Introduction to Lie Algebras

Table of contents :
Preface
Notation
Contents
Chapter 1
1. Definitions and Examples
2. Homomorphism Theorems
3. Lie p-Algebras
Chapter 2
1. Nilpotent and Solvable Lie Algebras
2. Cartan Subalgebras
3. Solvable Lie Algebras
Chapter 3
1. Semisimple Lie algeras
2. Jordan Decomposition in Lie Algebras
3. Cartan Subalgebras
4. Arbitrary Lie Algebras with Non-Degenerate Killing Form
5. Classical Lie Algebras
Chapter 4
1. Universal Enveloping Algebras
2. Irreducible Representations of Classical Lie Algebras
3. Complete Reducibility and Weyl's Theorem
4. Representation Theory in Prime Characteristic
Bibliography
Index

Citation preview

INTRODUCTION TO LIE ALGEBRAS

by

Richard D. Pollack

... Notes by Gordon Edwards

QUEEN'S PAPERS IN PURE AND APPLIED MATHEMATICS - NO. 23

QUEEN 1 S UNIVERSITY, KINGSTON, ONTARIO.

PREFACE

These notes constitute the contents of a course g-iven to graduate students at Uueen's University ~ring the sprin~ term of 1969.

An attempt was made to give a general over-

view of the sub.iect, with perhaps more .attention than usual

paid to a comparison between modular and non-modular Lie algebras; that is, those over fields of prime characteristic and those over fields of characteristic zero.

The only other

novelty is in giving Curtis' modification of the Cartan-Weyl classification of the finite-dimensional ir~educible representations for the classical Lie algebras of arbitrary characteristic (other than 2 or 3). Throu~twut, little more is required than standard linear algebra.

I am indebted to (~ordon Erlwa1·cts for writing, correct,-

in~, revising and clarifying these notes, and to .Mrs. Eileen Wight for typing them.

I have been much influenced in these

notes by the books of Jacobson [8] and Seligman [10], and by a course given by the latter at Yale University.

R.D. Pollack.

Note:

Operators, aside from linear functionals, are written on the right. Unproved results are preceded by an asterisk.

- i -

NOTATION

K

arhi1.rary field.

IN

natur·al numbers.

7L.

inte ,;rers

«l

rational numbers.

C

complex numbers.

L

Lie al~ebra over

Ln

Ln

L(n)

L(n)

L (a.)

L(a.) -

t,.l(B)

normalizer of

CL(B)

centralizer of

f.ndK (\')

al~ehra of endomorphisms of

gt ( V)

Lie al~ebra of endomorphisms of

,.,.t n .,..,

Lie algebra of

stn

matrjces of trace zero in

d

n

scalar matrices in

n

lower trian@:ular matrices in

n

lower nil triangular matrices in

T l1 €

=

K.

[L,L n-1] , L1

=

= [L,L ] •

[L(n-1),L(n-l)], L(l)..,, fL,L]

B

in

n

•

L.

L •

in

V • V

•

K.

matrices over

nxn

gtn •

gtn. @:.f,n • g,t

•

n identity matrix; identity endomorphism.

e. . _ _ _ _matrix ( i_,j) elsewhere. _l....J_ _ _ _ _with _ _ _ _1_ _as ___ _ _ _entry, _ _ _ _ _0_ _ _ _ _ __ DK(A)

derivation algehra of

A •

ad a

inner derivation defined by

a EL •

set of inner derivations of

L •

p-map in a Lie p-algebra.

, ] ( ' >p

t.rar.e form for a representation

< ' >

Killin~ form of

H

Cartan subalgebra of

[

Lie multiplication. p

L. L.

~}

II

dual space of

A

set of roots of

II. L

-

relat.ive t.o

ii -

11 •

of

L •

Tl

-~

fundamental system of roots.

*"

star sum of roots.

A A a. ' .., t(a)

Cartan integer (a ,B E b. , ~

, h

level of

a

element of ---'-a_______ _

E

6.

II. such that

I=

0)

•

'

=

a(h)

l1

universal enveloping algebra; u-algebra.

ii

u-algebra of

11(K,T,S)

standard monomial in

II

embedded in ll •

- iii -

lJ

•

•

CONTENTS

l'KHFACE

i

NOTA'f'lON

ii

CHAP'fEk. I 1. Definitions and Examples

1

2.

Homomorphism Theorems

10

3.

Lie p-al~ebras

15

CilAPTER II

1.

Nilpotent and Solvable Lie Algebras

24

2.

Cartftn Suhalgebras

45

3.

Solvable Lie Algebras

49

CHAPTER III

1.

Semisimple Lie Algebras

66

2.

,Jordan Decomposition in Lie Algebras

79

3. 4.

Cart,an Subalr.-ehras

94

)•

Arbitrary Lie Algebras with Non-de1~enerate KillinP. Forms

102

Classical Lie Algebras

127

CI IA PT E I{ IV

1.

Universal Enveloping Algebras

187

2.

Irreducible Representations of Classical Lie Algebras

200

Complete Reducibility and Weyl' s Theorem

231

3. 4.

Representation Theory in Prime Characteristic

BIBLIOGRAPHY

245 257

lNl>EX

259

- iv -

CHAPTEK I

I.l

Definitions and ExamQles.

Definition 1.

An alg~bra

A

over a field

space with a K-bilinear product

AXA--:+ A

K

is a vector

•

We do not require associativity of multiplication in our definition.

However we will assume that all vector

spaces mentioned in this paper are finite dimensional, unless otherwise specified.

As usual, the multiplication in a Lie algebra will be denoted by means of square brackets.

Definition 2.

An al~ebra

L

Lie algebra if the product

(i) (ii)

[x,x]

=

over a field [

for all

0

, ]

is called a

satisfies

x EL

[[x,y],z] + [[z,x],y] + [[y,z],x] for all

K

=

0

x,y,z E L •

The second condition in the above definition is called the Jacobi identity.

- 1 -

Note that any Lie algebra is ADticommutative.; that

=

Thie

llows

from

(i) and the bilinearity of multiplication:

=

[x+y,x+y]

for

x y.

