On the cohomology of restricted lie algebras

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On the cohomology of restricted lie algebras Jörg Feldvoss

a

a

Mathematisches Seminar der Universität, Bundersstr.SS, Hamburg 13, W-2000, Federal Republic of Germany Published online: 27 Jun 2007.

To cite this article: Jörg Feldvoss (1991): On the cohomology of restricted lie algebras, Communications in Algebra, 19:10, 2865-2906 To link to this article: http://dx.doi.org/10.1080/00927879108824299

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COMMUNICATIONS IN ALGEBRA, 1 9 ( 1 0 ) , 2 8 6 5 - 2 9 0 6

(1991)

ON THE COHOMOLOGY OF RESTRICTED LIE ALGEBRAS

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Jorg Feldvoss Mathematisches Seminar d e r Universitat, Bundesstr. 55, W-2000 Hamburg 13 Federal Republic of Germany

50. INTRODUCTION

In 1954, G. P. Hochschild CHo21 initiated t h e study of a cohomology theory f o r restricted Lie algebras. He established t h e usual elementary interpretations of low dimensional cohomology a s extensions of n1'3dules or algebras a n d , more surprisingly, a connection between restrictcsd and ordinary Lie algebra cohomology in f o r m of a six-term e x a c t sequence. Apart f r o m a few papers in t h e l a t e sixties [Ma, Ch and Pa21, r e s t r i c t e d cohomology has received considerable a t t e n t i o n only quite recently in connection with the cohomology theory of algebraic g r o u p s (cf. CHu, jan21), associative algebras CFa2-41 and in i t s own right CFal,6; FP 1-3; FS and J a n 11. In this paper, we prove s o m e general theorems a b o u t complete r e s t r i c t e d cohomology t h a t parallel t h o s e concerning t h e T a t e cohomology of finite

C o p y r g h t O 1Q91 by Marcel Dekker I n c

2866

FELDVOSS

groups. Aside f r o m a single occurrence in CPa21, this concept has apparently n o t been used in connection with Lie algebras. Our t h e o r e m s especially yield s t r u c t u r a l characterizations of t h e m o s t i m p o r t a n t classes of finite dimensional solvable restricted Lie algebras a s well a s new information a b o u t t h e block s t r u c t u r e of their reduced universal enveloping algebras CFe31.

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In more detail, t h e paper begins in t h e f i r s t section with a very s h o r t proof of t h e f a c t t h a t extensions of reduced universal enveloping algebras are f r e e Frobenius extensions of t h e second kind. This has quite recently been shown independently in CFSI. Among t h e many consequences are a homological version of t h e Frobenius reciprocity theorem and Shapiro's lemma f o r restricted and ordinary Lie algebra cohomology (cf. a l s o CFa4,6; FSI). This will be used in forthcoming papers by t h e a u t h o r t o develop s o m e homological f e a t u r e s of t h e representation theory of r e s t r i c t e d simple Lie algebras. The n e x t t w o sections provide t h e basic f a c t s of complete restricted cohomology which were n o t treated in [Pa21 and a r e mainly an adaptation of t h e well known properties of T a t e cohomology b u t affording different proofs. Moreover, we p r e s e n t a connection between complete reduced extension f u n c t o r s and complete restricted cohomology s p a c e s , thereby extending a r e s u l t in CFP31. In section 4, we investigate restricted Lie algebras with periodic cohomology by supplementing t h e early r e s u l t s obtained by B. Pareigis in CPa21. The recent classification of restricted simple Lie algebras [BWI is used t o generalize slightly an early r e s u l t of R. D. Pollack [Pol. Furthermore, by means of t h e geometry of

restricted

modules, we

characterize p-nilpotent restricted Lie algebras with periodic cohomology. In t h e l a s t section, we prove s o m e vanishing and non-vanishing t h e o r e m s f o r t h e complete restricted cohomology of restricted Lie algebras and in

2867

COHOMOLOGY O F R E S T R I C T E D L I E ALGEBRAS

particular a n s w e r a question raised by J . E. Humphreys (cf. CHu, Problem 7, p. 951). In a sequel t o t h e p r e s e n t paper, we will investigate p-nilpotent restricted Lie algebras by general ring theoretic m e t h o d s t o get. more insight into t h e f e a t u r e s of restricted cohomology.

