The Lie bialgebra structures on the Witt and Virasoro algebras

Table of contents :
A b s tr a c t ...................................................................................................................................... ii
A cknow ledgem ents .............................................................................................................. iii
D edication ............................................................................................................................... iv
1 . Introduction ...................................................................................................................... 1
2 . Q uantization of Lie b ia lg e b ra s ................................................................................. 4
2.1. Hopf algebras ........................................................................................................... 4
2.2. Quantization of Lie algebras ................................................................................ 8
2.3. Lie bialgebras ........................................................................................................... 10
2.4. Quasitriangular Hopf algebras ............................................................................ 12
3. Cohomology of graded Lie a lg e b ra s ................................................................... 16
3.1. Graded Lie alg eb ras ............................................................................................. 16
3.2. p-modules ................................................................................................................. 18
3.3. Duality of homology and cohomology of Lie algebras .................................... 22
3.4. Universal central extension of Lie subalgebras of V containing W . . . . 26
4. Lie bialgebra structures on the W itt and V irasoro a lg e b ra s ............... 33
4.1. Classical Yang Baxter equation ......................................................................... 34
4.2. Classification of finite dimensional subalgebras of the Witt and Virasoro
algebras ..................................................................................................................... 38
4.3. The Lie bialgebra structures on W \ .................................................................. 40
4.4. Some cohomology results .................................................................................. 41
4.5. Proof of Main R e su lts ......................................................................................... 47
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5. Lie bialgebra structures of the saturated Lie subalgebras of the W itt
a lg eb ra ....................................................................................................................................... 50
5.1. Saturated subalgebras of the Witt alg eb ra ...................................................... 50
5.2. Calculations of the cohomology group L(I) A L{I)) .................... 52
5.3. Lie bialgebra structures on the saturated Lie subalgebras of W ............. 55
6. Uniqueness of the Taft Lie Bialgebras In Characteristic p .................... 57
6.1. Uniqueness for the case i = 0 (m odp) ................................................................ 57
6.2. Uniqueness for the case i « 0 (m odp) ................................................................ 63
R eferences ................................................................................................................................ 72
V ita .............................................................................................................................................. 75
vi
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THE LIE BIALGEBRA STRUCTURES ON THE W ITT A N D VIRASORO ALGEBRAS B Y S I U -H U N G N G

A d isse r ta tio n s u b m itte d to th e G ra d u a te S ch o o l— N e w B ru n sw ick R u tg e r s, T h e S ta te U n iv e r s ity o f N ew J e r se y in p a rtia l fu lfillm en t o f th e req u irem en ts for th e d eg ree o f D o c to r o f P h ilo so p h y G ra d u a te P ro g r a m in M a th em a tic s W r itte n u n d er th e d ire ctio n o f E arl J . Taft an d a p p ro v ed b y

N e w B ru n sw ick , N e w J ersey O cto b er, 1997

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UMI Number:

9814115

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A B S T R A C T O F T H E D IS S E R T A T IO N

THE LIE BIALGEBRA STRUCTURES ON THE W ITT AND VIRASORO ALGEBRAS

b y S iu - H u n g N g D is s e r t a t io n D ir e c t o r : E a r l J . T a ft

This thesis begins w ith a brief expositions of th e theory of deform ation of Hopf algebras. Lie bialgebras are defined and th e relations betw een Lie bialgebras and quantizations of Lie algebras are described. Some cohomology results for inner graded Lie algebras are established. As an application, we prove th e central extension of Bloch’s algebra Q"’" corresponding to the projective representation on the Fock space is the universal central extension of 'D+ . After these prelim inary results are established, we show th a t T aft’s Lie bialgebra structures are all th e Lie bialgebra structures on the one-sided W itt algebra W \ over an algebraically closed field k o f characteristic zero up to isomorphism . For the W itt and Virasoro algebras over a field k of characteristic zero, we prove th a t every Lie bialgebra structure on such Lie algebras is of trian g u lar coboundary type associated to a solution r of th e CYBE of th e form r = a A 6 for some a, b in the underlying algebra. We also prove th a t the antilogous results hold for all the satu rated subalgebras of the W itt algebra. All these results are not true in finite characteristic case. Nevertheless, the sequence of T a ft’s bialgebras are distinct up to isom orphism when the characteristic is not equal to 3.

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Acknowledgements

I would like to express m y deepest thanks to my advisor, E arl J. Taft for his excellent guidance and instruction. W ithout his su pport, this work would never have been pos­ sible. I have been fo rtu n ate to have the opportunity to w ork with such an interesting and insightful person. I would also like to th an k M urray G erstenhaber, Jam es Lepowsky, Charles Weibel and R obert W ilson for consenting to be on my thesis com m ittee and for their helpful com m ents on this thesis.

iii

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Dedication

This thesis is dedicated to my parents.

iv

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Table of Contents

A b s t r a c t ......................................................................................................................................

ii

A c k n o w l e d g e m e n t s ..............................................................................................................

iii

D e d ic a t io n ...............................................................................................................................

iv

1.

I n tr o d u c t io n

......................................................................................................................

1

2.

Q u a n tiz a tio n o f L ie b i a l g e b r a s .................................................................................

4

2.1. Hopf a lg e b r a s ...........................................................................................................

4

2.2. Q uantization of Lie a lg e b ra s ................................................................................

8

2.3. Lie b ia lg e b ra s...........................................................................................................