[x,y]

0

-[y,x]

1

is,

=

[x,x] + [x,y] + [y,x] + [y,y]

= [x,y] + [y,x] ,

If [X,Y]

X

and

Y

so

[x,y] = -[y,x] .

algebra

are subsets of a

denotes the set

3.

A subspace

B

B

is

then

..!I.

of a Lie algebra [B,B] ~ B; and if

called a Lie sµbalgebr~ if then

9

r { I: [x. ,y.] : x~ E X , Y.; E Y} • · 1 l. 1 1= JI.

Definitipn

=

called an ideal of

L

is

[L,B] £ B •

L •

Examples

a)

Given an associative algebra

A , we can always

introduce a Lie algebra structure by defining a Lie multiplication denoted

[x,y]

AL.

= xy - yx ; the resulting Lie algebra is

This example is, in a sense, the universal

example, as will be explained later. called the commutator of

x

and

')

y •

[x,y]

=

xy - yx

is

h)

If

V

is any vector space over

of all endomorphisms of associat.ive algebra. is isomorphic to with entries in

K

K

n

V

(denoted

K , the set is

EndK(V))

811

It is well known that this algebra , the algebra of all

·( where

=

n

nXn

matrices

dim V) ; of course the

isomorphism depends upon a choice of basis for associated Lie algebra is denoted

g.t,(V)

or

V •

g.(, 0

,

The with

the commutator mult.iplication introduced above. c)

If

A

K

is any algehra over

(not necessarily

r

associative) let + x(yF,)

for all

DK(A)

= {F.i

x ,Y E A}

E EndK(A)

•

: (xy)6.

=

(xi)

)y +

This is not:. an algehra under

composition of mappings; in fact

DK(A)

is not even closed

we obtain a Lie algebra called the derivation algebra of the verification is trivial. derivation of the algebra

Definiti£!!..._4. .ruL.!_

for each

the map from

for all

L

Each

6 E DK(A)

A

is called a

A •

a

in a Lie algebra

L

into

defined by

x EL.

-

.1 ,..

L , denote by x(ad a)== [x,a]

a derivation of -----------

a f L •

L

Ptoof:

a

Evidently

is an endomorphism of

[x,y]ad a= [[x,y],a]

[[a,x],y] - [[y,a],x]

Since multiplication in a Lie algebra

have

ad a

is

(

.Jacob

ant, icommut at i ve ,

) ., wt:

adL(L)

f_t_:.~position

[x ad a,y] + [x, y ad a] •

is a derivation on

Def j nit ion 6. L

Moreover,

[x,y]ad a= [[x,a],y] + [x,[y,a]J

= Thus

L •

L.

The set of inner d e r i ~ § . of a Lie algehra

= {ad a : a

z.

If

L

EL}

•

is a Lie algebra over _a field

K

Proof: We first show that hy showing that

adL(L)

is a subspace of

ad a+ ad b = ad(a+b)

-

4 -

and

DK(L)

cad a - ad(cal

for all

a,b EL , c EK •

This follows immediately

from the identities below:

x ad(a+b)

and

x ad(ca)

= [x,a+b] = [x,a] + = [x,ca]

Next we show that 6 E DK(L)

x(ad a+ ad b)

[ad a,6] = ad(a6)

for any

a EL ,

, thus establishing the fact that

-

[x5,a]

= [x6,a] +

[ X,

[ X,

adL(L)

.

is an ideal in

Let

~: L1

=

--+ L2

[xq,,yq,]

DK(L) •

be Li~ algebras over

and

a Lie homomorphism [x,y]cp

80 ] - [x6,a]

a(>]

= x ad(a&)

Definition is-.

6(ad a ) )

= [x,a](>

=

such that

=

c[x,a] = ex ad a •

==

x[ad a, 6 J = x((ad a)6

Therefore

[x,b]

.

'

is a vector space homomorphism

for all

- 5 -

K

x,y E L •

Definition 9. ~pa,~e over

Thus if

yep ) -

L

be a Lie algebra and

A f',!:'presentatiorr of

K •

homomorphism

( XC!) )(

Let

q:>:

L ~ g.f. (V)

q:>

is a repN·:sentation,

L

in

V

a vector

V

is a Lie

•

[x,y]cµ

=

[xq,,ycp] ·-

( Yq:> ) ( Xq:> ) •

Propo_§ition 10. defined by

acp

C;iven a Lie al_g~.!'..!.

=

ad a

L , t'1!Lma,n

is _a re12resentation of

q:>

in

L

_itself.

Proof: In the proof of proposition 7 we have seen that

ad(a+h) and

=

ad a+ ad b

c EK

phism from

=~0

ad h

ad(ca) =cad a

in other words, L

into

[ad a,6] - ad(a&) 5

and

gt( L)

for every

, we obtain

thtis showing that

r·epresentat ion.

•

q:>

q:>

for

a,h EL

is a vector space homomor-

We have also seen that

a EL , 6 E DK(L)

[ ad a, ad b]

=

ad ( a ad h)

takint!

=

ad[ a, h

is a Lie homomorphism and hence a

J

t

:,i

Thus the centre of

,,:\. • 1 [ 1,.,./,.·.~.' L" 'RY ,

an

is

{c E

The d~'r'i\f~'ti ,algebra

Definition ,3:::3.

C 1 ~~r'.l.',y:,

L

L-J ·· 1c._" .[ U:1 ','L0

··.··J·

··_,

L:1(tl:Jli.L

gf'

.!.L ( ld)J ,

for all

O

1 a 1L11.'i'e_ .ia·'1' g·e b.>r a ' L

11 so':t " th ai'·t'·;:,

L{ l)

is

L •

is

an

r

Examples U~t h L·

space over

= K

g,t:(V1)

where'

V

'is ati' n-dimerisional .vector

with a given basis.

In accordance with the

isbhiorphisnf mehti'ohed. in example' (h) 'above' we may think of t.he ,~lements 01f.

L

I

as

nxn · matrices·.

- r-

d)

'l'hP.matrices of Lrace zero form an ideal

~tn •

o(

This follows directly from the al~ehraic

properties or the trace, namely that = Tr(a) + Tr(b)

and

Tr(a

Yr(ab) = T~(ba) •

± b) =

(Note that the

trace of an endomorphism, defined to be the trace of any .matrix representation of that endomorphism, is independent of the given basis of The centre of

e)

where as usual d

6. .

