lie algebra representations of dimension < p2€¦ · lie algebra representations of dimension...

transactions of theamerican mathematical societyVolume 319, Number 2, June 1990

LIE ALGEBRA REPRESENTATIONS OF DIMENSION < p2

HELMUT STRADE

Abstract. Various methods of representation theory of modular Lie algebras

are improved. As an application the structure of the Lie algebras having a

faithful irreducible module of dimension < p is determined. Applications to

the classification theory of modular simple Lie algebras are given.

0. Introduction

In this note we investigate finite dimensional Lie algebras over an algebraically

closed field F of characteristic p > 3 , which admit a faithful irreducible rep-

resentation of dimension < p . This problem arises in a more specific form

in the classification theories both of Block-Wilson [6] and Benkart-Osborn [2].

They derive the structure of these algebras under various additional assump-

tions, which naturally arise in the context of classification theory. Our more

general approach yields some structural insight, which in turn simplifies those

proofs and should also be useful for further progress in the classification theory.

In [10, 11, 12] semisimple Lie algebras having a faithful representation of

dimension less than p have been fully determined. One cannot expect to get

results of the same completeness in the more general situation, since the mass

of examples makes it hard to describe them by means of general concepts. This

reflects the fact that the strong argument on traces fails to hold, whenever the

module under consideration has dimension > p .

In §1 we list results and methods which have been developed during the last

twenty years and now enable us to attack the problem. Some of them were

improved. In §11 an appropriate (but unusual) concept of a socle is introduced.

We describe the structure of the investigated Lie algebras in terms of this socle.

In §111 we restrict ourselves to Lie algebras of toral rank one with respect to

some Cartan subalgebra (CSA) H and then obtain a complete determination

of all these algebras.

Received by the editors September 19, 1988.

1980 Mathematics Subject Classification (1985 Revision). Primary 17B05, 17B10, 17B50.Partially supported by NSF grant no. DMS-8702928.

©1990 American Mathematical Society

0002-9947/90 $1.00+ $.25 per page

689License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

690 HELMUT STRADE

This paper was written while the author was visiting the University of Wis-

consin. I would like to take the opportunity and thank Georgia M. Benkart and

J. Marshall Osborn for their hospitality and support.

I. General results

In this section we will for convenience state and prove some results which we

will refer to in the sequel. These results describe the structure of Lie algebras

and modules in the presence of some (quite natural) assumptions. We always

consider Lie algebras L finite dimensional over an algebraically closed ground

field F of characteristic p > 3 .

Theorem 1. Let Sx, S2 be ideals of L such that

L = SX+S2, [Sx,S2] = 0.

Assume that p: L —► gl(U) is an irreducible representation and let p¡: S¡ —>

gl(U¡) be irreducible subrepresentations for SX,S2. Then U = Ux <¡$> U2 and

p = px ® id + id <g> p2 .

Proof. Let A, B, C denote the associative subalgebras of End(L) generated

by p(L), p(Sx), p(S2), respectively. Our present assumptions imply that

(*) A = End(U), BC = A, [B,C] = 0.

Let R be the nilpotent radical of B. Then RA = RC is according to (*)

a nilpotent two-sided ideal of A. As A is simple this shows that R = (0),

proving that B is semisimple. (*) also shows that the center C(B) of B is

contained in the center of A , which is F id. Hence C(B) = F id and therefore

B is a central simple subalgebra of A with C(B) = C(A).

Put D := {x e A\[x, B] = 0}. The double centralizer theorem implies that

D is simple and that A = B ®F D. Since C <z D and BC = A = BD, a

dimension argument proves C = D and A = B ®F C .

Let Bx, Cx be minimal left ideals of B and C, respectively. Then Bx <g> C,

is a minimal left ideal of B ® C. As an irreducible B ® C-module, U is

module-isomorphic to BX®CX. With the same reasoning, any irreducible Sx-

submodule of U is an irreducible ß-module isomorphic to Bx as a 5-module.

This proves the result. D

In [4] Block and Zassenhaus proved a similar result for the direct sum of Lie

algebras.

Recall the Blattner-Dixmier theorem on induced representations. Although

this theorem is stated in [7] only for Lie algebras over a field of characteristic 0,

Block-Wilson [6] have pointed out that it is true with only the obvious changes

for the following situation:

Let L be a restricted Lie algebra, / an ideal of L and p: L —► gl(U) an

irreducible representation. Then p admits a character S e L* such that

p(x)p - p(x[p]) = S(x)pid VxeL.

Let U0 be an irreducible /-submodule of U and put Ufl) the sum of all

/-submodules of U isomorphic to UQ. Put

G = G(I, U0) := stab(L, Ûfl)) = {x e L\p(x)IffI) c Úfj)}.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

LIE algebra representations of dimension < p 691

For a restricted Lie algebra L and S e L* we consider the ideal J of U(L)

which is generated by the set of (central) elements {xp - x - S(x)p 1 \x e L} c

U(L). Put

U(L,S):= U(L)/J.

If L is finite dimensional then dim U(L, S) = pdimL . U(L, S) has the fol-

lowing universal property: Every representation p: L —> gl(U) which has the

character S (i.e. p(x)p - p(x[p]) = S(x)pidVx e L) can be uniquely extended

to an associative unitary representation

p: £/(L,S)-End(C/).

Let G be a restricted subalgebra of L of finite codimension. Then U(L, S)

is a free right U(G, 5'|G)-module of rank pdimL/ . (For the general notation

we always refer to [15].)

Theorem 2 (Blattner, Dixmier). Let L be a restricted Lie algebra, U an ir-

reducible L-module of character S, I an ideal of L, U0 an irreducible I-

submodule and G = G(I, U0). Then

( 1 ) G is a restricted subalgebra of L.

(2) U^U(L,S)®U{GSlG)ÚfI).(3) U0(I) is an irreducible G-module.

For further references we state the following corollary.

Theorem 3. Let L be a (nonrestricted) Lie algebra and I an ideal of L. Suppose

L has a faithful irreducible representation p: L -* gl(U) with dim U < p2. Let

U0 be an irreducible I-submodule of U. Then dimÚfl) p . then since dim U < p2 Theorem 2 yields U =

UQ(I). Therefore U = ¿~2W¡ is the sum of irreducible /-modules isomorphic

to U0 . If x e I annihilates U0, then

p(x)(U) = Y,Pix)iWl) = 0.