10

2.4. Q uasitriangular Hopf a lg e b ra s ............................................................................

12

3. C o h o m o lo g y o f g r a d e d L ie a l g e b r a s ...................................................................

16

3.1.

Graded Lie a l g e b r a s .............................................................................................

16

3.2.

p -m o d u le s .................................................................................................................

18

3.3.

Duality of homology and cohomology of Liea lg e b r a s ....................................

22

3.4.

Universal central extension of Lie subalgebras of V containing W .

...

26

4. L ie

b ia lg e b r a s tr u c t u r e s o n t h e W i t t a n d V ir a s o r o a l g e b r a s ...............

33

4.1.

Classical Yang B axter e q u a t i o n .........................................................................

34

4.2. Classification of finite dim ensional subalgebras of the W itt and Virasoro a l g e b r a s .....................................................................................................................

38

4.3.

The Lie bialgebra structures on W \ ..................................................................

40

4.4.

Some cohomology results

..................................................................................

41

4.5.

P ro o f of M ain R e s u l t s .........................................................................................

47

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5. L ie b ia lg e b r a stru ctu res o f th e s a tu r a te d L ie su b a lg eb ra s o f th e W itt a l g e b r a .......................................................................................................................................

50

5.1. S atu rated subalgebras of the W itt a l g e b r a ......................................................

50

L ( I ) A L { I ) ) ....................

52

5.2. C alculations o f the cohomology group

5.3. Lie bialgebra structures on the s a tu ra te d Lie subalgebras of W

.............

55

6. U n iq u e n e ss o f th e Taft Lie B ia lg eb ra s I n C h a r a c te r istic p ....................

57

6.1. Uniqueness for the case i = 0 ( m o d p ) ................................................................

57

6.2. Uniqueness for the case i « 0 ( m o d p ) ................................................................

63

R e f e r e n c e s ................................................................................................................................

72

V i t a ..............................................................................................................................................

75

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1

Chapter 1 Introduction

T he problem of classifying th e Lie bialgebra structures on W itt algebra was m otivated by W itten w hen he was stud y in g the left invariant symplectic stru ctu res of th e Virasoro group. In th is thesis, we are going to stu d y the Lie bialgebra stru ctu res on infinite di­ mensional Lie algebras including the W itt and Virasoro algebras. We will prove th a t all these Lie bialgebra structures adm it quantizations, by using the universal deform ation form ula given in [14]. We will also study the central extensions o f all Lie subalgebras of the Lie algebra of differential operators containing the W itt algebra. In the p ap er [34], W itten studied the Poisson structures on the V irasoro group which in tu rn is rela ted to the Lie bialgebra stru ctu res on the Virasoro algebra. He attem p ted to construct Lie bialgebra stru ctu res on V e c tp (5 x) by inverting 2-cocydes on V e c t^ fj1). The rigorous form ulation o f th e construction on V e c t^ S 1), the com plex vector fields on S 1, was done by Beggs an d M ajid (cf. [l]). By considering Gelfand-Fuks cocycles on the Lie algebra V ect"(5 x), Grabowski constructed another Lie bialgebra stru cture on V e c t"(5 x) by applying th e procedure described by Belavin and D rinfel’d in the finite dim ensional case (cf. [2], an d [15]). The choice of different function spaces for the Lie algebra of vector fields sheds a different light on the problem . In [21], Leitenberger considered th e formal vector fields of th e circle and obtained a class of Lie bialgebra structures on it. R elated resu lts are also studied independently by K upershm idt and Stoyanov in a different context (cf. [20]). In [32], T aft proved th a t, for any coboundary d(r) of a Lie algebra g w ith values in g A g, d(r) defines a Lie bialgebra stru ctu re on g if and only if r is a solution of the modified classical Y ang-B axter equation (M CYBE). Moreover, M ichaelis showed th a t (cf. [29]), for any two dim ensional Lie subalgebra f| of g, f| A b is a set of solutions of

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the classical Yang-Baxter equation (C Y B E) and hence one can construct Lie bialgebras using the two dimensional Lie subalgebras. In C hapter 4, we study th e Lie bialgebra structures on the polynomial vector fields W\ on the complex line, on the W itt algebra W of polynom ial vector fields on the circle, and on its universal central extension V. It is well known th at V is the Virasoro algebra. We prove th a t every Lie bialgebra stru ctu re on W x, W and V is of triangular coboundary type, associated to a solution of CYBE of the form a A 6 for two elements a, b whose linear span is a Lie subalgebra. In p articu lar, solutions of the CYBE for W\ are completely determined. Moreover, we prove th a t the Lie bialgebra structures on W\ obtained by Taft in [32] yield every Lie bialgebra stru ctu re on W\ up to isomorphism . There is a class of Lie subalgebras of th e W itt algebra W for which the analogue of th e results in C hapter 3 holds. We will stu d y this class of Lie subalgebras and their Lie bialgebra structures in C hapter 5. C hapter 6 will be devoted to the discussion of the T aft’s Lie bialgebra W[ l) over a field of finite characteristic p. We prove th a t th e underlying Lie coalgebra structures for th e T aft’s Lie bialgebras are distinct for p ^ 3. For p = 3, i < j , W[ l) and

are

isomorphic as Lie coalgebras if and only if i = 3r — 1 and j = 3r + 1 for some r > 0. In this case, W |3r-1> and W^3r+1> are isom orphic as Lie bialgebras. It is natural to ask whether the Lie bialgebra structures on the W itt and Virasoro algebras can be quantized. Using some basic results sta te d in C hapter 1 and the results in C hapter 3, we can conclude th a t every Lie bialgebra stru ctu re on