1J

=

n ' the set of all

in the centre.

for any

p

implies

ae

and

j

'

and

dn f)

e

pq .a

'

i 'j

=

p

1J

f9r any

q

i

(1)

=

6 ..

1J

1,1

for

1

( 6 .. ) .

~

€

'

i

a

=

e pq

(a .. ).

.

=

( 6. 6

.) .

which in turn implies

a 1 p6qj

=

with

Given

i

=

j

1 < i

and

'

to obtain i

j

+

j

Thus

•

j

Now

< .n •

.

1p QJ 1,J 0

. . a .. ; this aij =· 6 1J J.1

and also that and therea E dn '

gtn •

[ g"'n • ,g"'n • ]·~

6

-

•

tn •

8 -

·since

Tr(ab) =· Tr(bal

In order to show· the

.}

l.,J

j

:::

in

1J 1,J

=

-

= [gtn ,gtn] = . st . n

it is clear that

.

[a,epq]

1 < p,q < n •

is the centre of L

and

j

H:K

scalar matrices, is clearly contained

whenever

a .. ·= 0

a ... = ajj 11

fore

=

choose

shows that

nxn

= h.t::

is dn

+

i

for

0

with

q

for all

6. a . 1p QJ

= gtn

and consider the matrix

L

pq

L

To see the converse, take

the centre of

=

V .)

reverse inclusion we will make use of the matrix units e

introduced ahove.

pq

diag(>.. 1 , ••. ,>.. ) +

the form

st n

is of

where

n I: >...

Every element of

·

; a .. e .. . . l.J 1J

n

=

. 1 1 1=

J

1

0 ;

n-1 note that

t

diag(>.. 1 , ••. ,>,,n)= for

+j

i

>i,.(e .. - e ) . 1. 11 nn

i=l

it follows that [ e.

1n

for all

i ~ it follows that

g)

a

-

>.. € E st n

have n

'

a

=

>.. €

we may write and

.

>.. € E d n

with

Tr(a)

-

= (a

a

n1

K

= n>.. =

0

d n + n

and

s.t n n d JI

On the other hand if

n

=

{ o}

'

e 1. 1.

1J

l.,

e

-

nn

belongs

a E gt 0

a E flt

; hence

n ,

and

where

>.,E:) + >.. E:

Next, if

a. je ..

does not divide

is not a multiple of the characteristic.

= sl

=

,e . ]

First, if Tr(a) = >..

I:

i~j

diag(>,, 1 , ••• ,>..n)

If the characteristic of

Since

)

d n t1 n

=

0

Thus

1¥'e

since gtn

=

so the sum is direct.

is a multi12le of the characteristic

_t_h_e_n__T_r_(.,.\_t..,)_=_n_.>.,_= __o__f_o_r_a...,1_1_...e-..._E_-_K_____s_o_t_h_a_-t_,_d n s; s.f, 0

-

()

-

•

I.2

Homomorphism Theorems.

is an ideal in a Lie algebra

I

If

quotient space

[x,y]

::::

+

The c.anonical map

verify.

then the

'

has a Lie algebra structure given

L/I

[x + I' y + I]

by

L

I

as the reader may easily

'

L ~ L/I

TI;

(whece

Kn

=

X

I)

..j..

is then clearly a Lie homomorphism.

Prop~sition 1.

= ker

I

L/I

~

ls a Lie homonaor12hj..sm, l.!l.~

q:,: L--+ M

is an ideal of

q:i

H , and

of

If

L ,

=

N

im q,

is a Li~. subal_g_ebr:a

N •

Proof:

and

We know already that

I

If

a

respectively.

~l

[x,a}p

= rxco 'acp] = [xco,OJ =

an ideal of

.

because

L

.

E I 0

'

defined by

N

are subspaces of

N

and

so

Similarly we have

[L,L] £ L so '

: L/I ~ N

and

then

[L,I] £ I

and

[N ,N]

(x +

Ih =

xco

[x,y}p

= [x::p,yrp] = [(x+I)f , (y+Ih]

I

is

= [ L, L ]cp S. Lcp =

is a·subal~ebra of

be a vector space isomorphism, and since

=

E L

'

X

L

M

0

'l'he map

is already known to [x+I,y+I]iu

=

it follows that

~

is a Lie isomorphism.

I - 10 -

N

Corollary 2.

There is a one-to-one 2orrespondence

L

between tJwse subalgebras of all the subalgebras of

N

I

containing

( wit!!

L, I ,I\

and

as above).

Proof: We know there is a one-to-one correspondence between subspaces of

L

containing

I

, and subspaces of

It is therefore sufficient to show that if

L ,

algebra of

B~

is a subalgebra of

u~-l

is a sub-

N , and if

is a subalgebra of N

B

N •

D L

is a subalgebra of

lhese implications are both clear.

of a Lie algebra 13

n

I

L

is an ideal of

B+I

then

B , and

~f

is a sub.algebra 8/B

n

I -

13 + I/I

L (l!..§.

Lie algebras).

Proof: Since sec that

[ll+I, IHI]£ [B,B] + [H,I] + [I,I] £ B+I B+ I

is a subalgehra of

be the map defined by

xco

=

x+I

-

L for all

11 -

Let

qi:

x CB

n

, we

~ IHI/I

We know

aJ 1·eady that ·this is a vector space· homomorphism with

ker rf> = B n l

and

im cp =IHI/I.

Moreover

= rx,y] +I= [x+I, y+I) = [x~,y~] , so honiomorphism. 8

n

I

~

rx,y]cp

=

is a Lie

It follows from Proposition 1 above that

is an ideal in

B

and th~t

n

B/D

I~ B+I/I

(as Lie alaebras).

Definition

I.l.lJ). k > 1

4.

Recall that

~e now define

L(l) = [L,L]

(definition

L(k) = [L(k-l),L(k-l)]

for

•

Hy convention, we will write

easily seen that each L = L(o)

is~ and

L = L(o) •

is an ideal of

It is L , since

[L(k) ,L] = [[L(k~l) ,L(k-l)],L] £

£ [[L,L(k-1)], L(k-1)] £ [L(k-1), L(k-1)] = L(k) the Jacobi identity and induction.

by

The series is called

the derived series of the Lie algebra

Definition

5.