The faithfulness of p gives the result. Q

If in Theorem 2 the ideal / is assumed to be abelian we have the following

more detailed description. Let L denote the p-envelope of p(L) in gl(U).

Lemma 4. Let I be an ideal of a (nonrestricted) Lie algebra L and assume that

p: L —» gl(U) is an irreducible L-module.

(1) If I acts nilpotently on some nonzero I-submodule then I c ker(p).

(2) // r ' c ker(p) then there exists Eel* such that for all x e I, p(x) -

E(x) id ¿s nilpotent.

(3) Assume that I c ker(p). Then any irreducible I-submodule UQ is one-

dimensional and G(I, Uf) = {g e L \E([p(x), g]) = 0 Vx e 1}. Moreover,

L = G(I, U0) if and only if [L, I] c ker(p).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

692 HELMUT STRADE

(4) If I is abelian and [L, I] ct ker(p) then dim U > p dim U0(I).

Proof. ( 1 ) Since / acts nilpotently on some /-submodule U' ^ (0), there is

a nonzero vector annihilated by p(I). Put U0 = {u e U\p(x)u = 0 Vx e 1}.

This is an L-submodule and nonzero. Hence U0 = U, proving / c ker(p).

(2) Decompose U into weight spaces with respect to p(I). Since / is an

ideal and hence acts nilpotently on p(L) these weight spaces are L-submodules.

The irreducibility of U entails the existence of some mapping E: I —> F such

that p(x) - E(x) id is nilpotent for all x e I. Since p(I) is abelian there is a

common eigenvector u ^ 0. Then

E(x + ay)u = p(x + ay)u = p(x)u + ap(y)u = E(x)u + aE(y)u

for all x, y e I, a e F . Therefore E is linear.

(3) Clearly U0 is one-dimensional and

Úfl) = {ue U\p(x)u = E(x)u Vx e /}.

Take any g e G(I, Uf = {h e Lp\hUfI) c Ûfl)}. Then we have for u e

UfI) and xel

p(x)(g ■ u) = E(x)g -u = g-E(x)u = g ■ p(x)u

proving [p(x), g]u = 0 and E([p(x), g]) = 0. Next assume E([p(x), g]) =

0 for some g e Lp . Then [p(x), g]u = 0 Vx e I, u e U0(I) and hence

p(x)(g ■ u) = E(x)g ■ u . Therefore g -ue U0(I) and g e G(I, Uf .

If L = G(I, Uf) then [L, /] c ker(£). [L, I] is an ideal of L contained

in ker(.E). Hence it acts nilpotently on U. According to (1) we obtain the

result.

(4) Apply Theorem 2 and (3). D

As a result of Lemma 4 we obtain that whenever L has a noncentral abelian

ideal, only the first case of Theorem 3 can occur.

Actually, we want to derive some general information about both cases. To

do this we recall the following notation.

Any linear mapping tp: L —► End((7) is said to be aprojective representation,

if cp([x, y]) - [cp(x), tp(y)] eFid for all x, y e L.

Let ^4(zz; 1) denote the truncated polynomial ring

F[Xx,...,Xn]/(Xp,...,Xp)

and W(n; 1) = Der(A(n ; 1)) the restricted Jacobson-Witt algebra.

For any finite dimensional associative algebra A , Jac(A) denotes the maxi-

mal nilpotent ideal of A .

The following improves a result of [13].

Theorem 5. Let I be an ideal of the restricted Lie algebra L. Assume that

p: L -» gl(U) is an irreducible representation. Let UQ be an irreducible I-

submodule and put B equal to the subalgebra of End(L) generated by p(I).

The following cases can occur.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

LIE ALGEBRA REPRESENTATIONS OF DIMENSION ux) = (b ■ u0) <8> ux VbeB, u0eU0, uxeUx.

(2) B is not simple. Then B/Jac(B) is simple. In addition, L ^ G(I, Uf).

Proof. Let A = End(U) be the algebra generated by p(L). Since / is an

ideal of L, the multiplication in L extends to a mapping p: L —> Der(ß)

determined by p(x)(y) = [p(x), y] Vx e L, y e B.

(1) Assume that B is simple and C(B) = C(A). Then the double centralizer

theorem proves that C := {a e A\[a, B] = 0} is a simple algebra and A =

B <&C, i.e. BC = A, B nC = Fid. Since any derivation of B is inner, there

is a linear mapping

fi'.L^B

such that [p(x), b] = [fi(x),b] Vx e L, V/j e B. Then p(x) - f(x) e Cproving p(L) c B + C. As p(I) c B there is a Lie algebra homomorphism

tp : L/I - (B + C)/B = C/F id.

Let L' be the preimage of <p(L/I) in C. L' is an extension of cp(L/I) by

the one-dimensional center F id. C is generated by L' as L generates A .

Let Bx, C, be minimal left ideals of B and C, respectively. Then U =

BX®CX and U0 = Bx . The above also shows that C, is an irreducible projective

L//-module. Since B is simple, U decomposes into a direct sum of irreducible

modules isomorphic to U0 . Theorem 2 yields L = G(I, Uf .

(2) We now consider the general case. Let R denote the maximal nilpotent

/¿(L)-invariant ideal of B. Then RA is a two-sided nilpotent ideal of A,

whence R = 0. Therefore B is /z(L)-semisimple. From [5] we conclude that

there exist simple algebras Bx, ... , Bk and nx, ... ,nk such that

k

B = C$B¡®A(n¡;\).(=i

Moreover, any B¡®A(n¡; 1) is //(L)-invariant and

Der(ß,. ® A(n¡ ; 1)) = Der(B¡) ® A(n¡ ; 1) + F o W(n¡ ; 1).

Since B¡ is simple, we obtain p(L) c Der(Bj <g> A(n¡ ; 1)) = adi?, ® ^(zz(. ; 1) +

F®W(n¡; 1). This proves that p(L) annihilates ®k=xC(BA)®F . Since p(L)

generates A, this means ®/=1 C(B¡) ® F c C(A) = Fid. As a result, k = 1

and B = Bx®A(n; 1) (for zz = zz,) as well as C(BX) = C(A).

Put D := {x e A\[x, Bx] = 0}. As above A = Bx <g> D. Note that

Y,Bx®Fl\xsi'(sx + --- + sn>l)

is a nilpotent ideal of Bx ® A(n; 1) proving that Bx = B/Jac(B) is simple.