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51. FROBENIUS EXTENSIONS DEFINED BY RESTRICTED LIE ALGEBRAS

In this section, we derive a relative version of [SF, Cor. 4.3, F . 2181 and especially generalize t h e main theorem of [ B e r l . Let A b e an associative algebra with identity element IA over a n arbitrary commutative r i ~ gIK, R a unitary subalgebra of A and a s s u m e t h a t all l e f t (right) modules under consideration are unitary. Every l e f t (right) A-module M is via t h e canonical embedding fl-om R into A a l e f t (right) R-module. In particular, we can consider A a s s twosided R-module. If V is a l e f t ( r i g h t ) R-module and if r is a unitary algebra automorphism of R, we denote by .V (V, ) t h e l e f t (right) R-module with t h e induced action r;v

:=

~ ( r ) . v(resp. v.,r

:=

v.r(r)) Y r

E

R,v

a V. More-

over, t h e dual HomR(A,,R) is via (a.f)(x) : = f(xa) a s well a s ( f . r ) ( x ) : = f ( x ) r Y x,a

E

A; r

R; f

c

E

HomR(A,,R) a n (A,R)-module and in a corresponding

manner HomR(A,R,) i s an (R,A)-module.

Definition. The extension A

:

R is called a r-Frobenius extension if

A is a finitely generated projective l e f t R-module and t h e r e e x i s t s an isomorphism

L

of (A,R)-modules f r o m A o n t o HomR(A,,R). A r-Frobenius

extension is said t o be f r e e if A is a free l e f t R-module.

In t h e case R

8

C(A) and r = idR, A is a n R-Frobenius algebra (cf. [EN,

p. 31). According t o CNT, Prop. 1, p. 901, every t-Frobenius extension A

:

R

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FELDVOSS

possesses a n isomorphism

L'

of (R,A)-modules from A o n t o H O ~ , ( A , R , - ~ ) .

~ ) obtain A' = T-1 We consider X := ~ ( 1 a~s )well a s X' := ~ ' ( 1 and

A (cf.

CPal, p. 21). Moreover, f o r every element a of t h e centralizer CenA(R) of R in A there e x i s t s a uniquely determined element a' a CenA(R) such t h a t T-I

- ( a . A ) = 1'.a'. The induced

Nakayama automorphism of A

algebra automorphism v of CenA(R) i s called :

R and is unique up t o inner automorphisms

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(cf. CPal, p. 31). In t h e sequel, L will always d e n o t e a restricted Lie algebra over a commutative field IF of prime characteristic p. We shall identify L with i t s canonical image in t h e S-reduced universal enveloping algebra U(L;S) f o r any S

a

L* (cf. [SF, p . 2131).

(1.1) Theorem. Let L be a finite dimensional restricted Lie algebra, S

E

L*

and K a restricted subalgebra o f L. Then t h e extension UfL;S): U ( K ; q K ) i s a free r-Frobenius extension with twist r induced by Y

Y

-

t r ( a d ~ / ~.yI u) ( L ; s , ~ )

VY

6

K.

Proof. According t o [SF, Cor. 4.3, p. 2181, U(L;S) is a Frobenius algebra. The computation in CSch, Lemma 3, p. 11241 remains t r u e f o r arbitrary S a L* and s h o w s t ! ~ a t t h e Nakayama automorphism w L of U(L;S) is

x

induced by x

tr(adlx) . lU(LiS) V x c L. Owing t o t h e reduced

+

analogue of t h e PoincarC-Birkhoff-Witt-Theorem,

U(L;S) is a finitely

generated free U(K;S

I K )-module. This in conjunction with

ensures

the

applicability

U(L;S) : U(K;S lK) phism wL:K

of

CPal, Satz

7, p.

w

LIu(L:s,I~= 'K

61. We

obtain

that

is a f r e e T-Frobenius extension with Nakayama automor-

:= u L

I C ~ ~ ~ ( ~ ; ~ ) ( U ( K ; S ~ ~ ) ) and

twist

T

of the identity t r ( a d L I K y) = t r ( a d L y) - tr(adK y) V y d y ) = u ~ ( w K ( y )=) y

-

:= 'v; E

~

WK. By virtue

K, we conclude t h a t

t r ( a d L / K ~ ) . l U ( L , S , K )v y

6

K . 3

2869

COHOMOLOGY O F R E S T R I C T E D L I E A L G E B R A S

Remark. If one generalizes t h e proof of A.J. Berkson in t h e obvious way, one can s h o w t h a t Theorem 1.1 remains valid f o r arbitrary dimensional restricted Lie algebras L and K which satisfy dim L/K

< a.