If

for some

called a solvable Lie algebra.

- 12 -

L •

k > 0 ,

L

is

Exampl~.

a)

Let

matrices in

Tn

be the set of all lower triangular Thus

gtn •

i < j}

whenever

i < j + 1}

1J

=

=

{ (a .. ) . . : aij 1J 1,J

T(l) = [T , T l n n n

whenever

a ..

Tn

'

and

Lie algebra,, since

T(n) n

=

0 •

Hence

If

L

=

0

T(k) = {(a .. )i j: n l.J ' n

is a solvable

T(l) = u n n

is called the

T

set of all lower nil triangular matrices in

Proposition~.

0

{ ( aij).1,J. : aij

in general

i < j + k} •

0 whenever

=

=

gtn •

is a solvable_Lie algebr~, so is

every Lie suhalgebra and every Lie homomorphic image of

L •

Proof: This follows immediately from the observations that B £. L

implies

B(k) £ L(k) , and that

for every Lie homomorphism

(L(k))cp = (Lcp)(k)

cp •

I

- 13 -

Proposition algebra

l•

L

If

I

such that

is a solvable ideal of a Lie

is solvable, th~

L/I

L

itself is solvable.

Proof: We know there exist integers

(L/l)(j) = 0

and

I(k)

=

0 •

L(j) £ I , which implies

But

k

and

j

such that

=

(L/I)(j)

L(j+k) £ I(k)

=

0

0

implies

•

I

r1

Proposition b.

If

and

L , then so is

I1 + I2 •

I2

are solvable ideals of

Proof: Since

1 1 /1 1 n 1 2 ~ (I 1 +I 2 )/I 2

solvable, it follows that

I1 + I1

and

I2

are both

is also solvable.

I

Definition

9.

The radical

R

of a Lie algebra

the sum of all the solvable ideals of

..;. 14 -

L •

L

is

It follows from proposition 8 that the radical of a Lie algebra is the unique maximal solvable ideal.

Definition 10.

A Lie algebra having no nonzero solvable

ideals is called semisimple.

Thus if

I.3

R

is the radical of

L , L/R

is semisimple.

Lie p-Algebras.

Let

A

over a field

be a finite dimensional associative algebra K

of prime characteristic

p .

make some observations about the Lie al~ebra

We will AL

and use

them as a basis for the definition of Lie p-algebras.

Le~~·

If

X

and

Y

are commuting indeterminates

over a field

K

.Q.L._P.rime characteristic

-

l '1 -

p , then in

p-1

t

(X-Y)p-l -

(ii)

xivp-i-1 •

i=o

!'roof:

The first result follows inunedia.tely from Newtou' s

p

binomial formula, since efficient

(r) 1

divides each binomial co-·

O < i < p •

with

The second result

follows from the first, together with the identity

= (X-Y)

p-1

'

'

]

t X1Yp-1-.

i=o

£.Fopo_~ition_l.

field

K

If

A

is an associative algebra over a

~rime characteristic

( i)

(ii)

(ad a)P = ad(aP)

p

for all

'

where

coefficient of of ring

AL

we

a C A

p-1

(a+b)P = aP + bp + a,b E A

, then in

t

i:::l

s.(a,b) 1

isi(a,b) i-1

X

for

is the

in the expansion

a(ad(aX+b))p-l , in the polynomial

A[X] , considered as a Lie algebra. - 16 -

Proof: Consider the endomorphisms of

at: x ~ ax at

commute.

A

into

a : x ~ xa r

and

Evidently

and

By lemma 1 it follows that

(ad a)P

A

•

=

.

)P .. a p - ap = ad(aP) J',foreover (ad a)p-1 r .t p-1 i p-i-1 by lemma (a - at )p-1 l so I: a r at r ' i=o p-1 b(ad a)p-1 = t aib a p-i-1 Similarly in A [X] we ' i=o p-1 . have a(ad(aX+b))p-l = t (aX+b) 1 a(aX+b)p-i-l i=o

= ( ar

-

at

=

.

each

p-1 I: si(a,blXi i=l

(aX+b)P = aPxP +hp+

Clearly si(a,b)

is a polynomial in

a

b •

and p-1

derivatives of both sides we obtain

where

Taking

.

t (aX+b) 1 a(aX+b)p-i-l =

i=o p-1 ( ) i-1 =-I: i s.1 a,b X i=l the derivative of hut rather

(Since

and

a

(aX + b)P

need not commute,

b

p(aX + h )P-1 .a

is not

p-l . )p-i-1 I: (aX + h) 1 a(aX + h i=o

=

0

'

as inrlicated.)

Makin~ use of the identity from the previous para~raph, we see that

i

s.(a,h) 1 .

the expansion of

is the coefficient of

a(ad(aX+b))p-l

Xi~l

Finally, setting

in X

= 1

in the identity displayed in the openine; sentence of this paragraph, we have

(a+b) p

=

a

JJ

+ hp

.L

p-1 I: s.(a,b) •

1=

1·

1

•

I

- 17

~

Since all the important examples of Lie algebras in prime characteristic have properties similar to those indicated above, we introduce the followin~ definition:

Dcfi11itiou a field

K

"p-map"

a

J.

A Li~ p-algebr~

L

is a Lie algebra over

p

of prime characteri~tic

~ a [ P]

(ii) (iii)

such that

(ad a)P

=

ad(a[p])

(a+b)[p] = a[p for

]

for all .

+ bfp

a,h EL , where

coefficient of of

which has a

Xi-l

]

p-1

+

E s.(a,b)

. l 1=

1

is.(a,b) 1

is the

in the expansion

a(ad(aX+b))p-l , and

indetermiriate over

a EL;

X

is an

K •

p-1 Note that the term expansion of

Note also that if for then

s 1 (a,b) L

s. (a,h)

occurring in the

is independent of the given p-

(a+b)[p]

map, since each

t

. 1 1 1=

is defined intrinsically. (a+b)[p] = a[p] + bfp]

is ahelian

a(ad(aX+b))p-l = 0 .