Now assume that (1) is not true. Then zz # 0 and Jac(B) ^ 0. Since U is a

694 HELMUT STRADE

faithful A-module, it is not the direct sum of irreducible 5-modules. Therefore

U0(I) j=.U . According to Theorem 2 this yields L ^ G(I, U0). D

This theorem reduces the representation theory of Lie algebras with a non-

central ideal to representations of smaller algebras.

We also may apply the first part of the theorem to the second: Let p: L —>

gl(U) be an irreducible representation of the restricted algebra L and assume

that / is an ideal of L such that L f= G(I, U0) := G for some /-submodule

U0 . Then Theorem 2 proves that U is induced from the irreducible G-module

Ufl). Applying Theorem 5 to p: G —> gl(U0(I)), we see that the first case oc-

curs. Hence U0(I) is the tensor product of U0 with some irreducible projective

G//-module.

The problem, how to describe restricted Lie algebras having a suitable large

subalgebra (which occurs if L ^ G(I, Uf), was dealt with in [9]. Recall that

i=l |a|>Zr+l

denotes the canonical filtration of W(n; 1).

Definition [9]. A subalgebra 5 of W(n ; 1) is called essential, if S+ W(n ; 1)(0)

= W(n; 1).

Theorem 6 [9]. Let L be a restricted Lie algebra and G a restricted subalgebra

of codimension n . Then there exists a restricted homomorphism

tp: L^ Win; 1)

such that

(1) cp(L) is an essential subalgebra of W(n ; 1).

(2) ker(cp) is the unique maximal ideal of L which is contained in G.

(3) tp(L) n W(n ; 1)(0) = cp(G).

It will come out that in this note the essential subalgebras of W(l;\) are of

major importance. We improve the results on these algebras given in [9].

Proposition 7. Suppose that L is an essential subalgebra of W(l ; 1). Then

either L coincides with W(l; 1) or there exists an automorphism tp ofW(l; 1)

such that cp(L) c Fd + Fxd + Fx2d . In either case L is a restricted subalgebra

ofi 1V(1;1). '

Proof. We consider the filtration of L inherited by the filtration of W(l ; 1).

Let z := max{y'|L n W(l; 1),., f 0}. If z > 2 then, as L is essential and

therefore contains an element d + f, fi e W(l; 1)(0), and L + W(l; 1)(3) =

W(l ; 1). It is easily seen that L = W(l ; 1) in this case.

If i' = -1 , then L = Fe with e = d + Y%Zi a¡x'd . Applying an auto-

morphism exp(ady) for some y e W(l ; 1)(2), we may assume e = d + axd .

If i = 0, 1 there exists an element h e Lr\W(l; 1)(0), which does not act


LIE ALGEBRA REPRESENTATIONS OF DIMENSION < f 695

nilpotently on W(l;\). Fh is then a CSA of W(l ; 1). Adjusting h by some

nonzero scalar we can construct a proper basis fi_x, ... , fp_2 of W(l ; 1) such

that f0 = h and /_, e L. The mapping cp: f^ x'+ld defines an automor-

phism of W(l;l).

This proves the first assertion. Fd + Fxd + Fx d is a restricted subalgebra

of W(l;l) and isomorphic to 5/(2). The subalgebras of s I (2) are: one-

dimensional abelian, two-dimensional solvable nonabelian, three-dimensional

simple. Hence the p-closure of <p(L) has the same dimension as cp(L). This

shows that tp(L) is a restricted subalgebra of W(l;l). Since W(l; 1) is

centerless, every automorphism is a restricted mapping. D

II. Structure theory of Lie algebras with

FAITHFUL IRREDUCIBLE MODULE OF DIMENSION < p2

Semisimple Lie algebras having a faithful representation of dimension < p

have been determined in [10, 11, 12]. We improve on the results of those papers.

(We will refer to a previous chapter as usual: (1.4) means Lemma 4 in §1.)

Theorem 1. Let L be any Lie algebra and p: L —> gl(U) a faithful irreducible

representation of dimension < p. Then

(1) L = 0/=1 A¡ ® C(L) where either all A¡ are classical simple algebras or

rc = l and Ax = W(l; 1).

(2) Der(A¡) = adA¡ in all cases.

(3) Let U¡ be irreducible A^submodules (i = I, ... , k). Then U = Ux ®

-®Uk.

Proof. (I) Let x e L n C(L). As U is irreducible, x acts as a scalar,

p(x) = aid . Since x e L we have

0 = trace(p(x)) = a dim U,

i.e. a = 0. This shows L( ' n C(L) = (0). There exists an ideal S of L

containing L(1) such that L = S ® C(L), C(S) = 0. Assume that S has an

abelian ideal 1^0. According to (1.4) this implies dim U > p , a contradiction.

This proves rad(S) = 0. We can now apply the results of [10, 11, 12] and obtain

(!)•

(2) If Ax = W( 1 ; 1) the result is known. If A¡ is classical and by assumption

has a faithful irreducible module of dimension < p it has a nondegenerate trace

form. Then all derivations are inner [3]. (3) results from (1.(1)). D

Recall that a Lie algebra L is said to be indecomposable, if it is not a direct

sum of ideals:

L = Sx ® S2 => Sx = 0 or S2 = 0.

Corollary. Suppose p: L —> gl(U) is a faithful irreducible representation. Let S

be an ideal of L having a submodule U0 of dimension <p. Then S{ f)C(L) =

696 HELMUT STRADE

0. If moreover L is indecomposable and U0 irreducible faithful, then either

S = L and C(L) = 0 or dim S = 1.

Proof. Let V c U0 be an irreducible 5-submodule with representation p: S —>

gl(V). Theorem (II. 1) applies to S/ker(p), proving

C((S/ker(p)f]) = 0.

This implies C(L) f)S{l) c ker(p). Put in (1.4) I = C(L)nS{i}. Then (1.4(1))

yields C(L)nS{ ' = 0. Next assume that U0 is faithful. According to Theorem

(II. 1 ) we have S{i) = ©f=1 A¡, A¡ simple and 5 = S(1) © C(S). Let D be any

derivation of S. Then D(A¡) = D((A¡f]) c [DIA^A^ c A¡. Therefore

D\A¡ is a derivation, hence inner.

As a result, for any x e L there is an element y e S{ ' such that

[x-y,5(1)] = 0.

Put

R:={xeL\[x,Sw] = 0}.

Then L = S{l)®R . Since L is indecomposable, this implies S{1) = 0 or R = 0.

