)) = ω(s(x), j (s(y))) for all x, y ∈ V , (3) ω(s(x), s(y)) = ω(x, y) and ω(s(x), j (s(y))) = ω(x, j (y)) for all x, y ∈ V . Proof It is obvious that (1) implies (2) and (2) implies (3). If (3) holds, then ω(s(x), s( j (y)) − j (s(y))) = 0 holds for all x, y ∈ V . Thus s ◦ j = j ◦ s and (3) implies (1). Condition (1) of Lemma 1.6.4 amounts to say that s ∈ GlC ∩ Sp(V, ω), condition (2) amounts to say that s ∈ GlC ∩ O(V, b), where O(V, b) denotes the orthogonal transformation group of V with respect to the bilinear form b(x, y) = ω(x, j (y)), and (3) amounts to say that s keeps the Hermitian form h = b − iω invariant. If (V, ω) is the canonical symplectic space (R2n , ω), and j is the complex structure defined by −J2n , then b is the canonical Euclidean bilinear form b(ei , e j ) = δi j , i, j = 1, . . . , 2n, and h is the canonical Hermitian form h(ei , e j ) = δi j , , i, j = 1, . . . , n. Thus, we have the following proposition.

1.6 Symplectic Complex Structures

19

Proposition 1.6.5 We have Sp(2n) ∩ Gl(n, C) = O(2n) ∩ Gl(n, C) = U (n), where O(2n) denotes the orthogonal group, and U (n) denotes the unitary group. Corollary 1.6.6 The unitary group U (n) is a maximal compact subgroup of Sp(2n). Proof Let G be a compact subgroup of Sp(2n) and assume that G ⊃ U (n). Let b be the canonical Euclidean bilinear form on R2n . Since G is compact, there exists a positive definite symmetric bilinear form b˜ on R2n that is invariant under the action of G (see Ref. [5]). It is not hard to see that it is possible to choose λ ∈ R such that the rank of b − λb˜ < 2n. Since U (n) ⊂ O(2n), thus the kernel of b − λb˜ is a nonzero invariant subspace of U (n). Also, since U (n) acts transitively on the set of unitary ˜ = R2n , that is b = λb˜ and G ⊂ Sp(2n) ∩ O(2n) = U (n). vectors, so ker(b − λb) Thus G = U (n).  A B Exercise 1.6.7 Let S = be an element of Sp(2n), where A, B, C, D are C D n × n matrices. Let P be the set of all complex matrices Z such that Z is symmetric and 2i1 (Z − Z ) is positive definite. Prove that if Z ∈ P, then C Z + D is invertible. Prove that the map (S, Z ) −→ (AZ + B)(C Z + D)−1 , S ∈ Sp(2n), defines an action of the group Sp(2n) on P, that is, for all S ∈ Sp(2n) and Z ∈ P, (AZ + B)(C Z + D)−1 ∈ P. Prove that this action is transitive and find the stabilizer of i In ∈ P (see Ref. [22]).

Chapter 2

Symplectic Manifolds

In this chapter and the chapters that follow, “differentiable” means C ∞ -differentiable, “manifold” means C ∞ -differentiable real manifold, “differential form” and “vector field” mean exterior differential form and C ∞ vector field, respectively. Let M be a manifold. We denote by  p (M) the vector space of differential pforms on M and by (M) the graded algebra p≥0  p (M). For each vector field X on M, we can define two important operators on (M): the inner product through X , denoted by i(X ), and the Lie derivation θ (X ).The definition of i(X ) can be found in Chap. 1, and θ (X ) can be defined by the formula θ (X ) = d ◦ i(X ) + i(X ) ◦ d, where d is the exterior derivation of (M). We have the following relations between i(X ) and θ (X ) (see Ref. [12]): θ ([X, Y ]) = θ (X ) ◦ θ (Y ) − θ (Y ) ◦ θ (X ), i([X, Y ]) = θ (X ) ◦ i(Y ) − i(Y ) ◦ θ (X ), i(X ) ◦ i(Y ) + i(Y ) ◦ i(X ) = 0, where X and Y are vector fields on M. By the definition, the Lie derivation θ (X ) is a degree 0 derivation of the graded algebra (M), and the inner product i(X ) is a degree −1 Z2 -derivation. Let N be a submanifold of M and let α ∈  p (M). We denote the pullback of α in  p (N ) by α| N .

2.1 Symplectic Structures on Manifolds Definition 2.1.1 Let ω ∈ 2 (M) be a differential form. If ω satisfies the following two conditions: © Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_2

21

22

2 Symplectic Manifolds

1. for any x ∈ M, ωx is a symplectic structure on the tangent space Tx M at the point x, and 2. d ω = 0, then we call ω a symplectic structure on the manifold M, and call (M, ω) a symplectic manifold. Let (M1 , ω1 ) and (M2 , ω2 ) be two symplectic manifolds, and let ϕ : M1 −→ M2 be a differentiable map. If ϕ satisfies the condition ω1 = ϕ ∗ (ω2 ),

(2.1.1)

then we call ϕ a symplectic manifold homomorphism from (M1 , ω1 ) to (M2 , ω2 ). A symplectic manifold isomorphism from (M1 , ω1 ) to (M2 , ω2 ) is a diffeomorphism from M1 onto M2 that satisfies (2.1.1). If ϕ : (M1 , ω1 ) −→ (M2 , ω2 ) is a symplectic manifold homomorphism, then for any x ∈ M1 , the derivative of ϕ at the point x is a homomorphism between symplectic vector spaces: ϕxT : (Tx M1 , (ω1 )x ) −→ (Tϕ(x) M2 , (ω2 )ϕ(x) ). Thus, all symplectic manifold homomorphisms are immersion maps. 2.1.1 Basic properties. Let (M, ω) be a symplectic manifold. Then we have: 1. The dimension of M is an even number. 2. M is orientable. If dim M = 2n, then the nth exterior power ωn is a volume element on M. 3. If v ∈ Tx (M), then i(v)ωx ∈ Tx∗ (M). By using the map v −→ i(v)ωx , v ∈ Tx (M), a canonical isomorphism from the tangent bundle T M onto the cotangent bundle T ∗ M can be defined. Similarly, the map that sends a vector field X on M to a 1-form i(X )ω is a canonical isomorphism from the module of vector fields onto the module of 1-forms. 4. The fiber bundle of all order 1 coordinate frames on M contains a subbundle determined by the structure group Sp(2n), which is formed by the symplectic coordinate frames. Here, a symplectic coordinate frame means a coordinate frame ξ : R2n −→ Tx M such that A(ξ )ωx = ω, where ω is the canonical symplectic form on R2n (see Sect. 1.2). 5. If M is a compact manifold, then for i = 0, 1, . . . , n, the cohomology spaces (de Rham) H 2i (M, R) are all nonzero. In fact, the form ωi corresponds to the cohomology class [ωi ] = [ω]i ∈ H 2i (M, R). Since ωn is a volume element, [ωn ] = 0, thus [ωi ] = 0. Therefore there does not exist a symplectic structure on any sphere of dimension = 2. We can define a symplectic structure on the projective space C P 1 , which is homeomorphic to the 2-dimensional sphere.

2.1 Symplectic Structures on Manifolds

23

Remark 2.1.3 Let (M, ω) be a 2n-dimensional symplectic manifold. Then for any 0 = λ ∈ R, λω is a symplectic structure on M. If M is compact and |λ| = 1, then the symplectic manifold (M, ω) is not isomorphic to the symplectic manifold (M, λω), this is because          ωn  =  (λω)n  .     M

M

Examples 2.1.4 1. Let x1 , . . . , x2n be the natural coordinates on R2n , then the 2-form ω=

n 

d xi ∧ d xn+i

i=1

is a symplectic structure on the manifold R2n . We call this symplectic structure the canonical symplectic structure on R2n . 2. Let G be the group formed by the affine transformations of R2n such that their linear parts belong to Sp(2n). Then G contains the parallel translations, and it is the transitive automorphism group of the symplectic manifold (R2n , ω). If is a discrete subgroup of G, and it acts on R2n freely, then R2n \ is a manifold, and there exists a unique symplectic structure on R2n \ such that the canonical map R2n −→ R2n \ is a symplectic manifold homomorphism. If is the integer parallel translation group Z2n , then a symplectic structure can be defined on the torus R2n \Z2n such that it is invariant under the natural action of R2n on R2n \Z2n .  −→ M be a covering of M. 3. Let (M, ω) be a symplectic manifold, and let π : M ∗  Then ( M, π (ω)) is a symplectic manifold and π is a symplectic manifold homomorphism. 4. Let (M1 , ω1 ) and (M2 , ω2 ) be two symplectic manifolds, and let pri : M1 × M2 −→ Mi , i = 1, 2, be the canonical projections. Then pr1∗ (ω1 ) + pr2∗ (ω2 ) is a symplectic structure on M1 × M2 . We call the symplectic manifold (M1 × M2 , pr1∗ (ω1 ) + pr2∗ (ω2 )) the product of the symplectic manifolds (M1 , ω1 ) and (M2 , ω2 ), and denote it by (M1 , ω1 ) × (M2 , ω2 ). 2.1.5 Kähler structures. Let M be a 2n-dimensional manifold, and let J be a complex structure on M. As a tensor on M, J is of type (1, 1), thus we can view it as an endomorphism of the module of the vector fields on M. Then J satisfies the following conditions: (1) J 2 = −id, and (2) J [X, Y ] = [J X, Y ] + [X, J Y ] + J [J X, J Y ], where X and Y are two arbitrary vector fields on M. Let g be a differential symmetric 2-form on M such that g(J X, J Y ) = g(X, Y ) holds for all vector fields X, Y on M. Let ω(X, Y ) = g(J X, Y ), then ω ∈ 2 (M).

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2 Symplectic Manifolds

If the rank of g is equal to 2n at every point of M, that is, g is a pseudo Riemannian form, then for any x ∈ M, ωx is a symplectic form on the vector space Tx M and Jx is a symplectic complex structure on (Tx M, ωx ). The form h = g − iω is a pseudo Hermitian form on M. If in addition d ω = 0, then ω is a symplectic structure on M, and h is called a pseudo Kähler form. Finally, if d ω = 0 and g is positive definite on the tangent space at every point on M, then h is called a Kähler form on M. If h is a Kähler form, then for any x ∈ M, Jx is a suitable complex structure on (Tx M, ωx ). If h is a Kähler form on a complex manifold M, then for any complex submanifold N of M, the pullback of h on N is a Kähler form on N . In particular, all complex submanifolds of M are equipped with induced symplectic structures. Example 2.1.6 Let CP n be the complex projective space formed by all complex lines in Cn+1 . The tangent space at any point D ∈ CP n can be identified with the complex linear map space LC (D, Cn+1 /D). If η is the canonical Hermitian form on n+1 n+1 C , that is η = i=1 z i z i , where z 1 , . . . , z n+1 are the natural coordinates on Cn+1 , let D ⊥ be the hyperplane that orthogonal to D with respect to η, then the tangent space at D can be identified with the complex vector space LC (D, D ⊥ ). Thus, for any D ∈ CP n , we can define a Hermitian form on TD CP n as follows: η D (ϕ, ψ) =

η(ϕ(u), ψ(u)) , η(u, u)

where ϕ, ψ ∈ LC (D, D ⊥ ), u ∈ D\{0}. Let p be the canonical map from U = Cn+1 \{0} onto CP n given by u −→ Cu. Identify the tangent bundle T U with U × Cn+1 by using the following equality: (u, v) =

d (u + tv)|t=0 , u ∈ U, v ∈ Cn+1 . dt

Then for any (u, v) ∈ Tu U , we can define a vector p T (u, v) in LC (Cu, (Cu)⊥ ) such that η(v, u) u. ( p T (u, v))(u) = v − η(u, u) Then for (u, v), (u, w) ∈ Tu U we have ηCu ( p T (u, v), p T (u, w)) =

η(u, u)η(v, w) − η(v, u)η(u, w) , η(u, u)2

which is the value of the Hermitian form η˜ on U defined by  η˜ =

j

zjzj



  − z z z z z z d d d d j j j j j j j j j  2 j zjzj

2.1 Symplectic Structures on Manifolds

25

evaluated at the vector pair (u, v), (u, w). Thus, for any D ∈ CP n , η D is always the value of a Hermitian form η on CP n that satisfies p ∗ (η) = η˜ at the point D. Denote ˜ Let z j = x j + i y j , the imaginary part of η by ω, then p ∗ (ω) is the imaginary part of η. then p ∗ (ω) is equal to − where r =

 j

1 dr  1 ∧ (x j d y j − y j d x j ), d x j ∧ d yj + r j 2 r2 j

z j z j . Direct computation shows that p ∗ (ω) is a closed 2-form. Thus p ∗ (d ω) = d ( p ∗ (ω)) = 0.

Since p is also an immersion, d ω = 0. This proves that η is a Kähler form and ω is a symplectic structure on CP n . The action of the unitary group U (n + 1) on CP n is transitive and keeps the Kähler form η invariant. Let D ∈ CP n and let r D ∈ U (n + 1) be the reflection determined by the following condition:

r D (x) =

−x, x,

x ∈ D, x ∈ D⊥.

Then the action of r D on CP n fixes D, and for any ϕ ∈ TD CP n , (r D )T ϕ = −ϕ. Thus it is a “symmetry” centered at the point D. Thus for any differential p-form α on CP n , (r D∗ (α)) D = (−1) p α D . If p is an odd integer and α is invariant under U (n + 1), then α D = (r D∗ (α)) D = −α D holds for all D ∈ CP n . Thus α = 0. This implies that the U (n + 1) invariant differential forms on CP n are either even degrees or 0, and thus all are closed. Since η is a U (n + 1) invariant differential form, this also provides a different proof for the fact that the imaginary part of η is a closed form. It can be proved that all invariant differential forms on a homogeneous symmetric space are closed by using the same method (see Ref. [12]). Remark 2.1.7 There exist compact symplectic manifolds that do not have a Kähler structure, a 4-dimensional example was given by Thurston. The construction method is similar to that of 2.1.4 Example 2 (see Ref. [29]).

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2.2 Operators of the Algebra of Differential Forms on a Symplectic Manifold Let (M, ω) be a 2n-dimensional symplectic manifold. From the definition of a symplectic manifold, for any x ∈ M, (Tx (M), ωx ) is a symplectic space. Let x (M) be the algebra of the skew-symmetric forms on the vector space Tx M. As described in Sect. 1.3, there exists a canonical linear representation of the Lie algebra sl(2, R) on every x (M), which can be obtained by restricting a canonical linear representation of sl(2, R) on (M) to x (M). We define this canonical representation of sl(2, R) on (M) next. First, define an endomorphism X of (M) by X (β) = ω ∧ β, ∀ β ∈ (M). Second, define an endomorphism H of (M) by H (β) = ( p − n)β, ∀ β ∈  p (M). Let x ∈ M and let (e1 , . . . , e2n ) be a symplectic basis of (Tx (M), ωx ). Define an endomorphism Y of (M) such that (Y (β))x =

n 

i(e j ) i(en+ j )βx , ∀ β ∈ (M).

j=1

Locally, Y can be expressed as 1 i(E j ) i(E j ), 2 j=1 2n

Y =

where E 1 , . . . , E 2n and E 1 , . . . , E 2n are two collections of vector fields satisfying

ω(E i , E j ) = δi j . View (M) as a module over 0 (M) = C ∞ (M). Then it is easy to see that the operators X, H, Y are module endomorphism of (M). Note that this situation has been discussed in Sect. 1.3, and thus we have

[H, X ] = H ◦ X − X ◦ H = 2X, [H, Y ] = H ◦ Y − Y ◦ H = −2Y, [X, Y ] = X ◦ Y − Y ◦ X = −H.

(2.2.1)

Thus the operators X, H, Y define a canonical linear representation of the Lie algebra sl(2, R) on (M). This representation is important in the study of manifolds, in particularly in the study of Kähler manifolds (see Ref. [27]).

2.2 Operators of the Algebra of Differential Forms on a Symplectic Manifold

27

Now let ω be an exact form, that is, there is a 1-form α on M such that d α = −ω. We call the endomorphism of (M): β −→ α ∧ β, ∀ β ∈ (M), the left exterior product defined by α, and denote it by μ(α). Using α, we define two new operators P and Q on (M) as follows: P = d −μ(α) Q = [Y, P].

(2.2.2)

The operators P and Q are both order one differential operators. For example, for any f ∈ C ∞ (M) and any β ∈ (M), we have P( fβ) = f P(β) + d f ∧ β. Also, P and Q are degree 1 and degree −1, respectively, derivations, that is P( p (M)) ⊂  p+1 (M),

Q( p (M)) ⊂  p−1 (M).

(2.2.3)

Proposition 2.2.1 The operators X, Y, H, P and Q satisfy the following relations: (1) (2) (3) (4)

[H, P] = P, [X, P] = 0, [Y, P] = Q, P 2 = X, Q 2

[H, Q] = −Q; [X, Q] = −P; [Y, Q] = 0; = Y, P ◦ Q + Q ◦ P = H .

Proof The formulas in (1) can be derived from (2.2.3) directly. Now we prove (2). Since d ω = 0, thus [d, X ] = 0, and thus by (2.2.2) and the definition of α, [X, P] = 0. According to the definition, Q = [Y, P], so we have [X, Q] = [X, [Y, P]] = [[X, Y ], P] + [Y, [X, P]] = −[H, P] = −P. Thus (2) is proved. Also since d α = −ω, we have d ◦μ(α) + μ(α) ◦ d = −X, thus P 2 = (d −μ(α))2 = X. Since Q = [Y, P], so P ◦ Q + Q ◦ P = P ◦ Y ◦ P − P2 ◦ Y + Y ◦ P2 − P ◦ Y ◦ P = [Y, X ] = H. Now it remains to prove that [Y, Q] = 0 and Q 2 = Y . Let a = RX + RY + RH

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2 Symplectic Manifolds

be the subalgebra of the endomorphism Lie algebra gl((M)) of the space (M) generated by X, Y, H , then a is isomorphic to sl(2, R). Let ε be the subspace of gl((M)) spanned by the following elements: P, ad(Y )P, . . . , ad(Y )r P, . . . , where ad is the adjoint representation of gl((M)) (that is, ad(U )V = [U, V ], U, V ∈ gl((M))). Since ad(Y )r = 0 for integers r > n, the space ε is finite dimensional. Since for all r ≥ 0, ad(H ) ad(Y )r P = (2r + 1) ad(Y )r P, thus ε is invariant under the actions of both ad(Y ) and ad(H ). Since ad(X ) ad(Y )r − ad(Y )r ad(X ) = r (1 − r ) ad(Y )r −1 − r · ad(H ) ad(Y )r −1 , so ε is also invariant under ad(X ). Since ad(X )P = 0 and ad(H )P = P, so P is a weight 1 primitive element for the restriction of the adjoint representation ad of a to ε. We know that (see Refs. [13] or [31]), if this is the case, then ad(Y )2 P = 0, and thus [Y, Q] = ad(Y )Q = ad(Y )2 P = 0. Write Q 2 differently as Q 2 = Q ◦ Y ◦ P − Q ◦ P ◦ Y = Y ◦ Q ◦ P − Q ◦ P ◦ Y, and as Q 2 = Y ◦ P ◦ Q − P ◦ Y ◦ Q = Y ◦ P ◦ Q − P ◦ Q ◦ Y, then from what we have proved, we get 2Q 2 = [Y, H ] = 2Y .



By using the concept of “Lie superalgebras”, the relations in Proposition 2.2.1 can be well described in “Lie superalgebra” language. Next, we will define Lie superalgeras and provide some examples. Definition 2.2.2 (See Ref. [14]). A Lie superalgebra over a field k is a Z2 -graded vector space g = g0 + g1 over k, together with a bilinear map (called a bracket) [ , ] : g × g −→ g such that the following (1)–(3) hold: (1) [g p , gq ] ⊂ g p+q , p, q ∈ Z2 , (2) [x, y] = −(−1) p·q [y, x], ∀ x ∈ g p , y ∈ gq , (3) [x, [y, z]] = [[x, y], z] + (−1) p·q [y, [x, z]], ∀ x ∈ g p , y ∈ gq , z ∈ g.

2.2 Operators of the Algebra of Differential Forms on a Symplectic Manifold

29

Examples 2.2.3  1. Let V = p∈Z V p be a Z-graded vector space. Denote by gl(V )r the set of all elements α in gl(V ) such that α(V p ) ⊂ V p+r . Define a bracket on the Z-graded space gl(V )∗ = r ∈Z gl(V )r by [α, β] = α ◦ β − (−1) pq β ◦ α, ∀ α ∈ gl(V ) p , β ∈ gl(V )q . With this bracket, if a new gradation is defined by regrading the original gradation in Z according to modulo 2 congruence classes, then the space gl(V )∗ becomes a Lie superalgebra.  2. Let A = p∈Z A p be a Z-graded associative algebra (A p Aq ⊂ A p+q ), and let gl(A) p be defined as in Example 1. If α ∈ gl(A) p and α(x y) = α(x)y + (−1) pr xα(y), ∀ x ∈ Ar , y ∈ A, then we call α a Z2 -derivation of A. Let Der (A) p be the subspace formed by all Z2 -derivations in gl(A) p , then Der (A)∗ =



Der (A) p

p∈Z

is a Lie supersubalgebra of gl(A)∗ , that is, it is a graded subspace that is invariant under the bracket of gl(A)∗ . 3. Let (e−1 , e0 , e1 ) be the natural basis of R3 . Define a Z -gradation on R3 as follows: the grading of e p = p,

p = −1, 0, 1.

Let b be a bilinear form on R3 such that with respect to the basis (e−1 , e0 , e1 ), it has the matrix ⎛ ⎞ 0 01 ⎝ 0 1 0⎠ . −1 0 0 Then the restriction of b on Re−1 + Re1 is a symplectic form, while its restriction on Re0 is a positive definite symmetric form (b is called an orthogonal symplectic form). Let osp(2, 1) be the Z-graded subspace of gl(3, R) = gl(R3 )∗ such that its elements of grading p are those elements α ∈ gl(3, R) p satisfying b(α(x), y) + (−1) pr b(x, α(y)) = 0, ∀ x ∈ Rer , y ∈ R3 . Then osp(2, 1) is a Lie super subalgebra of gl(3, R) (cf. Example 1). It is of 5dimensional, and it has a basis given by the following elements:

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2 Symplectic Manifolds

⎛ ⎞ ⎛ 001 1 X = ⎝0 0 0⎠ , H = ⎝0 000 0 ⎛ ⎞ 010 P = ⎝0 0 1⎠ , 000

⎞ 0 0 0 0 ⎠, 0 −1 ⎛ 0 Q = ⎝1 0

⎛

⎞ 0 00 Y = ⎝ 0 0 0⎠ , −1 0 0 ⎞ 0 0 0 0⎠ . −1 0

The Z-gradings of the elements X , P, H , Q, Y are, respectively, 2, 1, 0, −1, −2. The subspace of osp(2, 1) spanned by the even degree elements is a subalgebra isomorphic to sl(2, R). Now let (M, ω) be a symplectic manifold, and let X, H, Y, P, Q be the operators on (M) defined by using an α satisfying d α = −ω as in the beginning of this section. Then it is not hard to prove (the reader can prove this as an exercise), that the map ρ from osp(2, 1) into gl((M)) defined by ρ(X ) = X, ρ(H ) = H, ρ(Y ) = Y, ρ(P) = P, ρ(Q) = Q, is a Lie superalgebra homomorphism. Thus a choice of α determines a linear representation of osp(2, 1) on the graded space (M). The Lie superalgebra osp(2, 1) is rather similar to sl(2, R) (see Refs. [14, 21]). For example, osp(2, 1) is a simple algebra, that is, it does not contain any ideal subalgebra other than {0} and itself. All finite dimensional representations of osp(2, 1) are semisimple (i.e. completely reducible). For any integer n ≥ 0, there exists a unique 2n + 1-dimensional simple linear representation (irreducible linear representation). Corresponding to this representation, the weights of H are all integers in the interval [−n, n]. If n > 0, then the representation decomposes into the direct sum of two simple representations of dimensions n and n + 1, respectively, of the subaglebra sl(2, R) ⊂ osp(2, 1). Finally, we point out that the dimensions of all finite dimensional simple representations of osp(2, 1) are odd numbers.