Lie p-algebras are some-

times called "restricted Lie algehras 11 (see Jacobson [ S J.

- lh -

chapter five).

Definiti01!__4_._

L

If

is a Lie p-al{!ehra. then hy a

p-suhalgehra (p-ideal) we mean a subal~ebra (ideal) of

L

which is closed under the p-map of

L.

Examples. a) ·

If

A

is an associative al{!ebra over

prime characteristic

p , then

algebra if we define

a[p]

AL

= ap

In particular

for all

gt(V)

the associative p-map, if field of characteristic

V

seen that every

(ab)~k =

~

~

a EA ; we ( see proposition

is a Lie p-algebra under

is a vector space over a

p > 0 •

Given an algebra

b)

of

becomes a Lie p-

will call this ''the associative p-map" 2 above).

K

A

over

K

,

it is easily

satisfies Leibniz' rule

E DK(A)

(~)(a~j)(b6k-j)

for

a. h E A •

If

j=o J

char ( K)

= 11 >

(ab)~P

= (a6P)b

of lemma 1). p-map

fl[p]

0

it follows that + a(bt,P)

DK(A)

=

tip

6 P E DK (A)

also, since

by Leibniz' rule (see the proof

becomes a Lie p-al~ebra under tl1e

•

- 19 -

c)

If

G

is an algebraic group over a field of

p > 0 , its Lie algebra

characteristic

is a

L(G),.,~

Lie p-al,:el>ra (see, for example, Aorel [15)).

Definition.__5.

If

characteristic

p

is a Lie algebra over

L

>

0

p - s em.i;..!.!_n~,.!U a .P of

f: L

~

for all

'

a,b EL

(a+b)f

=

af + bf

If

[p\ a~ a cent_££•

'

then by a

(>..a )f

and

=

A.P(af)

A EK •

and

p-maps, b~~h makin_g

C

into its centre we mean a map

L

such that

C

having centre

of

K

a~a

rP J,.·)

are two

into a Lie p-.!,_~ebra, !.!!..en

L

[pll - a

is_"'! p-!t_enaili..!!,ear map of

Moreover if

f

L

i11to_j.ts

is..!.,!!Y p-semilinear map of

L

[p]l

+ af

a---+ a for

is another p-m~p

L •

Proof:

[ p] l

We first show that map of

L

Since

a---+ a

[p ]2 -a

is a p-semilinear

into its center• ad(a

fp\ - a

[ pl') [p]l .... ) == ad(a )

-

20 -

[p]2 ad (a

)

=

::.-=

centre for every

we have

EL.

a

.

[p],,

"' )

.

f pl.,

(ad a) 11 - (ad a)fl = 0

- a . [p]l (la) · -

Clearly

[p]l

[p]2 (la)

•

[p]2

(a+b)

Also

in the·

=

h1+b)

[p]2 b

[ p]. 1

(a+b)

=

[p

Ji

(.p].

+ b

a

, because

)

p-1

t s.(a,b) j=l J

+

1

= 1,2)

(i

and

p-1

t

as previously remarked, the term

s.(a,b)

is independent

J

j=l of the choice of p-map.

On the other hand if

f

i~ a p-semilinear map of a [p]

into its centre, let us write

=

a[p]l

+ af.

L

Then we

have (i)

= (ii)

=

af

(a+b)[p]

[p]l

+ (la)f

(>..a)

a[p]

>.. p

au(a[p])

since (iii)

=

(>..a)[p]

for

a ~ a[ P]

>..P(a

+ af)

=

[p]

+ af)

[p]l ad(a

is in the centre of

= (a+b)

[p]l

=

E K

[p]l

ad(a

+ (b Hence

)._

=

1 + (a+b)f

)

= (ad a)P

L;

=

[p]l (a

+ af) +

p--1

[p]l +hf)+

"" s.(a,b) • ,., 1 . i=l

is a p-map as asserted.

I

- 21 -

As an Jm.mediate consequence we have

!f

Co_rollary_l. QY_~

L

!!LJ!..

of characte~isti,.£

K

.Q!~ p-!,llaP making

L

lie alge_pra with zero centre

p

> 0 , there is at most

into a Lie p-algebra.

Pro()f: This is so since the only p-semilinear map of

L

into its centre is the zero map.

I

Def3:.nitiot~-~. p-algebra (a[p])~

L

A representation

0 •

( Jacobso_f!) }

Let

(ad xi)P

yi EL

(i

= l, •.• ,n)

= ad yi , then there is a unigu~ p-

x.[p] = y.

map such that

he a Lie algebra

over a field of characterist~E_

If there exist elements

such that

L

1

1

(i

= 1 , ... , n) , makj...!!,g

L

into a Lie p-algebr~.

(Ad~-Iwasawa)

Every_Lie algebra has a

faithful (i.e., 9ne-to-one) finite dimensional represent~ti_o_!!.

Thus, every Lie algebra is "essentially" a suhalgebra of some

istic

g-t 0

p > 0

•

~

be embedded in a Lie p-algehra.

Proof: This follows immediately, since algehra for any positive integer

n

gt 0

is a Lie p-

(see example

a).

I

- 23 -

CHAPTER II

II.l

Nilpotent _and Solvable Lie Al_g_ebras.

Definition~.

If

L

1s a Lie algebra we define

L0

Clearly, the

are ideals of

L.

It is important

to distinguish these ideals from the ideals in

defined

r.2.4.

The series

Oefinition 2.

L

L2

>

>

the descending cen~ral series of Ln

in zero, i.e. ,

=

0

L

for some

L3

'

. ..

>

is called

and if it terminates

n > 2

L

'

is called

!:!!-lpotent.

Example. a)

Let

matrices in

so is

ad,-

Un gt 0

be the set of all lower nil triangular •

In general, if

; for we can write

,- E gt 0

is nilpotent,

,-t. as in the

ad

proof of proposition I.J.2, and so

= "

.

0

for

k

sufficiently large.

We will see (Corollary 8) that in ~eneral, a Lie algebra L

is nilpotent if and only if each

nilpotent endomorphism of

ad a, a EL , is a

un

That

L •

is nilpotent

also follows directly from the fact that

ll k ={(a .. ) . . : a .. n 1J 1, J 1J

so

U

Lemma

n

= o

n =

0

J.