In the latter case L = S and C(L) = 0. In the former case S is abelian, and

as it has a faithful irreducible representation this yields dim S = 1 . D

Throughout this section we assume from now on

(A) L is an indecomposable Lie algebra having a faithful irreducible repre-

sentation p: L —> gl(U) of dimension <p .

We are going to construct some kind of socle S(L) for L .

Lemma 2. Assume that C(L) f 0. Suppose that L has an ideal Kx with the

properties

(a) K[l) = Kx.

(b) C(L) c Kx .(c) KX/C(L) is a minimal ideal of L/C(L).

Then(1) C(L) n K{01) = 0 holds for any ideal KQ with KxçtKQ.

(2) rad(L) = C(L).

Proof. We prove both assertions simultaneously, by putting K0 = rad(L) in

case of (2). Since K{ is not solvable, we also have Kx çf KQ in this case.

Suppose K0 is an ideal with Kx çt K0. The minimality of KX/C(L) implies

that ^n^c C(L). Then

[K0,KX] = [K0,K[])] C [[K0, Kx],Kx] c [C(L),KX] = 0.

Let V be an irreducible (A^0 + Kx )-submodule of U and U¡ irreducible K¡-

submodules of F (i = 0, 1). According to (1.1) and our general assumption

this yields

p2 > dim U > dim V = (dim c70)(dim Í7, ).

LIE ALGEBRA REPRESENTATIONS OF DIMENSION < p2 697

As 0 ^ C(L) c k\1) the corollary to 11.(1) proves dimUx > p and therefore

dim UQ < p. Next consider the ideal I := KQ + Kx , and apply (1.3). Since

dim V > p this implies that F is a faithful (K0 + A^,)-module. According to

(1.1) i/„ is a faithful A:0-module. (II. 1, corollary) yields K¡¡i)nC(L) = 0 and

we are done in case (1).

Next assume K0 = rad(L). Then dim UQ is a p-power; hence dim U0= 1 .

Since U0 is faithful this implies dim K0 = 1. Moreover, since V is a faithful

irreducible (A:o + Kx )-module, we have dim C(KQ + Kx ) < 1. As C(K0 + Kx)d

C(L) + K0 , this proves K0 C C(L). D

Let Kx® ■ ■ ■ ® Kt be the sum of all minimal, nonabelian ideals of L/C(L)

and K¡ the preimage of K¡ in L. Note that dim C(L) < 1. If there is some

j, say j = I, such that K^l) n C(L) / 0 then (II.2) and the minimality of all

K¡ prove that C(L) cí K¡1] for all iA¿ 1. In addition, rad(L) = C(L) C Kx in

this case. We then put Sx := Kx, S¡ := Kf]i ¿ 1. If tf;(1) n C(L) = 0 for all

z, we put Sx = rad(L) and Sj+X = K{¡l)i > 1 .

Finally define

S(L):=-£S¡.

By construction S(L) has the following properties.

Proposition 3.

(1) S¡l)=S¡, C(L)f)S¡ = 0, S i is a minimal ideal of L (i > 1).

(2) Either Sx = rad(L) or C(L) cS[l), S[l) = Sx, SX/C(L) is L-irreducible.

(3) S(L) = 0S,..

Proof. (1) For z > 1 we have KJl) n C(L) = 0. K¡/C(L) is a nonabelian and

minimal ideal of L/C(L). Therefore K¡ = K]l) ® C(L) = S¡ ® C(L). Thisproves (1).

(2) Is true by definition.

(3) Let n: L —► L/C(L) be the canonical homomorphism. If Sx ^ rad(L)

then n(S(L)) = 0zr(5;) proving that

s,nJ2Sj = (0) vi.

If 5", = rad(L) is solvable, then n(Sx n X),>i ^,) is a solvable ideal contained

in 07>1 7t(5-) = Kx ® ■ ■ ■ ® Kt and therefore has to be zero. Then

SxnJ2sjcC(L)n\J2SJ)=0.j>\ \j>i J

Since n: X];>i Sj —» £;>1 ^(5;.) is an isomorphism we are done. □

In order to derive the structure in more detail, we first settle two particular

cases.

698 HELMUT STRADE

Lemma 4. Let R' be a simple Lie algebra and R := R' ® A(r; 1). Put Qr =

¿2R'® FI\xsA (s, + ■ • • + sn > r). Then

(1) Qx is the unique maximal ideal of R.

(2) Qx is nilpotent.

(3) ßf'/O z/2r<zz(p-l).

(4) Qr = (ad Qxf(R).(5) Qr is an ideal of R.

(6) R = R' ®F®QX.

Proof. Qx is an ideal of R satisfying R = R' ® F ® Qx . Since R' s R/Qx is

simple, it is maximal. Note that

R' ®Fl[x?, R' ® fHx'>] = R'[1) ® fYIxsA+,A

This proves (4) and hence (5) and (2). Moreover, it also implies that Q^p = Q2,

which yields (3). In order to prove (1) we consider any ideal J. If J <f. Qx

then R = Qx+ J and hence R/J is nilpotent. Since R = R this is possible

only if R/J = 0. Therefore Qx is the unique maximal ideal of R . □

Lemma 5.(1) Assume that a Lie algebra R has a faithful irreducible module of

dimension < p . Suppose that I is a nilpotent ideal. Then I = 0.

(2) Let R be one of the direct summands in S(L). If R has a faithful

irreducible module of dimension p then VQ is a faithful irre-

ducible /-submodule of dimension < p . From [9, Corollary 3.3] we conclude

/<2) = 0. If dim V0 < p then, as it is a p-power, dim V0 = 1 . /(1) annihilates

V0. (1.4) then yields 7(1) = 0.

(2) Assume that R/C(L) is neither simple nor solvable. Then by definition

R/C(L) is L-irreducible. Hence [5] yields the existence of zz > 1 such that

R/C(L) = R' ® A(n ; 1) for some simple algebra R'.

According to (II.4) Qx is a nilpotent ideal and Q(2) ^ 0. Let / be the

preimage of Qx in R. Then / is nilpotent and 7( ' ^ 0, contradicting (1). D

Lemma 6. Let R be one of the direct summands of S(L) but R ^ L and R

not solvable. If U has an irreducible R-submodule U0 of dimension < p , then

R = R' ®A(l; 1)

where R' is simple and has a faithful irreducible module of dimension <p.