W or V can be

quantized. Moreover, we can w rite down a n explicit quantization for each of these Lie bialgebra structures by using a universal deform ation form ula discovered by G iaquinto and Zhang (see [14]). We will also prove th a t isom orphic quantizations of a Lie algebra have isomorphic infinitesimal Lie bialgebras in C hapter 1 . In C hapter 2, we establish some results for com puting the cohomology of an inner graded Lie algebra p w ith coefficients in an inner graded p-module. Using the coho­ mology results for W , we prove th a t for every Lie subalgebra p of the Lie algebra D of differential operators on the circle annihilating 1 such th a t W C p C U , H 2( g, k) = k where the base field k is of characteristic zero. As a consequence, the central extension

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3

of B loch’s algebra 'D+ (cf. [3]) corresponding to the projective representation on the Fock space is the universal central extension of D + . In fact, X)+ is th e unique Lie sub­ algebra of D such th a t W C X)+ C X>. The analogous results for th e whole algebra cl of differential operators on the circle and its completion were also stu d ied by Li and W il­ son w ith different technique (see [22] and [23]). Feigin also com puted the cohomology groups of the Lie algebra of differential operators on th e line (see [8 ]).

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4

Chapter 2 Quantization o f Lie bialgebras

In this chapter, we will recall some basic definitions about Hopf algebras and q uantiza­ tion of Lie algebras. We will prove th at isom orphic quantizations of a Lie algebra over a field of characteristic zero have isomorphic infinitesimal Lie bialgebras. Some related results will also be sta te d w ithout proof for later reference.

2 .1

H o p f a lg e b r a s

D efin itio n 2 .1 .1 Let fc be a com m utative ring.

(i) An algebra over k is a triple (A, m , u ) w ith A a fc-module and ^-linear m aps m:

A — -A called multiplication and u: k — >A called the unit map such th a t

the following diagram s are com m utative : a

,

,

A ® .4 ® A m®lj

l®m

. A ® .4 .

, _ . «®i . _ , i®« . „ . k 0 .4 ----->■A ® .4 -*------ .4 ® k .

jm

A ® A ----- 1:1— - .4 Associativity of m

Unitary property

Let (.4, 772,1, U4 ), ( B ,m f l,u g ) be algebras over k. A fc-linear m ap / : .4— ’B is called an algebra hom om orphism if / o

= mg o ( / ® / ) and / o

= ug.

(ii) A coalgebra over & is a triple (C, A ,e ) w ith C a /b-module an d ^-linear m aps A : C — *C ®jt C called diagonalization or comultiplication and e: C — -k called the augm entation or counit which satisfies com m utative diagram s dual to those

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for an algebra, nam ely :

A01

c0c Coassociativity

Counitary property dt;f

C is called cocommutative if A = A 09 = r o A, where r is the transposition map. Let (C i, A i , f i ) and (C 2 ,A 2 ,e 2) be coalgebras. A coalgebra map f : C \ — -C 2 is a fc-linear m ap such th a t A 2 ° f = ( / 0 / ) o A i a n d e2 0 / = *iE x a m p le s 2 .1 .2 ( T e n s o r p r o d u c ts o f a lg e b ra s a n d c o a lg e b ra s ) (i) Let A, B be algebras over a com m utative ring k. T hen A 0 a.. B is an algebra over k w ith the u n it 1,4 0 I s and m ultiplication given by (cti 0 6 i)(fl 2 ® ^2 )

UiU2 ® ^1^2 •

for a i , a 2 € A and &i, 62 € B. (ii) Let (Cx, A i, e\ ) and (C 2, A 2, e2) be coalgebras. T h en C i 0 fc C 2 is a coalgebra with counit £i 0 e2 and com ultiplication (1 0 tc,.c-, ® 1) 0 (A i 0 A 2) where t cx.c.. : C\ 0 C 2 — >C2 0 C i is the transposition m ap. P r o p o s i tio n 2 .1 .3 (cf. [31], P r o p o s i tio n 3 .1 .1 ) Let ( H , m , u ) be an algebra and let ( H , A, e) be a coalgebra over a com m utative ring k. The followings are equivalent : (i) m and u are coalgebra maps, (ii) A and e are algebra maps. D e f in itio n 2 .1 .4 A bialgebra H = ( H , m , u , A ,e) over a com m utative ring k is a 5tuple such th a t ( H , m, u) is an algebra and ( H , A, e) is a coalgebra and b oth A and e are algebra m aps. We will call H a H opf algebra if there exists a fc-linear m ap S : H — ~H (called the antipode) such th at m o (5 0 1 )o A = m o (1 0 S )o A = u o f .