For any Lie algebra

i < j + k}

whenever

'

•

( i)

(ii)

[Lj Lk] c Lj+k j

L (k)

-

s.;;_ L

2k

we have

L

for al],_

for all

k

j

>

0

and

k

.

Proof:

We will establish property (i) by induction on For

j = 1

£ Lj+k+l

we have

[L,Lk] = Lk+l

by definition 1.

This yields the result.

We now establish property (ii).

=

L

.

using definition 1, the Jacobi identity, and

the induction hypothesis.

L (l)

j

2

•

'"' 0uppose that

For

L(k) £;: L2k

k

k

k

= 1

for some

s;: [L 2 ,L 2 ] c L

2 k+l

we have k

>

1

.

by property (i).

I ... ,.

Proposition

4.

Every ni~_~nt Lie algebra is s9lvabJ.e.

Proof: This is an immediate consequence of property (ii)

above. Note that the converse is not true, since, for example T

n

T k = U

is solvable, while

n

n

for all

k

>

2

9

(as is

easily checked).

Definition~.

If

normalizer of

B

B

is a Lie subalgebra of

in

It is clear that that

B

is

L

NL(B)

is an ideal of

NL(B)

= {x

L , the

E L: [x,B)11 ~ B}

is a Lie subalgebra of

NL(B) , and that

NL(B)

L

,

is maximal

with respect to these properties.

If

L

is a Lie subalgebra of

consisting of nilpotent endomorphisms, where

V

space of dimension

> 0 , then there is a nonzero

such that

for all

VT= 0

TEL •

gt(V)

is a Kt

v EV

Proof: When

L

is of dimension

O

or

,.

obvious; in the latter case, if

,. n XT

=

0

n-1

,. n-1

'

+

+

0

'

and let

0

=

X 'T

E V

X

L

and

such that

n-1

dim L"" n > 1

Now suppose

the theorem is

generates

simply choose V

l

, and assume the theorem

is true for all Lie algebras having dimension less than and consisting of nilpotent endomorrhisms of some K-

n

space. It suffices to find an ideal For then

OJ\C •

L

=

KT

0

$

B

where

B

in T

0

L

i

B , and by the

induction hypothesis there is a nonzero

=

wo

0

a E B}

for all

=

0

Thus

ideal.

.

If

a E B

for any

Consider

a EB

for all

'

=

K ,-

0

we

9

[,-o,o] E n

E \\

V

with VT

=

V 'T

Let

adL ( n O

):

I~ L

0

---3,-·

=/=

L

0

==

0

W(J

WCTT

since

0

·- 0

0

.

for all

0

show the existence of an ideal

one,

E V:

{w

WT

such that

13

+ wr-r 0 ,a]

0

B

D

is a.n

+

w

0

,

Since T

C L

of codimension

be any proper suhalgebra of leaves

0

--

is a nilpotent endomorphism on

TO

, it follows that

B

w EV

W

we have

because

so there is a non-zero L

E w

w

of codimension

L •

Since

fixed, it induces a Lie algebra

0

of nilpotent endomorphisms of

L/B

-

27 -

0

(by example

a

)

which

we denote by

adL/B (B 0

)

Since the dimension of

•

0

adL/D (B 0

)

is less than the dimension of

L , it follows

0

from the induction hypothesis that there is a

of

B

cr

t

so

B

Now take a maximal proper subalgebra

B

other words

and

E NL{Bo)

p

we have ,seen, there exists that

E B0

for every

, such that

0

NL (B)

=

L

= Ko

0

0

0

E NL(B)

Hence

E9 B •

B

B

'

o EL ,

In

+NL(Bo) .

0

L ; as

of

with

0

t

B

'

so

is an ideal of co-

dimension one, and this concludes the proof.

I

morphism...§__of

, . then there is a basis of

V

~-e..rx.. endomorphi_.filll_ in

L

V

such that_

has a lower nil triangular matrix

representation.

Proof: If L

dim V

=

1

the result is trivially true, for then

contains only the zero endomorphism. Suppose

dim V

=

K-spaces of dimension

n

> 1

and the result is true for all

n-1 •

-

By Engel's theorem there is a

?~

-

nonzero

v 1 EV

such 'that

lt is clear that

L

endomorphisms on

V/Kv 1 •

V

1

'f = 0

for all

'f

E L •

induces a Lie al~ehra of nilpotent

Applyin~ the induction hypothesis to conclude that there is, a basis of

V/Kv 1

V/Kv 1 , we relative to

whicJ1 all of these endomorphisms are represented by lower rd 1 triangular matrices o

so that

If

v 2 , ••• , v n

{v 2 + Kv 1 , ••• ,v 0 + Kv 1 }

every endomorphism in

L

t V

are chosen

is such a basis, then

will have a lower nil triangular

matrix representation relative to the basis of

V

•

I

if

ad a

is_!lilpotent for all

a E L •

Proof: To B8.Y that

for some X

H.

'a·l , , a)m-l =

l)

L

is nilpotent is to say that

Civen any

a ( L

f or a 11

x l.: L, ;

Lm -

0

therefore we have

· 111

nilpotent.

- 29 -

o th er wor d s

cHj

a

is

Conver·sely if

ad a

is nilpotent for every

then hy corollary 7 there is a basis of

Let

a 1 ,a 2 , ••• ,an+l E L

=

n

ad a

are lower

dim(L) ; given arbitrary

it follows from example II.l.a that

= O •

a 1 a d a 2 a d a 3 ••• a d •n+l

L f ore T11ere

Ln+l

=

0

is nilpotent.

L

so

In the next two propositions we consider a given ,. ~ EndK(V)

having all of its characteristic roots k.

m

;\. 1 , ••• ,;\.m

in

K.

=

i

=

T

,

Il (X-)..)

i=l

and let

=

1 , ... , m

t

(D

i=l

f. (X) l.

l.

=

k. (X-).. ) l.

'

then

V.l. = {v . EV: vf.(,-) = o} l.

vi,.

C

V.

l.

for each

i

V.1

Proof: V.,-

be the

l.

m

V

1

1, .•• ,m •

If i

p(X) =

Let

minimal polynomial of for

c- V.l.