Proof, (a) According to (II. 1, Corollary) we may assume that U0 is not a faith-

ful module and that R{i) n C(L) = 0. Therefore R = R{1) is L-irreducible.

[5] yields the existence of zz > 1 such that R = R' <g> ,4(zz ; 1) for some simple

algebra R'.

The imbedding R' = R' ® F gives rise to a faithful representation of R' in

U0 of dimension < p .

(b) We may consider L as contained in gl(U) and put Lp the p-envelope

of L in gl(U). Lp acts on R as an algebra of derivations. Since according

to (a) (II.l) is applicable we obtain Der(R') = adR'. As Der(R') = R' (II. 1 )

and Der(R) = Der(R') ® A(n ; 1) + F id ® W(n ; 1) [5], there exists a mapping

f:Ln-*R such thatj p

ad(b-f(b))\ReFid® W(n; I) VbeLp.

Using (11.4(5)) this yields inductively

(*) (adbx)o---o(adb,)(R'®xp~l ■■■rf'1) c^R'®Rx[< ■■■x'j

where r, + • •■ + rn > n(p - 1) - í, Vb¡ E Lp .

(c) Next we apply (1.2) with G = G(R, Uf ,

U=U(Lp,S)®U{GSlG)UjR).,

Since U0 is assumed not to be faithful, we obtain dim UfR) < p and dim L IG= 1.

Note that in accordance with (II.l) and Ä(1) = R, the solvable ideal Qx

annihilates U0 and therefore U0(R). Since Qx is a maximal ideal of R it is

the exact annihilator. U0(R) is a (/-module. We obtain [G, Qx] c Qx and

therefore [G, Qr] c Qr Vr. Moreover, /?' ® Fxp~l ■ ■ xp~l = (adQx)"{p~l](R)

is G-invariant.

We now combine this result with (*). Take any y e L \G. Then (as L

and G are restricted),

J:=^(ady)fR'®Fxp-l---xPn-l) = J2(ady)J(R'®FxPx-l-.-xPn-1).

j>0 >=0

From this one proves by (*) that J c Qn,p_X)_,p_x) ■ J is invariant under y

by construction and by induction it is seen that J is G-invariant. This implies

(zz - l)(p - 1) < 0 and therefore zz = 1. D

Theorem 7. Let L be an indecomposable Lie algebra having a faithful irreducible

representation p: L —> gl(U) of dimension < p . Then S(L) is one of the

following :

(1) S(L) = C(L) ®S2,S2 simple.

(2) S(L) = Sx, C(L) cS[l),S[l) =SX, SX/C(L) simple.

(3) S(L) = Sx ® (S1 ® A(l ; 1)), Sx abelian, S' is semisimple and has a

faithful irreducible representation of dimension < p.

(4) S(L) = rad(L).

Proof. Put S(L) = Sx ® ■ ■■ ®Sr. Let (70 be an irreducible S(L)-submodule of

U.

700 HELMUT STRADE

(a) Consider first the case that dim U0(S(L)) > p . (1.3) proves that U0 is a

faithful 5'(L)-module. (1.1) shows that we have for any irreducible S -module

V¡ of U0

U0 = (g}V¡.

Since U0 is faithful, all V¡ are faithful S-modules. If r = 1 we are done by

(II.5). So assume r > 1. If, for some j > 1, dim V p Vy > 2 then r = 2 and dim Vx p. According to (1.4)

this implies Sx = C(L). (II.5) shows that S2 is simple.

(b) Consider the case that dimUfS(L)) <p. Let Vj be irreducible S¡-

submodules of U0(S(L)), dim V¡ < p. The corollary of (II.l) yields Sf1 n

C(L) = 0, proving (by definition of S(L)) that Sx = rad(L). Then Sx is

solvable, and hence dim Vx is a p-power. We conclude dim Vx = 1 and Sx

annihilates V{ . (1.4) yields S\l) = 0.

If, for some j > 1, S- = L then S is simple. This is case (1). We may

therefore assume that S¡ ^ L for all j > 1. According to Lemma 5 there are

simple algebras S'¡ such that S¡ = Si® A(l;l). K isa faithful irreducible

^-module. Put S' = J]Sj. a

While cases (l)-(3) of Theorem 7 give considerable information about S(L)

and by this describes L in terms of smaller pieces, case (4) allows further

reduction.


representation p: L —► gl(U) of dimension < p . Assume that S(L) = rad(L).

Then either dimL = 1 or L has an abelian ideal I ct C(L) or L is the

six-dimensional split extension of 5/(2) by the three-dimensional Heisenberg

algebra.

Proof, (a) Assume that C(L) is the only abelian ideal of L. If S(L) = C(L)

then by definition L/C(L) is a semisimple algebra having no minimal non-

abelian ideal, i.e. L/C(L) = (0). We may therefore assume that rad(L) ^

C(L). Since C(L) is assumed to be the only abelian ideal we have C(L) ^ 0.

In this situation we can apply the methods developed in [13] as follows.

(b) Let / be the ideal of L maximal with the property /(1) C C(L) and

take U0 to be an irreducible /-submodule. Since / is nonabelian, we have

dimt/0 > p and hence UQ is faithful (1.3). This implies dimC(7) = 1 and

hence C(I) = C(L). Consider the bilinear mapping

a:lxl -^C(L) = F

given by a(x, y) = [x, y]. Put J := rad(a) = {x e I\a([x, I]) = 0} . Since by

definition of a J c C(I) = C(L), a is nondegenerate on I/C(L) x I/C(L).

Note that [x, y] acts nilpotently on U0 if a([x, y]) = 0 and invertibly other-

wise. General representation theory of nilpotent algebras now proves

dimc/0>p(1/2)d,m//C(L),

hence dim I/C(L) < 2. As / is not abelian, we obtain dim I = 3 and 7 is a

Heisenberg algebra.

(c) Put K := {x e L\[x, I] = 0}. K is an ideal of L. Let F be an

irreducible (7C+ /)-submodule of U. As any 7-submodule has dimension >p,

F is a faithful irreducible (K + 7)-module (1.3). According to (1.1) K has a

faithful irreducible module of dimension < p. Then (II.l, Corollary) implies

dimK = 1 . Therefore (I + K){[) c I{1) c C(L). The maximality of / yields

K c I and hence K c C(I) = C(L).

(d) Let M be any ideal of L such that [M, I] c C(L). a can be extended

t0 a: Mxl ^C(L) = F, a'(x, y) = [x, y].