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( 2 .1)

6

Let H i, H 2 be bialgebras. A fc-linear m ap / : H \— 'H 2 is called a bialgebra map if / is an algebra map as well as a coalgebra map. If H\ and H 2 are H opf algebras w ith antipodes S i and S 2 respectively, a bialgebra map / : E \ — >H2 is called a Hopf algebra map i f S 2 o f = f o S 2. E x a m p le s 2.1.5 Let G be a m onoid and let kG be the m onoid algebra of G. Let A (z) = x x and e(x) = 1 for x £ G. Extending A and e to be algebra m aps gives rise to a bialgebra structure on kG. I f G is a group, then the group algebra k G is a Hopf algebra w ith antipode S ( x ) = i - 1 for x 6 G. D e fin itio n 2.1.6 A Lie algebra g over a field k is a ^-vector space equipped w ith a bilinear product [-, •]: g ® g— -g, called the Lie bracket, which satisfies S kew -sym m etry :

[®,xj = 0

Jacobi I d e n tity :

[x, [y, z}} + [y, [z, x]] + [z, [r, y]] = 0

for x. y, z € g. A Lie subalgebra h of g is a vector subspace such th a t [h, h] C h, th a t is. h is closed under Lie bracket. An ideal £ of g is a subspace which satisfies [g, t] C E. Note th a t an ideal £ is g is a Lie subalgebra in its own right and th a t quotient g/'E inherits a n a tu ra l Lie algebra stru ctu re from g. A linear map between two Lie algebras is a Lie algebra homomorphism if it preserves the Lie brackets. Let Ug be an associative algebra w ith a linear map i: g— >Ug. The pair (Ug, z') is called a universal enveloping algebra for g if for any associative algebra A the map / • — / o f gives a gives a one-one correspondence between algebra m aps / from Ug to A and Lie m aps from g to .4, where the Lie bracket on A is given by [a, 6 ] = ab — ba. E x a m p le s 2 .1 .7 Let g be a Lie alg eb ra over a field k w ith universal enveloping algebra 17(g). T hen 17(g) is a H opf algebra w ith comultiplication, counit and antipode defined by A ( r ) = 1 ® i + i ® 1 , e(x) = 0 and 5(ar) = - x for x € g. R e m a r k 2 .1 .8 Let H be a bialgebra and let M, N be left H -m odules. Then M 0 N is also a left 77-module w ith the Tf-action given by h( x ® y) d= ^ 2 hix ® h'{y

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where

ht-®/i'- = A (h). The category F - m o d of left H -modules is actually a m onoidal

category. In some situations, th e ground ring k is equipped w ith a topology. The notion of topological algebra, coalgebra, bialgebra or H opf algebra can be defined similarly. In th a t case, one can replace the algebraic fc-module A by a topological fc-module and the algebraic tensor product A ®k A by a suitable com pletion, requiring th a t the structure maps are continuous. The m ost im p o rtan t example of such a situ atio n for us is where th e ground ring is K = &[[/i]], the ring of form al power series in an indeterm inate h over a field k. Every A-module V has the h-adic topology, which is the linear topology w ith { h nV | n > 0} the basic open neighborhoods of 0 in V . W ith respect to this topology, the A-m odule structure m ap K x V — *F is continuous. Moreover, any hf-linear m ap / : V — - W is continuous. For any K -module V', let V denote the topological com pletion of V . Algebraically, V = Urn V/ { h nV ) . n

We write V @ W for the completion of V k IF , and call it the complete tensor product of V and W . E x a m p les 2 .1 .9 Let V be a vector space over a field k. Let F[[/i]] denote the form al power series in an indeterm inate h w ith coefficients in V . One can easily see th at F[[/i]] is complete and HausdorfF. Note th a t V[[h]] is “bigger” th a n V ®jt Ar[[/i]] unless V is finite dimensional. In fact, F[[/i]] is th e topological com pletion of V Ug ® Ug be such th a t Ah

=

A -i- h A i

(m od h2)

=

A + h&[

{m od h r ) .

By definition, the associated Lie bialgebra stru ctu re maps on g are given by 8

= Ai - A ^

and

By Lem m a 2.2.3, /olp is a Lie bialgebra Let gh be th e

8'

=

- ( A ') ^ .

map.

inverse of fh and ffh - go + hg\ + h~gn + • ■■

on Ug for som e pt-€ Endjt(17g), a Liebialgebra

(z= 0 , 1 , 2 , - - •

By the same argum ent, □

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: g— -g is

12

D efin itio n 2 .3 .4 (C la ssic a l Y a n g -B a x te r E q u a tio n ) Let g be a Lie algebra over a field k and r =

a,- ® 6t- £ g ® g. Let Ug be the universal enveloping algebra

(i) Consider r as

of g

an element in Ug ® Ug. We define C( r ) = [r12, r 13] £ [r12, r 23] + [r13, r 23]

where r 12 = r ® 1, r 23 = 1 ® r, r 13 =

cq ® 1 ® 6 ,-. It is clear th a t if r £ g A g,

then C{r) £ / \ 3 g. (ii) We say th a t r is a solution to th e classical Y ang-B axter equation (CY BE) if C (r) = 0. (iii) We say th a t r is a solution to th e modified classical Y ang-B axter equation (M CYBE) if C( r) is a g-invariant (i.e. [ z ,C ( r ) ] = 0, where [z,

^ ® b{ ® ct-] = ^ [ z , Oi] ® b{ ® a 4- a, ® [x, 6 ,-] ® c,- -I- a, ® 6 t- ® [z. c,]). I t

There is a correspondence betw een solutions of th e M CYBE w ith invariant sym m etrization and coboundary Lie bialgebra structures on a given Lie algebra. This correspon­ dence was stated in [5]. A com plete proof this result can be found in [32] and i4] (see also [29],[24] and [5]). We will s ta te this as the following proposition. P r o p o sitio n 2.3.5 Let g be a Lie algebra a n d r £ g®g. Then d ( r ) gives a Lie bialgebra structure on g if and only if r is a solution o f the M C Y B E and r

r 2i is g -in va n a n t

where r 21 is the transposition o f r. D efin itio n 2.3 .6 Let (g ,d (r)) be a coboundary Lie bialgebra for some r £ g ® g. We say th a t (g ,d (r)) is quasitriangular if r is a solution of th e CYBE. In addition, if r £ g A g, we say th a t (g ,d (r)) is a triangular coboundary Lie bialgebra.