Let K b e an arbitrary restricted subalgebra of L and S L-module

M with character S is a K-module

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Conversely

E

L? Then any

MIK with character SIK.

t h e r e a r e t w o possibilities t o g o f r o m K-modules

with

character SIK t o L-modules with character S by means of t h e s o - c a l l e d induced and coinduced modules, respectively. Both play a central t o l e in t h e representation theory of restricted Lie algebras (cf. e.g. [SF o r FP31) and a r e essentially t h e same, a s will be shown below. When conkenient, we shall consider ( l e f t ) L-modules with character S a s unitary ( l e f t ) U(L;S)-modules and vice versa (cf. [SF, p . 2121). For every K-module V with character SIK we l e t

I~~:(v;s):=

U(L;S) @ U'K;slK) V

denote t h e K-induced L-module with character S and action u . ( r . @ v ) := u x @ v V u,x

U(L;S), v

r

r

V. Note t h a t V can be mapped (as a K - m ~ d u l e )

into I ~ ~ ; ( v ; sby ) means of v

+ +

l @ v . By virtue of i t s universal p m p e r t y

(cf. CHig, p. 4911), we obtain t h a t t h e induction f u n c t o r ~nd:(.;~) 1s l e f t adjoint t o t h e restriction f u n c t o r .lK. Dually, we l e t ~ o i n d ; ( ~ ; ~ HomK(U(L;S)),V) ):= denote t h e K-coinduced L-module with c h a r a c t e r S and action (u,Fl(x) := f ( x u ) V u,x mapped via f

E

U(L;S), f

r

HomK(U(L;S),M). Note t h a t ~ o i n d : ( V ; ~ ) i s

f(1) i n t o V (as a K-module). By virtue of i t s ur~iversal

++,

property (cf. CHig, p. 4931), we obtain t h a t ~ o i n d k ( . ;is ~ )right adjoint t o t h e restriction f u n c t o r

.lK.

(1.2) Corollary. Let L be a finite dimensional r e s t r i c t e d Lie alpebra, S

r

a and

K a r e s t r i c t e d subalgebra o f L. Then f o r every K-module V

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FELDVOSS

with character SIK we have the following natural isomorphism: a) Ind; (V;Sl

="

L

Coind;

(,

V;SJ,

=" Ind; (,-I V f -s), L CoindK (V;SI* L=" c o i n d k (,-I V* -S) i f dim V < a .

bl Ind; (v;s) * C)

Proof. a) is an immediate consequence of Theorem 1.1 and CNT, (18),

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p. 971. b ) : As in CBla, Prop. 1, p. 458/4591, one can s h o w t h a t t h e dual of a K-induced L-module of V with character S and t h e K-coinduced L-module of V* with character -S are isomorphic a s L-modules and by virtue of [SF, Th. 2.7, p. 2111 have t h e same character. Hence t h e assertion follows f r o m a ) . Finally, we obtain c ) by combining a) and b ) .

Another application of Theorem 1.1 a r e t h e twisted p a r t s of t h e following homological version of a Frobenius reciprocity theorem (cf. CNT 6 0 , Lemma 7, p. 971):

(1.3) Corollary. Let L be a finite dimensional restricted Lie algebra,

S c L* and K a restricted subalgebra o f L. Suppose that M is a n L-module with character S and V i s a K-module with character SIK. Then for every n

r

No we have the following natural isomorphisms:

L a) ~ x t G ( (M,IndK ~ ; ~ ) (V;S)) ="

n EX^^(^,^^^) (MIK ,,V),

b) ~ x t f l ( ~ ,(Indk S ) (v;s),M)

EX^ U ( K ; S , K ) fV,MIK ),

n

="

G(L;S)( M , ~ o i n d ;(V;S)J =" Ext G(K;SIK)(vK, VI, EX^ ;(L;S, ( c o i n d k ( V ; ~MJ ) , -' Ext G(K,S,K)(,--I V,MIK 1.

cl EX^ dl

Note t h a t t h e assumption o n t h e finite dimension of L ( o r more generally L/K) is only necessary in t h e cases a) and d) which a r e consequences of t h e more general s t a t e m e n t s b) and c)!

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COHOMOLOGY O F R E S T R I C T E D L I E ALGEBRAS

In t h e special case S = 0, t h e reduced universal enveloping algebra Up(L) := U(L;O) is a n augmented associative algebra with augmentation map

i.e., t h e trivial L-module IF has character 0 . Following LHo21, we

E,

define f o r any restricted L-module M t h e restricted cohomology spaces o f

L with coefficients in M by means of H ~ ( L , M := ) ~ x t & ~ ~ , ( l V~ n, M E No. ) Then we finally obtain Shapiro's lemma for restricted cohomo1o~:y (cf.