L ,

such that all

L

matrices corresponding to the linear maps nil triangular.

a l

because

,-f 1.(T) = f 1.(T)T

- 10 -

•

and

1

ancI

DI

r.v.l

v-

To see that

q. (X) ~

• l':"t

i

.

Since the

for each

1

i=I

qi(X} are relatively prime, there exist polynomials m

r.(X) i.: K[X]

E r. (X )q. (X) i=l 1 J.

such that

1

=

1 •

Thus for every

E V '

V

m

t

i;:l

V

r.(T)q.(T) 1

v r. ( -r )q, ( T) E V.

But

1

1

1

V

•

~

, since

1

fnr ~~rh

tn

In other words

V

=

i: V. i=l 1

This sum is in fact dir~ct, for if

v Ev. n

then

V

and

1

f. (X)

,, q.(-:-)

=

v(

Q. (

1

= f 1 (X)r(X)

Consequently

X)

n

j=/=i

J.

and

1

l

f .(,J)

ct

v.)

j=J.i

J.

f .1.(7)) = 0. - ,

J

But

are relAtively p~i~~, so

~

q 1 (X)s(X)

V

= "

r(X), s(X)

for soroe

f.;(,}r(-r) +

v

-'·

q.(T)s(,-) 1

=

0

~

KfX] .

,

and

m V ""

r. i.=l

ID V.

l.

•

I

- 31 -

is the algebraic closure of K-space with basis

{v 1 ® 1, ••. ,v 0

induces~

T = -r&.i.d

'T

®

=

(v 1 -r)

V

= V ®K K is

1) •

~ E End_(V) K

(i.e. (vi® l)i=

E f:nd_(V)

K , then

® 1

a

Every

by the rule for

=

:l

1, ••• ,n)

.

is diagonalizableo

K

Examples. b)

The only endomor,Ehism which is both nilpotent

and semisimple is the zero endomorEhism 9 because th0 only diagonal matrix that is nilpotent is zero. c)

The sum of

is semisimplc.

~ commuting semisimple

endomorphisms

This follows from the fact that if two

diagonalizable matrices commute, they can be simultaneously diagonalized.

Note too that the sum of~o comntuti~ n;L~-

potent endomorphisms is nilpoten~, as follows immediately from Newton's binomial formula.

---------·--·

As in proposition 9, let

-r C EndK(V)

have ~11 its Ol

eigenvalues

x1- ." .•

,).. m

the minimal polynomial of

and let

'T'

v. = {.v Ev: v f.(.,.) 1

1

Let

K

in

(where

=

o} •

p(X) f.(X) 1

=

0: f.(X) i=l l. k.

=

(X-A,) l.

1 )

he

l'ro_g_osition 11. belong to

K , then

,.s +·,-n and

If all the e~nvalues of

[,- 8 ,Tn]

=

can be unigue~.Eressed in ti~!!

T

where 0 •

T

s

= s('i)

'i

n

'is semisimple, ,-

s

n

is ni~potent,

Moreover, there exist polynomials s(X}, n(X) E K[X]

without constant terms

,.

,- E EndK(V)

=

n('i)

such that

•

Proof: The only characteristic root of

Vi

is

Ai •

For if

f. (X)

prime to

1

and

q(X), r(X) E K[X] •

Thus

,. -1-l f:

restricted to

(X-~)

~+Ai , then 1

,.

is relatively

= (X-~)q(X) + fi(X)r(X)

If

v E V.

, we have

1

is invertible on

V.

,.

on

is the only eigenvalue of

for some

1

'

and consequently

V.

1

.

It is therefore possible to choose a basis for so that

,.

( restricted to

X.i

V.

1

V.) has a matrix representation 1

of the form 0

A.= 1 X. 1. If we let

- 33 -

and

where

is semisimple,

IL

1.

[B.,C.] l.

=

then we have

l.

=

Since

0

C.

1s

J.

B. + (:. l.

1

nilpotent, and

. V

-

m

t

EB

i=l

together a basis for

by proposition 9, we can piece

V.l.

so that

V

has a matrix represents-

T

tion of the form

A

=

cl·

0

A

m 0

Bl

n =

Setting

we obtain

and

[B,C]

A= B+C

=

0 •

corresponding to

where If

B

'f s

and

~nd

.

C

=

B

0

is semisimple,

\

is nilpotent,

are the endomorphisms

and

C

C

'

cJ

0

m

B

0

cl

respectively, then

is the desired decomposition of

T

T

=

T8

+

T0

•

By the Chinese remainder theorem there is a polynomial

s(X)

satisfying- the congruences

- 34 -

s(X) -

0 (mod X)

=

s(X)

A . ( mod f . ( X )) 1

for

1

=

i

\ . =

(These congruences are consistent, because k. f.(X) =X J .)

For any

vs(T)

=

i

each

V.

J

\.v , for 1

=

VE V.1

1, ... ,m •

implies

0

J

therefore, we have

1, ••. ,m.

Thus

=

s(T)

T

on

S

m

'

1

=

s(T)

and since on

T

s

.

V

V

I:: EB V.1 i=l

=

Now if

=

n(X)

'

it follows that

X-s(X)

'

--

n(T) = 'r-S(T)

We must still show the uniqueness of the decompositiono Accordingly, suppose is nilpotent, and

commute with -r

n

T8

--

n(T)

-

a=

T

,

From v

-

T0

T

= cr+v

where

[o,vJ = 0 •

Since

o

is semisimple, o

they also commute with

+

T8

=

T0

, with

Ts -

T

cr

=a+ v

T

s

both

v

= s(T)

and

it follows that

semisimple and

Therefore

nilpotent (see example c).

and

\J

v

-

Tn

an

the weigh!_ spaces).

Proof: ( i) and

u

\:

must

\v~

To show that. L

implies

produce

V

co

L £ V c,o

. E V rp

VO

an integer

In other words, ~iven

0

such that

n

p = r - cp( ,- )(£

is nilpotent because

L

(ad p ) ·i == 0

V

(

V

,p

'

14

j

it follows that

sufficiently large.

V

~r

\.

(vJ)(r-w(Tk l 11

note that

Op

n

adp =ad,.