2 .4

Q u a s itr ia n g u la r H o p f a lg e b r a s

D efin itio n 2.4.1 (cf. [5]) Let

A ,e) be a bialgebra. We call H alm ost co-

commutative if there exists an invertible element 1Z = Y li ai ® b{ £ H ® H such th a t A rjp(z ) = 7£A (x)TZ- 1

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13

for any x £ H. An alm ost cocom m utative Hopf algebra ( Hj l Z) is said to be (i) coboundary if 1Z satisfies 7?.12(A 3 1)(7£) = 1Z23( l 1 . Then th e series T =

“ r-^m

is a UDF based on Ug, where ® E r H {m~r) r=0

\ r

/

R e m a r k 2 .4 .7 Let A be a H opf algebra over a field k and let H be a subH opf algebra of A. If T is a U D F based on H , th en !F is a UDF based on A. In p articular, if g is a Lie algebra over a field k o f characteristic zero such th a t g adm its a 2-dimensional non-abelian Lie subalgebra b, th e n th e form ula in 2.4.6 (ii) defines a U DF T based on Ug. Moreover, (U g^f/ij]^ is a q u an tizatio n of g by Proposition 2.4.3. It is natu ral to ask which quantizations of a Lie algebra g have an associated Lie bial­ gebra which is coboundary, quasitrian g u lar or triangular. We say th a t a quantization UhQ of g is coboundary, quasitriangular or triangular if there is an element 7Z ~ Uhg®UhQ such th a t (U/,g, TZ) satisfies the analogous property in 2.4.1 as a topological Hopf algebra and 1Z = 1 ® 1 (m od h). P r o p o s itio n 2 .4 .8 (cf. [4], 6 .3 .2 ) Let g be a Lie algebra over a field o f characteristic zero and let (U^g, 1Z) be a coboundary quantization o f g. Define r ~ U g ® U g by r = Uy - 1 ^ 1 h

(m o —1, is called the one-sided W itt algebra W \. One can see easily th a t W\ is the Lie algebra of linear derivations on the polynom ial algebra

It is clear th at W i is also inner graded.

(iv) Consider th e algebra k[t, £- 1 ,

of differential operators on the circle. T hen £'£>-'

{ i , j € Z an d j > 0) form a basis for k[t, t ~l , Jj] where D = t ^ (cf. [3] Proposition 1.1). Let £ be the Lie algebra associated to k{t , t ~l ,

i.e. £ = fc[£,£- 1 , ^ ] as

vector space and for x, y 6 £, [i,y ] = xy - yx . For i £ Z , let £,• be th e linear subspace spanned by th e differential operators of th e form t 1f ( D ) for some polynom ial / . Then we have th e form ula (cf. [3] p481) [£’/(£>), tj g( D) \ = £,+J ( f ( D + j ) g ( D) - g ( D + i ) f ( D) )

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(3.1)

18

for any i , j £ Z and polynom ials f , g. In p articular, [D , t Jg{D)\ = jt-ig(D) for i , j £ Z. Hence, £ is inner Z -graded w ith th e degree operator D. Moreover, the subspace spanned by the elem ents tl D {i € Z ) is the W itt algebra and hence W is a graded Lie subalgebra of £ . Let "0 denote the set of all differential operators E on th e circle such th a t £ ( 1 ) = 0. Then V is a graded Lie subalgebra of £ where T>i the linear span of the differential operators of the form tl D-’ , j > 0 . (v) There is an interesting graded Lie subalgebra 'D+ C D containing W (cf. [3]). 'D+ adm its the basis = tm D r+l { - D - m )r | m € Z and r for n > 0 and

= v.

Since V is invariant under the action o f z u, u,r| £ V for r > 0 . Notice th a t N U(r)

Zr V i .

=

i= -N

Therefore, we have

/

1

-N

1 - N

N

+ 1

\ ( - N ) 2N ( ~ n + i )2N

\ /

•

N 2N

\

(

w 0. The differential d for the cochain complex C '( g , M ) is a g-module m ap. Therefore, for any g £ g, g ■C ' ( g , M ) is a subcom plex of C~(g, M) . be a graded Lie algebra. By 3.2.1 (iv), / \ q g is a

D efin itio n 3 .3 .2 Let g = ® t € 2

graded g-module for q > 0. Let ( f \ q g)j denote the degree j homogeneous space o f f \ q g. Let .4 = ® ig 2

a right graded g-module. For r € Z, let C,|r ,(g, .4) denote the

subspace of Cq(g, A) generated by

Y A*®(/\ * + t= r

One can easily th a t 5 (C ,r ,(g, A)) C C ^ l^ g , A). Therefore, C ir | (g ,A ) is a subcom plex of C .(g ,A ) and we will write Z^r)(g ,.4 ), B^r)(g,.4) and Hqr\ g , A ) for the ^-cycle, qboundary and th e g-homology spaces of the complex C ir ,(g, .4) respectively. Let M =

Mi be a graded left g-module. For r 6 Z, define

C ’r | ( g , M ) = ( / 6 C ’ ( g , t f ) ! / ( ( / \ " 0 M C M .,.r f o r a l l s £ Z } . It is obvious th a t d ( C qr)(g, M) ) C C (,i^ 1 (g, Af) for any q > 0. Therefore, C ‘r)(g,iVf) is a subcomplex of C ‘ (g, M) for r £ Z. We w rite Z qr ^(g, M) , B ’r)(g, M ) and H qr ^(g, M) for the g-cocycle, ^-coboimdary an d th e g-cohomology spaces of th e complex C “r |(g, M ) respectively. R em a rk 3 .3 .3 Let g be a graded Lie algebra and .4 a graded right g-module. Notice that C .(g ,j4 ) = ® rl= -C ir,(g, A) as complexes of graded vector spaces. H . ( g . A ) = ® r e 3 fl"ir *(g, A).