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also CFS, Theorem 2.41):

(1.4) Corollary. Let L be a finite dimensional restricted Lie algebra and

K a restricred subalgebra o f L. Then for every restricted K-module V and every n

E

No we have the following natural isomorphjsms:

Hp"l~.lnd; (v;o))=" H ~ " ( K,v), , HH~R(L, ~ o i n d (v;o)) b =" H:(K, V ) . 1 1

52. COMPLETE RESTRICTED COHOMOLOGY

Let M b e an arbitrary L-module with character S. By virtue o f [ S F , Cor. 4.3, p 2181, t h e S-reduced universal enveloping algebra U(L;S) of a finite dimensional restricted Lie algebra L is a Frobenius algebra ( c f . 51). Hence every module I" of an injective resolution 0

- M

'l

dl do 10 + 11 +

12 ---+

...

of M over U(L;S) i s projective. (In f a c t , this is a consequence o f t h e special case K := 0 in Corollary 1.2a)!) Let

...

-

p,

d d 4 p, -& p,

I , M

4

be a projective resolution of M over U(L;S) and p u t P-, V n

E

N and do := .rl

E.

0 :=

In,-' d-,, := dn-'

If we splice t h e t w o resolutions t o g e t h e r , we

obtain t h e following acyclic chain complex of projective U(L;S)-modules:

FELDVOSS

The diagram (*) is referred t o as a complete projective resolution o f M

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over U(L;S).

Remark. I t can readily be verified t h a t every acyclic projective chain complex (Pn)n,Z together with an epimorphism o n t o M , which satisfies Im d l = Ker

E,

of L-modules f r o m Po

E

is necessarily a complete projective

resolution of M.

If we apply HomL(.,N) t o (*) f o r any L-module N with character S and d e n o t e Sn(cp) := cp

d n + l V cp

F

HomL(P,,N), n

E

Z , we obtain t h e chain complex

We define t h e complete extension functors o f U(L;S) with coefficients

in M and N a s A

n

E x ~ ~ ~ ( ~ , ~ , := ( MHn(HornL(P.,N)) ,N) := Ker Sn/lm

sn-l

V

n

c

k

As usual, one can show t h a t a complete projective resolution is unique up t o homotopy and therefore t h e above definition is independent of t h e chosen complete projective resolution. Since t h e trivial L-module IF has character 0 (cf. §I), we can introduce t h e complete restricted cohomology of L in t h e common way:

Definition. Let M b e a restricted L-module and n

A;(L,M)

A n

:= Ext,

(,,(IF,M)

c

Z. Then

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COHOMOLOGY O F R E S T R I C T E D L I E ALGEBRAS

is called t h e n-th complete restricted cohomology space o f L with coeffi-

cients in M. The number n is referred t o a s t h e degree of t h e periaining cohomolog,y space. Moreover, we p u t

A;(L,M)

@

:=

n~ Z

A;(L,M).

The long exact homology sequence (cf. [Rot, Th. 6.3, p. 1721) yields

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immediate1.y t h e long exact complete restricted cohomology sequecce:

(2.1) Theorem. Let L be a finite dimensional restricted Lie algebra. Then

-

the following statements hold: a ) For any short exact sequence 0

--)

M'

M

M" --' 0 o f

restricted L-modules there is a natural homomorphism 3.= (3"),,,:

-

-AJL,M*)

$(L,M+Y

- I~,"(L,M)-I~,"(L,MY-

o f graded vector spaces o f degree 1 such that the sequence

...

~>;(L,MY

A;+l(~,~ ---, * ) ... .

is exact. n

b) I f P is a projective Up(L)-module then Hi(L,PI = 0.

For t h e conveniencr of t h e reader we n o t e t h e following useful f'acts.

(2.2) Proposition. For every finite dimensional restricted Lie algebra

L the following statements hold: a ) Suppose that MI,...,Mk are restricted L-modules. Then we htave the

following isomorphisms: k

H;(L,

@ M ~ 2)

i =I

k

i=1

H;(L,MJ

v 17

Z.

b) Suppose LE is a field extension o f lF and M is a restricted L-module. Then we have the following isomorphisms: ~?;jpn(~

E,M

LE)~A;(L,M) @

E

i ~ n

r

I

2874

FELDVOSS

In t h e proof of t h e t w o main r e s u l t s of this section we need t h e following generalization of a well-known r e s u l t ( c f . CPa2, Lemma 2.5, p. 3051). By virtue of [SF, Th. 2.7, p. 2111, t h e proof is t h e same a s in t h e restricted case.