2.3 Symplectic Coordinates By using a theorem of Darboux, the following conclusion can be made: two symplectic manifolds with the same dimension is locally isomorphic. In this section, we use induction on the dimension to give a proof for this conclusion. Moser (see Ref. [29])

2.3 Symplectic Coordinates

31

pointed out an alternative proof1 by using transformation, we will describe his result in Sect. 2.6. Let M be a manifold, and let α be an arbitrary p-form on M. Consider the elements v of the tangent bundle T M that satisfy the condition: if v ∈ Tx M, then i(v)αx = 0. The set of these v is called the kernel of α, and is denoted by ker α. If ker α is a sub-vector bundle of T M with the dimensions of the fibers equal to dim M − r , then α is said to have constant rank r . Lemma 2.3.1 Let M be a manifold, let ϕ : M −→ Rs be a submersion, and let α ∈  p (M). If ker ϕ T ⊂ (ker α) ∩ (ker d α), then for any point x 0 of M, there exists an open neighborhood U of x 0 and a form β in (ϕ(U )), such that α|U = ϕ ∗ (β). Proof Let n = dim M, let y1 , . . . , ys be the natural coordinates of Rs , and let xi = yi ◦ ϕ. Since ϕ is a submersion, there exist an open neighborhood V of x0 in M and differentiable functions xs+1 , . . . , xn on V , such that x1 , . . . , xn is a coordinate system on V . For any i = s + 1, . . . , n, the vector field ∂∂xi is a cross section of the fiber bundle ker ϕ T on V . Let S( p, n) be the set of all maps τ : [1, p] −→ [1, n] such that τ (1) < τ (2) < · · · < τ ( p), and let the coordinate expression of α|V be α|V =



f τ d xτ (1) ∧ · · · ∧ d xτ ( p) .

τ ∈S( p,n)

Since for j > s, i



∂ ∂x j



α = 0, so if τ ( p) > s, then f τ = 0. Therefore,

α|V =



f τ d xτ (1) ∧ · · · ∧ d xτ ( p) .

τ ∈S( p,s)

 Also for j > s, i ∂∂x j d α = 0, so for all j > s, ∂∂x j f τ = 0. Thus, there exists an open neighborhood of x0 , say U , such that in this neighborhood, the function f τ has the form f τ = ϕ ∗ (gτ ), where gτ is a differentiable function on ϕ(U ). Let β=



gτ d yτ (1) ∧ · · · ∧ d yτ ( p) ,

τ ∈S( p,s)

then we have α|U = ϕ ∗ (β). 1 Added



by the authors of the Forewords. Originally, Moser introduced his method for manifolds endowed with a volume form. The use of this method for a proof of Darboux’ theorem is due to the French mathematician Jean Martinet (Sur les singularités de formes différentielles, thèse de doctorat d’État, Grenoble, 1969, and Ann. Inst. Fourier Grenoble, 20, 1970, 95–178). Symplectic coordinates are often called canonical coordinates or Darboux coordinates in other texts.

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Proposition 2.3.2 Let M and N be two manifolds, and let ϕ : M −→ N be an onto submersion. If the fibers of ϕ are connected and α is a differential form on M such that ker ϕ T = (ker α) ∩ (ker d α), then there exists a unique differential form β on N such that α = ϕ ∗ (β). Proof According to Lemma 2.3.1, for any x ∈ M, there exists an open neighborhood Ux of x and a differential form βx on ϕ(Ux ) such that α|Ux = ϕ ∗ (βx ). Consider βx as a cross section of the fiber bundle on the space N , then for arbitrary x, y ∈ M, we have βx = β y on ϕ(Ux ∩ U y ). Also since the fibers of ϕ are connected, βx = β y on ϕ(Ux ) ∩ ϕ(U y ). Thus there exists a differential form β on ϕ(M) = N such that the equation βx = β|ϕ(Ux ) holds for all x ∈ M. Then α = ϕ ∗ (β). Also, since ϕ T : T M −→ T N is an onto map, β is uniquely determined by the required equation.  Let M be a manifold. A sub-vector bundle E of the vector bundle T M is called integrable if: (1) it is a differentiable manifold, and (2) for arbitrary differentiable cross sections X, Y on E, [X, Y ] = X ◦ Y − Y ◦ X is also a cross section of E. For example, if ϕ : M −→ N is a constant rank differentiable map, then the subfiber bundle ker ϕ T ⊂ T M is integrable. The Frobenius theorem described below shows, locally, all integrable sub-bundles of T M can be obtained by using the same method of the above example. The reader can find the proof of this theorem in any book about manifolds. Theorem 2.3.3 (Frobenius’ theorem) Let M be a manifold, and let E be an integrable sub-bundle of T M. Let the dimension of the fibers of E be r , and let the dimensions of the fibers of T M be n. Then locally, there exist coordinates x1 , . . . , xn on M such that  ker d xi . E= r 0. Since for all i, {xi , a} ⊂ a, thus a is invariant under the action of ∂∂xi for all i. Thus, a contains a polynomial f = f 0 + f 1 , where f 0 ∈ P0 , f 1 ∈ P1 \{0}. Then for any g ∈ P, we have { f, g} = { f 1 , g}. So since the map g −→ { f 1 , g}, g ∈ P,  is onto (Lemma 2.5.1), a = P. Also, {0} and P0 are obviously ideals. 2.5.3 The Lie algebra of formal symplectic vector fields. Let (M, ω) be a symplectic manifold and let x ∈ M. Let m ∞ x be the maximal ideal formed by the functions of p C ∞ (M) whose values at the point x are zero, and let m x the maximal ideal formed p by the functions of C (M) whose values at the point x are zero. Then m∞ x =



m xp .

p>0

According to Proposition 2.4.11, for any f ∈ C ∞ (M), the map g −→ { f, g}, g ∈ p p−1 C ∞ (M), is a derivation of C ∞ (M). Thus for any p > 0, { f, m x } ⊂ m x , so ∞ ∞ { f, m ∞ x } ⊂ m x . This implies that m x is not only an ideal for the associative algebra ∞ structure on C (M), but it is also an ideal for the Lie algebra structure defined by the Poisson bracket. Thus, we can define a quotient algebra by using m ∞ x . We call the quotient algebra Jx (M) = C ∞ (M)/m ∞ x the jet algebra at the point x. It is a Lie algebra. We also call the Lie algebra bracket on Jx (M) a Poisson bracket and denote it by { , }. Proposition 2.5.4 {0}, R, and Jx (M) are the only ideals in the Lie algebra Jx (M). Proof {0} and R are obviously ideals of Jx (M). We show that except {0} and R, the only ideal in Jx (M) is Jx (M) itself. For this, choose a symplectic coordinate system on some neighborhood of x that is zero at the point x. Then the case under 2n consideration is reduced to the case of the jet algebra J0 (R n ) of functions of the 2n symplectic manifold (R , ω) at the origin, where ω = i=1 d xi ∧ d xn+i is the canonical symplectic structure. As an associative algebra, J0 (R2n ) is equal to the algebra of formal series on x1 , . . . , x2n . If f =

 p≥0

f p,

f p ∈ Pp ;

g=

 p≥0

g p , g p ∈ Pp ,

2.5 Poisson Brackets Under Symplectic Coordinates

then { f, g} =



47

{ f p , gq }.

p,q≥0

Let a be an ideal of the Lie algebra J0 (R2n ) that is not contained in R = P0 . Since {xi , a} ⊂ a, i = 1, . . . , 2n, a is invariant under the action of the partial  derivatives ∂ , i = 1, . . . , 2n. Thus, in a, there exists at least one element h = p≥0 h p , h p ∈ ∂ xi Pp , such that h 1 is a nonzero element in P1 . Then the map g −→ {h, g}, g ∈ J0 (R2n ), is an onto endomorphism of J0 (R2n ). In fact, if g=



g p ∈ J0 (R2n ),

p≥0

then the term of {h, g} contained in Pr is {h, g}r = {h 1 , gr +1 } + {h 2 , gr } + · · · + {h r +1 , g1 }. Thus by Lemma 2.5.1, for any f =



f p ∈ J0 (R2n ),

p≥0

we can define g p recursively such that fr = {h 1 , gr +1 } + {h 2 , gr } + · · · + {h r +1 , g1 } holds for any r . Let g = arbitrary, a = J0 (R2n ).

 p≥0

g p , then {h, g} = f . But h ∈ a, so f ∈ a. Since f is 

Let X be a vector field on a manifold M. Then the Lie derivation θ (X ) : f −→ X f keeps m ∞ x invariant. By taking quotient, it induces a derivation Jx (X ) on the jet algebra Jx (M), which we call a jet of the vector field X at the point x. If (M, ω) is a symplectic manifold, then the composition map of the following two maps f −→ H f

and

X −→ Jx (X )

is a homomorphism from the Lie algebra C ∞ (M) into the Lie algebra formed by the derivations of Jx (M). Since any symplectic vector field coincides with a Hamiltonian vector field on some neighborhood of x, the image of the homomorphism map f −→ Jx (H f ) is the jet Lie algebra of the symplectic vector fields at the ∞ ∞ point x. If f ∈ m ∞ x , then H f C (M) ⊂ m x , so Jx (H f ) = 0. By taking quotient, we obtain an onto homomorphism from the Lie algebra Jx (M) to the Lie algebra of the jets of the symplectic vector fields at the point x. If f ∈ C ∞ (M) and ∞ ∞ Jx (H f ) = 0, then H f C ∞ (M) ⊂ m ∞ x , thus for all g ∈ C (M), { f, g} ∈ m x . This

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implies f − f (x) ∈ m ∞ x . Thus Jx ( f ) is the jet of the constant f (x). Summarize, we obtain an exact sequence {0} −→ R −→ Jx (M) −→ Jx (S(M, ω)) −→ {0}.

Proposition 2.5.5 The Lie algebra formed by the jets of all symplectic vector fields on a symplectic manifold at a given point is a simple Lie algebra. Proof Since Jx (S(M, ω)) is isomorphic to Jx (M)/R, thus this proposition is actually a different version of the statement of Proposition 2.5.4. 

2.6 Submanifolds of Symplectic Manifolds Let (M, ω) be a symplectic manifold and let N be a submanifold of M. If x ∈ N , then Tx N is a vector subspace of the symplectic vector space (Tx (M), ωx ). Definition 2.6.1 A submanifold N of a symplectic manifold (M, ω) is called an isotropic (or coisotropic, Lagrangian, symplectic, respectively) submanifold, if for all x ∈ N , the space Tx N is an isotropic (or coisotropic, Lagrangian, symplectic, respectively) subspace of the symplectic space (Tx M, ωx ). For an immersion of a manifold P into a symplectic manifold (M, ω), we use the same terminologies. For example, an immersion ϕ : P −→ M is called a Lagrangian immersion, if for all y ∈ P, ϕ T always maps Ty P to a Lagrangian subspace of (Tϕ(y) , ωϕ(y) ). Regardless whether a submanifold N of (M, ω) is isotropic, coisotropic, Lagrangian, or symplectic, the pullback ω| N of ω on N is always a constant rank 2-form. If N is coisotropic, then the rank of ω| N is equal to 2 dim N − dim M. If N is a Lagrangian submanifold of M, then dim N = 21 dim M. Let N be a submanifold of M. We define a vector bundle TN M over the base space N by letting the fiber of TN M over the point x ∈ N be Tx M. So each fiber has a symplectic vector space structure. Consider T N as a vector subbundle of TN M, and denote the subbundle of TN M whose fiber at the point x is (Tx N )⊥ by T N ⊥ . Then, in order for the submanifold N to be isotropic (or coisotropic), it is necessary and sufficient that T N ⊂ T N ⊥ (or T N ⊃ T N ⊥ ). In order for N be Lagrangian, it is necessary and sufficient that T N = T N ⊥ . If T N ∩ T N ⊥ = {0}, then N is a symplectic submanifold. Lemma 2.6.2 (A. Weinstein’s extension lemma) Let M be a 2n-dimensional manifold, and let N be a closed submanifold of M. Let ω0 , ω1 be two symplectic structures on M such that their restrictions on 2 (TN M) are equal. Then there exist two open neighborhoods U0 and U1 of N in M, and a diffeomorphism ϕ1 from U0 onto U1 , such that the restriction of ϕ1 on N is the identity map and ω0 = ϕ1∗ (ω1 ).

2.6 Submanifolds of Symplectic Manifolds

49

Proof We need to quote a general version of the following Poincaré Lemma (see Ref. [29]): Since N is a closed submanifold, there exist an open neighborhood V of N in M and a 1-form β on V such that β is zero on TN M and d β = (ω0 − ω1 )|V . Denote the projection from R × M onto R by t, view the differential forms on R and their pullbacks on R × M by the projection as the same, then we can assume that ω˜ = ω0 + t (ω1 − ω0 ) ∈ 2 (R × M). The rank of this differential form is 2n at every point of R × N . Thus its rank is also 2n on some open neighborhood of R × N in R × M, we let V be such an open neighborhood. Since R × V is an open neighborhood of R × N , we can choose V such that V ⊂ R × V . According to the definition of β and the choice of V , we know that the kernel of β|V is a sub-vector bundle of T V generated by the vector field ∂t∂ .   Since i ∂t∂ β = 0, there exists a vector field X on V such that i(X )ω˜ = β|V . Also, we can choose X such that i(X ) d t = 0. This is possible since it amounts to choosing a vector field on a neighborhood of N so it does not depend on the parameter t. Since β is zero on TN M, so the vector field X is identically zero on R × N . Thus, there exists an open neighborhood W of [0, 1] × N in R × M and a differentiable map ϕ : W −→ R × M, such that (1) ϕ T ◦ ∂t∂ = (X + ∂t∂ ) ◦ ϕ, and (2) ϕ(0, x) = x, ∀ (0, x) ∈ W . Note that in (ϕ(W )), condition (1) is equivalent to  θ

∂ ∂t



∗

∗

◦ϕ =ϕ ◦θ



∂ X+ ∂t

 .

Apply this to the function t to get ∂t∂ (t ◦ ϕ) = 1. Thus t ◦ ϕ = t on some neighborhood of [0, 1] × N (refer to the figure).

For any x ∈ N , by the definition of W , there exists an open neighborhood Vx of x in M such that [0, 1] × Vx ⊂ W . Thus, there exists an open neighborhood U0 of N

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in M such that [0, 1] × U0 ⊂ W . For any a ∈ [0, 1], define a map ϕa from U0 into M by ϕ(a, x) = (a, ϕa (x)), ∀ x ∈ U0 . ˜ on U0 does not We show that the family of differential forms ϕa∗ (ω)  depend on a. ˜ Note that this family of differential forms is actually given by i ∂t∂ (d t ∧ ϕ ∗ (ω)),   so we apply θ ∂t∂ to get  θ

∂ ∂t

        ∂ ∂ ∂ i (d t ∧ ϕ ∗ (ω)) ϕ ∗ (ω) ˜ =i ˜ dt ∧θ ∂t ∂t ∂t     ∂ ∂ ϕ ∗ (ω) ω˜ ˜ = ϕ∗θ X + =θ ∂t ∂t = ϕ ∗ (d i(X )ω˜ + ω1 − ω0 ) = d β + ω1 − ω0 = 0.

Thus ϕ ∗ (ω) ˜ does not depend on the parameter a ∈ [0, 1], and thus ϕ1∗ (ω1 ) = ϕ0∗ (ω0 ). Since ϕ0 is the identity map, so we have ϕ1∗ (ω1 ) = ω0 , that is, ϕ1 is an isomorphism from (U0 , ω0 ) onto (ϕ1 (U0 ), ω1 ). Also since X is 0 on R × N , so the restriction of  ϕ1 on N is the identity map. Remarks 2.6.3 1. Since we assume that ω0 and ω1 are equal not only on 2 T N but on 2 TN M, so we are able to choose a 1-form β on some neighborhood of N such that d β = ω0 − ω1 . Moreover, we can choose such a β, so it is not just that βx = 0 for any x ∈ N , but also that the first order jet flow of β is zero at any point of N . When β has these properties, the corresponding homeomorphism ϕ1 provides a tangent bundle map ϕ1T which is equal to the identity on TN M (see Ref. [29]). 2. By applying Lemma 2.6.2 to N = {x}, it implies that there exist symplectic coordinates on some neighborhood of x ∈ M. Proposition 2.6.4 Let (M, ω) be a symplectic manifold, and let N be a closed Lagrangian submanifold of (M, ω). Then for any point x ∈ N , there exist an open neighborhood V of x and symplectic coordinates y1 , . . . , y2n on V , such that N ∩ V is the set of points in V satisfying y1 = . . . = yn = 0. Proof In fact, let U be an open neighborhood of x, and let x1 , . . . , x2n be a coordinate system on U such that for the points in N ∩ U , x1 = x2 = . . . = xn = 0 hold. n Then by applying Lemma 2.6.2 to the symplectic structures ω0 = ω|U and ω1 = i=1 d xi ∧  d xn+i , we obtain the desired symplectic coordinates yi = ϕ1∗ (xi ).

2.6 Submanifolds of Symplectic Manifolds

51

The Lagrangian submanifolds in a symplectic manifold are important in symplectic manifold theory, they are related to the discussions of many topics. We first consider a couple of examples, then give a basic Lagrangian submanifold construction method. Examples 2.6.5 1. Let (M, ω) be a 2-dimensional symplectic manifold. Then its Lagrangian submanifolds are the 1-dimensional submanifolds, in other words, all its embedded curves. 2n 2. We now consider the Lagrangian submanifolds n of (R , ω). Let x1 , . . . , x2n be 2n symplectic coordinates on R such that ω = i=1 d xi ∧ d xn+i . Let U be an open subset of Rn , let ϕ : x −→ (ϕ1 (x), . . . , ϕn (x)) be a differentiable map from U into Rn , and let Nϕ = {(x, ϕ(x)) : x ∈ U }. Then Nϕ is called the graph of ϕ in U × Rn ⊂ R2n . In order for Nϕ to be a Lagrangian submanifold of (R2n , ω), it is necessary and sufficient that the immersion x −→ n(x, ϕ(x)), x ∈ U, is an isotropic immersion. This condition can be expressed as i=1 d xi ∧ d ϕi = 0. State it differently, in order a Lagrangian submanifold of (R2n for Nϕ to be , nω), it is necessary and sufficient that n ϕi d xi is a closed form. If i=1 ϕi d xi is the total differential of the 1-form i=1 some function f : U −→ R, then of course it is closed. In this spacial case, we call f a generating function of the Lagrangian submanifold Nϕ . Let ψ : (x, y) −→ (ψ1 (x, y), . . . , ψn (x, y)) be a differentiable map from U × Rn into Rn . We define another map  by using ψ:  : (x, y) −→ (x, y + ψ(x, y)), (x, y) ∈ U × Rn . Then  is a differentiable map from U × Rn into itself. In order for  to be an endomorphism of the symplectic manifold (U × Rn , ω), it is necessary and sufficient that n  d xi ∧ d ψi = 0. i=1

n ψi d xi This amounts to the requirement that ψ(x, y) does not depend on y and i=1 is a closed form on U . If ψ satisfies these two conditions,the  is an automorphism n ψi d xi is an arbitrary of the symplectic manifold (U × Rn , ω). Conversely, if i=1 closed differential 1-form on U , then we can define a  as above by using the coefficients ψi of the d xi , i = 1, . . . , n, then obtain an automorphism of (U × Rn , ω). Thus, there exists a canonical isomorphism between the additive group of the closed

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differential forms on U and the group of automorphisms of the symplectic manifold (U × Rn , ω) that keep the fiber Rn invariant. The restriction of an element of this automorphism group on the fiber {x} × Rn is a parallel transformation. The corresponding Lagrangian submanifold Nϕ is the image of the Lagrangian submanifold U × {0} under the map . Proposition 2.6.6 Let (M1 , ω1 ) and (M2 , ω2 ) be two symplectic manifolds, and let ϕ : M1 −→ M2 be a differentiable map. Then ϕ is a homomorphism of symplectic manifolds if and only if the graph {(x, ϕ(x)) : x ∈ M1 } is an isotropic submanifold of the symplectic manifold (M1 , ω1 ) × (M2 , −ω2 ). Proof In fact, let ϕ˜ be the immersion from M1 into M1 × M2 defined by x −→ (x, ϕ(x)), x ∈ M1 . Then we have

ϕ˜ ∗ ( pr1∗ (ω1 ) − pr2∗ (ω2 )) = ω1 − ϕ ∗ (ω2 ).

Thus ϕ is a symplectic manifold homomorphism is equivalent to the graph of ϕ being  an isotropic submanifold of (M1 , ω1 ) × (M2 , −ω2 ). If dim M1 = dim M2 , then the graph of a symplectic manifold homomorphism from (M1 , ω1 ) into (M2 , ω2 ) is a Lagrangian submanifold, because in this case, the graph of this homomorphism is a dimension dim M1 isotropic submanifold of (M1 , ω1 ) × (M2 , −ω2 ). But a Lagrangian submanifold of (M1 , ω1 ) × (M2 , −ω2 ) is not necessary a graph of some homomorphism. 2.6.7 Constructing Lagrangian submanifolds using the contraction method. Let (M, ω) be a symplectic manifold, and let N ⊂ M be a coisotropic submanifold of M. Moreover, assume that (M , ω ) is another symplectic manifold, and ϕ : N −→ M is a submersion such that ϕ ∗ (ω ) = ω| N . Since N is a coisotropic submanifold, ω| N has constant rank. Thus by the Corollary of Proposition 2.3.5, we can always find a symplectic manifold (M , ω ) and a submersion ϕ such that they satisfy the condition above at some neighborhood of any point of N . Let L be a Lagrangian submanifold of (M, ω). We assume that L and N have a “good intersection”, that is (1) L ∩ N is a submanifold of M, and (2) T (L ∩ N ) = T L ∩ T N .