Therefore,

However, for a graded left g-module M , C~ { g , M) is

usually not th e direct sum of th e subcomplexes C ‘r j(g, M) . P r o p o sitio n 3 .3 .4 Let g =

&e a graded Lie algebra and let M = 0 ,^ - - Mi be

a graded left g-module. Then fo r r £ Z and q > 0, we have a natural isomorphism Hqr*(g, M 1)' = H qr^(g. M 1) as go-modules. Moreover, ® re -r H qr \ g , M £)‘ is naturally isomorphic to ® rc 2

,( 0 , M ‘)

as graded g-module where the degree I homogeneous space o f ^ r e Z H^q \ g , M 1)' and

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24

® r e z H qr ^(g, M' ) are Hq l\ g , M l)~ and H q_^( g, M' ) respectively.

In particular, if

dim M i < oo fo r all i 6 Z. then H .l r){g, M '*)■ ^ 5 \'r)(g. M). Proof.

T here is a natural linear isom orphism 4>q : ( M ® f \ q g )m— -H o m ^ A ' g, M ~).

nam ely (® (/)(z ))(™) = f(™ ® x) for / € ( M 0 / \ q g)“, x £ f \ q g and m 6 M . M oreover, the diagram

C q( g , M - ) - ^ C q+l( g , M - ) com m utes. Hence. 0 induce an isom orphism betw een the complexes {(C (/(g, M *))“. d ‘ } and {C '^g , M "), d}. Since C\.(g, M ) =

C ir , ( 0 , M ) as complexes, we can n atu rally

>(>•)/ identify {{C qr (g, M 4))*, z0 • C ' { g , L ) is null homotopic. By Lem m a 3.3.6, z () is an isom orphism . Therefore, th e homology groups of the complex z^ -C '(g , L ) are zero.

□

C o r o lla r y 3 .3 .8 Let g be an inner graded Lie algebra over a field k with chark — 0 and M an inner graded left g-module. Then H q(g, M~) = H q{g, M ’) = £r‘°»(g, M 1) ' . Proof. Since M is inner graded, M ' is also ixmer graded. Moreover, M ’ = M ' as g-modules. Therefore, by Theorem 3.3.7, and Proposition 3.3.4 H q( g , M" ) = H?0 )( g , M ' ) =

□

Theorem 3.3.7 is also sta te d in [9] w ithout proof.

3.4

U n iv e r s a l c e n tr a l e x t e n s io n o f L ie s u b a lg e b r a s o f D c o n t a in in g W.

In this section, we study the IT-modules

^ a.o and "3 of Examples 3.2.1 and 3.1.1.

P r o p o s itio n 3 .4 .1 Let W be the W itt algebra over the field k o f characteristic zero. 2

fo r

A= 0

1

fo r

A = —1 , - 2

0

Proof. Suppose first th a t A

0. Let / c Z ^ ^ W ,

otherwise u). Since

n f ( e n ) = e 0 /( e n ) - e „ / ( e 0) = n f { e Tl) - e „/(e„) for n € Z, and H °{W , IFx.-t) = 0 for any A = 0 , we have f ( e u) = 0. Let / ( e n ) = a „ / „ for n 6 Z where a n € k and a 0 = 0. Since / is a derivation, applying / to [e„, em] yields the relation (m - u ) a m+„ = (m - A n)am - (n - mA )an

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(3.3)

for m , n G S . In p articu lar, by applying m — —n to th e above equation, we have (1 -f A)q_„ = —(1 + A )a„

(3.4)

for all n, because a 0 = 0 . 0, j d ( f u) ( e i ) = j ( e x • f 0 ) = f i .

For A ^

Therefore, / is cohomologous to a cocycle

which vanishes ex. Thus, we m ay assume a x = 0. By relatio n (3.3), we have a recursive relation for { a n } for n > 2 : (n-A) n —1

( n - A) • • • ( 2 - A)

(n — 1 )!

for n > 2. If we fu rth er assum e th a t A 5= —1 , equation (3.4) yields a _ Tl = —a „ for n > 1 . Therefore, dim 5 ’1(VF, JFA, 0) < 1 for A =£ 0 , - 1 . Applying equation (3.3) for n = 2 and m = 3, we have 0

= ( ( 4 ~ A ) ( 3 ~ A ) ( 2 ~ A) - (3

- 2A)(2 - A) + (2 - 3A)j

a 2 . (3.5)

Suppose a 2 == 0. Elem entary algebra shows th a t eq u atio n (3.5) holds if and only if A = - 2 . M oreover, for A = - 2 , / ( e n ) = n ( n 2 —1 ) /n is a non-trivial cocycle. Therefore, 1 for

A = —2

0 for

A 5= 0 , - 1 , —2 .

For A - - 1 , we apply relation (3.3) for m = r + 1 an d n = —r 0 = Qr + 1 -

where r > 0. Therefore,

a _ r .