(2.3) Lemma. Let P be a projective U(L;SI-module a n d M an arbitrary

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U(L;SF)-module with S , S ' < L? Then PC3 M i s a projective U(L;S+Se)-module. C

The n e x t r e s u l t s h o w s how t o c o m p u t e complete extension f u n c t o r s of reduced universal enveloping algebras by means of complete restricted cohomology spaces and allows, in c o n t r a s t t o t h e special case t r e a t e d in CFP3, Cor. 5.3, p. 10791, modules of arbitrary dimension:

(2.4) Theorem. Let M a n d N be arbitrary L-modules with character S. Then we have t h e following n a t u r a l isomorphisms:

Proof. Let (Pn)n,Z b e a complete projective resolution of IF over Up(L) = U(L;O). On account of Lemma 2.3 and [SF, Th. 2.7, p. 2111, we conclude t h a t (P, 8 M)n,Z is a complete projective resolution of M over yields according t o U(L;S). Application of H O ~ ~ ( ~ , ~ , =( . HomL(.,N) ,N) t h e adjointness of Horn und 8 ( c f . [Rot, Th. 2.11, p. 371) t h e following commutative diagram

COHOMOLOGY OF RESTRICTED LIE ALGEBRAS which establishes f o r every n

r

2875

Z t h e following natural isomorphism

Remark. If we define t h e complete Hochschild cohomology f i n ( u ( ~ ; s ) ; . ) a s in CNal and consider any two-sided U(L;S)-module M via x*m := x m- m.x

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V x r L, m r M a s a restricted L-module

M*, we could e s t a b l i s h in a

similar manner t h e following natural isomorphisms

A"(u(L;s),M)

2

A;(L,M*)

v n

r

E.

Let X d e n o t e t h e image of t h e identity element of Up(L) under t h e Frobenius isomorphism f r o m Up(L) o n t o i t s dual space u,(L)*. e x i s t s a uniquely determined element s which satisfies

E

E

The? t h e r e

Up(L), called t h e trace element,

= s . 1 . The mapping 1 defined by q(u) := u s t' u

E

IF

is a Up(L)-module monomorphism f r o m IF i n t o Up(L) (cf. CPa2, p. 3131) and i s referred t o a s coaugmentation map of Up(L). We p u t u ~ ( L ) + Ker : =E and Up(L)+ := Coker 1. Hence t h e following s h o r t sequences of Up(L)modules a r e exact:

Upon tensoring (over IF) with a n arbitrary restricted L-module M , we obtain t h e following s h o r t exact sequences:

We finally p u t M('):= @ 'u,(L)+@ M f o r r for r

2

0 and := @"'u,(L)+@

M

0 and obtain t h e following Reduction theorem for complete

restricted cohomology :

2876

FELDVOSS

( 2 . 5 ) Theorem. Let M be a restricted L-module.

Then we have the

following natural isomorphisms: H;(L, M )

2

Y n,r < Z .

H;-'~L, M ' ~ ) )

Proof. Application of t h e long exact cohomology sequence 2.1 t o t h e t w o s h o r t exact sequences of (**) yields according t o Lemma 2.3:

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(+)

A;(L,M)

A",'(L,u,(L)+

2

Then t h e assertion

B M ) , A;(L,M)

follows by

2

A;-*(L,u,(L)+

repeated application of

B M)

vnr

Z.

t h e natural

isomorphisms in ( + ) . 0

Remark. Note t h a t U,(L)=

u,(L)+!

The Reduction theorem 2.5 will be very useful t o t r a n s f e r s t a t e m e n t s a b o u t any complete restricted cohomology space t o analogue s t a t e m e n t s f o r a higher o r lower degree (cf. s 5 ) . This method is called degree-shifting o r dimension-shifting and is well known. In particular, t h e computation of t h e higher restricted cohomology spaces could b e reduced t o t h e computation of t h e 0 - t h c o m p l e t e restricted cohomology space (cf. t h e remarks a f t e r Proposition 2 . 6 ) . Therefore, in t h e n e x t s t e p we will provide a simple description of t h e latter. Put ML := M / v ~ ~ ( u , ( L ) + ) Mand l e t

oM d e n o t e t h e mapping from ML into M L which i s induced by t h e action of t h e trace element s o n M .

( 2 . 6 ) F'roposition. Let M be a restricted L-module. Then the sequence

o

-

&;'(L,M)

- ML

OM

M ~ -

i s exact and the following statements hold: L a) A;(L, M ) 2 M / Zrn ( d M , b) &;'(L, M )

2

Ker ( S ) M / ~ ~ l ( ~ p ( ~ ) ? ~ .

A$L,M)

-

o

2877

COHOMOLOGY O F R E S T R I C T E D L I E ALGEBRAS

Proof. Let P.

:=

(Pn)n,Z d e n o t e a complete projective resolution of IF

over Up(L) and p u t C ' : = (HomL(Pn,M)),,,

- -

-

, C;

:=

(HomL(P,,M)) n r o a s

well a s C1 := ( H O ~ ~ ( P , , M ) ) , , ~ Then . we have t h e s h o r t e x a c t sequence C;

C'

Hn(C;) = 0 V n


~.