2.6 Submanifolds of Symplectic Manifolds

53

Let x ∈ L ∩ N . Since Tx N is a coisotropic subspace of (Tx M, ωx ), the subspace Tx L ∩ Tx N + Tx N ⊥ is a Lagrangian subspace of (Tx N , (ω| N )x ) (Proposition 1.2.7). Since ϕxT : Tx N −→ Tϕ(x) M is an onto map and ϕ ∗ (ω ) = ω| N , the subspace ϕxT (Tx L ∩ Tx N + Tx N ⊥ )

). But is a Lagrangian subspace of (Tϕ(x) M , ωϕ(x)

T N ⊥ ⊂ ker ϕ T and Tx L ∩ Tx N = Tx (L ∩ N ),

). thus for any x ∈ M, ϕ T (Tx (L ∩ N )) is a Lagrangian subspace of (Tϕ(x) M , ωϕ(x) 1

This shows that the restriction of ϕ on L ∩ N is of constant rank 2 dim M . Locally on N , this restriction factorizes into a submersion followed by a Lagrangian immersion into M . Thus, for any x ∈ L ∩ N , there exists a neighborhood V of x in N such that ϕ(V ) is a Lagrangian submanifold of (M , ω ). We say that a Lagrangian submanifold of (M , ω ) constructed this way is obtained by contracting the Lagrangian submanifold L. We consider an example next.

Example 2.6.8 Let z j = x j + i y j , j = 1, . . . , n + 1, be the natural coordinates on  Cn+1 and let h = n+1 d z j d z j be the canonical Hermitian form. The imaginary n+1 j=1 part of h is − j=1 d x j ∧ d y j . This is a symplectic structure on Cn+1 . If we denote by p the canonical projection from Cn+1 \{0} onto the projective space CP n , then by Example 2.1.6 in Sect. 2.1, there exists a symplectic structure ω on CP n such that p ∗ (ω) = −

1 1 dr  (x j d y j − y j d x j ), d x j ∧ d yj + r j 2 r2 j

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2 Symplectic Manifolds

 where r = j z j z j . Let S 2n+1 be the sphere defined by r = 1. Since its codimension  is 1, it is a coisotropic submanifold of (Cn+1 , − j d x j ∧ d y j ). We have p ∗ (ω)| S 2n+1 = −



d x j ∧ d y j | S 2n+1 .

j

By restricting p on S 2n+1 , we obtain the needed conditions of constructing Lagrangian 3 submanifolds of CP n in the contraction method  by using the Lagrangian submann+1 ifolds of the symplectic manifold (C , − j d x j ∧ d y j ). For example, if E is a  Lagrangian vector subspace of the symplectic space (Cn+1 , j x j ∧ y j ), then it is a  Lagrangian submanifold of the symplectic manifold (Cn+1 , − j d x j ∧ d y j ), and E ∩ S 2n+1 is an n-dimensional sphere. The restriction of the canonical projection p on E ∩ S 2n+1 is an immersion, by taking quotient, a homeomorphism from the real projective space determined by E onto some Lagrangian submanifold of CP n can be obtained from it. 2.6.9 Poisson bracket and coisotropic submanifolds. Let (M, ω) be a symplectic manifold and let N be a coisotropic submanifold of (M, ω). If the restriction of a function f ∈ C ∞ (M) to N is a constant, then for any vector v ∈ T N , (i(H f )ω)(v) = d f (v) = 0. Thus for any x ∈ N , (H f )(x) ∈ T N ⊥ . Since N is coisotropic, T N ⊥ ⊂ T N , thus for any x, (H f )(x) ∈ T N . Therefore, the Hamiltonian vector fields defined by the functions whose restrictions to N are constants are tangent to N . Now let f, g ∈ C ∞ (M) and assume that both their restrictions on N are constants. Then since H f is tangent to N , for any x ∈ N , { f, g}(x) = (H f g)(x) = 0. This shows that all differentiable functions on M whose restrictions to N are constants form a Lie subalgebra of C ∞ (M) with respect to the Poisson bracket. We now further assume that the codimension of N is k. Let f 1 , . . . , f k ∈ C ∞ (M) be functions that are zero on N and for all the points on N , d f 1 ∧ · · · ∧ d f k = 0. Then for any x ∈ N , the vectors (H f1 )(x), . . . , (H fk )(x) form a basis of Tx N ⊥ , and Tx N is the orthogonal complement of the subspace of (Tx M, ωx ) spanned by (H f1 )(x), . . . , (H fk )(x). Exercise 2.6.9 Let (M, ω) be a symplectic manifold and let E be an integrable subbundle of T M. We further assume that E is a Lagrangian subbundle, that is, for all x ∈ M, E x is a Lagrangian subspace of (Tx M, ωx ). Let P be a submanifold of M. If P satisfies the two conditions below: (1) Tx P = E x , ∀ x ∈ P, and 3 Added by the authors of the Forewords. The contraction method for the construction of Lagrangian

immersions is often called the use of Morse families of functions in other texts. It is due to the Swedish mathematician Lars Hörmander.

2.6 Submanifolds of Symplectic Manifolds

55

(2) under the set inclusion relation, P is a maximal element in the set of all submanifolds of M that satisfy condition (1), then we call P an integral leaf of E. Since we assume that E is a Lagrangian subbundle, every integral leaf of E is a Lagrangian submanifold of (M, ω). We say that E defines a Lagrangian foliation on (M, ω). Prove that if X, Y are two differentiable cross sections of E, then there exists a unique vector field  X Y such that for any vector field Z on M, ω( X Y, Z ) = X ω(Y, Z ) − ω(Y, [X, Z ]), and that  X Y is a cross section of E. Also, prove that for two differentiable cross sections X, Y of E (see Ref. [28]),  X Y − Y X = [X, Y ],  X (Y Z ) − Y ( X Z ) = [X,Y ] Z , where Z is a vector field on M. Thus derive that, for any integral leaf P of E, there exists a linear connection , whose curvature and torsion are zero, and it can be characterized as follows4 : if X and Y are two differentiable cross sections of E, then  X | P (Y | P ) = ( X Y )| P . We will give examples of Lagrangian foliations by using cotangent bundles in Sect. 3.1.

4 Added by the authors of the Forewords. The existence of a flat torsionless connection on the leaves

of a Lagrangian foliation of a symplectic manifold was observed by the French mathematician Paulette Libermann in her thesis (Sur le problème d’équivalence de certaines structures infinitésimales régulières, thèse de doctorat d’État, Strasbourg, 1953, and Ann. Mat. Pura Appl. 36, 1954, 27–120).

Chapter 3

Cotangent Bundles

3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles In this section, we denote by P a manifold, and denote the cotangent bundle on P by T ∗ P. The fiber Tx∗ P of T ∗ P at any point x ∈ P is the dual space of the vector space Tx P, and the elements in Tx∗ P are the cotangent vectors at the point x. We use π and π∗ to denote the projections of T P and T ∗ P on P respectively. We use T (T ∗ P) to denote the tangent bundle of the cotangent bundle T ∗ P and use π0 to denote the projection of T (T ∗ P) on the base space T ∗ P. Denote the tangent space map defined by π∗ as π∗T , then π∗T : T (T ∗ P) −→ T P. It is not hard to see that the following diagram is commutative T (T ∗ P) T∗P

π0

π∗T

TP

π∗

π

P Thus the map ϕ1 defined by ϕ1 : v −→ (π0 (v), π∗T (v)), v ∈ T (T ∗ P), is a differentiable map from T (T ∗ P) into the fiber bundle product T ∗ P × T P. Define a map ϕ2 from T ∗ P × T P into R by the usual method:

P

P

© Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_3

57

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3 Cotangent Bundles

ϕ2 : (ξ, u) −→ ξ(u), (ξ, u) ∈ T ∗ P × T P. P

Then ϕ2 is also a differentiable map. The composition map ϕ2 ◦ ϕ1 of ϕ1 and ϕ2 is a differentiable map from T (T ∗ P) into R, its restriction on every fiber Tξ (T ∗ P) (ξ ∈ T ∗ P) is a linear map. Thus it is a differential 1-form on the manifold T ∗ P, and it is a very important differential 1-form. For our convenience, if (ξ, u) ∈ T ∗ P × T P, P

we denote the real number ξ(u) as ξ, u.

Definition 3.1.1 Define a differential 1-form α on the cotangent bundle T ∗ P as follows. For any v ∈ T (T ∗ P), let α(v) = π0 (v), π∗T (v).

(3.1.1)

We call α the Liouville form on T ∗ P. It is obvious that if α is the Liouville form on T ∗ P, then ker α ⊃ ker π∗T . We call a vector v ∈ T (T ∗ P) that satisfies π∗T (v) = 0 a vertical vector. Then the Liouville form is zero on the vertical vectors of T (T ∗ P). Let X be an arbitrary vector field on P. We use X to define a differentiable function f X on T ∗ P as follows: f X (ξ ) = ξ, X (π∗ (ξ )), ∀ ξ ∈ T ∗ P, where X (π∗ (ξ )) denotes the value of the vector field X at the point π∗ (ξ ) ∈ P. Thus f X = 0 if and only if X = 0. Thus the map F : X −→ f X ,

X is vector field on P,

is a real linear inclusion map. Its image is a subspace of C ∞ (T ∗ P), the restrictions of every function of this subspace on the fibers of T ∗ P are linear. If g ∈ C ∞ (P) and X is a vector field on P, then f g X = (g ◦ π∗ ) f X . Let U be an open subset of P, let x1 , . . . , xn be coordinates on U , and let yi = f ∂∂x , i = 1, . . . , n. i

Then the functions (x1 ◦ π∗ ), . . . , (xn ◦ π∗ ), y1 , . . . , yn form a coordinate system on π∗−1 (U ) = T ∗ U . To simplify our notation, we identify an element g in C ∞ (P) with the element g ◦ π∗ in C ∞ (T ∗ P). According to this assumption, x1 , . . . , xn , y1 , . . . , yn form a coordinate system on π∗−1 (U ). If the expression of a vector field X on U is  n ∂ −1 i=1 ai ∂ xi , then on π∗ (U ), we have fX =

n  i=1

ai yi .

(3.1.2)

3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles

59

For any x ∈ P, ξ ∈ Tx∗ P, and v ∈ Tξ (T ∗ P), we have π∗T (v) =

n 

 d xi (π∗T (v))

i=1

∂ ∂ xi

 . x

Thus since π0 (v) = ξ , α(v) = π0 (v), π∗T (v) =

n 

yi (ξ ) d xi (π∗T (v)),

i=1

where yi (ξ ) = ξ,



∂ ∂ xi

 x

. But

d xi (π∗T (v)) = (π∗∗ d xi )(v) = (d(xi ◦ π∗ ))(v) = d xi (v), thus on π∗−1 (U ) we have α=

n 

yi d xi .

(3.1.3)

i=1

Proposition 3.1.2 Let α be the Liouville form on the cotangent bundle T ∗ P, then the 2-from ω = − d α is a symplectic structure on T ∗ P. Proof In fact, according to (3.1.3), if x1 , . . . , xn are coordinates on a coordinate neighborhood U in P, then there are coordinates x1 , . . . , xn , y1 , . . . , yn on π∗−1 (U ) such that n  ω= d xi ∧ d yi . i=1

Thus the rank of ω at every point of T ∗ P is always 2n. Also, since it is a closed form, it is a symplectic structure.  We call the 2-form ω = − d α the canonical symplectic structure on T ∗ P. Let U ⊂ P be a coordinate neighborhood in P, and let x1 , . . . , xn be coordinates on U . Then xi = xi ◦ π∗ and yi = f ∂∂x , i = 1, . . . , n, i

are symplectic coordinates on π∗−1 (U ). We call them local symplectic coordinates on T ∗ P induced by x1 , . . . , xn . Canonical symplectic structures play an important role in classical mechanics. By using the Legendre transformation, we can transform a Lagrangian dynamical system on a tangent bundle T Q (see 2.4.4 Example 1) to a Hamiltonian vector field on the cotangent bundle T ∗ Q that is equipped with the canonical symplectic structure (see reference [2]).

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We know that any differential 1-form on a manifold P corresponds to a differentiable cross section of the vector bundle T ∗ P. If β ∈ 1 (P), we denote the corresponding cross section by β. Proposition 3.1.3 The Liouville form α on T ∗ P can be characterized by the followT ∗ ing property: for any differential 1-form β, β = α ◦ β (= β (α)). Proof Let x ∈ P, u ∈ Tx P, and β ∈ 1 (P). Then since π∗ ◦ β is the identity map T T on P, we have (π∗T ◦ β )(u) = u. On the other hand, since π0 ◦ β = β ◦ π, by formula (3.1.1), T

T

T

(α ◦ β )(u) = (π0 ◦ β )(u), (π∗T ◦ β )(u) = β(x), u = β(u), (x = π(u)). T

This proves β = α ◦ β . Furthermore, for any point ξ ∈ Tx∗ P, the tangent space T Tξ (T ∗ P) is generated by the collection of vectors in all β (Tx P), where β ∈ 1 (P) T and β(x) = ξ . Thus α is uniquely determined by the equality β = α ◦ β .  Let P and Q be two manifolds, and let ϕ : P −→ Q be a differentiable map. If we assume that ϕ is a local diffeomorphism, then for any x ∈ P, ϕxT is an isomorphism from Tx P onto Tϕ(x) Q. Thus we have an inverse isomorphism (ϕxT )−1 : Tϕ(x) Q −→ Tx P. The conjugate map of the isomorphism above defines an isomorphism ∗ Q. Tx∗ ϕ : Tx∗ P −→ Tϕ(x)

Thus we can define a differentiable map T ∗ ϕ from T ∗ P onto T ∗ Q, such that the restriction of T ∗ ϕ on each fiber Tx∗ P is the Tx∗ ϕ above. Then, for any x ∈ P, u ∈ Tx P, and ξ ∈ Tx∗ P, we have (3.1.4) (T ∗ ϕ)(ξ ), ϕ T (u) = ξ, u. If β ∈ 1 (Q), then (T ∗ ϕ)((ϕ ∗ β)x ), ϕ T (u) = (ϕ ∗ β)x , u = βϕ(x) , ϕ T (u). Therefore,

(T ∗ ϕ) ◦ (ϕ ∗ β) = β ◦ ϕ.

Proposition 3.1.4 Let P and Q be two manifolds, and let α P and α Q be the Liouville forms on T ∗ P and T ∗ Q respectively. Let ϕ : P −→ Q be a differentiable map, and assume that ϕ is a local homeomorphism, then α P = α Q ◦ (T ∗ ϕ)T . Proof In fact, by (3.1.1) and (3.1.4), for any ξ ∈ T ∗ P and v ∈ Tξ (T ∗ P),

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61

(α Q ◦ (T ∗ ϕ)T )(v) = (π0 ◦ (T ∗ ϕ)T )(v), (π∗T ◦ (T ∗ ϕ)T )(v) = (T ∗ ϕ)(ξ ), (ϕ T ◦ π∗T )(v) = ξ, π∗T (v) = π0 (v), π∗T (v) = α P (v). 

Therefore the equality holds. ∗

∗

Corollary 3.1.5 Assumptions are as in Proposition 3.1.4. Then T ϕ : T P −→ T ∗ Q is a symplectic homomorphism. Example 3.1.6 By using cotangent bundles as symplectic manifolds, we can obtain a lot of examples of Lagrangian foliations. In fact, if P is a manifold, then we can take the canonical symplectic structure so that T ∗ P is a symplectic manifold. Take a coordinate neighborhood U in P, let x1 , . . . , xn be coordinates on U . Then on the submanifold π∗−1 (U ) of T ∗ P, there exist the symplectic coordinates x1 , . . . , xn , y1 , . . . , yn induced by x1 , . . . , xn . Consider π∗−1 (U ) as a symplectic manifold, and let E=



ker d xi .

1≤i≤n

Then E is an integrable Lagrangian subbundle of T (π∗−1 (U )). Each fiber space of π∗−1 (U ) is both an integrable leave of E and a Lagrangian submanifold of π∗−1 (U ), thus E defines a Lagrangian foliation on π∗−1 (U ).

3.2 Symplectic Vector Fields on a Cotangent Bundle In this section, P denotes a manifold, α denotes the Liouville form on T ∗ P, and ω denotes the canonical symplectic structure − d α. Let U be a coordinate neigh, . . . , yn be borhood in P, let x1 , . . . , xn be coordinates on U , and let x1 , . . . , xn , y1 n yi ∂∂yi . the symplectic coordinates on π∗−1 (U ) induced by x1 , . . . , xn . Set C = i=1 Then C is a vector field on π∗−1 (U ) and π∗T ◦ C = 0, that is, C is a vertical vector field.1 It is obvious that we can define a vector field on P such that locally, this vector field has the expression given above. We again denote this vector field by C. Lemma 3.2.1 Keep the above notation. We have i(C)ω = −α, i(C)α = 0, θ (C)α = α, θ (C)ω = ω. Proof To verify the first  formula, choose local symplectic coordinates x1 , . . . , xn , n yi d xi (see Sect. 3.1) and compute directly. By using y1 , . . . , yn such that α = i=1 the first formula, we have 1 Added

by the authors of the Forewords. This vector field, which can be defined on the total space of any vector bundle, is often called the Liouville vector field in other texts.

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3 Cotangent Bundles

i(C)α = − i(C)2 ω = −ω(C, C) = 0. Thus the second formula holds. Then by using the already proved first and second formulas, θ (C)α = d i(C)α + i(C) d α = − i(C)ω = α. Thus proves the third formula. Finally, by using θ ◦ d = d ◦θ , θ (C)ω = −θ (C) d α = − d θ (C)α = − d α = ω.



Lemma 3.2.2 For any vector field X on P, C f X = f X , where f X is defined as in Sect. 3.1. Proof We use formula (3.1.2). Choose local coordinates x1 , . . . , xn on P and local symplectic coordinates x1 , . . . , xn , y1 , . . . , yn on T ∗ P, such that locally, f X =  n ∗ i=1 ai yi , where the ai ’s are the pullbacks on T P of some functions on P. By the definition of C, Cai = 0, thus C fX =

n 

ai C yi = f X .

i=1



Proposition 3.2.3 Let X be a symplectic vector field on T ∗ P. Then: (1) X + [C, X ] is a Hamiltonian vector field and X + [C, X ] = Hα(X ) . (2) [C, X ] is a symplectic vector field. Proof According to Lemma 3.2.1, θ (C)ω = ω and i(C)ω = −α, thus we have i(X + [C, X ])ω = i(X )ω + θ (C) i(X )ω − i(X )θ (C)ω = θ (C) i(X )ω = d i(C) i(X )ω + i(C) d i(X )ω, where in the first step, we have used the following formula in Chap. 2: i([X, Y ]) = θ (X ) ◦ i(Y ) − i(Y ) ◦ θ (X ). Also, since d i(X ) = θ (X ) − i(X ) d, and X is a symplectic vector field, i(C) d i(X ) ω = 0. Moreover, by using the identity i(C) i(X ) + i(X ) i(C) = 0, i(X + [C, X ])ω = − d i(X ) i(C)ω = d i(X )α = d(α(X )). Thus X + [C, X ] = Hα(X ) is a Hamiltonian vector field. But according to the properties of a Hamiltonian vector field (Sect. 2.4), it is a symplectic vector field, thus [C, X ] is a symplectic vector field. 

3.2 Symplectic Vector Fields on a Cotangent Bundle

63

Corollary 3.2.4 (Fist corollary of Proposition 3.2.3) For any function f ∈ C ∞ (T ∗ P), we have [C, H f ] = H(C f − f ) . α(H f ) = C f, Proof In fact, direct computation shows α(H f ) = i(H f )α = − i(H f ) i(C)ω = i(C) i(H f )ω = i(C) d f = C f. Then by Proposition 3.2.3, [C, H f ] = Hα(H f ) − H f = H(C f − f ) .



Corollary 3.2.5 (Second corollary of Proposition 3.2.3) For any f, g ∈ C ∞ (T ∗ P), we have C{ f, g} = {C f, g} + { f, Cg} − { f, g}, where { , } is the Poisson bracket. Proof In fact, C{ f, g} = C(H f g) = [C, H f ]g + H f Cg = {C f, g} − { f, g} + { f, Cg}.



For any integer r , let Er be the set of all functions f in C ∞ (T ∗ P) such that C f = (r + 1) f. According to the definition, the elements in Er are those differentiable functions on T ∗ P whose restrictions on every fiber Tx∗ P are homogeneous polynomials of degree r + 1. Thus, if r < −1, then we have Er = {0}. While the vector space E −1 is the space formed by all functions that are constants on every fiber, that is, the space formed by all pullbacks of the functions of C ∞ (P) on T ∗ P. Given any two integers r, s, we have Er E s ⊂ Er +s+1 , and by second corollary of Proposition 3.2.3, for ∀ f ∈ Er , ∀ g ∈ E s , C{ f, g} = (r + 1){ f, g} + (s + 1){ f, g} − { f, g} = (r + s + 1){ f, g}. Thus {Er , E s } ⊂ Er +s . Thus, the sum of all Er is a subalgebra of C ∞ (T ∗ P), whether for the associative algebra structure or for the Lie algebra structure defined by the Poisson bracket. Furthermore, it is a Z-graded Lie algebra. Let S(T ∗ P, ω) be the space formed by the symplectic vector fields on the symplectic manifold (T ∗ P, ω). For any integer r , let Sr be the space spanned by the symplectic vectors X in S(T ∗ P, ω) such that [C, X ] = r X.

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Let U be an open neighborhood in P, let x1 , . . . , xn be coordinates on U , and let x1 , . . . , xn , y1 , . . . , yn be the symplectic coordinates on π∗−1 (U ) induced by x1 , . . . , xn . Let X ∈ Sr , and assume that on π∗−1 (U ), X has the expression X=

n  

fi

i=1

∂ ∂ + gi ∂ xi ∂ yi

 .

Then in the open set π∗−1 (U ) of T ∗ P, by the definition of C, [C, X ] =

n n   ∂ ∂ (C f i ) + (Cgi − gi ) . ∂ xi ∂ yi i=1 i=1

But [C, X ] = r X , thus C fi = r fi ,

Cgi = (r + 1)gi ,

i = 1, . . . , n.

Therefore the restrictions of f i and gi on the fibers of T ∗ P that contained in π∗−1 (U ) are homogeneous polynomial functions of degree r and r + 1, respectively. In particular, this shows that if r < −1, then Sr = {0}. Proposition 3.2.6 For any r ≥ −1 and any f ∈ Er , we have H f ∈ Sr . Proof In fact, by the first corollary of Proposition 3.2.3, [C, H f ] = H(C f − f ) = Hr f = r H f . Thus H f ∈ Sr .



Proposition 3.2.7 For any integer r > −1, the map (refer to Sect. 2.4) H : f −→ H f ,

f ∈ C ∞ (T ∗ P),

induces an isomorphism from Er onto Sr . The inverse isomorphism of this isomorphism is 1 α(X ), X ∈ Sr . X −→ r +1 Proof In fact, if f ∈ Er , then α(H f ) = C f = (r + 1) f. Since r > −1, r + 1 = 0, thus this map is injective. It is also onto. This is because if X ∈ Sr , then Cα(X ) = θ (C) i(X )α = i([C, X ])α + i(X )θ (C)α = (r + 1)α(X ), thus α(X ) ∈ Er , and by Proposition 3.2.3, Hα(X ) = X + [C, X ] = (1 + r )X .