^_i.u) < 1. One can check th a t f ( e n ) = n(n - 1 )/„

is actually a non-trivial cocycle. Thus d im 5 ’1(lF ,^ -x .u ) = 1For A = 0, we cannot assume a x = 0 or ao = 0. N evertheless, by relation (3.3), we have the recursive relation (n - l ) a n+1 = n a n - a x a Ti-(-i

=>

ocn

-------- = ------- r + (

n

n —1

1

n

1

N

r )« i

n —1

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28

for n > 2. Hence, a n = (n — l ) a 2 — (n — 2)ax for n > 2. By equation (3.3) with m = —n, 2 a n = a _ n + a „ for all n. If we apply m = 2 and n = —1 to equation (3.3), we have 3ax = 2 a 2 + a _ x. This inaplies th a t 2 ct.\ — ql2 ~F cto •

Therefore, d im Z^0 ){W ,!F\ .o) < 2 . On th e other hand, f ( e n ) = n f n an d g{en ) -- f n define two linearly independent cocycles. Hence, d i m Z ^ ^ W , Fa.u) - 2. Since / (J £ H °(W ,^o.u), Bf u) = 0. Therefore,

Fj.u) = 2.

□

Sa_b is the K ronecker delta function. We now tu rn to the module 'D of 3.1.1 (iv). As a W -m odule, D adm its a filtration F iD C F 2 T) C - • • by graded W -subm odules where F„D is the subspace o f "D generated by the set { tr D* |

t

£ Z and 1 < s < n} .

Note th a t F iD = W . By equation (3.1), one cam see th a t as W -modules FnX) j Fn- i Q = F a.u for all n > 1. Hence the filtration has sim ple quotients; see 3.2.1 (ii).

L e m m a 3 .4 .2 For any W-submodule M o f T l / W , H U( W , M~) = H \ W , AT) = 0 .

Proof. By C orollary 3.3.5, H q(W, M~) = H q(W , M 1) ' . Therefore, it suffices to show HU(W, M ‘) = H i(W , AT*) = 0. Notice th a t 0 = ( F iD ) /W C (F 2 D ) / W C ••• is a Wmodule filtratio n of V / W . Let Fn M = M n (F nD )/W . Then, F \ M C F2M C • • • is a W -m odule filtratio n of M . The inclusion m ap i induces a W -module m onom orphism i: Fn M f F n - \ M — ► FriD /F fl_ 1'D for n > 2. Since F „D / Fn_iXl is a simple W -module for n > 2, either Fn M / F n- XM = 0 or Fn M / Fn^ \ M

= F^.u as W -m odules. The

complex C .(W , M l ) inherits a filtration from A f, nam ely FnC. = FnC .(W , Af*) := C .(W ,(F „ M )£) .

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29

for n > 1 . Therefore, H q(FnC ./F n^ C . ) = Hq{W, (FnM /

i

s

0

or Hq( W , F ^ )

for n > 2. Prom example 3.2.2, th e graded dual T'ni) is F - n-i.u-

By Proposition

3.4.1, this yields H l {W ,F'n Q) = 0 for n > 2. Hence, by Corollary 3.3.8 and 3.3.5, # i W ^ . u ) = 0 for n > 2 . M oreover, Since !F-n- \ u is a simple W -m odule for n > 2 . H °{W , T'n 0) = 0 and so H 0 (W ,

= 0- By induction Hq( W, ( F nM Y ) = 0 for n > 1 ,

q = 0 .1 . Therefore, H 0 (W, M l ) = H ^ W , M l ) = 0.

□

Let { U a } a be a family of W -m odules and let V be their direct sum

Ua. We will

WTite Ur, A Ufl for the subspace of V A V spanned by the set { Xr, A x 0 \Xr, £ Ur, and x 0 £ U0 ) . Notice th a t Ua A U0 = Ua ® Ub if a d. 0. Consider the family {^aJacA: o f W -modules. For any Ai,A2 £ k, F \ lA) A iF\2A) is also an inner graded W -module b u t all of its homogeneous subspaces are infinite dimensional.

L e m m a 3 .4 .3 Let r, s be integers, not both zero and r + s + 1 = 0. Then H Q( W , ( F r UA ^ uy ) = 0 . Proof. Since F tAA A

= F a.o A F r Q as W -modules, we may assume th a t r = 0. Recall

th at H °{W , (fr. o A ^ .o ) - ) = H o m « r(JrJ) A JF ^ , k ) . Let { /n }n6 2 and { 0, a , = 0 for i < r or i > 2r. Taking / = —1 for the equation (3.6), we can obtain an o th er recursive relation i- 1 - s a ,_ i = ——------Qti i+ r This implies th a t

... . (i = - r ) .

= 0 for r < £ < 2 r . Therefore, a t- = 0 for all i £ Z and so q = 0.

Similarly, one can prove th a t a = 0 for r < 0.

□

Recall from [11] th a t H ° [ W, k ) = k, H l { W, k ) = 0 and H 2 { W, k ) = k. L e m m a 3 .4 .4 The natural map H 2( V , k ) — >H2 (W , fc) is onto. Proof. Consider th e bilinear form B : D x D — >k defined by

( 3 ' T )

for r, s £ N and m ,n £ Z . One can check directly B is a 2-cocycle (cf. [18],[221 and [23]). Since dim H 2 (W , &) = 1 and B \ w « w is the non-trivial 2 -cocycle of W , the result follows.