(3.6) Corollary. Let L be a finite dimensional restricted Lie algebra. Then

the following statements are equivalent: a ) L is a torus.

b) Up(L) is semisimple. c) F i s a projective Up(L)-module. dl

E(s,)

#

0.

e) H;(L,MI vanishes for every restricted L-module M.

Proof. a)

-

b): For every t o r u s L t h e Fitting-0-component

Mo of any

restricted L-module M i s a trivial L-module and t h e r e f o r e an application of CFa3, Cor. 1.4, p. 971 in conjunction with Proposition 2.7 implies t h e vanishing of H;(L,M)

f o r every finite dimensional restricted L-module M.

Thus every finite dimensional restricted L-module, and especially Up(L), is completely reducible. F r o m this we obtain t h a t Up(L) is semisimple.

COHOMOLOGY OF R E S T R I C T E D L I E ALGEBRAS

b) c)

--

c) follows immediately f r o m CCE, Th. 4.2, p. 111. d): According t o Proposition 2.6, we have

where s L . ac = s ( s L ) a V a

E

A;(L,IF)

=I F / I ~ ( S ~ ) ~ ,

IF. Hence t h e assertion E(sL)

*

0 fc) l o w s

directly from o u r assumption and Theorem 2.1 b ) . d) = e) i s a direct consequence of Theorem 3.5, applied t o t h e ideal

e)

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-

{o}.

I :=

a): From t h e exact sequence o n p. 575 in CHo21 and H ~ ( L , I I : = 0,

we obtain by a dimension consideration t h a t L = < L [ P ] > ~ ,i.e., especially H ~ ( L , I F=) 0 Then t h e above mentioned exact six-term sequence yields by a dimension consideration t h a t L is abelian, a s required. We consiclw an arbitrary element x

r

L and p u t X := ,,.

our assumption we deduce H ~ ( x , I F )

From Shapiro's lemma 1.4. and

H;(L,~oind:(IF;0))

= 0 . We now

apply Proposition 2.7 in o r d e r t o see t h a t x

E

< x [ P ] >c~10 I F X [ P ] ~

8



* 0, we

vp(L).

Next, we give a cohomological characterization of nilpotent restricted Lie algebras t h a t involves i t s complete r e s t r i c t e d cohomology with irreducible modules a s coefficients and has a n analogue in t h e cohomology

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theory o f finite g r o u p s (cf. CStal).

(5.6) Theorem. For every finite dimensional restricted Lie algebra L the

following statements are equivalent: a ) L is nilpotent.

b) H ~ ( L , M )vanishes for every irreducible restricted L-module M c)

A;(L,MIvanishes for

-

every irreducible restricted L-module M

*

IF.

+ IF.

Proof. By virtue of Proposition 5.5, it is sufficient t o prove t h e implication b)

a) in t h e case n = 1. From M L = H ~ ( L , M )= 0 f o r every

non-trivial irreducible restricted L-module M o n e deduces o n account of t h e e x a c t sequence o n p. 575 in CHo21 and CDz, Th. 2, p . 1311 t h e vanishing of H'(L,M) f o r every non-trivial irreducible L-module. If we modify t h e proof of CBa2, Th. 4, p. 2961 in an obvious way, w e obtain an analogous cohomological characterization f o r ordinary nilpotent Lie algebras. This concludes t h e proof of o u r theorem.

Remark. The proof of Theorem 5.6 provides a n analogous cohomological characterization f o r ordinary modular nilpotent Lie algebras if we utilize ordinary Lie algebra cohomology in conditions b) and c ) . Furthermore, in t h e non-modular

c a s e t h e s e conditions are, according t o a classical

vanishing theorem of J . H . C . Whitehead, equivalent t o L being t h e direct product of a semisimple and a nilpotent Lie algebra.

2897

COHOMOLOGY OF RESTRICTED LIE ALGEBRAS

By virtue of Corollary 3.6 and Proposition 5.5, we obtain f o r t h e trivial irreducible r e s t r i c t e d L-module IF, t h a t i s n o t treated in Theorem 5.6, a t l e a s t t h e non-vanishing of every complete restricted cohomology space:

(5.7) Corollary. Let L be a finite dimensional nilpotent restricted Lie

algebra and suppose that L is not a torus. Then we have

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~;(L,IF) ++

o vn

6

Z.

Without additional assumptions o n L, i t i s n o t possible t o prove s t r o n g e r r e s u l t s , since t h e case dim

A;(L,IF)

= 1 V n

c

Z c a n occur, a s

t h e example of a p-nilpotent cyclic r e s t r i c t e d Lie algebra s h o w s (cf. Proposition 2.6).