3.2 Symplectic Vector Fields on a Cotangent Bundle

65

3.2.8 The spaces E −1 and S−1 . We consider two special spaces: E −1 and S−1 . From the definition of E −1 , it is not hard to see that, the local constant function space H 0 (T ∗ P, R) on T ∗ P is included in E −1 . By using the exact sequence (2.4.1) H

{0} −→ H 0 (T ∗ P, R) −→ C ∞ (T ∗ P) −→ S(T ∗ P, ω) −→ H 1 (T ∗ P, R) −→ {0}, we obtain a sequence H

{0} −→ H 0 (T ∗ P, R) −→ E −1 −→ S−1 −→ H 1 (T ∗ P, R) −→ {0}.

(3.2.1)

We have the following proposition. Proposition 3.2.9 The sequence (3.2.1) is an exact sequence. Proof According to Proposition 3.2.3, for any X ∈ S−1 , Hα(X ) = 0. Thus locally, the function α(X ) is a constant. Since α is zero on the zero cross section of T ∗ P, α(X ) = 0. If X ∈ S−1 and the image of X in H 1 (T ∗ P, R) is 0, then there exists f ∈ C ∞ (T ∗ P) such that H f = X . Furthermore, by the first corollary of Proposition 3.2.3, C f = α(H f ) = 0. Thus, the sequence (3.2.1) is exact at S−1 . Now we show that S−1 −→ H 1 (T ∗ P, R) is onto. Let σ be the zero cross section of the vector bundle T ∗ P. The map σ ◦ π∗ is homotopic to the identity map on T ∗ P. Thus the homomorphism H i (π∗ ) : H i (P, R) −→ H i (T ∗ P, R) is in fact an isomorphism for any i. Thus each cohomology class that shows up in H 1 (T ∗ P, R) contains a form r , which is the image of some closed 1-form under the map (π∗ )∗ . Since C is a vertical vector field, θ (C)r = 0. Therefore, if X is a symplectic vector field on T ∗ P satisfying i(X )ω = r , then since θ (C)ω = ω (Lemma 3.2.1), i([C, X ])ω = θ (C)r − i(X )θ (C)ω = − i(X )ω, thus [C, X ] = −X and X ∈ S−1 . Thus the cohomology class of r is the image of some element of S−1 . The exactness of the sequence at E −1 and H 0 (T ∗ P, R) are clear.  Remark 3.2.10 The exact sequence (3.2.1) is isomorphic to the following exact sequence {0} −→ H 0 (P, R) −→ C ∞ (P) −→ Z 1 (P) −→ H 1 (P, R) −→ {0}, d

where Z 1 (P) is the space of the closed 1-forms on P, and the map Z 1 (P) −→ H 1 (P, R) is the map that maps a closed 1-form to the cohomology class it represents. In fact, we can define a canonical isomorphism from the space 1 (P) to the space of vector fields on T ∗ P such that for ∀ Y ∈ T (T ∗ P), [C, Y ] = −Y. To do that, for

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any β ∈ 1 (P), define a vector field X β on T ∗ P such that i(X β )ω = (π∗ )∗ (β). This map induces an isomorphism from Z 1 (P) onto S−1 , denote it by ρ. Then it is not hard to verify that the diagram below is a commutative diagram (exercise): {0} −−−−−−→ H 0 (T ∗ P, R) −−−−−−→ ⏐ 0 ⏐ H (π∗ ) {0} −−−−−−→

H

E −1 −−−−−−→ ⏐ ∗ ⏐(π∗ )

S−1 ⏐ρ ⏐

−−−−−−→ H 1 (T ∗ P, R) −−−−−−→ {0} ⏐ 1 ⏐ H (π∗ )

H 0 (P, R) −−−−−−→ C ∞ (P) −−−−−−→ Z 1 (P) −−−−−−→

H 1 (P, R) −−−−−−→ {0}.

3.2.11 The Lie algebras E 0 and S0 . Let D(P) be the Lie algebra formed by the vector fields on P. In Sect. 3.1, we defined a linear map from D(P) into C ∞ (T ∗ P): F : X −→ f X ,

X ∈ D(P).

This is a vector space isomorphism from D(P) onto to E 0 . Lemma 3.2.12 Identify C ∞ (P) with E −1 . Then for any X ∈ D(P) and any g ∈ C ∞ (P), { f X , g} = Xg. ∗ Proof In fact, let x1 , . . . , xn , y1 , . . . ,  yn be local coordinates on Tn P induced from n ∂ local coordinates on P, and let X = i=1 ai ∂ xi . Then f X = i=1 ai yi . Thus (see (2.5.1)) n  ∂g ai = Xg. { f X , g} = ∂ xi  i=1

Since {E 0 , E 0 } ⊂ E 0 , E 0 is a Lie algebra with respect to the algebra structure defined by the Poisson bracket. Proposition 3.2.13 The isomorphism F : X −→ f X from D(P) onto E 0 is a Lie algebra isomorphism. Proof In fact, if X, Y ∈ D(P), g ∈ C ∞ (P), then {{ f X , f Y }, g} = { f X , { f Y , g}} − { f Y , { f X , g}} = X (Y g) − Y (Xg) = [X, Y ]g = { f [X,Y ] , g}, where, as in Lemma 3.2.12, we identified C ∞ (P) with E −1 . From the definition of F, it can be seen that under local coordinates, { f X , f Y } − f [X,Y ] =

n  i=1

ξi

∂ . ∂ xi

Thus, according to the computation above, { f X , f Y } − f [X,Y ] is contained in the center of the Lie algebra C ∞ (T ∗ P), and thus it is a local constant function on

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67

T ∗ P. Since it is also contained in E 0 , it is equal to 0, that is, F is a Lie algebra isomorphism.  For X ∈ D(P), we define a Hamiltonian vector field T ∗ (X ) on T ∗ P by T ∗ (X ) = H f X . According to Propositions 3.2.7 and 3.2.13, the map T ∗ : X −→ T ∗ (X ),

X ∈ D(P),

is an isomorphism from the Lie algebra D(P) onto the Lie algebra S0 . For any g ∈ C ∞ (P), we have T ∗ (X )g = H f X g = Xg,

X ∈ D(P).

Thus T ∗ (X ) is a projectable vector field on P, and its projection is X . We call the vector field T ∗ (X ) an extension of X on T ∗ P. By the first corollary of Proposition 3.2.3, for any X ∈ D(P), α(T ∗ (X )) = C f X = f X . Proposition 3.2.14 Let Y be a vector field on T ∗ P. Then Y is an extension of some vector field on P if and only if (1) [C, Y ] = 0, and (2) θ (Y )α = 0. Proof Assume that Y is an extension of a vector field X on P, that is Y = T ∗ (X ). Then Y ∈ S0 , and thus [C, Y ] = 0. Also, θ (Y )α = d i(Y )α + i(Y ) d α = d(α(Y )) + i(Y ) d α = d f X − i(H f X )ω = 0. Thus the necessary part is proved. Conversely, if Y is a vector field on T ∗ P that satisfies conditions (1) and (2), then (cf. the proofs of Propositions 3.2.3 and 3.2.7) Cα(Y ) = (θ (C)α)(Y ) + α([C, Y ]) = α(Y ), thus α(Y ) ∈ E 0 . On the other hand, since θ (Y )α = 0, we have d α(Y ) = i(Y )ω. Thus Y = Hα(Y ) ∈ S0 , and therefore Y is an extension of some vector field on P.  Example 3.2.15 Let x1 , . . . , xn be the natural coordinates on Rn , and let x1 , . . . , xn , y1 , . . . , yn be the symplectic coordinates on T ∗ Rn induced by x1 , . . . , xn . If X =  n ∂ n ∗ n i=1 ai ∂ xi is a vector field on R , then its extension on T R is

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⎞ n  ∂ ∂a ∂ i ⎝ai ⎠. T ∗ (X ) = − yi ∂ x ∂ x ∂ y i j j i=1 j=1 n 

⎛

Remark 3.2.16 If X ∈ D(P), then any flow generated by T ∗ (X ) is a set of symplectic isomorphisms of T ∗ P . These symplectic isomorphisms keep the Liouville form invariant. If ϕt is a flow generated by X , then it can be seen from the discussions in Sect. 3.1 that T ∗ ϕt is a flow on T ∗ P generated by T ∗ (X ). 3.2.17 The spaces E 1 and S1 . The space E 1 is spanned by those functions on T ∗ P whose restrictions on every fiber Tx∗ P are degree two homogeneous polynomials. If for any x ∈ P, the restriction of an element g ∈ E 1 on Tx∗ P is always nondegenerate, then g can be viewed as a pseudo Riemannian metric on P, and we can use it to define an isomorphism from the cotangent bundle T ∗ P onto the tangent bundle T P as follows. Let ξ ∈ T ∗ P and let π∗ (ξ ) = x. Define an element gξ in Tx P by gξ : v −→ g(ξ, v), ∀ v ∈ Tx∗ P. Then the map

ϕ : ξ −→ gξ , ξ ∈ T ∗ P,

is an isomorphism from T ∗ P onto T P. If T P is identified with T ∗ P via the isomorphism ϕ, then the Hamiltonian vector field Hg (∈ S1 ) is called a spray induced by the metric g. The geodesics on P are the projections of the orbits of Hg on P. Note that Hg g = {g, g} = 0, it can be seen that the vector field Hg is tangent to the hypersurface defined by the equation g = a (a ∈ R).

3.3 Lagrangian Submanifolds of a Cotangent Bundle Let P be a manifold, and let β ∈ 1 (P). As in Sect. 3.1, we denote by β the cross section on the cotangent bundle that corresponds to β. Then β is an immersion from P into T ∗ P. Proposition 3.3.1 The immersion β : P −→ T ∗ P is a Lagrangian immersion if and only if β is a closed 1-form. ∗

Proof In fact, according to Proposition 3.1.3, we have β = β (α), where α is the ∗ ∗ Liouville form on T ∗ P. Thus d β = β (d α) = −β (ω), and the immersion β is an isotropic immersion if and only if d β = 0. Also, since dim P = 21 dim T ∗ P, an isotropic immersion from P into T ∗ P must be a Lagrangian immersion. Therefore, β is a Lagrangian immersion if and only if d β = 0.  A special case of Proposition 3.3.1 is the following: For any function f ∈ C ∞ (P), there exists a corresponding Lagrangian submanifold of T ∗ P, that is, the image of

3.3 Lagrangian Submanifolds of a Cotangent Bundle

69

d f . We call the image of d f a Lagrangian submanifold generated by f , and call f a generating function of this submanifold. Let L be a Lagrangian submanifold of T ∗ P, and let ξ be a point of L such that Tξ L traverses the vertical subspace of Tξ (T ∗ P). Then the map π∗ : T ∗ P −→ P restricts to L giving a map π∗ | L = π1 , whose tangent space map (π1T )ξ at the point ξ is a tangent space isomorphism. Thus, it follows that there exists an open neighborhood U of ξ in L such that the restriction map π1 |U is a homeomorphism from U onto the open set π∗ (U ) of P (see the figure below). If β is a differential 1-form on π∗ (U ) such that the map β ◦ π∗ is the identity map on U , then β is a Lagrangian immersion from π∗ (U ) onto U . According to Proposition 3.3.1, d β = 0. Thus there exists an open neighborhood V of the point π∗ (ξ ) in P and a function f ∈ C ∞ (V ) such that d f = β|V . Thus d f (V ) is a neighborhood of ξ in L, and in this neighborhood, f is a generating function. It is natural to ask, is it true that any Lagrangian submanifold in T ∗ P can be generated by a generating function locally? The answer is negative. There exist Lagrangian submanifolds in T ∗ P which are not cross sections of T ∗ P, thus cannot, even locally, be generated by some functions. For example, every fiber of T ∗ P is a Lagrangian submanifold of T ∗ P. In fact, the pullback of the Liouville form on each fiber is zero, thus the pullback of ω on each fiber is also zero. Since the dimension of each fiber is half of the dimension of T ∗ P, it is a Lagrangian submanifold. These Lagrangian submanifolds cannot be generated by functions.

The method of constructing Lagrangian submanifolds of T ∗ P by using the functions of P can be generalized as follows. We can assume without loss of generality that there exist global coordinates x1 , . . . , xn on P, otherwise, we can consider a coordinate neighborhood of P. Let x1 , . . . , xn , y1 , . . . , yn be the symplectic coordinates on T ∗ P induced by x1 , . . . , xn . Let s be an integer, and let a1 , . . . , as be the natural coordinates on Rs . We consider the product manifold P × Rs . Let pr1 : P × Rs −→ P be the projection map, and let F : P × Rs −→ R be a differentiable function on P × Rs . We define a lift γ : P × Rs −→ T ∗ P of the projection pr1 such that

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π∗ ◦ γ = pr1 ;

yi ◦ γ =

∂F , i = 1, . . . , n. ∂ xi

Then γ is a differentiable map. Let N F be the set of points in P × Rs defined by the following equations ∂F ∂F ∂F = = ... = = 0. ∂a1 ∂a2 ∂as We assume that on N F there is always 

∂F d ∂a1





∂F ∧ ··· ∧ d ∂as

 = 0.

Then, N F is an n-dimensional closed submanifold of P × Rs , and the restriction of γ on N F is an immersion. In fact, for any x ∈ N F , the condition that a vector field X on P × Rs is tangent to N F at the point x is  Xx

∂F ∂ai

 = 0, i = 1, . . . , s.

That is to say, X x belongs to the intersection of the kernels of the following 1-forms:  d

∂F ∂a1



 , ..., d

∂F ∂as

 .

Thus, according to the definition of γ , the kernel of (γ | N F )Tx is the intersection of the kernels of the following 1-forms:  d xi , d

∂F ∂ xi



 , d

∂F ∂a j

 , i = 1, . . . , n; j = 1, . . . , s,

that is, the intersection of the kernels of the following 1-forms: s s   ∂2 F ∂2 F d xi , d ak , d ak , i = 1, . . . , n; j = 1, . . . , s. ∂ xi ∂ak ∂a j ∂ak k=1 k=1

  ∂F , j = 1, . . . , s, are linearly independent, the rank of the folSince on N F , d ∂a j lowing s × (n + s) matrix  2  ∂ F ∂2 F , ∂ xi ∂ak ∂a j ∂ak

3.3 Lagrangian Submanifolds of a Cotangent Bundle

71

is identically equal to s on N F . Thus the kernel of (γ | N F )Tx is the intersection of the kernels of the 1-forms: d x1 , . . . , d xn , d a1 , . . . , d as . Therefore the kernel of (γ | N F )Tx contains only the zero vector, that is, γ | N F is an immersion. n On the other hand, for the Liouville form α = i=1 yi d xi , of T ∗ P, we have ∗

γ (α) = γ

∗

 n 

 yi d xi

i=1

=

n  ∂F d xi . ∂ xi i=1

Thus the pullback of γ ∗ (α) on N F is equal to the pullback of d F on N F . Thus γ ∗ (ω)| N F = γ ∗ (− d α)| N F = − d γ ∗ (α)| N F = 0. Thus the restriction of γ on N F is a Lagrangian immersion from N F into T ∗ P. In general, this immersion is not injective (see Examples 3.3.3 2 below). If it is an injective map, then its image L is a Lagrangian submanifold of T ∗ P, and α| L is an exact 1-form. If this is case, we also call F a generating function of L. Remark 3.3.2 Consider the restriction pr1 | N F of the projection pr1 : P × Rs −→ P on N F . It is easy to see that the kernel of ( pr1 | N F )T is the intersection of the kernels of the following 1-forms:  d x1 , . . . , d xn , d

∂F ∂a1



 ,...,d

∂F ∂as

 .

Thus it can be at most s-dimensional. If we compose the Lagrangian immersion γ : N F −→ T ∗ P with the natural projection π∗ : T ∗ P −→ P, then we obtain a map whose rank at any point is at least n − s. Examples 3.3.3 1. Lets = n. Let p ∈ P, let the coordinate of the point p be ( p1 , . . . , pn ), and let n (xi − pi )ai . Then F = i=1 N F = { p} × Rn ⊂ P × Rn , and yi ◦ γ = ai , i = 1, . . . , n. That is, γ | N F is an isomorphism from N F onto the fiber T p∗ P. 2. Let P = R, let s = 1, and let F = F(x, a) =

a3 + (x 2 − 1)a. 3

Then ∂∂aF = x 2 + a 2 − 1. Thus N F is the unit circle contained in P × R = R2 . For any ( p, t) ∈ R2 , we have γ ( p, t) = ( p, 2 pt), and γ (0, 1) = γ (0, −1) = (0, 0). Thus the map γ : N F −→ T ∗ R is not injective.

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We know in general that a Lagrangian submanifold of T ∗ P is not necessary generated by some function of C ∞ (P) even locally, that is, it is not necessary the image of some d f even locally, where f ∈ C ∞ (P). However, if L is a closed Lagrangian submanifold of T ∗ P, then locally, it can be constructed by using the method we have just described. In fact, let ξ ∈ L be an arbitrary point. According to Proposition 2.6.4, there exist an open neighborhood U of ξ in T ∗ P and symplectic coordinates u 1 , . . . , u n , v1 , . . . , vn on U such that v1 , . . . , vn are all equal to 0 on L ∩ U . According to Lemma 2.6.2, we can assume that the coordinates u 1 , . . . , u n , v1 , . . . , vn have the property that at any point of U , d x1 , . . . , d xn , d u 1 , . . . , d u n are independent, where x1 , . . . , xn are coordinates on P. Let x1 , . . . , xn , y1 , . . . , yn be the symplectic coordinates on P induced by x1 , . . . , xn . Then since both u 1 , . . . , u n , v1 , . . . , vn and x1 , . . . , xn , y1 , . . . , yn (restricted to U ) are symplectic coordinates on U , the form n  (yi d xi − vi d u i ) i=1

is a closed form on U . Since every closed form is exact locally, we can assume that there exists a differentiable function F on T ∗ P such that on U (shrink U if necessary), n  (yi d xi − vi d u i ). d F|U = i=1

Thus on U ,

 ∂u j ∂F =− vj , i = 1, . . . , n. ∂ yi ∂ yi j=1 n

Since all v1 , . . . , vn are 0 on L ∩ U , thus on L ∩ U , ∂∂ yFi = 0, i = 1, . . . , n. Moreover, 

∂F d ∂ yi thus on L ∩ U ,

 d



∂F ∂ yi

=−

n  ∂u j j=1

 =−

n 



∂u j vj d d vj − ∂ yi ∂ yi j=1

n  ∂u j j=1

∂ yi

 ,

d v j , i = 1, . . . , n.

  ∂u But since d x1 , . . . , xn , d y1 , . . . , d yn are independent on U , the n × n matrix ∂ yij   is invertible at every point of U , and thus the forms d ∂∂ yFi , i = 1, . . . , n, are independent on L ∩ U . Identifying T ∗ P with P × Rn by using the symplectic coordi-

3.3 Lagrangian Submanifolds of a Cotangent Bundle

73

n ∗ nates x1 , . . . , xn , y1 , . . . , yn , then the constructed  γ : P × R −→ T P is a  map Lagrangian immersion. Also, since on L ∩ U , ∂∂ xFi = yi , i = 1, . . . , n, thus the map γ is the identity map on the open set L ∩ U of N F . As explained by Weinstein in reference [29], the method of constructing a Lagrangian submanifold by using a differentiable function on P × Rs is a special case of another more intuitive geometric construction method. Next, we give a brief description of this method, the reader can refer to the relevant references for more details. Let Q be a manifold, and let ϕ : Q −→ P be a submersion. If N is the set of the cotangent vectors of the form ξ ◦ ϕ T (ξ ∈ T ∗ P) on T ∗ Q, then N is a subvector bundle of T ∗ Q. If dim P = n and dim Q = n + s, then dim N = 2n + s. Let μ : N −→ T ∗ P be a differentiable map such that μ(η) ◦ ϕ T = η holds for all η ∈ N . Then it can be proved that:

1. If α P and α Q are the Liouville forms on T ∗ P and T ∗ Q respectively, then α Q | N = μ∗ (α P ). Also, if ω P and ω Q are symplectic structures on T ∗ P and T ∗ Q respectively, then μ∗ (ω P ) = ω Q | N . 2. With respect to the symplectic structure ω Q , N is coisotropic. Let F be a differentiable function on Q, and let L be the image of d F (the cross section of T ∗ Q corresponding to d F) in T ∗ Q. Then L is a Lagrangian submanifold of T ∗ Q. If L and N has a “good intersection”, then a Lagrangian submanifold of T ∗ P can be constructed by using the contraction method (see Sect. 2.6). For any η ∈ L ∩ N , there exists an open neighborhood V of η in L ∩ N such that μ(V ) is a Lagrangian submanifold of T ∗ P.

Chapter 4

Symplectic G-Spaces

In this chapter, we denote by G a Lie group, that is, G is simultaneously an abstract group and a real analytic manifold, and the group operation of G and the analytic structure of G are compatible. By compatible, it is meant that the map defined by (x, y) −→ x y −1 , x, y ∈ G, from G × G onto G is a real analytic map. Definition 4.0.1 Let M be a manifold. If a differentiable map γ : G × M −→ M satisfies the following two conditions: (1) γ (e, x) = x, ∀ x ∈ M, where e is the identity of G; and (2) if s1 , s2 ∈ G, then γ (s1 , γ (s2 , x)) = γ (s1 s2 , x), ∀ x ∈ M, then we say that G acts differentially on M. We usually write an action γ of G on M in a product form: γ (s, x) = sx, s ∈ G, x ∈ M. If T (G × M) is identified with T G × T M, then the tangent space map γ T of a differentiable action γ of G on M can be viewed as a differentiable map from T G × T M into T M. It is easy to see that this is an action of the group T G on the manifold T M. We also write it as a product. We identify M (or G) with the image of the zero cross section of T M (or T G). Under these assumptions, if s ∈ G, x ∈ M, a ∈ Ts G, and v ∈ Tx M, then sv = γ T (s, v) ∈ Tsx M, ax = γ T (a, x) ∈ Tsx M.

© Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_4

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Denote the tangent space of G at the identity e by g, then for any a ∈ g and x ∈ M, we have ax ∈ Tx M. The map x −→ ax, x ∈ M, is a vector field on M for a fixed a, we denote it by a . In particular, if M = G and γ : G × G −→ G is the product map of the Lie group G, then for any a ∈ g, a : s −→ as, s ∈ G, is a right invariant vector field on G, we use a special notation Ra to denote it. Obviously, Ra (e) = a. Define a bracket operation [ , ] on g such that for any a, b ∈ g R[a,b] = [Ra , Rb ] (= Ra Rb − Rb Ra ), then the vector space g becomes a real Lie algebra, we call it the Lie algebra of G. Definition 4.0.2 Let M be a manifold, let G be a Lie group, and let h be an arbitrary finite dimensional real Lie algebra. (1) If G acts differentially on M, we call M a G-space. (2) If there exists a Lie algebra homomorphism from the Lie algebra h into the Lie algebra of vector fields on M, we call M an h-space. If g is the Lie algebra of G, then for any G-space M, the map  : a −→ a , a ∈ g, is a homomorphism from g into the Lie algebra D(M) of vector fields on M, and thus M is a g-space. We call this g-space structure of M the adjoint g-space structure. Remark 4.0.3 We can also start from the left invariant vector fields on G and then define the Lie algebra of G, there is no essential difference between the Lie algebra of G thus defined and the Lie algebra of G defined by using the right invariant vector fields.