Since a is nondegenerate on I/C(L) x I/C(L), M decomposes

M = K + I, [K, I] = 0.

According to (c) K = C(L), and hence M = I.

(e) Let R be an ideal of L such that R{ el. Consider the representation

p:L^gl(I/C(L)).

Since I/C(L) is two-dimensional and L has no abelian ideals except C(L),

I/C(L) is an irreducible L-module. The property Ril) c ker(p) and (1.4)

ensure the existence of E e R* such that

p(x) - E(x) id is nilpotent for all x e R.

Then (adx)"^) - E(x)"y e C(L) VxeR,yeI and

0 = (adx)"([yx , y2]) = 2E(xf[yx , y2] Vx e R, yx , y2e I.

Since 7(1) ̂ 0, this yields E(x) = 0Vx6Ä. R acts nilpotently on I/C(L),

proving R c ker(p). Then [R, I] c C(L). From (d) we obtain R c I. As a

result, L/I acts faithfully on the two-dimensional irreducible module I/C(L)

and is semisimple. This is possible only if L/I = 5/(2). Considering eigen-

values of some semisimple element in 5/(2), we conclude that the extensionsplits. G

The representation theory of this six-dimensional split extension algebra is

completely developed in [13].


representation p: L —> gl(U) of dimension < p2. Put

K:={xeL\[x,S(L)] = 0}.

Then K = C(L) or K = rad(L) is abelian.

702 HELMUT STRADE

Proof. Assume that C(L) § K. Put as above S(L) = Sx® ■■■ ® Sr. Since

S¡ is (for i > 2) a minimal nonabelian ideal of L and K centralizes S , we

have Kí)S¡ = (0) (i = 2, ... , r). By construction of the socle then K/C(L)

intersects SX/C(L) nontrivially. Hence I := KnSx is a noncentral abelian ideal

of L. Let U0 be an irreducible 7-submodule. As I çt C(L), (1.4(4)) proves

that dim Úfl) < p. LJfl) is a 7C-module because [I, K] c [Sx, K] = 0.

K ] nSx is contained in C(K) and therefore acts nilpotently on any irreducible

/C-submodule of U0(I). Then K{1)nSx vanishes according to (1.4). As a result,

K n S(L) = 0 and therefore K is abelian. This in turn implies by definition

that Sx = rad(L) and I = K. Then Ufl) is a rad(L)-module of dimension

< p , proving that rad(L) is abelian and K = rad(L). D

This theorem shows that K = C(S(L)) is contained in S(L) and L/K may

be considered a subalgebra of Der(S(L)). Therefore L is described in terms

of its socle only.

Combining the aforementioned results we obtain some structural insight for

the general case.

Theorem 10. Let L be a Lie algebra having a faithful irreducible representation

of dimension I, L2, ... , Ls_x are

simple and Ls is simple or Ls = C(L). They all have a faithful irreducible

representation of dimension < p . L, is indecomposable having a faithful irre-

ducible representation of dimension < p .

Any CSA H of L decomposes 77 = 0(7/ n LA), and 77 n L¡ is a CSA of L¡.Any ideal J of L decomposes J = 0(7 n LA).

Proof. Decompose L = 0¿=1 L; into indecomposable ideals. According to

(1.1) all summands, except possibly one, have faithful irreducible representa-

tions of dimension < p . In particular, all derivations of these are inner. This

observation immediately yields that H and / are decomposable as wanted and

that 77 n L¡ is a CSA of L(. D

III. Applications to classification theory

In [6] of Block-Wilson, in which they classified the restricted simple Lie alge-

bras, the problem which we have discussed in this note occurs in the following

manner. Starting with a restricted simple Lie algebra L, they consider two-

sections A = J2^i„+j8 where a, ß are G F (p)-independent roots with respect

to some torus. These algebras are restricted, every torus has rank at most two

and every CSA acts triangulably on A . They choose some particular torus ("op-

timal") T of A and some maximal subalgebra A0 ("distinguished") which

contains the CSA CA(T). By means of an irreducible /l0-submodule of A/AQ

one defines a standard filtration and considers the associated graded algebra G.

G inherits the following properties from A : There is some restricted subal-

gebra G0 which contains a CSA CG(T) of G. T then is a maximal torus

of G0 of dimension 2 and CG(T) acts triangulably on G. In addition, G0

acts faithfully and irreducibly on G_, . In Chapter 7 of [6] Block-Wilson deal

with the case that G0 has toral rank 1 with respect to T (or equivalently, with

respect to the CSA CG (T)) and dim G_x<lp < p - 3p .

Similarly, the above-mentioned problem occurs in some forthcoming paper

of Benkart-Osborn [2]. They start with a CSA 77 of a simple (nonrestricted)

Lie algebra L and consider a two-section K = £) L. . „ where a, ß are roots

with respect to 77 which are GF(p )-independent but /"-dependent. They, too,

consider a graded algebra G, obtained in the same way as Block-Wilson, and

obtain in [2] Lemma 3.3 (iv(b)) as a major case

G0 acts faithfully on G_x; dimG_x = p - 1,

G0 is semisimple.

Our present results yield with only little extra work that G0 is indecomposable

and S(G0) is simple, thereby reducing this case to a very specific situation:

Remark. Assume that L is semisimple and suppose that p: L -> gl(U) is a

faithful irreducible representation of dimension p2 - 1. Let 77 be a CSA of

L such that L = J2¡€f ^¡„{11) is ine root space decomposition with respect to

77. Assume that p(77 ) consists of nilpotent transformations. Then

( 1 ) L is indecomposable.

(2) S(L) is simple.

(3) L c Der(5(L)).

Proof, (a) Since p is faithful and 77(1) acts by nilpotent transformations on

U, 77(1) acts nilpotently on L. Then a(h) = 0 V/z e 77(1). Therefore a is

linear. Put 770 = ker(a). As 77(1) c 770 , 770 is an ideal of 77 of codimension

1.Let I, J be ideals of L and suppose that 7 n 77 ç? 770 . Then [Lja , I n 77] =

Lia n 7 Vz t¿ 0, proving

J cH + inJ.

(b) Suppose L = 0'=1 L;. According to (11.10) 77 n L¡ is a CSA of L¡. If

L( n 77 c 770 , then L( n 77 acts nilpotently on L(. L; is therefore nilpotent,

contradicting rad(L) = 0. We conclude L, n 77 çt 770 . According to the above

remark L( c 77 Vz > 1, and L¡ c rad(L) = 0. Therefore L is indecomposable.