A restricted Lie algebra L is called strongly solvable if t h e r e i s a t o r u s

T such t h a t L = T @ Radp(L). E.g., every Bore1 subalgebra of a classical r e s t r i c t e d Lie algebra is s t r o n g l y solvable. According t o [SF, Th 5.3(2), p. 2211, every irreducible restricted L-module i s a n irreducible r e s t r i c t e d L/Radp(L)-module and therefore t h e conditions in t h e n e x t r e s u l t make sense.

(5.8) Proposition. For every finite dimensional restricted Lie algebra over

a perfect field lF the following statements are equivalent: a ) L i s strongly solvable. b) There is an integer n

E

Z such that ~ ( L / R ~ ~ ~ ( L vanishes ) , M ) for

every irreducible restricted L-module M. C)

l ? i f ~ / ~ a dM~) fvanishes ~), for every irreducible restricted L-module M.

Proof. a) =. c) i s a direct consequence of Corollary 3.6 and c) * b) i s trivial. The long e x a c t cohomology sequence 2.1 in conjunction w i t h t h e

2898

FELDVOSS

Reduction theorem 2.5 and Corollary 3.6 s h o w t h a t L / R ~ ~ ~ is ( La ) t o r u s , i.e., L(')

G

Radp(L). Since IF is assumed t o b e perfect, one can conclude

t h e proof of b )

=,

a) a s in [SF, Lemma 8.7a), p. 2461.

A Lie algebra i s called supersolvable if t h e r e is a chain of ideals such

t h a t all f a c t o r s are one-dimensional.

One readily verifies t h a t every

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subalgebra and every f a c t o r algebra of a supersolvable Lie algebra is supersolvable. We a r e going t o derive a s l i g h t generalization of a restricted analogue of a r e s u l t due t o D. W. Barnes (cf. CBal, Th. 3, p. 3461):

(5.9) Proposition. Let L be a finite dimensional r e s t r i c t e d supersolvable Lie algebra. I f t h e r e s t r i c t e d L-module M d o e s n o t contain a onedimensional L-submodule,

then

the complete restricted

cohomology

H ~ ( LM, ) vanishes.

Proof. Let

d e n o t e t h e algebraic closure of IF,

M := M @IF IF. Since of

i such t h a t T

ideal of

i(') is

i :=

L @IF

-

IF and

by assumption nilpotent t h e r e i s a t o r u s

= T @ N i l n ) , where N i l n ) d e n o t e s t h e l a r g e s t nilpotent

Then Proposition 2.2b) in conjunction with Proposition 3.7 yields

A;(L,M) @IFF2 fI;K,M)

-

fi;(~il(~),M)~

If M does n o t contain a one-dimensional submodule,

v

n

E

Z.

does n o t contain

t h e trivial N i l n ) - m o d u l e and t h e assertion follows f r o m Proposition 5.5. C

By means of Proposition 5.9, i t i s possible t o derive t h e following cohomological characterization of supersolvable r e s t r i c t e d Lie algebras:

(5.10) Theorem. F o r every finite dimensional r e s t r i c t e d Lie algebra L t h e following s t a t e m e n t s are equivalent:

COHOMOLOGY OF RESTRICTED L I E ALGEBRAS

a ) L i s supersolvable.

bl H ~ ( L , M vanishes ) for every irreducible restricted L-module M that is not one-dimensional. C)

H;(L,M)

vanishes for every irreducible restricted L-module M that is

not one-dimensional.

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Proof. a ) * c) is an immediate consequence of Proposition !i9 and C)

* b) is trivial. b) * a ) : From M = = H ~ ( L , M =) 0 , we deduce by means

of t h e e x a c t sequence o n p. 575 in CHo21 t h e vanishing of H'(L,M) f o r every irreducible restricted L-module M such t h a t dim M

*

1. B j virtue

of CDz, Th 2, p. 1311 and CBa2, Th. 4, p. 2961, we obtain t h e assertion.

Put A

:=:

f o r every X

{X e

E

L*: X(L('))

= 0 , X(x[pl) = X(x)p V x

E

A. IF, is a r e s t r i c t e d L-module via x.1,

and therefore IF* :=

L} and IF, := IF.1, := X(xblx t x e L

@ IF, is a l s o a r e s t r i c t e d L-module. On account

,€A

of Theorem 5.10 and Proposition 2.2a) in conjunction with t h e Relduction theorem 2.5 and Corollary 3.6, w e obtain t h e following non-vanishing theorem f o r supersolvable restricted Lie algebras:

(5.11) Corollary. Let L be a finite dimensional supersolvable restricted

Lie algebra and suppose that L is not a torus. Then we have

H~R(L,F*)+ o v n

E

Z.

o

Without additional assumptions o n L and IF t h e conclusion in Corollary 5.11 could n o t b e sharpened, a s t h e non-abelian two-dimensional Lie algebra over a field of characteristic 2 shows. The n e x t r e s u l t i s similar t o Proposition 5.8 and is well-known in t h e case of finite g r o u p s (cf. CStal).