4.1 Definitions and Examples Definition 4.1.1 Let M be a G-space. If there exists a symplectic structure ω on M that is invariant under the action of G, then we call M a symplectic G-space. By this definition, if a symplectic manifold (M, ω) is a symplectic G-space, then for any s ∈ G, the diffeomorphism s : x −→ sx, x ∈ M, is a symplectic automorphism of (M, ω). Definition 4.1.2 Let h be a finite dimensional real Lie algebra, and let M be an h-space. If there exists a symplectic structure ω on M such that for any a ∈ h, the vector field a : x −→ ax, x ∈ M, is a symplectic vector field of (M, ω), then we call the h-space M a symplectic h-space.

4.1 Definitions and Examples

77

Let g be the Lie algebra of a Lie group G, let M be a G-space, and let ω be a symplectic structure on M. Since M is a G-space, M is a g-space. If (M, ω) is a symplectic G-space, then it is also a symplectic g-space. In fact, choose an arbitrary a ∈ g, and let F : R −→ G be the one parameter subgroup of G corresponding to a, that is, F is a Lie group homomorphism from the additive real number Lie group R into G. For any x ∈ M, let ϕ : (t, x) −→ F(t)x, t ∈ R. Then the map ϕ : R × M −→ M satisfies the following conditions: 1. ϕ(0, x) = x, ∀ x ∈ M, and 2. the map: t −→ ϕ(t, x), t ∈ R, for any x ∈ M, is an integral curve of the vector field a : x −→ ax, x ∈ M. In fact, for any t ∈ R, a : ϕ(t, x) −→ a F(t)x. Also, since F : R −→ G is the = a F(t). Therefore one parameter subgroup corresponding to a, d F(t) dt d F(t) d ϕ(t, x) = x = a F(t)x. dt dt Thus ϕ is a flow generated by a . Since ω is G-invariant, ϕt∗ (α) does not depend on the parameter t ∈ R. According to Lemma 2.4.1, θ (a )ω = 0. That is, a is a symplectic vector field for any a ∈ g. This proves that (M, ω) is also a symplectic g-space. Conversely, if (M, ω) is a symplectic g-space, then when G is a connected Lie group, (M, ω) is also a symplectic G-space. Examples 4.1.3 1. let P be a G-space, and let γ : G × P −→ P be the corresponding action. For any s ∈ G, γs : x −→ sx, x ∈ P, is a diffeomorphism of P. Using the notation in Sect. 3.1, the map T ∗ γs : T ∗ P −→ T ∗ P satisfies the relation: (T ∗ γs )(ξ ), γ T (u) = ξ, u, ∀ ξ ∈ Tx∗ P, u ∈ Tx P. Thus the map T ∗ γ : (s, ξ ) −→ (T ∗ γs )(ξ ), s ∈ G, ξ ∈ T ∗ P, defines an action of G on T ∗ P, which we also write in the form of a product: T ∗ γ : (s, ξ ) −→ sξ, s ∈ G, ξ ∈ T ∗ P. Then for any (ξ, v) ∈ T ∗ P × T P and s ∈ G, sξ, sv = ξ, v. Thus, the action of P

G on T ∗ P defined above is in fact the only action that satisfies this equality. By

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Proposition 3.1.4, we know that the Liouville form α P on T ∗ P is invariant under this action of G. Therefore, if ω P = − d α P is the canonical symplectic structure on T ∗ P, then (T ∗ P, ω P ) is a symplectic G-space. n 2. Let ω = i=1 d xi ∧ d xn+i be the canonical symplectic structure on R2n . Under the natural action of the symplectic group Sp(2n), the symplectic manifold (R2n , ω) is a symplectic Sp(2n)-space. Similarly, consider R2n as a Lie group, then under the parallel translation action, (R2n , ω) is a symplectic R2n -space. If (M, ω) is a symplectic G-space, then the pullback of ω on every orbit of G is always a constant rank closed 2-form. But in the general case, this closed form is not necessary a symplectic structure on the orbit. We have the following lemma. Lemma 4.1.4 Let (V, ω) be a real symplectic vector space, let H be a compact subgroup of Sp(V, ω), and let V H be the subspace of V formed by the vectors fixed by H . Then V H is a symplectic subspace of (V, ω). Proof Let (x1 , . . . , x2n ) be a symplectic basis of V . Define a linear map j of V such that j (xi ) = xn+i , j (xn+i ) = −xi , i = 1, . . . , n. Moreover, define a bilinear form b on V by b(x1 , x2 ) = ω(x1 , j (x2 )), ∀ x1 , x2 ∈ V. Then b is a positive definite symmetric bilinear form on V . Since H is a compact subgroup of Sp(V, ω), so by the result of Sect. 1.5, b is invariant under the action of H , that is, for any s ∈ H , b(x1 , x2 ) = b(sx1 , sx2 ), ∀ x1 , x2 ∈ V. Let F be the subspace of V spanned by {sx − x : s ∈ H, x ∈ V }. Since H ⊂ Sp(V, ω), thus for any x, y ∈ V and s ∈ H , ω(sx − x, y) = ω(x, s −1 y − y). In particular, if y ∈ V H , then ω(sx − x, y) = 0. Thus, F is the orthogonal complement of V H in V with respect to ω. Similarly, since b is invariant under H , we also have b(sx − x, y) = b(x, s −1 y − y), ∀ x, y ∈ V, s ∈ H.

4.1 Definitions and Examples

79

Thus, F is also orthogonal to V H with respect to b. Let the orthogonal complement of V H in V with respect to b be (V H )⊥ , then (V H )⊥ = F, and V H ∩ (V H )⊥ =  V H ∩ F = {0}. Thus V H is a symplectic subspace. Proposition 4.1.5 Let (M, ω) be a symplectic G-space. If G is a compact Lie group, then the fixed point set of G: M G = {x ∈ M : sx = x, ∀ s ∈ G}, is a symplectic submanifold of (M, ω). Proof Since G is compact, a G-invariant Riemannian metric g can be defined on M. For this g, we can define an exponential map on an open neighborhood U of the zero cross section of T M (see Ref. [12]): exp : U −→ M. Since g is invariant under the action of G, thus for any s ∈ G and v ∈ U , exp(sv) = s(exp(v)). Let x ∈ M G , and let Wx be an open neighborhood of x in Tx M such that Wx ⊂ U and the restriction of exp on Wx is a diffeomorphism from Wx onto exp(Wx ). Then M G ∩ exp(Wx ) = exp((Tx M)G ∩ Wx ), where (Tx M)G is the G-fixed point subspace of Tx M. Thus M G is a submanifold of M, and for any x ∈ M G , Tx (M G ) = (Tx M)G . Since (M, ω) is a G-space, for any s ∈ G and x ∈ M G , the linear transformation of Tx M: v −→ sv, v ∈ Tx M, is an element of Sp(Tx M, ωx ). Thus by Lemma 4.1.4, Tx (M G ) = (Tx M)G is a symplectic  subspace of (Tx M, ωx ). Thus M G is a symplectic submanifold of (M, ω).

4.2 Hamiltonian g-Spaces and Moment Maps We denote by g a finite dimensional real Lie algebra below. Definition 4.2.1 Let (M, ω) be a symplectic g-space. If for any a ∈ g, the vector field a : x −→ ax, x ∈ M, is always a Hamiltonian vector field, then (M, ω) is called a Hamiltonian g-space. If (M, ω) is a symplectic manifold, then there exists an exact sequence of Lie algebras (2.4.1): H

ρ

{0} −→ H 0 (M, R) −→ C ∞ (M) −→ S(M, ω) −→ H 1 (M, R) −→ {0}, and the Hamiltonian vector fields on M are exactly the symplectic vector fields on M that belong to the image of H . Thus, a Hamiltonian g-space structure on (M, ω) is in fact a homomorphism  : g −→ S(M, ω) that satisfies ρ ◦  = 0. If

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H 1 (M, R) = {0}, then this equality always holds, and all symplectic g-structures on (M, ω) are Hamiltonian g-space structures. Since as a Lie algebra, the bracket operation of H 1 (M, R) is the zero operation, so if [g, g] = g, then the only Lie algebra homomorphism from g into H 1 (M, R) is the zero homomorphism. Thus, if [g, g] = g, then all symplectic structures on (M, ω) are also Hamiltonian g-space structures. Assume the map  : g −→ S(M, ω) is a Lie algebra homomorphism. Then the necessary and sufficient condition for  to be a Hamiltonian g-structure on the symplectic manifold (M, ω) can also be described as: there exists a linear map ˜ ˜ from g into C ∞ (M) such that  = H ◦ . If ˜ is an arbitrary linear map from g into C ∞ (M), then we can define a differentiable map μ from M into the dual space g∗ of g as follows: for any a ∈ g, the value ˜ of μ(x) (x ∈ M) at a is defined by μ(x), a = (a)(x). It is easy to see that μ(x) is uniquely determined. Thus, we can view the image of an element a ∈ g under ˜ as a function on M defined by the following map: μ, a : x −→ μ(x), a, ∀ x ∈ M. If  is a Hamiltonian g-space structure, then Hμ,a = a , ∀ a ∈ g. Definition 4.2.2 Let (M, ω) be a Hamiltonian g-space. We call any differentiable map μ : M −→ g∗ that satisfies the equality Hμ,a = a , ∀ a ∈ g, a moment map of the g-space (M, ω). Lemma 4.2.3 Let μ : M −→ g∗ be a moment map of a Hamiltonian g-space (M, ω). Then the following equalities hold: (1) dμ, a = d μ, a = i(a )ω, (2) d μ(ax), b = ω(bx, ax), (3) d μ(ax), b = {μ, a, μ, b}(x), where a, b ∈ g and x ∈ M are arbitrary. Proof By the definition, i(a )ω = i(Hμ,a )ω = dμ, a = d μ, a, thus (1) holds. From (1), d μ(ax), b = dμ, b(ax) = i(b )ω(ax) = ω(bx, ax), thus (2) holds. Also by Lemma 2.4.10 {μ, a, μ, b} = ω(Hμ,b , Hμ,a ) = ω(b , a ), thus {μ, a, μ, b}(x) = ω(bx, ax), and (3) holds.



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Let μ1 and μ2 be two moment maps of a Hamiltonian g-space (M, ω). Then for any a ∈ g, d(μ1 − μ2 ), a = 0 always holds. Thus d(μ1 − μ2 ) = 0, which implies μ1 − μ2 is a constant map locally. Therefore, the moment maps of (M, ω) differ only by some local constants. If μ is a moment map, and ϕ : M −→ g∗ is a locally constant map, then since Hϕ,a = 0, ∀ a ∈ g, we have Hμ+ϕ,a = Hμ,a = a , ∀ a ∈ g. Thus μ + ϕ is a moment map. If (M, ω) is a connected Hamiltonian g-space, then any two moment maps of (M, ω) differ only by a parallel translation of g∗ . Proposition 4.2.4 Let μ be a moment map of a Hamiltonian g-space (M, ω). Then for any a, b ∈ g, the function {μ, a, μ, b} − μ, [a, b] is a locally constant function on M. Proof In fact, by Lemma 2.4.10, Hμ,[a,b] = [a,b] = [a , b ] = [Hμ,a , Hμ,b ] = H{μ,a,μ,b} , thus the statement holds.



According to Proposition 4.2.4, for any moment map μ of a Hamiltonian gspace (M, ω), we can define a skew symmetric bilinear form cμ on g with values in H 0 (M, R) as follows: cμ (a, b) = {μ, a, μ, b} − μ, [a, b], ∀ a, b ∈ g. Since the Poisson bracket satisfies the Jacobi identity, thus cμ ([a, b], c) + cμ ([b, c], a) + cμ ([c, a], b) = 0, ∀ a, b, c ∈ g. View H 0 (M, R) as a trivial g-module, then the above equation says that cμ is a closed 2-cochain1 on g that takes values in H 0 (M, R). If μ = μ + ϕ is another moment map of M, where ϕ : M −→ g∗ is a locally constant map, then for any a, b ∈ g, cμ (a, b) = cμ (a, b) − ϕ, [a, b]. This says that cμ − cμ is the coboundary of the 1-cochain f on g that takes values in H 0 (M, R) defined by the equation f (a) = ϕ, a, ∀ a ∈ g. Thus, as an element of H 2 (g, H 0 (M, R)), the cohomology class that corresponds to the closed 2-cochain cμ does not depend on the choice of μ, it is determined by the g-space structure on (M, ω). We denote it by c(M, ω).

1 Added

by the authors of the Forewords. A closed cochain is often called a cocycle in other texts.

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Definition 4.2.5 Let (M, ω) be a symplectic g-space. If the homomorphism  : g −→ S(M, ω) ˜ where is of the form H ◦ , H : C ∞ (M) −→ S(M, ω) is as in (2.4.1), and ˜ is a homomorphism from g into the Lie algebra C ∞ (M) (with the Poisson bracket), then (M, ω) is called a Poisson g-space or a strong Hamiltonian g-space. In order for a symplectic g-space to be Poisson, it is necessary and sufficient that there exists a moment map μ of (M, ω) such that cμ = 0, that is, c(M, ω) is 0. If g is a semisimple real Lie algebra, then any symplectic g-space is Poisson. In fact, in this case, any symplectic g-space is Hamiltonian. This is so, because by the second lemma of Whitehead (see Ref. [13]), H 2 (g, R) = {0}, thus H 2 (g, H 0 (M, R)) = H 2 (g, R) ⊗ H 0 (M, R) = {0}, and so c(M, ω) = 0. Proposition 4.2.6 Let (M, ω) be a symplectic g-space. If there exists a differential 1-form α on M such that ω = − d α and θ (a )α = 0 for any a ∈ g, then (M, ω) is a Poisson g-space. Proof In fact, for any a ∈ g we have d(α(a )) = d i(a )α = θ (a )α − i(a ) d α = i(a )ω. ˜ where ˜ Thus a is a Hamiltonian vector field, and we can write  as  = H ◦ , is a linear map from g into C ∞ (M) satisfying ˜ a = α(a ), ∀ a ∈ g. Thus, by the formulas that we quoted at the beginning of Chap. 2, ˜ [a,b] = i([a,b] )α = i([a , b ])α = θ (a ) i(b )α − i(b )θ (a )α = θ (a )α(b ) = −ω(a , b ) = {˜ a , ˜ b }, ∀ a, b ∈ g. So ˜ is a Lie algebra homomorphism and (M, ω) is Poisson.



Corollary 4.2.7 Let P be a g-space. Denote the extension of a (a ∈ g) on T ∗ P by T ∗ (a ), then the map  ∗ : a −→ T ∗ (a ), a ∈ g, is a Poisson g-space structure on the cotangent bundle T ∗ P. Proof Use Proposition 3.2.14 and Proposition 4.2.6.



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Examples 4.2.8 1. Let g = R2 , let x, y be the natural coordinates on R2 , and define the bracket operation on g to be the zero operation. Define a symplectic g-space structure on (R2 , d x ∧ d y) by using the equation (a,b) = a

∂ ∂ + b , a, b ∈ R. ∂x ∂y

Then because (a,b) = Hay−bx , this is a Hamiltonian g-space structure. The map from g = R2 into g∗ = R2 defined by the equation μ(x, y) = (y, −x) is a moment map. For arbitrary (a1 , b1 ), (a2 , b2 ) ∈ g, we have cμ ((a1 , b1 ), (a2 , b2 )) = {μ, (a1 , b1 ), μ, (a2 , b2 )} = ω((a2 ,b2 ) , (a1 ,b1 ) ) = a2 b1 − a1 b2 . This is because g is an commutative Lie algebra. Also, cμ is independent of the choice of the moment map, thus cμ is nonzero, and the g-space (R2 , d x ∧ d y) is not Poisson. n 2. Let ω = i=1 on R2n , and take d xi ∧ d xn+i be the canonical n symplectic structure 2n the skew symmetric bilinear form ω0 = i=1 xi ∧ xn+i on R . For any endomorphism a of R2n , let a be a linear vector field on R2n such that for any linear form f on R2n , a f = f ◦ a. Then for arbitrary a, b ∈ End(R2n ) and f, [a , b ] f = a b f − b a f = f ◦ b ◦ a − f ◦ a ◦ b = (b◦a−a◦b) f, thus [a , b ] = (b◦a−a◦b) . For any endomorphism a of R2n , by the definition of a , i(a )ω =

n  ((xi ◦ a) d xn+i − (xn+i ◦ a) d xi ). i=1

Since d i(a )ω = θ (a )ω, thus i(a )ω is a closed 1-form if and only if a belongs to the Lie algebra sp(2n) of the symplectic group Sp(2n), that is, for arbitrary x, y ∈ R2n (cf. Sect. 1.4), ω0 (a(x), y) + ω0 (x, a(y)) = 0. Thus the map  : a −→ a , a ∈ sp(2n), defines a symplectic sp(2n)-space structure on (R2n , ω). Since sp(2n) is a real semisimple Lie algebra,  is also a Hamiltonian sp(2n)-space structure and a Poisson sp(2n)-space structure. Let e1 , . . . , e2n be the natural coordinates on R2n . Then for any a ∈ sp(2n),

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⎛ i(a )ω = d ⎝

⎞

2n 

ω0 (a(e j ), ek )x j xk ⎠ .

j,k=1

This also proved directly that (R2n , ω) is a Hamiltonian sp(2n)-space. Also, we have a quadratic moment map μ : R2n −→ (sp(2n))∗ , such that for any x ∈ R2n , μ(x), a = ω0 (ax, x). 3. Let Q = R3 be the 3-dimensional Euclidean space, and let q1 , q2 , q3 be the natural coordinates on Q. Then there are the following coordinates on the tangent bundle T Q (see 2.4.4 Example 1): q1 , q2 , q3 , q˙ 1 = d q1 , q˙2 = d q2 , q˙ 3 = d q3 .  If we identify T Q with T ∗ Q via the Riemannian metric i d qi2 , then the Liouville  3 form on T ∗ Q can be written as α Q = i=1 q˙i d qi . Take the symplectic structure −m d α Q on T Q, where m ∈ R. Let gl(3) be the Lie algebra formed by the endomorphisms of Q, then we have a natural gl(3)-space structure on Q. For arbitrary a ∈ gl(3), we have 3  ∂ (qi ◦ a) . a = ∂qi i=1 By the corollary of Proposition 4.2.6, for any a ∈ gl(3), there exists an extension T ∗ (a ) of a in T ∗ Q = T Q, and a Poisson space structure can be defined on (T Q, −m d α Q ) such that ∗

∗

i(T (a ))(−m d α Q ) = m d i(T (a ))α Q = m d

 3 

(qi ◦ a)q˙i .

i=1

Thus we have a moment map μ : T Q −→ gl(3)∗ determined by the following formula: 3  μ, a = m (qi ◦ a)qi , ∀ a ∈ gl(3). i=1

If we only consider the Lie algebra so(3) of the orthogonal group of Q, that is, consider only the set of elements of gl(3) of the form ⎛

⎞ 0 −a3 a2 ⎝ a3 0 −a1 ⎠ , a1 , a2 , a3 ∈ R, −a2 a1 0 then we have a Poisson so(3)-space structure on T Q, and for any a ∈ so(3), μ, a = m(a1 (q2 q3 − q3 q2 ) + a2 (q3 q1 − q1 q3 ) + a3 (q1 q2 − q2 q1 )).

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85

Where the coefficients of ai (i = 1, 2, 3) are the so-called components of the kinetic torque. Kinetic torque means the torque of a particle of mass m, whose position and speed can be represented by a point in T Q, about the origin. The term moment map we adopted comes from this special case. 4. Let z j = x j + i xn+ j , j = 1, . . . , n, be the natural coordinates on Cn , and let G = T n be the subgroup of U (n) formed by the diagonal matrices of the form: ⎛ ⎜ ⎜ ⎜ ⎝

e−ia1

⎞ ⎟ ⎟ ⎟ , a1 , . . . , an ∈ R. ⎠

e−ia2 ..

.

e−ian

Then the Lie algebra of G can be identified with the Lie algebra Rn with the zero bracket. Corresponding to the natural action of G on Cn , there exists an adjoint g-space structure on the manifold Cn (g = Rn is the Lie algebra of G). If a = (a1 , . . . , an ) ∈ g, then a =



∂ ∂ . a j xn+ j − xj ∂x j ∂ xn+ j j=1

n 

 If we take the symplectic structure ω = nj=1 d x j ∧ d xn+ j on Cn , then since ω is invariant under the action of U (n), it is invariant under the action of G. For any a ∈ g, we have ⎞ ⎛ n 1 ⎝ a j (x j d x j + xn+ j d xn+ j ) = d a j |z j |2 ⎠ . i(a )ω = 2 j=1 j=1 n 

Thus (Cn , ω) is a Hamiltonian g-space. If g∗ is also identified with Rn , then we have a moment map μ(z) =

1 (|z 1 |2 , . . . , |z n |2 ), z = (z 1 , . . . , z n ) ∈ Cn . 2

Let

1 (xn+ j d x j − x j d xn+ j ). 2 j=1 n

α= Then d α = −ω, and

1 a j |z j |2 , ∀ a ∈ g. 2 j=1 n

i(a )α =

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Thus θ (a )α = d i(a )α − i(a )ω = 0. Therefore by Proposition 4.2.6, (Cn , ω) is a Poisson g-space. Proposition 4.2.9 Let (M, ω) be a Hamiltonian g-space, and let μ : M −→ g∗ be a moment map of (M, ω). Then for any x ∈ M, the kernel of d μx is the orthogonal complement of the subspace gx = {ax : a ∈ g} in the symplectic vector space (Tx M, ωx ) with respect to ωx . In order for μ to be an immersion at the point x, it is necessary and sufficient that gx = Tx M. Proof By Lemma 4.2.3, for any a ∈ g and v ∈ Tx M, d μ(v), a = ω(ax, v), thus the proposition holds.  If a g-space structure on (M, ω) is the adjoint structure of some symplectic Gspace structure, then the space gx is the tangent space at x of the orbit G(x) of the point x. In order for gx = Tx M, it is necessary and sufficient that the orbit G(x) is an open subset of M. In order for a point x ∈ M to be a stationary point of the moment map μ, that is d μx = 0, it is necessary and sufficient that gx = {0}. If gx = {0}, then the point x is a fixed point of the identity component of G. Thus the moment map is a constant map on any connected submanifold of M that is formed by the fixed points of G. Proposition 4.2.10 Let (M, ω) be a Hamiltonian g-space, and let μ : M −→ g∗ be a moment map. Then for any x ∈ M, the image of d μx in g∗ is the annihilator of gx = {a ∈ g : ax = 0}, i.e., the vector subspace of g∗ formed by all vectors that vanish on gx . In order for μ to be a submersion at the point x, it is necessary and sufficient that gx = {0}. Proof As in the proof of Proposition 4.2.9, the proposition can be proved by using d μ(v), a = ω(ax, v).