□

T h e o r e m 3 .4 .5 For any Lie algebra 21 o f X) such that W C 2 1 C D , the natural map H q{%, k ) — >Hq(W. k) is an isomorphism fo r q = 0 , 1 , 2. Proof. Consider th e Hochschild-Serre spectral sequence for the pair (21, W ), as it is described on p .593 o f [16]. We have E” By Lem m a 3.4.2,

=

tf« (W ,H o m * (/\P(21/W ),fc)).

= fl }-1 = 0. Therefore, H q(W , k) = H q( % k ) for q = 0 . 1 .

We claim th a t E^'° = 0. By Corollary 3.3.5, we have Since H U(W, H o m jt(/\ 2 (2l/W ), k) = Hu(W , ( / \ 2 (21/W ))4)* , so it suffices to show th a t HU( W , (A ^ ^ l/W ))4)- Let M = 21/ W . We saw in the proof of Lemma 3.4.2 th a t M adm its a filtration of W -subm odules 0 = F \ M C FnM C • • • such

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31

th a t g rn = FnM / Fn- \ M is either F n_o or 0, for n > 2. This filtratio n of W -submodules induces a filtration 0 = F> C F 3 C Fj C - • • of W '-submodules on M A M where Fn is the linear span of th e set {x A y | x G F iM , y € F jM and i + j — n} . It is easy to see th a t Fn/F n- i = ^

yri^grj.

i+ j = n t . j > l

By Lem m a 3.4.3, H 0 (W, (gr{ A grj)1) = 0. Hence, H U(W, ( F „ /F „ _ i) ‘ ) = 0 for n > 2. Therefore, H a{W, ( M A M ){) = 0. This implies our claim th a t E 2U = 0. Since

= E I '1 = 0, the edge m ap /T 2 (2l , k ) — *H 2( W , k ) is injective. By Lemma

3.4.4, th e com posite H 2{ V , k ) — - H 2(51, Jb)— >H2(W, ifc) is onto. Therefore, th e m ap f f 2(2l. k ) — >H2 { W , k ) is also onto, hence an isomorphism.

□ R e m a r k s 3 .4 .6 (i) By Theorem 3.4.5 and Corollary 3.3.5, for any Lie subalgebra ‘21 of T) containing W,

k) = f f 1^ , k ) M= 0 and H 2{% k) = H 2 (21, * )“ = k. Hence. '21 adm its a

1 -dim ensional

universal central extension (see 7.9.2 of [33]). Moreover, the non­

triv ial 2 -cocycle of 21, with coefficients in in fe, is given by FIsiAa where B is defined in (3.7). In particular, the nontrivial central extension for D 4- obtained in the paper [3] is universal. It was proved by Bloch [3] th a t there is only one graded Lie subalgebra X>+ such th a t W C 2 + C D(ii) The filtration on FjD of V satisfies the relation [F,D , FjD ] C Fi+y-i'D . Therefore, there is a graded Lie algebra gr ( Q) associated to this filtratio n nam ely gr(V) = 0 0 ; t >0 where

= Fi+i'Q/Fi'D and for any

£ Fj+i and x j € F j +1 ( i , j > 0 )

[xi + Fi Q, Xj + FjD ] := [x;, xj] + Fi+j .

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(3.8)

32

In p articu lar, i50 = W . Since th e Lie bracket [•, -]: g ® g— -g is a g-m odule m ap for any Lie algebra g, th e Lie bracket on gr(1d) induces a ik'-module m ap from - 1 ) is shown to consist of solutions of the

CYBE. Moreover, the Lie bialgebra structures associated to these solutions are distinct when characteristic k = 0 [32]. Rem arkably, T aft’s Lie bialgebra structures on W \ are all of the Lie bialgebra structures on W i up to isom orphism when k is algebraically closed of characteristic zero. We will prove the following result in section 4.5. T h e o r e m 4 .0 .7 Let

be the Lie bialgebra on W \ associated to the solution

e lt

A en

o f the C YBE. Every Lie bialgebra structure on W \ is isomorphic to W [ n^ for some n > —1 when k is algebraically closed o f characteristic zero, where the Lie cobracket 5 on

is given by 6 (ei)

= (n - i)e 0 A en+i - ie { A

e rl

(i > - 1 ) .

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34

However, a sim ilar result does not hold for th e W itt algebra W . There are m any more Lie bialgebra structures on W distinct from those associated to the solutions e„ A. e,L of the CYBE ( n £ Z ) [32]. It was proved in [29] th a t for every 2-dimensional subalgebra spanned by a, 6 , of a Lie algebra, a A b is a solution of the CYBE. The converse holds for W itt and V irasoro algebras when characteristic k = 0 [29]. We will prove the following Theorem in section 4.5. T h e o r e m 4 .0 .8 Let g be the W itt or Virasoro algebra over a field o f characteristic zero. Every Lie bialgebra structure on g is a triangular coboundary Lie bialgebra associated to a solution r o f the C Y B E o f the form r = a A 6 fo r some a , 6 € g. In particular, for any solution r £ g A g o f the C Y B E , we have r = a A b fo r som e a,b £ g.

4 .1

C la s s ic a l Y a n g B a x t e r e q u a tio n

Let k be a field and g be a Lie algebra over k. By proposition 2.3.5, for any solution r of the M CYB E in g A g, (g, d(r)) is a coboundary Lie bialgebra. We let g,r| denote the Lie bialgebra stru ctu re on the Lie algebra g. P r o p o s itio n 4 .1 .1 Let g i , g 2 be Lie algebras and on above equation, we have 0

=

{®