(c) Let {X. be an irreducible SYLVmodule. Since dim U is not divisibleu- „-•

by p, U is not induced by U0(S(L)). Therefore dim U0(S(L)) > p. In

combination with rad(L) = 0 the proof of (II.7) shows that S(L) = S2 is

simple. Theorem 9 implies (3). D

To deal with the former application we will consider the following situation.

Let 77 be a CSA of L and L = ffLa(H) be the root space decomposition

of L with respect to 77. L is said to have toral rank one with respect to 77

704 HELMUT STRADE

if the roots span a one-dimensional G77(p)-vector space. Note that in this case

all roots are contained in GF(p)a for any nonzero root a. This in turn means

that an element h e 77 acts nilpotently on L if a(h) = 0 for some nonzero

root a .

Assumption B. Suppose that 77 is a CSA of L such that L has toral rank one

with respect to 77. Assume that L has a faithful irreducible representation

p: L —► gl(U) of dimension <p . In addition, p(77( ') is supposed to consist

of nilpotent transformations.

Let 77 be a CSA of L such that L has toral rank 1 with respect to 77.

Then Theorems (II.7) and (11.10) reduce to a much simpler structure. We refer

to [14] for the following.

Theorem 1. Let L be a Lie algebra of toral rank 1 with respect to some CSA

77. Assume that 77 ' acts nilpotently on L.

Let J be a nonsolvable ideal of L. Then(1) L = J + 77. J n 77 is a CSA of J . J has toral rank 1 with respect to

JC\H.

(2) Any ideal I of L can be written as I = IC\J +1 C\H.

Corollary. Let L be a Lie algebra satisfying (B). Then L = Lx® L2 where L,

Z5 indecomposable and L2 c C(L).

One of the following is true:

(1) L/C(L) is simple or dimL = 1.

(2) L is the six-dimensional split extension of 5/(2) by a Heisenberg algebra.

(3) L has an abelian ideal I çf C(L).

Proof. According to (11.10) L = 0¿=1 L¡ decomposes, and if s > 1, Ls is not

solvable. Put in Theorem (III. 1) J := Ls and 7 := 0*1,%. Then 7 c 77

(as 7 n J = 0). Therefore 7 is an ideal acting nilpotently on U . Hence

7(1) = 0, whence [7, L] c 7(1) + [7, J] = 0 and 7 c C(L). This proves the

first assertion.

If rad(L) = C(L), then L/C(L) is simple (III.l). Therefore we may assume

that rad(L) ^ C(L) and C(L) is the only abelian ideal of L. Note that in

this case C(L) c rad(L)(1) c L, and therefore L is indecomposable. Since

rad(L) is a direct summand in S(L) and is not abelian, only case (4) in (II.7)

can occur. Then (II.8) gives the result. D

In order to determine the full structure of L we first deal with the case that

L has a noncentral abelian ideal. Let Lp denote the p-envelope of p(L) in

gliU).

Proposition 2. Assume that L satisfies (B) and that L has an abelian ideal

I çf C(L). Then there is a restricted homomorphism cp: Lp —► W(l ; 1), such

that <p(L ) is an essential subalgebra of W(l ; 1) and J := ker(tp) is abelian.

Put G := tp~\w(\ ; 1)(0)). Then [G, /] acts nilpotently on U.

Proof. We use the notation of (1.2) and (1.4). Since 7 t¿ C(L) we have Lp ^

G(I, Uf=:G and hence

U = U(Lp,S)®mGS¡G)Ujl).

As a result, dim L/G = 1 . Let J denote the maximal ideal of L in G. (1.6)

yields the existence of a restricted homomorphism

tp:Lp^W(l;l)

such that cp~\\V(l; 1)(0)) = G(7, Uf. By definition [G, 7] acts nilpotently

on U. We next want to prove that J is abelian.

If J is not solvable, then neither is 7(1) c L. According to (III. 1 ) this

implies L = 77 + J{ '. Consider the root space decomposition with respect

to 77. Since L and J have toral rank 1 with respect to 77 and 77 n J ,

respectively, there is an h e HC\JW and a root a , such that a(h) / 0. Hence

Y^Iiac[I,J]c[I,G]ckcT(E).<¥0

Since 77 J acts nilpotently on LA this yields

[7, L] c [7, 77] + [7, 7(1)] c XX + 77(1) n 7 + [7, JW] C ker(E),

hence L c G and L c G, a contradiction. Therefore J is solvable. U0(I) is

a /-module of dimension < p . As every irreducible /-module has dimension

a p-power, there is a one-dimensional /-submodule. This is annihilated by

7(l). (1.4) yields /(1) = 0. Choose some one-dimension /-submodule VQ of

U . As above G(J, Vf) / L and dimL IG(J, Vf) = 1. On the other hand,we have 7 c J and by definition G(7, Uf) D G(J, Vf). This proves equality.

By definition, [G(J, Vf), J] acts nilpotently on U . D

We apply this result and use in the next theorem the same notation.

Theorem 3. Assume that L satisfies (B) and that L has an abelian ideal I <£

C(L). Then L is one of the following:

(1) L = 7r/z©0;GScG/r(p) Fy¡, [h, y¡] = iy¡, [y¡, yf = 0, p(y.) is invertible

for all i, S\{0} f 0.

(2) L = Fx® 0toí>,> \x,y¡] = y,_x, \y,, y.] = 0, 1 < k < p - 1,Fy0 = C(L) ¿ (0).

(3) L = Fh®Fx® ©*=0 Fy¡, [h , y¡] = iy¡, [h,x] = -x, [x, y¡] = y,_, ,

Lv,■ j y¡] = 0, 1 < k < p — 1. p(y¡) is nilpotent for I < i < k .

(4) L is the split extension of W(l; 1) by the restricted W(l ; \)-module

A(l;\).

(5) L is the split extension of s I (2) (considered as an essential subalgebra of

W(l;\)) by A(l;\).

Proof. We prove first that (l)-(5) of the theorem are true for L instead of L

and finally will show that the same results are true for L.

706 HELMUT STRADE

( 1 ) Assume first that L /J is one-dimensional and not [p]-nilpotent. Then

there exists a toral element h e L \J (which then acts semisimple on J).