If a g-space structure is the adjoint structure of some symplectic G-space, then gx is the Lie algebra of the isotropy subgroup of the point x: G x = {s ∈ G : sx = x}. In order for gx = {0}, it is necessary and sufficient that G x is a discrete subgroup of G. Remark 4.2.11 Let G be a compact Lie group, and let (M, ω) be a connected Hamiltonian G-space. Then the set formed by all those points x on M such that their G orbits G(x) have maximal dimensions is a dense open subset of M, let it be U . Then any moment map of (M, ω) has constant rank on U , its rank is equal to the dimension of the orbits of the points in U . If G is also commutative, and its action on M is

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87

effective, then for any x ∈ U dim G(x) = dim G. Thus the restrictions on U of the moment maps of (M, ω) are submersions (cf. Example 1 and Example of 4.2.8). If G is a connected compact commutative Lie group (that is, a torus), and (M, ω) is a connected compact Hamiltonian G-space, then the fixed point set M G of G has only finitely many connected components, let them be K 1 , . . . , K s . If μ is a moment map of (M, ω), then it is constant on each K i . It can be proved that in this case (see Refs. [3, 8, 11]), the image of μ is the convex hull of the point sets μ(K i ), i = 1, . . . , s, in g∗ .

4.3 Equivariance of Moment Maps Let G be a Lie group and let g be the Lie algebra of G. Let Ad be the usual adjoint representation of G on g, that is Ad(s)a = sas −1 , s ∈ G, a ∈ g. Let ad be the adjoint representation of g on g, that is ad(a)b = [a, b], a, b ∈ g. We define a coadjoint representation Ad∗ of G on g∗ as follows: Ad∗ (s) = t Ad(s −1 ), s ∈ G. Since the bracket of g is defined by using the bracket of the right invariant vector fields on G, thus Ad(exp a) = exp(− ad(a)), ∀ a ∈ g, where exp denotes the exponential map of G. Thus Ad∗ (exp a) = exp(t (ad(a)), ∀ a ∈ g. Let (M, ω) be a symplectic G-space. If the adjoint g-space structure is Hamiltonian (or Poisson), then we call (M, ω) a Hamiltonian (or Poisson) G-space. If M is a G-space and s ∈ G, we denote by s M the diffeomorphism x −→ sx, x ∈ M, from M onto itself. Lemma 4.3.1 Let (M, ω) be a Hamiltonian G-space and let μ : M −→ g∗ be a moment map. Then for any s ∈ G, the map μ ◦ s M − Ad∗ (s) ◦ μ : M −→ g∗ is a locally constant map.

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Proof For any a ∈ g, dAd∗ (s) ◦ μ, a = Ad∗ (s) d μ, a = d μ, Ad(s −1 )a. Also, by (1) of Lemma 4.2.3, for any x ∈ M and v ∈ Tx M, we have d μ(v), Ad(s −1 )a = ω(s −1 asx, v) = ω(asx, sv) = d μ(sv), a = (dμ ◦ s M , a)(v). Thus for any a ∈ g,

dAd∗ (s) ◦ μ, a = d(μ ◦ s M ), a,

and therefore the map μ ◦ s M − Ad∗ (s) ◦ μ is locally constant.



Proposition 4.3.2 Let (M, ω) be a connected Hamiltonian G-space and let μ : M −→ g∗ be a moment map of (M, ω). Then we have: (1) For any s ∈ G, ϕμ (s) = μ(sx) − Ad∗ (s)μ(x) is an element of g∗ independent of the point x ∈ M. (2) For any s, t ∈ G, ϕμ (st) = ϕμ (s) + Ad∗ (s)ϕμ (t). (3) For any a, b ∈ g, cμ (a, b) = d ϕμ (a), b, where cμ is defined as in Sect. 4.2. Proof Since M is connected, (1) can be proved by using Lemma 4.3.1. By (1), ϕμ (st) = μ(st x) − Ad∗ (st)μ(x) = ϕμ (s) + Ad∗ (s)μ(t x) − Ad∗ (s) Ad∗ (t)μ(x) = ϕμ (s) + Ad∗ (s)ϕμ (t), ∀ s, t ∈ G, thus (2) holds. Differentiate the formula that defines ϕμ to get d μ(ax) = t ad(a)μ(x) + d ϕμ (a), x ∈ M, a ∈ g. Thus for arbitrary x ∈ M and a, b ∈ g, dμ (ax), b = μ(x), [a, b] + d ϕμ (a), b. But by Lemma 4.2.3, dμ (ax), b = {μ, a, μ, b}(x). Thus cμ (a, b) = {μ, a, μ, b} − μ, [a, b] = d ϕμ (a), b, ∀ a, b ∈ g.



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89

Corollary 4.3.3 The map from G × g∗ into g∗ defined by (s, ξ ) −→ sξ = Ad∗ (s)ξ + ϕμ (s), s ∈ G, ξ ∈ g∗ , is an affine action of the Lie group G on the vector space g∗ . For this G-space structure of g∗ , the moment map μ : M −→ g∗ is G-equivariant, that is, μ(sx) = sμ(x) = Ad∗ (s)μ(x) + ϕμ (s), ∀ s ∈ G, x ∈ M. Proof Let e be the identity of G, then by definition, (e, ξ ) −→ eξ = Ad∗ (e)ξ + ϕμ (e) = ξ + μ(x) − μ(x) = ξ, ∀ ξ ∈ g∗ , also from formula (2) of the proposition, (s1 s2 )ξ = Ad∗ (s1 s2 )ξ + ϕμ (s1 s2 ) = Ad∗ (s1 ) Ad∗ (s2 )ξ + ϕμ (s1 ) + Ad∗ (s1 )ϕμ (s2 ) = Ad∗ (s1 )(Ad∗ (s2 )ξ + ϕμ (s2 )) + ϕμ (s1 ) = s1 (s2 ξ ), ∀ s1 , s2 ∈ G, ξ ∈ g∗ . Thus the first half of the corollary holds. While the second half comes from the  definition of ϕμ . Remark 4.3.4 Proposition 4.3.2 shows that ϕμ is a closed 1-cochain on G with values in g∗ . These closed cochains defined by the moment maps form a special class of closed cochains (see Sect. 5.3). Generally speaking, an action of G on g∗ cannot be obtained from some moment map even if its linear part is Ad∗ . Proposition 4.3.5 Let (M, ω) be a connected Hamiltonian G-space. A sufficient condition for (M, ω) to be a Poisson space is that there exists a moment map μ of (M, ω) such that for any s ∈ G, μ ◦ s M = Ad∗ (s) ◦ μ. If G is connected, then this condition is also necessary. Proof In fact, if there exists a moment map μ of M such that μ ◦ s M = Ad∗ (s) ◦ μ, ∀ s ∈ G, then for any s ∈ G, ϕμ (s) = 0. Thus by part (3) of Proposition 4.3.2, cμ = 0. Thus (M, ω) is a Poisson G-space. Conversely, if (M, ω) is a Poisson Gspace, then there exists a moment map μ of (M, ω) such that cμ = 0. By part (3) of Proposition 4.3.2, d ϕμ (a) = 0, ∀ a ∈ g. Then, by part (2) of Proposition 4.3.2, d ϕμ (sa) = Ad∗ (s) d ϕμ (a) = 0 holds for arbitrary s ∈ G and a ∈ g, thus d ϕμ = 0. If G is connected, then the map ϕμ is a constant value map. But by part (2) of Proposition 4.3.2, if e is the identity of G, then

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ϕμ (e) = ϕμ (ee) = ϕμ (e) + Ad∗ (e)ϕμ (e) = 2ϕμ (e). Thus ϕμ (e) = 0, which implies ϕμ = 0 and μ ◦ s M = Ad∗ (s) ◦ μ.



Remark 4.3.6 1. If G is a commutative Lie group, then for any s ∈ G, Ad∗ (s) = id. Thus, for any moment map μ, ϕμ is always a differentiable homomorphism from G into the additive group g∗ . If further G is compact and connected (that is, G is a torus), then ϕμ = 0 and μ(sx) = μ(x), ∀ s ∈ G, x ∈ M. Since a moment map is constant on each G orbit, for any x ∈ M, gx = {ax : a ∈ g} ⊂ ker d μx . By Proposition 4.2.9, ker d μx = (gx)⊥ . Thus the orbits of G are isotropic submanifolds. 2. Let (M, ω) be a homogeneous Hamiltonian G-space, that is, for any two points x1 and x2 of M, there exists an element s of G such that sx1 = x2 (G acts transitively on M). If μ is a moment map of M, then by Proposition 4.2.9, μ is an immersion. By the corollary of Proposition 4.3.2, in this case, the map μ : M −→ μ(M) is a covering of an orbit of the affine action of G on g∗ (defined by μ). 3. For related materials of this section, the reader is referred to Refs. [17, 23].

Chapter 5

Poisson Manifolds

5.1 The Structure of a Poisson Manifold 5.1.1 The Schouten–Nijenhuis Bracket Let M be a manifold. Denote the degree p skew symmetric contravariant differentiable tensor field space on M by D p (M).Then each space D p (M) is a differentiable cross section space of the vector bundle p T M. We have D0 (M) = C ∞ (M). With the operation defined by the wedge product ∧, D∗ (M) =



D p (M)

p≥0

is a Z-graded associative algebra. For any u ∈ D p (M) and v ∈ Dq (M), the wedge product ∧ satisfies the Z2 -commutative law: u ∧ v = (−1) pq v ∧ u. We define a bracket [ , ] on D∗ (M), that is, a bilinear map [ , ] : D∗ (M) × D∗ (M) −→ D∗ (M), (u, v) −→ [u, v], u, v ∈ D∗ (M), by requiring it satisfy the following conditions: (1) [ f, g] = 0, (2) [ f, X 1 ∧ · · · ∧ X p ] = (−1) p [X 1 ∧ · · · ∧ X p , f ] =

p 

(−1)i (X i · f )X 1 ∧ · · · Xˆ i · · · ∧ X p ,

i=1

(3) [X 1 ∧ · · · ∧ X p , Y1 ∧ · · · ∧ Yq ] =

p  q  (−1)i+ j [X i , Y j ] ∧ X 1 ∧ · · · Xˆ i · · · ∧ X p ∧ Y1 ∧ · · · Yˆ j · · · ∧ Yq , i=1 j=1

© Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_5

91

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where f, g ∈ D0 (M) = C ∞ (M) and X 1 , . . . , X p , Y1 , . . . , Yq ∈ D1 (M), are all arbitrary, and the notation Xˆ means X is deleted. The bracket thus defined is unique, and we call it the Schouten–Nijenhuis bracket (see Ref. [20]). Its restriction to D1 (M) × D1 (M) agrees with the usual bracket on the vector fields. If X is a vector field and f ∈ D0 (M), then [X, f ] = X f. For any integers p, q ≥ 0, if u ∈ D p (M) and v ∈ Dq (M), then [u, v] ∈ D p+q−1 (M), and [u, v] = −(−1)( p−1)(q−1) [v, u]. For any u ∈ D p (M), the map ad u : v −→ [u, v], v ∈ D∗ (M), is a Z2 -derivation of degree p − 1 of the graded associative algebra D∗ (M), that is, for any v ∈ Dq (M) and w ∈ D∗ (M), [u, v ∧ w] = [u, v] ∧ w + (−1)( p−1)q v ∧ [u, w]. Schouten–Nijenhuis bracket satisfies a sign change Jacobi identity: [u, [v, w]] = [[u, v], w] + (−1)( p−1)(q−1) [v, [u, w]], ∀ u ∈ D p (M), v ∈ Dq (M), w ∈ D∗ (M). Shift the gradation of D∗ (M) by 1, that is, define the degree of the elements in D p (M) to be p − 1, then we obtain a Lie superalgebra structure on D∗ (M). This is a Z -graded algebra (see Definition 2.2.2). If u ∈ D p (M) and f ∈ D0 (M), then −[ f, u] = i(d f )u. Let p > 0. If for a collection of f 1 , . . . , f n ∈ C ∞ (M) such that d f 1 , . . . , d f n generate the module 1 (M) of 1-forms, we have [ f i , u] = 0, i = 1, . . . , n, then u = 0. In the discussions that follow, w always denotes an element in D2 (M), unless stated otherwise, w is assumed to be fixed. By the properties of Schouten–Nijenhuis bracket, for any f ∈ C ∞ (M), H f := [ f, w] = [w, f ]. Thus the map H : f −→ H f ,

f ∈ C ∞ (M),

is a linear map from C ∞ (M) into the vector field space D1 (M). Also, for any X ∈ D1 (M) and f ∈ C ∞ (M), [X, H f ] = [X, [ f, w]] = [X f, w] + [ f, [X, w]].

(5.1.1)

5.1 The Structure of a Poisson Manifold

93

Thus [X, H f ] = H X f + [ f, [X, w]].

(5.1.2)

Define a bilinear map { , } : C ∞ (M) × C ∞ (M) −→ C ∞ (M) by

{ f, g} = H f g (= [[w,f],g]), ∀ f, g ∈ C ∞ (M).

Then for any f, g, h ∈ C ∞ (M), { f, gh} = { f, g}h + g{ f, h}.

(5.1.3)

Since for any f, g ∈ C ∞ (M), [ f, g] = 0, { f, g} + {g, f } = [H f , g] − [ f, Hg ] = [[w, f ], g] − [ f, [w, g]] = [w, [ f, g]] = 0. Thus { f, g} = −{g, f }.

(5.1.4)

If X, Y are two vector fields on M, then for any f ∈ C ∞ (M) and α ∈ 1 (M),

α, [ f, X ∧ Y ] = α, [X ∧ Y, f ] = α, (Y f )X − (X f )Y  = α ∧ d f, X ∧ Y  . Thus for any f ∈ C ∞ (M) and α ∈ 1 (M),

α, [ f, w] = α ∧ d f, w .

(5.1.5)

Thus for any f, g ∈ C ∞ (M), { f, g} = [ f, w]g = d g, [ f, w] = d g ∧ d f, w , that is { f, g} = d g ∧ d f, w .

(5.1.6)

Let U be a coordinate neighborhood of M, let x1 , . . . , xn be coordinates on U , and let the coordinate expression of w on U be n ∂ ∂ 1  ci j ∧ , ci j = −c ji . w= 2 i, j=1 ∂ xi ∂x j

Then for any f, g ∈ C ∞ (M), the following hold on U :

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Hf =

n  i, j=1

ci j

n  ∂f ∂ ∂ f ∂g , { f, g} = ci j , ∂ x j ∂ xi ∂ x j ∂ xi i, j=1

and ci j = {xi , x j }, i, j = 1, . . . , n. Lemma 5.1.2 The following conditions are equivalent: (1) (2) (3) (4)

For any f, g, h ∈ C ∞ (M), { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0. For any f, g ∈ C ∞ (M), H{ f,g} = [H f , Hg ]. For any f ∈ C ∞ (M), [H f , w] = 0. [w, w] = 0.

Proof Compute directly, { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = H f (Hg h) + Hg (−H f h) − H{ f,g} h = [H f , Hg ]h − H{ f,g} h. Thus (1) and (2) are equivalent. If f ∈ C ∞ (M), then [H f , w] ∈ D2 (M). If [g, [H f , w]] = 0 for any g ∈ C ∞ (M), then [H f , w] = 0. From [H f , Hg ] − H{ f,g} = [H f , [g, w]] − [H f g, w] = [g, [H f , w]], it follows that (2) and (3) are equivalent. Finally, from the equality [ f, [w, w]] = [[ f, w], w] − [w, [ f, w]] = 2[[ f, w], w] = 2[H f , w], it follows that (3) and (4) are equivalent.



Remark 5.1.3 Let B ⊂ C ∞ (M) be a collection of functions such that the differentials of the functions of B generate the module 1 (M) (for example, the collection formed by the coordinate functions of M). Then (1), (2), and (3) of Lemma 5.1.2 only need to hold for the functions f, g, h in B. If [w, w] = 0, then by (5.1.4) and Lemma 5.1.2, the bracket { , } defines a Lie algebra structure on C ∞ (M). If M is a symplectic manifold and w = ω, formula (5.1.3) shows that the action of this bracket on the product of the functions in C ∞ (M) is the same as the Poisson bracket. Definition 5.1.4 Let M be a manifold, and let w be a degree 2 skew symmetric contravariant tensor on M. If w satisfies [w, w] = 0, then w is called a Poisson structure on M. If w is a Poisson structure on M, then (M, w) is called a Poisson manifold. Any Poisson structure w on M defines a bracket { , }w on C ∞ (M). Let (M1 , w1 ) and (M2 , w2 ) be two Poisson manifolds, and let ϕ : M1 −→ M2 be a differentiable map. If for any f, g ∈ C ∞ (M2 ), ϕ ∗ { f, g}w2 = {ϕ ∗ ( f ), ϕ ∗ (g)}w1 ,

5.1 The Structure of a Poisson Manifold

95

then ϕ is called a homomorphism from (M1 , w1 ) into (M2 , w2 ). Remark 5.1.5 If w is a Poisson structure on a manifold M, then for any f ∈ C ∞ (M), by Lemma 5.1.2, [H f , w] = 0. This equality shows that the diffeomorphism flows on M generated by the vector field H f keeps the tensor w invariant.

5.2 The Leaves of a Poisson Manifold Let w be a Poisson structure on a manifold M. For any x ∈ M, define a linear map γx : Tx∗ M −→ Tx M, such that for any ξ, η ∈ Tx∗ M, ξ, γx (η) = ξ ∧ η, wx  . Denote the image of γx by L x . Then for any x ∈ M, L x is a vector subspace of Tx M, its dimension is equal to the rank of wx . In general, the dimension of L x depends on the point x, but it is always an even number. We keep the notation of Sect. 5.1. Lemma 5.2.1 For any f ∈ C ∞ (M) and x ∈ M, H f (x) ∈ L x . Proof In fact, by (5.1.6), for any g ∈ C ∞ (M), (H f g)(x) = { f, g}(x) = d g ∧ d f, w (x) = d g, γx (d f x ) , thus H f (x) = γx (d f x ) ∈ L x .



If the rank of w is always equal to 2 p on M, then for any x ∈ M, the vector space L x is a fiber of the differentiable subbundle of T M generated by all H f ( f ∈ C ∞ (M)). In particular, if f 1 , . . . , f n ∈ C ∞ (M) are independent on some neighborhood of a point x ∈ M, then for any point y in this neighborhood, L y is always generated by H fi (y) , i = 1, . . . , n, as a vector subspace of Ty M. Since for any f, g ∈ C ∞ (M), by Lemma 5.1.2, [H f , Hg ] = H{ f,g} , the vector subbundle of T M generated by all H f , denoted by L, is an integrable subbundle of T M. Thus, M is the union set of the integrable leaves of L. That is, M is the union set of those submanifolds F such that Tx F = L x , ∀ x ∈ F, and with respect to the set inclusion relation, they are the maximal ones among the submanifolds of M that satisfy this relation (see Ref. [4]). Similarly, in the cases where the ranks of w are not constant, we also call any maximal submanifold F of M that satisfies the relation Tx F = L x , ∀ x ∈ F, a leaf of the Poisson manifold (M, w). Proposition 5.2.2 Let F be a leaf of a Poisson manifold (M, w). Then there exists a unique 2-form ω F ∈ 2 (F) such that for any f, g ∈ C ∞ (M),

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5 Poisson Manifolds

ω F (H f | F , Hg | F ) = {g, f }| F . Furthermore, ω F is a symplectic structure on F, and for any f ∈ C ∞ (M), H f | F is the Hamiltonian vector field on F corresponding to the function f | F . Proof For any x ∈ F and any ξ, η ∈ Tx∗ M, according to the definition of γx , we have

ξ ∧ η, wx  = ξ, γx (η) = − η, γx (ξ ) . Thus, if at least one of ξ and η belongs to the kernel of the map γx : Tx∗ M −→ Tx M, then ξ ∧ η, wx  = 0. Since L x is the image of γx , we can define a skew symmetric bilinear form ωx on L x as follows. For any u, v ∈ L x , let u = γx (ξ ), v = γx (η), ξ, η ∈ Tx∗ M, then the value of ωx at (u, v) is defined by ωx (γx (ξ ), γx (η)) = ξ ∧ η, wx  . Then this value is independent of the choices of ξ, η, and it only depends on u = γx (ξ ) and v = γx (η). Since for any x ∈ F, L x = Tx F, there exists a unique differential 2-form ω F such that for any x ∈ F, (ω F )x = ωx . In fact, by (5.1.6), for any f, g ∈ C ∞ (M), we have ω F (H f | F , Hg | F ) = d f ∧ d g, w | F = { f, g}| F . Since the module of the vector fields on F is generated by the vector fields of the form H f | F ( f ∈ C ∞ (M)), thus the above equality determines ω F uniquely. In addition, the equality shows that ω F is a differential form, that is ω F ∈ 2 (F). For any x ∈ M, by the definition of γx , ker γx is the kernel of the bilinear form b : (ξ, η) −→ ξ ∧ η, wx  , ξ, η ∈ Tx∗ M. Thus the rank of ω F is always equal to dim F = dim L x at any point x of F. Next we prove that ω F is a closed differential form. Choose arbitrary f, g, h ∈ C ∞ (M). To simplify our notation, write H f , H g , H h for the restrictions of H f , Hg , Hh on F respectively. Then (θ(H f )ω F )(H g , H h ) = H f ω F (H g , H h ) − ω F ([H f , H g ], H h ) − ω F (H g , [H f , H h ]) = { f, {h, g}} − {h, { f, g}} − {{ f, h}, g} = 0.

Thus for any f ∈ C ∞ (M), we have θ (H f )ω F = 0. On the other had, for any f, g ∈ C ∞ (M) we have (i(H f )ω F )(H g ) = {g, f }| F = (d f (Hg ))| F .

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97

Thus for any f ∈ C ∞ (M), i(H f )ω F = (d f )| F = d( f | F ). Thus

(5.2.1)

i(H f ) d ω F = θ (H f )ω F = 0, ∀ f ∈ C ∞ (M).

Since the set of all H f , f ∈ C ∞ (M), generates the module of vector fields on F, the above equality shows that d ω F = 0, that is, ω F is closed. We have proved that ω F is a symplectic structure on the leaf F. Furthermore, equality (5.2.1) shows that  H f is the Hamiltonian vector field on F corresponding to the function f | F . Corollary 5.2.3 Let (M, w) be a Poisson manifold. Then we have { f, g}| F = { f | F , f | F } F , ∀ f, g ∈ C ∞ (M), where the bracket on the left hand side is the Poisson bracket on M defined by w, while the bracket on the right hand side is the Poisson bracket on the leaf F defined by the symplectic structure ω F . Proof In fact, { f, g}| F = ω F (H f | F , Hg | F ) = { f | F , g| F } F .



If w is a Poisson structure on a manifold M such that its rank is always equal to dim M, then L = T M, and M is the only leaf of the Poisson manifold (M, w). Thus ω M is a symplectic structure on M. The bracket operation defined by this symplectic structure is the same as the bracket operation defined by the Poisson structure. Conversely, if ω is a symplectic structure on M, then it can be used to define an isomorphism from T M onto T ∗ M: ϕ : v −→ i(v)ω, v ∈ T M. Thus induces an isomorphism from D2 (M) onto 2 (M). Under this isomorphism, the inverse image of ω is a Poisson structure w on M whose rank is always equal to dim M, and the symplectic structure on the leaf M induced by w coincides with ω. Thus, symplectic structures are special cases of Poisson structures. The corollary of Proposition 5.2.2 shows that, for any leaf F of a Poisson manifold (M, w), the inclusion map i : F −→ M is a Poisson manifold homomorphism (choose the Poisson structure on F defined by the symplectic structure ω F ). It can be proved (see Ref. [15]), that any Poisson manifold is the union set of its integrable leaves.1 1 Added

by the authors of the Forewords. Even when the distribution L is not of constant rank, it is completely integrable in a generalized sense. Its maximal integral manifolds, called here the leaves of the Poisson manifold (M, w), are called, in other texts, the symplectic leaves of that Poisson manifold. The leaves of (M, w) are in general immersed, not embedded, submanifolds of M. The proof that for each point x ∈ M there exists a unique symplectic leaf which contains that point can

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Example 5.2.4 Let X, Y be two vector fields on a manifold M, and let w = X ∧ Y ∈ D2 (M). Then [w,w] = [X ∧ Y, X ∧ Y ] = X ∧ [Y, X ] ∧ Y + Y ∧ [X, Y ] ∧ X = 2[X, Y ] ∧ X ∧ Y.