2900

FELDVOSS

(5.12) Proposition. For every finite dimensional restricted Lie algebra L

t h e following statements are equivalent: a) L is solvable. b ) H ~ ( L / A ~ ~ ~ ( Mvanishes ) , M ) for every irreducible restricted L-module M. C)

9; ( L / A ~ ~ ~ (MIM vanishes ), for

every irreducible restricted L-module M.

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Proof. a) * c) i s a direct consequence of Propositions 5.5 and 3.9. Since t h e implication c ) * b) i s trivial, i t remains t o s h o w b) * a). In this case t h e f i r s t p a r t of t h e proof of CBa2, Th. 4 p. 2961 suffices t o obtain t h e assertion.

In t h e solvable c a s e t h e r e is t h e following analogue of a theorem due t o H. Onishi [On] and in f a c t , o u r proof i s very close t o t h e proof given by him.

(5.13) Proposition. Let L be a finite dimensional solvable restricted Lie

algebra and M a restricted L-module. I f f o r s o m e integer n , H;(K,MI

is

vanishing for all restricted subalgebras K o f L, then M is cohomologically trivial.

Proof. We proceed by induction o n t h e dimension of L. Since every one-dimensional restricted Lie algebra i s abelian, t h e assertion is a special case of Proposition S.S. Suppose now dim L > 1 and a s s u m e t h e t r u t h of t h e assertion f o r all restricted subalgebras K such t h a t dim K < dim L. In particular, we may assume t h a t

A;(K,M)

vanishes f o r every proper restricted subalgebra K

of L. Then, a s usual, w e have a proper restricted ideal I of L s u c h t h a t t h e f a c t o r algebra L/I i s abelian. Since A;(I,M)

vanishes, we obtain by

COHOMOLOCY OF RESTRICTED L I E ALGEBRAS

virtue of Proposition 3.9:

A:(L,M)

(*)

A;(L/I,M')

v k

Z.

Since L/I is abelian, o u r assumption in conjunction with (+) and Proposition 5.5 yields t h e vanishing of (*)

A;(L/I,M~) and

finally a n o t h e r application of

implies t h e assertion. 0

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We conclude this section with a non-vanishing

theorem similar t o

Corollary 5.7 and 5.11, which is valid f o r an arbitrary restricted Lie algebra. A Lie algebra L is called unimodular if t r ( a d L x ) = 0 V x r L.

(5.14) Theorem. Let L be a finite dimensional r e s t r i c t e d Lie algebra a n d s u p p o s e t h a t L is n o t a torus. Then t h e r e a r e infinitely m a n y positive i n t e g e r s a n d if L is unimodular there a r e a l s o infinitely many negative i n t e g e r s n s u c h t h a t HJL,IFI

*

0.

Proof. Assume t h a t there a r e only finitely many positive integers n such t h a t H;(L,IF)

* 0.

Then t h e associative IF-algebra H := HL(L,IF) is

finite dimensional and according t o EFP1, Prop. 1.3(c), p. 291, the l e f t H-module Hb(L,E) is also finite dimensional f o r every irreducible r e s t r i c t e d k

L-module E. Hence there e x i s t s a n integer k such t h a t Hp(L,E) = 0 f o r every n

k and every irreducible restricted L-module E. By v i r t ~ ~ofc t h e

Reduction theorem 2.5 and t h e long exact cohomology sequence 2.1, H,(L,M) vanishes f o r every finite dimensional restricted L-module M and a s in t h e proof of Corollary 3.6, w e obtain t h a t L is a t o r u s in contradiction t o o u r assumption. Finally, according t o t h e Duality theorem f o r r e s t r i c t e d cohomology (cf. CPa2, Satz 3.11, p. 3191), t h e non-vanishing in t h e negative case f o r unimodular Lie algebras is a direct consequence of the positive case.

FELDVOSS

ACKNOWLEDGMENTS

This paper is an augmented version of a p a r t of t h e author's doctoral dissertation w r i t t e n under t h e guidance of Professor H e l m u t S t r a d e a t t h e University of Hamburg. The a u t h o r would like t o express his deep gratitude t o him f o r critisism and encouragement. He a l s o thanks Rolf

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Farnsteiner f o r valuable suggestions concerning t h e presentation of this paper and sharing with t h e author preprints of his very r e c e n t work.

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Received:

February 1991