=

But

is sufficiently large, and s.:ince

vp

n-j

=

0

For large enough

_ 1x _

provided n

O·-.i

is

, therefore,

i.

()

is nilpotent (see corollary 8).

we have

provided

(

,~)

For this purpose let

By lemma

,,

we must show that

'

(va)pn = o •

llence

Vci>L ~ Vcp

vo ( Vcp , so

m

(ii)

t

\\'e now show t.hat the sum

V

is direct

i=l cpi

cp 1 , ... ,cpm

whenever

are distinct weights; from this it

follows that there are only a finite number of distinct weights in all. m = 1

If m > 1

'

the assertion is trivially true.

and assume the result is true for

-

m

1

Suppose

.

We

m-1 V

will show that

cpm

n (

}: V

i=l

co.1

=

)

every other such

0 ;

intersection just amounts to a relabelling of the weights. Accordingly, let

0

t

m-1

v E V

com

m-1

= }:

V

V,

i=l

say, that , ( 'T")

'+-'}

=t

v. ( V

with

1

1

v1

=t O .

for

)

1, ... ,m-1

,. EL

Now choose

V

•

Write

i=l cpi

=

i

coi

n ( t

, and assume,

such that

( we are assuming that the weights are

q m ( ,- .)

distinct).

v ( V

Since

=

0

v(T - CO (,-)€) 8

1

V

co.1

=

m

v.(,- - co (,-)€) 6

m

L c V coi

and

there is an integer

com

EV

coi

(T -

such that ~ote that

for

i

=

l, •.. ,m-1

.:.p (,-)e;)S m

because

is a polynomial in

T

m-1

t

By the induction hypothesis

i=l therefore

s

v 1 (,-

- r,pm(T)t)s

=

0 •

- 39 -

V

cpi

is a direct sum;

•

011 the other hand

t

inte12,"er choice of

such that

E L

T

(X - ~ (T)) 6 polynomials c.0 1

-

v 1 (,-

col

qil('!")&)t

r(X)

(.,-))

t

and

q(X)

there is an

=

.

0

By our

(X - cp.1 ( 'f) )t

it follows that

are relatively prime.

m

1 == (X -

vl E V

since

and

Hence there exist

such th.at

r(X) + (X - qim(r))

s

q(X); this implies

v 1 = v 1 (,- - cp 1 (,-)c }tr(,-)+ v 1 (T - com(,-)t) 6 q(1) ""' 0 ,

that

which contl'adicts the choice of V

Consequently

cpm

n

y

1 •

m-1 ) ( I: V cp. i=l 1

m

=

0

and

t

V

is

Cf).

i=l

Bl.

.l.

direct sum.

this we shall do by induction on

If

n

V =

It only remains to show that

(iii)

= l

n

V

1: co

=t- 0

V cp

= dim(V) •

the whole situation is trivial, since eveP}

endomorphism has exactly one eigenvalue; there is exactly one weight

~ , and exactly one weight space

V

q)

= V •

Let

us suppose that the result has been established for all vector spaces potent l f

t

lien

W

of dimension less than

Lie suhalgebras of ever·y

T

l"

L

g,t (\\')

k ;

, and all nil-

•

has just one eigenvalue, say

the minimal polynomial of

fur some integer

n

,-

An

T}

•

( X-ro ( ,- ) )

is of the form

from this it follows that

co(

co

is the

k

one and only weight and

V

Suppose there is a

CD

=

V •

TEL

with einenvalues "'

r

> 1 •

r

where

i

-

(X

i=l

,.

polynomial of

'

then

== 1 , ..• , r

n

p(X) =

Let

.) 1

k.

1

-

f. (X) = (X

and let

V =

),,

1

r I: EB V.1 i=l

' , /\.1,••••I\ . r

he the minimal ),,

.)

k.

1

for

1

(proposition 9 ) •

Uy an argument exactly similar to the one that shows V L c .:p

'Now

it can he seen that

V

-

rp

dim V.

since

n

1 , and the elements of

L

form a nilpotent Lie subal€£ebra of

1

L.1.

which we shall denote by

g_t{V.) . 1

for

V. L c V.

By the induction

co . • : L . - ~ K

hypothesis there exist weights

1J

such that

1

r.

1

v.1 =

EB (V.)

~

1

j=l

for

CO, ,

= l, ... ,r , where the

i

r.

are

1

1J

certain positive integ-ers. Let

n.: 1

for every

be the restriction map:

L~L.

1

a (- L •

are wei~hts for

We assert that the maps

L , and that

=

V

=

~--

and

j lt

, because it is known that

1

(V. )

is clear that

1,1) . .

1J

is the set of all for· all

o t L.1

v

t V.1

and

V

c -

(V. ) 1

V

, ..

-

CJ •.

1=

,1

l

-

,,

1 1J

To see i

0

1J

, since

v(o -

(V. ) J_

CJ) • •

1,1

co .. {c )e ) 0 = O 1J

is the set of all

Ill .. 1J

I1

1

for all

c.p •• 1J

1J

such that

rJ

1,1

(V.)

1J

=

.. =rr.cp ..

1J

VI¥ ..•

E © i, j

V

this it is sufficient to show

~

OTT.1

V

E V

such

that

v{a -

~- .(a)E:)n 1J

=

for all

O

Conversely, :it is evident that .si11ct•

\. l

, .. (-r) 1J

,.

' ..

v ( V.

V

(

V

implies

l.

l.J

1¥1·j·

c(V.) -

1

cp 1,1 •.

~

tv,

>... = q, .. ( 1

I

1J

restricted to

is Uw only eigenvalue of

it follows that V

o EL •

V.]

, and hence that

•

This concludes the proof.

Dt.:..fjn.:i:!:_ion l.].

A Lie subalg;ebra

act indecomposajll;y on

V

if

V

L

of

r;t (V)

is said to

is not the direct sum of

two L-invariant subspaces.

Corollary 18. gt (V)

If

L

is a nilpotent Lie subalgeb~_cL.9f

over an algehrai~_ly closed field which acts

one eigenvalu~.

The converse of Zassenhaus' theorem follows readily

from Engel's theorem; we drop the requirement that al~ehraically closed.

- 42 -

K

he

1=..2..

Corollary

If

L

Let

V L