If [X, Y ] = 0, then (M, w) is a Poisson manifold. If this is the case, for any f, g ∈ C ∞ (M), { f, g} = (Y f )(Xg) − (X f )(Y g), and H f = (Y f )X − (X f )Y. In general, there exist two types of leaves on M: the leaves that degenerate to points and the 2-dimensional leaves. The first type of leaves are the points x ∈ M such that (X ∧ Y )x = 0, and the second type of leaves form a foliation on the open set U = {x ∈ M : (X ∧ Y )x = 0} of M.

5.3 Poisson Structures on the Dual of a Lie Algebra In this section, we denote by g an n-dimensional real Lie algebra, and denote by (a1 , . . . , an ) a basis of g. We identify g with its double dual (g∗ )∗ , then we can view the space g as a subspace of C ∞ (g∗ ), and view a1 , . . . , an as coordinates on g∗ . Let w=−

n 1  ∂ ∂ [ai , a j ] ∧ . 2 i, j=1 ∂ai ∂a j

(5.3.1)

To avoid notation confusions, we use [ , ] S to denote the Schouten–Nijenhuis bracket in D∗ (g∗ ), and still use [ , ] to denote g’s own bracket. Lemma 5.3.1 The tensor w defined by formula (5.3.1) is a Poisson structure on g∗ .2 Furthermore, for the Poisson bracket { , } on C ∞ (g∗ ) defined by w and any b, c ∈ g, {b, c} = [[w, b] S , c] S = [b, c]. be done either by working with local quotients of that Poisson manifold, or by application of a generalization of Frobenius’ theorem proven around 1973 independently by P. Stefan (Integrability of systems of vectorfields, J. London Math. Soc., 2–21(3), pp. 544–556, 1980) and H. Sussmann (Orbits of families of vector fields and integrability of systems with singularities, Bull. Amer. Math. Soc., 79(1):197–199, 1973). 2 Added by the authors of the Forewords. This Poisson structure which exists on the dual space of a Lie algebra was noticed by Sophus Lie and rediscovered, much later, independently by A. Kirillov, B. Kostant and J.-M. Souriau. It is often called the canonical Lie-Poisson structure, or the KirillovKostant-Souriau structure.

5.3 Poisson Structures on the Dual of a Lie Algebra

99

Proof Let (ξ1 , . . . , ξn ) be the basis of g∗ dual to (a1 , . . . , an ). Then for any b ∈ g, ∂ b = ξi , b , i = 1, . . . , n. ∂ai Thus   n  n   ∂ 1  ∂ ∂ [ai , a j ] ξ j , b = − [ai , a j ] ξi , b [b, ai ] . [w, b] S = − 2 i, j=1 ∂ai ∂a j ∂ai i=1 Thus for any b, c ∈ g,

{b, c} = [[w, b] S , c] S = [b, c].

Then the Jacobi identity of g shows that { , } also satisfies the Jacobi identity, thus Lemma 5.1.2 and the remark after it show that [w, w] S = 0. So w is a Poisson  structure on g∗ . We can characterize the tensor w by using the following condition (cf. (5.1.6)): for any b, c ∈ g,

d b ∧ d c, w = [c, b]. Thus w does not depend on the choice of the basis (a1 , . . . , an ), and we call it the canonical Poisson structure on g∗ . Let g1 , g2 be two Lie algebras, and let ψ : g1 −→ g2 be a Lie algebra homomorphism. If we choose canonical Poisson structures on both g∗1 and g∗2 , then the map ψ t : g∗2 −→ g∗1 defined by  

ψ(a), ξ  = a, ψ t (ξ ) , ∀ a ∈ g1 , ξ ∈ g∗2 , is a Poisson manifold homomorphism. In particular, if g is the Lie algebra of a Lie group G, then the canonical Poisson structure on g∗ is invariant under the coadjoint representation of G. Proposition 5.3.2 Let g be the Lie algebra of a Lie group G and assume that G is connected. Then all the orbits of the coadjoint representation of G are all the leaves of g∗ with respect to the canonical Poisson structure. Proof For arbitrary a, b ∈ g and ξ ∈ g∗ , we have   b(Ad∗ (exp(ta))ξ ) = exp(t (ad(ta))ξ, b = ξ, b + t ξ, [a, b] + t 2 (· · · ). For a ∈ g, let a be the vector field on g∗ that corresponds to a under the coadjoint action of G: a : ξ −→ aξ, ξ ∈ g∗ .

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Then for any b ∈ g (view b as a function on g∗ ), a b = [a, b]. But from Lemma 5.3.1, [a, b] = {a, b} = Ha b. Thus for any a ∈ g, a = Ha . For any ξ ∈ g∗ , the vector space L ξ formed by all vectors of the form H f (ξ ), f ∈ C ∞ (g∗ ), coincides with the vector space formed by all vectors of the form a (ξ ), a ∈ g, that is, coincides with the tangent space of the orbit Ad∗ (G)ξ at the point ξ . This shows that every leaf of the canonical Poisson structure is invariant under the action of Ad∗ (G), and for any orbit θ of Ad∗ (G) and any ξ ∈ θ , Tξ θ = L ξ . Since G is connected, all the orbits of Ad∗ (G) are all the leaves.  Corollary 5.3.3 Let G be a connected Lie group and let F be an orbit of the coadjoint representation of G. Then there exists a unique symplectic structure ω F on F, such that the identity immersion i : F −→ g∗ is a Poisson manifold homomorphism for the canonical Poisson structure on g∗ . Furthermore, ω F is a G-invariant symplectic structure. Proof This is a direct application of Proposition 5.2.2 and its corollary.



In the following discussions, we call the symplectic structure ω F in the above corollary the canonical symplectic structure on the orbit F. Proposition 5.3.4 Let G be a connected Lie group, let F be an orbit of the coadjoint representation of G, and let ω F be the canonical symplectic structure on F. Then the symplectic G-space (F, ω F ) is a Poisson G-space, and the identity immersion μ : F −→ g∗ is a moment map of (F, ω F ). Proof In fact, according to Proposition 5.2.2, for any a ∈ g, we have H μ,a = Ha | F = a | F . Thus for any ξ ∈ F, H μ,a (ξ ) = a (ξ ). This shows that (F, ω F ) is a Hamiltonian G-space and μ is a moment map. Also, it is obvious that μ is  G-equivariant, thus by Proposition 4.3.5, (F, ω F ) is a Poisson G-space. Proposition 5.3.5 Let (M, ω) be a Poisson G-space, and let μ : M −→ g∗ be a moment map of (M, ω) such that for any s ∈ G and x ∈ M, μ(sx) = Ad∗ (s)μ(x), Then for the canonical Poisson structure on g∗ , μ is a Poisson manifold homomorphism. Proof In fact, μ is G-equivariant by assumption, so by Proposition 4.3.2, the 2dimensional closed cochain cμ = 0. Thus for any a, b ∈ g, {μ∗ (a), μ∗ (b)} = { μ, a , μ, b} = μ, [a, b] = μ∗ [a, b] = μ∗ {a, b}. Since a basis of g can be viewed as a coordinate system on g∗ , thus by the above equality, μ∗ { f, h} = {μ∗ ( f ), μ∗ (h)}, f, h ∈ C ∞ (g∗ ). Thus the proposition follows.



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101

Next, we will prove that if (M, ω) is just a Hamiltonian G-space but not necessary a Poisson space, then by replacing the canonical Poisson structure on g∗ by a Poisson structure determined by the G-space (M, ω), the conclusions above still hold (see Refs. [17, 23]). As before, let (a1 , . . . , an ) be a basis of g. We call the map ϕ : ξ −→

n  i=1

ξ, ai 

∂ , ξ ∈ g∗ , ∂ai

a canonical linear map from g∗ into the vector field space of g∗ . It follows from the definition that ϕ is invariant under the parallel translations of g∗ . The map ϕ can be extended to a canonical injective homomorphism from the exterior algebra (g∗ ) into the skew symmetric contravariant tensor algebra D∗ (g∗ ) of g∗ , such that the image of p (g∗ ) under this homomorphism is the degree p skew symmetric contravariant vector field space that is invariant under parallel translations of g∗ . In fact, if p (g∗ ) is identified with the skew symmetric p-form space A p (g) of g, then a p-form β ∈ A p (g) corresponds to the tensor field β˜ ∈ D p (g∗ ) with the following coordinate expression: β˜ =

1  ∂ ∂ β(ai1 , . . . , ai p ) ∧ ··· ∧ , p! i ,...,i ∂ai1 ∂ai p 1

p

where the indices i 1 , . . . , i p run through all integers 1, . . . , n. ˜ S = 0. We point out that for the For any α ∈ A p (g) and β ∈ Aq (g), we have [α, ˜ β] degree +1 Z2 -derivation d of the algebra A(g), we have (d β)(a, b) = −β([a, b]), where β ∈ A1 (g) and a, b ∈ g are all arbitrary. In fact, this identity is a direct consequence of the following well-known formula: (d β)(a, b) = aβ(b) − bβ(a) − β([a, b]), where d is the coboundary operator of the cochain complex on g with values in R (see Ref. [13]). ˜ S = −(d β), Proposition 5.3.6 For any p ≥ 0 and and β ∈ A p (g), we have [w, β] where w is defined by formula (5.3.1). Proof Since the map ρ : u −→ [w, u] S , u ∈ D∗ (g∗ ), is a Z2 -derivation of the al˜ β ∈ A(g), is a homomorphism from the algebra gebra D∗ (g∗ ), and ϕ : β −→ β, ∗ A(g)

into D∗ (g ), it suffices to verify the equality for β = ξ ∈ g∗ . If ξ ∈ g∗ , then n ˜ξ = i=1

ξ, ai  ∂a∂ i , and by the discussions just before the proposition,

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[w, ξ˜ ] S = −[ξ˜ , w] S =

n  ∂ 1  ∂ ξ, [ai , a j ] ∧ 2 i, j=1 ∂ai ∂a j

=−

n 1  ∂ ∂ ∧ = −(d ξ ). d ξ(ai , a j ) 2 i, j=1 ∂ai ∂a j

Therefore the proposition follows.



Remark 5.3.7 We have ˜ S ] S = [[w, w] S , β] ˜ S = 0. 2 d2 β = 2[w, [w, β] Thus the equality d2 = 0 is just another form of [w, w] S = 0. Corollary 5.3.8 Let β ∈ A2 (g) be a 2-cochain on g. Then in order for w − β˜ to be a Poisson structure on g∗ , it is necessary and sufficient that β is a closed 2-cochain, that is, d β = 0. Proof In fact, we have ˜ S − [β, ˜ w] S = −2[w, β] ˜ S = 2(d ˜ w − β] ˜ S = [w, w] S − [w, β] β), [w − β, ˜ w − β] ˜ S = 0 if and only if d β = 0. Thus [w − β,



According to the above corollary, every closed 2-cochain β on g corresponds to a Poisson structure w − β˜ on g∗ . We denote the bracket of C ∞ (g∗ ) defined by w − β˜ by { , }β . This bracket can be characterized by the following equality: ˜ a] S , b] S = [a, b] + β(a, b), ∀ a, b ∈ g. {a, b}β = [a, b] − [[β, Let G be a Lie group, let g be the Lie algebra of G, and let β be a closed 2-cochain on g. Then in general, the Poisson structure w − β˜ is not necessary invariant under the coadjoint representation of G. We will see that if G is simply connected, then w − β˜ is invariant under an affine action of G on g∗ , such that the linear part of this affine action is the coadjoint representation of G. Lemma 5.3.9 Let β ∈ A2 (g) be a closed 2-cochain on g with values in R. Define a linear map β : g −→ g∗ such that 

β(a), b = β(a, b), ∀ a, b ∈ g,

and use the dual relation to define a representation ad∗ of g on g∗ by   − ad∗ (a)ξ, b = ξ, ad(a)b , ∀ ξ ∈ g∗ , ∀ a, b ∈ g. Then for any a, b ∈ g,

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103

ad∗ (a)β(b) − ad∗ (b)β(a) − β([a, b]) = 0. Proof In fact, by the definitions, for any c ∈ g, 

ad∗ (a)β(b) − ad∗ (b)β(a) − β([a, b]), c



= −β(b, [a, c]) + β(a, [b, c]) − β([a, b], c) = (d β)(a, b, c) = 0. 

Thus the equality holds.

This lemma shows that if β is a closed 2-cochain, then β is a closed 1-cochain on ∗ ∗ g with values  in g , which has the g-module structure defined via ad . Note that for any a ∈ g, β(a), a = 0. In general, there exist closed 1-cochains on g with values in g∗ that do not satisfy this condition, for example, when g is a commutative Lie algebra, then there are such closed 1-cochains on g. Lemma 5.3.10 Let g be the Lie algebra of a simply connected Lie group G, and let χ be a closed 1-cochain on g with values in the g-module g∗ (with respect to the representation ad∗ ). Then there exists a unique differentiable map f : G −→ g∗ such that for any s, t ∈ G and a ∈ g, (1) f (st) = f (s) + Ad∗ (s) f (t), (2) d f (a) = χ (a). Proof For any a ∈ g, let L a be the left invariant vector field on G that corresponds to a: L a : s −→ sa, ∀ s ∈ G. Let α be a differential 1-form on G with values in g∗ such that for any s ∈ G and any a ∈ g, α(sa) = Ad∗ (s)χ (a). Then we have α(s exp(ta)b) = Ad∗ (s) Ad∗ (exp(ta))χ (b) = Ad∗ (s)χ (b) − t Ad∗ (s) ad∗ (a)χ (b) + t 2 (· · · ). Thus at the point s ∈ G, L a α(L b ) − L b α(L a ) is equal to − Ad∗ (s)(ad∗ (a)χ (b) − ad∗ (b)χ (a)). Since χ is a closed 1-cochain, we have L a α(L b ) − L b α(L a ) = −α(L [a,b] ) = α([L a , L b ]). Thus (d α)(L a , L b ) = L a α(L b ) − L b α(L a ) − α([L a , L b ]) = 0. This equality holds for arbitrary a, b ∈ g, thus d α = 0. Also, since G is simply connected, there exists a unique differentiable map f from G into g∗ such that

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d f = α and f (e) = 0 (e is the identity of G). Now for arbitrary s ∈ G and a ∈ g, d f (sa) − Ad∗ (s) d f (a) = α(sa) − Ad∗ (s)α(a) = 0. This shows that f (st) − f (s) − Ad∗ (s) f (t) is independent of t. But it is 0 when t = e, thus f satisfies conditions (1) and (2). Conversely, if f satisfies (1) and (2), then d f (sa) = Ad∗ (s) d f (a) = Ad∗ (s)χ (a), and f (e) = 0, thus the uniqueness follows.



Lemma 5.3.11 Let g be the Lie algebra of a Lie group G, and let f : G −→ g∗ be a differentiable map such that for any s, t ∈ G, f (st) = f (s) + Ad∗ (s) f (t). Then: (1) For any a ∈ g and t ∈ G, ad∗ (a) f (t) = d f (a) − Ad∗ (t) d f (Ad(t −1 )a). (2) For the affine action of G on g∗ defined by the equation sξ = Ad∗ (s)ξ + f (s), we have a b = [a, b] + d f (a), b , ∀ a, b ∈ g. Proof Take derivatives of the equality f (st) = f (s) + Ad∗ (s) f (t) with respect to s and t respectively to get: d f (at) = d f (a) − ad∗ (a) f (t), d f (sa) = Ad∗ (s) d f (a), ∀ s, t ∈ G, ∀ a ∈ g. Replace s by t and replace a by Ad(t −1 )a in the second equality to get d f (at) = Ad∗ (t) d f (Ad(t −1 )a). Then apply the first equality to obtain (1). View the elements of g as functions on g∗ , then from sξ = Ad∗ (s)ξ + f (s), it follows that   b(sξ ) = sξ, b = Ad∗ (s)ξ + f (s), b , ∀ s ∈ G, ξ ∈ g∗ . Take derivative with respect to s to get   ( a b)(ξ ) = d b(aξ ) = − ad∗ (a)ξ + d f (a), b = ξ, [a, b] + d f (a), b = ([a, b])(ξ ) + d f (a), b , which holds for arbitrary ξ ∈ g∗ and a, b ∈ g. Thus (2) is proved.



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Now assume that g is the Lie algebra of a simply connected Lie group G. Let β be a closed 2-cochain on g with values in R, and let β be the 1-cochain on g with  values in g∗ such that for any a, b ∈ g, β(a), b = β(a, b). According to Lemma 5.3.10, there exists a unique differentiable map f β from G into g∗ such that for any s, t ∈ G and a ∈ g, we have f β (st) = Ad∗ (s) f β (t) + f β (s) and d f β (a) = β(a). Thus we can use the equality sξ = Ad∗ (s)ξ + f β (s), ∀ s ∈ G, ξ ∈ g∗ , to define an affine action of G in g∗ . We call an affine action of G in g∗ thus defined the action associated with the closed 2-cochain β. On the other hand, according to the corollary of Proposition 5.3.6, we know that from a closed 2-cochain β, a tensor β˜ ∈ D2 (g∗ ) can be determined such that w − β˜ is a Poisson structure on g∗ . Proposition 5.3.12 Let g be the Lie algebra of a Lie group G, and let β be a closed 2-cochain on g with values in R. If G is simply connected, then under the affine action of G in g∗ associated with β, the Poisson structure w − β˜ on g∗ is invariant. Proof For any s ∈ G, let sg∗ : ξ −→ Ad∗ (s)ξ + f β (s), ∀ ξ ∈ g∗ , be the affine automorphism of g∗ that corresponds to s. Let { , }β be the bracket ˜ Then for any s ∈ G, a ∈ g, and of C ∞ (g∗ ) defined by the Poisson structure w − β. ξ ∈ g∗ ,   sg∗∗ (a)(ξ ) = Ad(s −1 )a(ξ ) + f β (s), a . Note that in the equality above, we have viewed sg∗∗ (a) as a function on g∗ , and have used the definition (see Sect. 4.3): Ad∗ (s) = t Ad(s −1 ), s ∈ G. Thus, according to part (2) of Lemma 5.3.10, {sg∗∗ (a), sg∗∗ (b)}β = {Ad(s −1 )a, Ad(s −1 )(b)}β = Ad(s −1 )[a, b] + β(Ad(s −1 )a, Ad(s −1 )b)  = Ad(s −1 )[a, b] + β(Ad(s −1 )a), Ad(s −1 )b   = Ad(s −1 )[a, b] + Ad∗ (s) d f β (Ad(s −1 )a), b . Also, by part (1) of Lemma 5.3.11,

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Ad∗ (s) d f β (Ad(s −1 )a) = d f β (a) − ad∗ (a) f β (s). Thus for any s ∈ G and a, b ∈ g,     {sg∗∗ (a), sg∗∗ (b)}β = Ad(s −1 )[a, b] + d f β (a), b + f β (s), [a, b]   = sg∗∗ ([a, b]) + d f β (a), b = sg∗∗ ([a, b] + β(a, b)) = sg∗∗ {a, b}β . This shows that for any s ∈ G, sg∗ is an automorphism of the Poisson manifold ˜ Thus the Poisson structure w − β˜ is invariant under the affine action of (g∗ , w − β).  G on g∗ associated with β. Proposition 5.3.13 The assumptions are the same as in Proposition 5.3.12. All the leaves of g∗ defined by the Poisson structure w − β˜ are all the orbits of the action of G associated with β. Proof In fact, according to Lemma 5.3.10, we have a b = [a, b] + d f (a), b = [a, b] + β(a, b) = {a, b}β = Ha b, ∀ a, b ∈ g. This implies a = Ha , ∀ a ∈ g. The rest of the proof is identical to the proof of Proposition 5.3.2.  Corollary 5.3.14 For any orbit F of g∗ under the affine action of G associated with β, there exists a unique symplectic structure ω F such that the identity immersion μ : F −→ g∗ is a Poisson homomorphism with respect to the Poisson structure w − β˜ on g∗ . This symplectic structure ω F is invariant under the action of G. Furthermore, the symplectic G-space (F, ω F ) is Hamiltonian, and μ is a moment map. Proof The proof is similar to the case β = 0 (Corollary of Proposition 5.3.2 and Proposition 5.3.4).  Let (M, ω) be a Hamiltonian G-space, and let μ : M −→ g∗ be a moment map. Recall that in Sect. 4.2, we defined a closed 2-cochain cμ on g with values in R by using μ: cμ (a, b) = { μ, a , μ, b} − μ, [a, b] , ∀ a, b ∈ g. Proposition 5.3.15 Choose the Poisson structure w − c˜μ on g∗ , then the moment map μ : M −→ g∗ is a Poisson manifold homomorphism (see Refs. [15, 17]). Proof In fact, for any a, b ∈ g, we have {μ∗ (a), μ∗ (b)} = { μ, a , μ, b} = μ, [a, b] + cμ (a, b) = μ∗ ([a, b] + cμ (a, b)) = μ∗ {a, b}cμ . Thus the claim follows.



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107

We have already known (Sect. 4.3) that if G acts on g∗ by sξ = Ad∗ (s)ξ + ϕμ (s), s ∈ G, ξ ∈ g∗ , where

ϕμ (s) = μ(sx) − Ad∗ (s)μ(x), x ∈ M,

then the moment map μ : M −→ g∗ is G-equivariant. In addition, by Proposition 4.3.2 we have cμ (a, b) = d ϕμ (a), b , ∀ a, b ∈ g. Thus d ϕμ = cμ . This shows that if G is connected, then the affine action of G on g∗ can be obtained from the affine action of the simply connected covering group of G associated with the close cochain cμ by taking the quotient. Exercise 5.3.16 Complete the proofs of Proposition 5.3.13 and its corollary. Exercise 5.3.17 Let g be the Lie algebra of a simply connected Lie group G and let β be a closed 2-cochain on g with values in R. (1) Let f β (G) be the orbit given by the origin of g∗ under the affine action of G on g∗ associated with β. Prove that the dimension of f β (G) is equal to the rank of β. (2) Assume that the rank of β is equal to dim g. Choose a left invariant symplectic structure ω on G such that for any a, b ∈ g, ω(a, b) = β(a, b). Prove that (G, ω) is a Hamiltonian G-space and the map f β : G −→ g∗ is a moment map of (G, ω). (3) Again assume that the rank of β is equal to dim g. Define a product in g: (a, b) −→ ab, a, b ∈ g, such that β(ab, c) = −β(b, [a, c]), ∀ a, b, c ∈ g. Prove that for any a, b, c ∈ g, ab − ba = [a, b], a(bc) − b(ac) = (ab)c − (ba)c. That is, for this product, g is a “left symplectic” algebra according to Vinberg’s definition (see Ref. [25]).

Chapter 6

A Graded Case

6.1 (0, n)-Dimensional Supermanifolds In this section, we generalize the concept of symplectic structures to supermanifolds (see Refs. [18, 19]). We first give the definition of supermanifolds, or according to B. Kostant (see Ref. [18]), graded manifolds. Definition 6.1.1 Let M0 be an n 0 -dimensional manifold, and let A be a variety of R-algebras on M0 with the following properties: 1. for any open set U ⊂ M0 , A(U ) is a Z2 -graded algebra: A(U ) = A(U )0 + A(U )1 , where 0 and 1 represent the two elements in Z2 ; 2. the variety A is locally isomorphic to the tensor product over R of the variety of differentiable functions on M0 and the exterior algebra (Rn 1 ) (for some n 1 ∈ Z+ ) considered as a Z2 -graded algebra. Then we call M = (M0 , A) an (n 0 , n 1 )-dimensional supermanifold, and call M0 the base space. According to this definition, when U is sufficiently small, we have an isomorphism from A(U ) onto C ∞ (U ) ⊗ (Rn 1 ), which maps A(U ) p ( p ∈ Z2 ) onto C ∞ (U ) ⊗



q (Rn 1 ).

q≡ p (mod 2)

In the following discussions, we will only consider the special case where the base manifold M0 shrinks to a point e. In this case, the dimension of M = (e, A) has the form (0, n). Such a supermanifold is formed by a point e and a Z2 -graded algebra A that is isomorphic to (Rn ). For this type of supermanifolds, the involved questions are purely algebraic, and thus we can replace R by any field k of characteristic 0 in the discussions. © Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5_6

109

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We call a Z2 -graded k-algebra that is isomorphic to (k n ) a rank n Grassmann algebra over k. Thus a Grassmann algebra is a Z2 -commutative algebra of dimension 2n , it is generated by a collection of grading 1 elements x1 , . . . , xn such that x1 x2 · · · xn = 0. Let M = (e, A) be a (0, n)-dimensional supermanifold. We call a collection of elements x1 , . . . , xn in A1 a coordinate system on M if x1 x2 · · · xn = 0. If x1 , . . . , xn form a coordinate system on M, then x1 , . . . , xn generate the algebra A. Thus choosing a coordinate system amounts to giving an isomorphism from A onto (k n ). Let m be the maximal ideal of A, that is, the ideal generated by a coordinate system of M (or A1 ). We call the vector space m/m 2 the covector space on M, and denote it by Te∗ M. The space Te∗ M is an n-dimensional space, all its elements have grading 1. The dual space Te M = (m/m 2 )∗ of Te∗ M is called the vector space on M. Let DerA be the left A-module formed by the Z2 -derivations of A, then the elements of DerA are called the vector fields on M. For any X ∈ Der A and a ∈ A, denote by Xa the image of a under X . As a vector space over k, DerA is also Z2 graded. If p ∈ Z2 , then (Der A) p is the set of the k-endomorphisms X of A that satisfy 1. X Aq ⊂ A p+q , q = 0, 1, and 2. X (ab) = (Xa)b + (−1) p·q a(X b), ∀ a ∈ Aq , b ∈ A. As a left A-module, DerA is a graded A-module, that is, for any p, q ∈ Z2 , A p (Der A)q ⊂ (Der A) p+q . We define a bracket [ , ] on DerA as follows: [X, Y ] = X ◦ Y − (−1) p·q Y ◦ X, ∀ X ∈ (Der A) p , Y ∈ (Der A)q . Then DerA becomes a Lie superalgebra over k (see Definition 2.2.2). Let x1 , . . . , xn be a coordinate system on M = (e, A). Then for any 1 ≤ i ≤ n, there exists a unique Pi ∈ (Der A)1 such that Pi x j = δi j . We denote this Pi by ∂∂xi . Then the Z2 -derivations: ∂ ∂ ,..., ∂ x1 ∂ xn from a basis of the A-module DerA, and for i, j = 1, . . . , n,  ∂ ∂ ∂ ∂  ∂ ∂ = , − = 0. ∂ xi ∂ x j ∂ xi ∂ x j ∂ x j ∂ xi Let A be a Grassman algebra over k (according to our definition), and let E be a Z2 -graded left A-module. Since A is a Z2 -commutative algebra, we can define a right A-module structure on E such that xa = (−1) p·q ax, ∀ x ∈ E p , a ∈ Aq .

6.1 (0, n)-Dimensional Supermanifolds

111

Then for any a, b ∈ A and x ∈ E, a(xb) = (ax)b. Therefore, all Z2 -graded left Amodules can be defined to be two sided (A, A)-modules. Let E be a Z2 -graded left A-module. We use H om A (E, A) to denote the Z2 -graded vector space over k of linear maps from E to A. The elements of H om A (E, A) of grading p are the k-linear maps ϕ from E into A such that: 1. ϕ(E q ) ⊂ A p+q , and 2. ϕ(ax) = (−1) p·q aϕ(x), ∀ q ∈ Z2 , a ∈ Aq , x ∈ E. We can define a Z2 -graded left A-module structure on H om A (E, A) such that (aϕ)(x) = a(ϕ(x)), ∀ a ∈ A, x ∈ E, ϕ ∈ H om A (E, A). We call the Z2 -graded A-module 1 (M) = H om A (Der A, A) the differential 1form module of the supermanifold M = (e, A). It can be proved that there exists a unique k-linear map d : A −→ 1 (M) such that (d a)(X ) = (−1) p·q Xa, ∀ a ∈ A p , X ∈ (Der A)q . Then d A p ⊂ 1 (M) p . For a ∈ A p and b ∈ Aq , set (d a)b = (−1) p·q b(d a), then d(ab) = (d a)b + a(d b), a, b ∈ A. It can be shown that if x1 , . . . , xn form a coordinate system on M, then (d x1 , . . . , d xn ) is a basis of the A-module 1 (M). Now we define supermanifolds T M and T ∗ M. If A is a Grassmann algebra, then any Z2 -graded left A-module E is a two-sided module, thus for any p ≥ 1, we can define the tensor power ⊗ p E = E ⊗ E ⊗ · · · ⊗ E . p copies

Denote by ⊗E the direct sum of all ⊗ p E and equip it with the usual product, then ⊗E is also called a tensor algebra. This is a doubly graded algebra, its gradation is in Z × Z2 . The elements of grading ( p, p) are the elements of the vector subspace of ⊗ p E spanned by the elements u 1 ⊗ u 2 ⊗ · · · ⊗ u p , where u i ∈ E p(i) such that 

p(i) = p.

1≤i≤ p

Let I be the two-sided ideal of the tensor algebra ⊗E generated by the following elements: u ⊗ v − (−1) p·q v ⊗ u, u ∈ E p , v ∈ E q ,

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and let S(E) = ⊗E/I. We call S(E) the Z2 -graded symmetric algebra of the Amodule E. Since E is doubly graded, S(E) is also a doubly graded algebra, it has both a gradation in Z and a gradation in Z2 . For the Z2 -gradation, it is Z2 -commutative. If A is a rank n Grassmann algebra, and E is a free Z2 -graded A-module with a basis consists of r elements of grading 1 (Z2 -gradation), then the symmetric algebra S(E) is a rank n + r Grassmann algebra with respect to the Z2 -gradation. In particular, S(Der A) and S(H om A (Der A, A)) are both rank 2n Grassmann algebras. Let M = (e, A) be a (0, n)-dimensional supermanifold. We call the supermanifold (e, S(1 (M)) the tangent supermanifold of M and denote it by T M, and call the supermanifold (e, S(Der A)) the cotangent supermanifold of M and denote it by T ∗ M. Both supermanifolds T M and T ∗ M are (0, 2n)-dimensional. The canonical inclusion homomorphism A −→ S 0 (Der A) ⊂ S(Der A) can be viewed as a homomorphism between the supermanifold M and the supermanifold T ∗ M, while the canonical inclusion map A −→ S 0 (1 (M)) ⊂ S(1 (M)) can be viewed as a homomorphism between the supermanifold M and the supermanifold T M. We now define the linear complex of differential forms (M) on the supermanifold M = (e, A). Let 0 (M) = A, 1 (M) = H om A (Der A, A). For p > 1, define  p (M) as follows: 1.  p (M) is submodule of H om A (⊗ p Der A, A), and 2. for 1 ≤ i ≤ p, if ϕ ∈  p (M), then ϕ(X 1 ⊗ · · · ⊗ X i ⊗ X i+1 ⊗ · · · ⊗ X p ) = −(−1) pi · pi+1 ϕ(X 1 ⊗ · · · ⊗ X i+1 ⊗ X i ⊗ · · · ⊗ X p ) holds for arbitrary X 1 , . . . , X p ∈ Der A (the Z2 -grading of X i is pi , 1 ≤ i ≤ p). We view ϕ ∈  p (M) as a map from (Der A) p into A and let ϕ(X 1 ⊗ · · · ⊗ X p ) = ϕ(X 1 , . . . , X p ). The A-module  p (M) is a Z2 -graded submodule of H om A (⊗ p Der A, A). We denote p the subspace of  p (M) formed by the elements whose Z2 -grading are p by  p (M). p Then the elements in  p (M) are ( p, p)-grading differential forms. We can define an associative product in (M) = p  p (M) by

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(ϕ, ψ) −→ ϕ ∧ ψ, ϕ, ψ ∈ (M). This definition is similar to the definition of the exterior product of two skewsymmetric forms (see Refs. [18, 21]). Under this product, (M) is a Z × Z2 -graded algebra, that is p+q

 pp ∧ qq (M) ⊂  p+q (M). If a ∈ A p = 0p (M), ψ ∈ qq (M), then we have a ∧ ψ = (−1) p·q ψ ∧ a = aψ. If ϕ ∈  pp (M), ψ ∈ qq (M), then we have ϕ ∧ ψ = (−1) p·q+ p·q ψ ∧ ϕ. Let x1 , . . . , xn be coordinates on M. As an algebra over k, (M) is generated by the following 2n elements: x1 , . . . , xn (∈ 01 ) and d x1 , . . . , d xn (∈ 11 ). They satisfy the following relations: xi x j + x j xi = 0, xi d x j + d x j xi = 0, d xi ∧ d x j − d x j ∧ d xi = 0. These relations show that as a left A-module, (M) is isomorphic to A ⊗k k[T1 , . . . , Tn ], where k[T1 , . . . , Tn ] is the polynomial algebra with indeterminates T1 , . . . , Tn over k. It is not hard to see that if n = 0, then for any p ≥ 0,  p (M) = {0}. If a kendomorphism γ of (M) satisfies p+q

γ (qq (M)) ⊂  p+q (M), ∀ (q, q) ∈ Z × Z2 , then we call it a degree ( p, p) endomorphism of (M). If a degree ( p, p) endomorphism γ satisfies γ (ϕ ∧ ψ) = γ (ϕ) ∧ ψ + (−1) p·q+ p·q ϕ ∧ γ (ψ), ∀ ϕ ∈ q (M), ψ ∈ (M),

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then we call it a Z × Z2 -derivation of (M). It can be proved that there exists a unique Z × Z2 -derivation d of degree (1, 0) on (M) such that 1. d2 = 0, and 2. (d a)(X ) = (−1) p·q Xa, ∀ a ∈ A p , X ∈ (Der A)q . This derivation is the extension of the map d : A −→ 1 (M) defined before. The chain complex defined by d has zero homology, that is, the sequence d

d

0 −→ k −→ A = 0 (M) −→ 1 (M) −→ · · · is an exact sequence.

6.2 (0, n)-Dimensional Symplectic Supermanifolds Let M = (e, A) be a (0, n)-dimensional supermanifold, and let ω ∈ 20 (M) be a differential form satisfying the following conditions: (1) if X ∈ Der A such that ω(X, Y ) = 0, ∀ Y ∈ Der A, then X = 0; and (2) d ω = 0. Then we call ω a symplectic form on M. Let X ∈ (Der A) p , denote by i(X ) the degree (−1, p) endomorphism of (M) defined by the following formula: (i(X )ϕ)(Y1 , . . . , Yq−1 ) = (−1) p·q ϕ(X, Y1 , . . . , Yq−1 ), where ϕ ∈ q (M) and Y1 , . . . , Yq−1 ∈ Der A. It can be proved that i(X ) is a Z × Z2 derivation of degree (−1, p). Condition (1) above implies that for a symplectic structure ω, the map X −→ i(X )ω, X ∈ Der A, is an injection. Thus, if condition (1) is satisfied, then the map X −→ i(X ) is an isomorphism from the A-module DerA onto the A-module 1 (M). If ω be a symplectic structure on M, then it induces a skew symmetric bilinear form ωe on Te M, and by condition (1), ωe is nondegenerate. Proposition 6.2.1 Let ω be a symplectic structure on M = (e, A). There exist a . . . , xn on M and an n × n symmetric matrix (λi j ) over the coordinate system x1 ,

field k, such that ω = i,n j=1 λi j d xi ∧ d x j and det(λi j ) = 0. This conclusion is similar to Darboux Theorem (Sect. 2.3), it is a basic and special case of Theorem 5.3 in [18]. A vector field X ∈ Der A is called a symplectic vector field if it satisfies d i(X )ω = 0. The symplectic vector fields on M form a Lie sub-superalgebra S of the Lie superalgebra DerA. It can be proved (see Ref. [14]) that if n ≥ 4, then S is a simple Lie superalgebra, that is, S and {0} are the only ideals of S.

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115

For any a ∈ A, there exists a unique Ha ∈ Der A such that i(Ha )ω = d a. This Ha is a symplectic vector field. The sequence H

{0} −→ k −→ A −→ S −→ {0} is an exact sequence. Define a Poisson bracket on A by {a, b} = Ha b, a, b ∈ A. Then A becomes a Lie superalgebra with center k, and the map H : a −→ Ha , a ∈ A, is a Lie superalgebra homomorphism. For any a ∈ A p , b ∈ Aq and c ∈ A, {a, bc} = {a, b}c + (−1) p·q b{a, c}.

Let x1 , . . . , xn be coordinates on M such that ω = i,n j=1 d xi ∧ d x j . Then for any j,  1 ∂ ∂ ω = 2 d x j and Hx j = i . ∂x j 2 ∂x j Thus for 1 ≤ i, j ≤ n, {xi , x j } = 21 δi j .

6.3 The Canonical Symplectic Structure on T ∗ P Let P = (e, A) be a (0, n)-dimensional supermanifold. Any X ∈ Der A = S 1 (Der A) can be identified with an element of the Grassmann algebra S(Der A). On the other X of Z2 -grading p hand, for any X ∈ (Der A) p , there exists a unique Z2 -derivation in S(Der A) such that (1) X (a) = Xa, ∀ a ∈ A = S 0 (Der A), (2) X (Y ) = [X, Y ], ∀ Y ∈ Der A = S 1 (Der A). The derivation X has degree zero with respect to the Z-gradation. Also, since T ∗ P = (e, S(Der A)),

X is a vector field on T ∗ P (i.e. an extension of X ). If x1 , . . . , xn are coordinates on P, we can view each ∂∂xi as an element in S(Der A) and let yi = ∂∂xi , i = 1, . . . , n. Then x1 , . . . , xn , y1 , . . . , yn are coordinates on T ∗ P and   ∂ ∂ x j = δi, j , y j = 0, i, j = 1, . . . , n. ∂ xi ∂ xi

116

6 A Graded Case

There exists a unique differential form α ∈ 10 (T ∗ P) such that for any X ∈ Der A,

n yi d xi α( X ) = X. This form is similar to the Liouville form, and we have α = i=1 under the coordinates x1 , . . . , xn , y1 , . . . , yn . The differential form ω = −dα =

n 

− d yi ∧ d xi

i=1

is a canonical symplectic structure on T ∗ P. The form ωe is the neutral form of duality on the space Te (T ∗ P), which is canonically isomorphic to the space Te P + (Te P)∗ .

Bibliography

1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Benjamin Cummings, Reading (1978) 2. Arnold, V.I.: Mathematical methods of classical mechanics. Nauka, Moscow (1974) 3. Atiyah, M.F.: Convexity and commuting hamiltonians. Bull. Lond. Math. Soc. 14, 1–15 (1982) 4. Bourbaki, N.: Variétés différentielles et analytiques. Hermann, Paris (1971) 5. Chevalley, C.: Theory of Lie Groups. Princeton Univ, Press (1946) 6. Darboux, G.: Sur le probléme de Pfaff. Bull. Des Sc, Math (1882) 7. Duistermaat, J.J.: Fourier Integral Operators. Courant Institute of Mathematical Sciences, New York (1973) 8. Duistermaat, J.J.: On the momentum map. In: Proceedings of the IUTAM-ISIMM, Symposium on Modern Developments in Analytical Mechanics, Torino (1982) 9. Godbillon, C.: Géometrie différentielle et mécanique Analytique. Hermann, Paris (1969) 10. Guillemin, V., Sternberg, S.: The momentum map and collective motion. Ann. Phys. 127, 220–253 (1980) 11. Guillemin, V., Sternberg, S.: Convexity properties of the momentum map. Invent. Math. 67, 491–513 (1982) 12. Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962) 13. Jacobson, N.: Lie Algebras. Interscience Publishers, Wiley, New York (1962) 14. Kac, V.: Lie superalgebras. Adv. Math. 26, 8–96 (1977) 15. Kirillov, A.A.: Local Lie algebras. Uspekhi Math. Nauk 31(4), 55–76 (1976) 16. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Publishers, New York (1969) 17. Kostant, B.: Quantization and Unitary Representations. Lecture Notes in Mathematics, vol. 170. Springer, Berlin (1970) 18. Kostant, B.: Graded Manifolds, Graded Lie Theory and Prequantization. Lecture Notes in Mathematics, vol. 570. Springer, Berlin (1977) 19. Leites, A.: Introduction to the theory of supermanifolds. Uspekhi Math. Nauk 35(1), 3–57 (1980) 20. Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geom. 12, 253–300 (1977) 21. Scheunert, M.: The Theory of Lie Superalgebras. Lecture Notes in Mathematics, vol. 716. Springer, Berlin (1979) 22. Siegel, C.L.: Symplectic geometry. Am. J. Math. 65, 1–86 (1943) 23. Souriau, J.M.: Structure des systèmes dynamiques, Dunod Paris (1969) © Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5

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24. Symes, W.W.: Hamiltonian group actions and integrable systems. Pysica 1(D), 339–374 (1980) 25. Vinberg, E.B.: The theory of convex homogeneous cones. Moscow Math. Soc. 12, (1963) 26. Wallach, N.R.: Symplectic Geometry and Fourier Analysis. Mathematical Science Press, Brookline (1977) 27. Weil, A.: Introduction à l’étude des variétés kählériennes. Hermann, Paris (1958) 28. Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329– 346 (1971) 29. Weinstein, A.: Lectures on Symplectic Manifolds. In: C.B.M.S. Regional Conference Series, vol. 29, AMS, Rhode Island (1977) 30. Weyl, H.: Classical Groups. Princeton University Press, Princeton (1946) 31. Yan, Z.: Representation Theory of Semisimple Lie Groups and Lie Algebras. Science and Technology Press, Shanghai (1963). (In Chinese)

Index

Symbols 3-dimensional Lie algebra, 8 G-equivariant, 89 G-space, 76 Z × Z2 -derivation, 114 Z × Z2 -graded algebra, 113 Z2 -commutative, 1 Z2 -derivation, 2 Z2 -graded symmetric algebra, 112 h-space, 76 i(x), 2 μ(α), 8 osp(2, 1), 29 p-linear forms, 1 sl(2, k), 8

A Action, differentially, 75 Adjoint g-space structure, 76 Adjoint representation of g, 87 Adjoint representation of G, 87 Affine action of G, 89

B Bijection, 4

C Canonical Poisson structure on g∗ , 99 Canonical symplectic form, 7 Cayley parametrization, 16 Closed 2-cochain, 81 Coadjoint representation of G, 87 Coboundary, 81 Coboundary operator, 101

Cochain complex, 101 Coisotropic submanifold, 48 Coisotropic subspace, 3 Complex of differential forms (M), 112 Complex projective space, 24 Complex structure, 16 Contraction method, 52 Cotangent bundle, 57 Covering of an orbit, 90 Critical point, 38 Curvature, 55

D Darboux theorem, 33 DerA, 110 de Rham group, 36

E Exact sequence, 37 Exponential map, 16 Exterior algebra (g∗ ), 101 Exterior product, 1

F Flow, 35 Formal symplectic vector fields, 46

G Generating function, 51 Generating function of a submanifold, 69 Good intersection, 52 Graded algebra, 1 Graded A-module, 110

© Springer Nature Singapore Pte Ltd. and Science Press 2019 J.-L. Koszul and Y. M. Zou, Introduction to Symplectic Geometry, https://doi.org/10.1007/978-981-13-3987-5

119

120 Graded manifold, 109 Grassmann algebra, 110

H Hamiltonian g-space, 79 Hamiltonian G-space, 87 Hamiltonian vector field, 37 Homogeneous Hamiltonian G-space, 90

I Inner product, 2 Integrable subbundle, 32 Integral curve, 35 Integral leaf, 55 Isotropic complement vector subspace, 5 Isotropic submanifold, 48 Isotropic subspace, 3

J Jet, 47 Jet algebra, 46

K Kähler form, 24 Kählar structure, 23 Kernel (of a p-form), 2 Kinetic torque, 85

L Lagrangian complement subspace, 7 Lagrangian foliation, 54 Lagrangian submanifold, 48 Lagrangian subspace, 3 Left exterior product, 27 Left invariant vector field, 76 Left symplectic algebra, 107 Lie algebra of G, 76 Lie derivation, 21 Lie group, 75 Lie superalgebra, 28 Lift, 69 Linear connection, 55 Liouville form, 58

M Manifold, 21 Moment map, 80

Index O Orthogonal group, 19 Orthogonality, 3

P Poisson bracket, 40 Poisson g-space, 82 Poisson G-space, 87 Poisson manifold, 94 Poisson structure on M, 94 Primitive element of a representation, 10 Pseudo-Hermitian form, 17 Pseudo Kähler form, 24 Pseudo Riemannian form, 24 Pullback of a form, 2 Pushforward of a form, 2

R Rank (of a p-form), 2 Right invariant vector field, 76

S Schouten–Nijenhuis bracket, 91 Semisimple representation, 30 Simple algebra, 30 Simple submodule, 10 Spray, 68 Stabilizer of L, 13 Stationary point of a moment map, 86 Strong Hamiltonian g-space, 82 Suitable complex structure, 17 Supermanifold, 109 Symplectic basis, 7 Symplectic complex structure, 16 Symplectic coordinates, 34 Symplectic form, 6 Symplectic group Sp(2n), 11 Symplectic G-space, 76 Symplectic h-space, 76 Symplectic manifold, 22 Symplectic manifold homomorphism, 22 Symplectic manifold isomorphism, 22 Symplectic space, 6 Symplectic structure on a manifold, 22 Symplectic submanifold, 48 Symplectic vector field, 36

Index T Tensor algebra, 111 Torsion, 55 Torus, 90 U Unitary group, 19

121 V Variety of R-algebras, 109 Vector field extension, 67 Vertical vector, 58