Decompose J = ^ffJ¡ into root spaces with respect to adh\} . Since this is a

semisimple endomorphism, J¡ is in fact the eigenspace. Since / is abelian, /

admits an eigenvalue function E e J* and J¡ n ker(L') is an ideal of Lp . This

acts nilpotently on U, and therefore is zero. As a result, dim J¡ < 1. This is

case (1).

(2) If the essential subalgebra cp(L ) is not of the above type we have

cp(Lp) = Fx®tp(G), xbl = 0.

If moreover tp(G) -f 0, we even may choose elements h e G and x e LAG

such that h = h, x = 0 and [h, x] = -x (observe that ad/z is a semisim-

ple endomorphism).

Consider / := {y e J\(adx)j(y) e ker(E) Vf > 0}. Since [G, J] C ker(£)

and Lp = Fx + G, f is an ideal of Lp satisfying [Lp, J1] c ker(Ti). As a

resuit, f = C(Lp).

(3) Consider the nilpotent subalgebra K := J + Fx of L . Let V be an

irreducible ÁT-submodule of U and 7? the kernel of this representation. Note

that RnJcJ1. According to (2) we obtain

7<n/cC(Lp)nker(£) = 0.

The assumption R ^ 0 would therefore imply K = R + J and hence K{ ' c

Ril) + RnJ + J(l) c R. But then [x, /] c ker(£) which leads to the contra-

diction x e G. The 7C-module V is therefore faithful.

Decompose J = 0 /; into indecomposable T^x-modules. As / is abelian,

every J. is an ideal of K and as such it intersects C(K) nontrivially. Since

V is faithful irreducible, we have dim C(K) < 1 . Then / itself is indecom-

posable. The Jordan normal form of adx|7 provides a basis (y0,... ,yk) for

/ with [x, y¡] = y¡_x . Note that according to (2) Fy0 = C(L ) ^ 0. Since

x^ = 0 we also have (adx)p|/ = 0, proving k 1 .

(4) If cp(L ) = Fx the above shows that we are in case (2). If tp(L ) = Fx®

F h we have in addition (see (2)) [h , x] = -x . Then any y¡ is an eigenvector

with respect to ad/z. As [h , yQ] e [h , C(L)] = 0 this yields [h, y¡] = iy¡. We

also see that for i ^0 y¡ e[G, J]c ker(£). This settles the case dimLp// <

2.

(5) We endup with: tp(L) is isomorphic to 5/(2) or W(l; 1). Applying the

results of (3) we obtain that C(Lp) f 0. J/C(L ) is a restricted cp(Lp)-module

of dimension k < p . Computing traces one gets

0 = trace(ad/z| J) = ¿ i = M^-til ;

i.e. k = p - 1 . A direct computation shows that there is a realization J =

A(l;l) and Lp/J = W(l;\) or Lp/J = Fd ® Fxd ® Fx2d .

(6) It remains to prove that the extension

0 - J -» Lp -> Lp/J - 0

splits.

Consider first the case that L JJ = 5/(2). Since ad/z is semisimple, there is

an eigenvector y e L \J such that u = [x, y] - h e J, [h, y] = y . Therefore

[h , u] = 0 proving that u e C(L ). Put h' := h + \u . Fx + Fh' + Fy is

isomorphic to 5/(2). The extension splits.

Next consider L JJ = W(l ; 1). Using the former result we put e_x := x ,

e0 := h , ex := y . There are elements c2, ... , e 2 e L \J with the properties

[e_x, e¡] = (i+ l)e¡_x, [e0, e¡] = ie¡. One proves inductively [e¡, ej\ = (j -

i)ei+i (f°r l<i,j,i + j<p-2) and =0 if i + j > p -2 . The extension

splits.

(7) We now derive the structure of L. Note that Lpl) c L. In cases (4) and

(5) this shows L = L . Assume in the other cases L ^ L . Case 1. Since

7 çf C(L), L is not abelian. Then there exist a, ß e F, a ^ 0, such that

h' := ah + ßy0 e L . Then L = Fh' + ^f Fy¡ having the same structure. C«35C 2.

There exist a, ß e F such that x := ax + ßyk e L. Then L = Fx ®@j=0Fy¡

having the same structure. Case 3. L = L = Fx ® 0/=o Fy¡. Then L acts

nilpotently on x , but L does not, a contradiction. D

For all algebras occurring in Theorem 3 A(l ; 1) is a faithful irreducible

module, as follows:

Case 1. p(h) = (1 + x)d , p(y¡)(xj) = (1 +x)V.

Case 2-5. L may be considered as a subalgebra of the semidirect sum W( 1 ; 1)

+A(l ; 1), and then containing d e W( 1 ; 1) nL. ^4( 1 ; 1) is an irreducible L-

module by restricting the representation

p: W(l;\) + A(l;\)^gl(A(l;\)),

p(x'd+xJ)(xk) := x'd(xk) + xJ+k

to L.

Theorem 4. Assume that L satisfies (B) and that S(L)/C(L) is simple. Then

L/C(L) can be embedded into Der(S(L)/C(L)). The following cases can occur:

(1) L = sl(2)®C(L),

(2) L= W(l;\)®C(L),

(3) W(l;2)® C(L) cLc Der(W(l ; 2) ® C(L)),

(4) S(L)/C(L) e {77(2 ; 1 ; <t>)(2), 77(2 ; (2, 1) ; <p)(2)} .

Proof. Put 7 := {x e L|[x,5(L)] c C(L)}. Since S(L) is not solvable,

(III. 1 ) yields that 7 = 7 n S(L) + I n 77. The simplicity of S(L)/C(L) shows7 c rad(L). By definition of S(L) we have rad(L) = C(L). Then the ad-

representation defines an embedding L/C(L) —> Der(5(L)/C(L)).

70X HELMUT STRADE

From [16] we conclude that S(L)/C(L) is one of the following simple alge-

bras:

5/(2) ,W(l;l),W(l;n)(n>2),H(2;n; <p){2).

(1) S(L)/C(L)^ 5/(2): Then L/C(L)=sl(2) and L = sl(2)®C(L).(2) S(L)/C(L) = W(l;\): Then L/C(L) = Der(W(l ; 1)) s W(l ; 1).It is proved in [ 13] that the faithful irreducible modules of the nonsplit central

extension of W(l ; 1) have dimension > pp~[/ which is at least p2 (as p > 3).

Therefore the extension splits.

(3) S(L)/C(L) = W(l;n) (n > 2). [8] shows that zz = 2 and S(L) =

Sx ®C(L) splits. Consider S(L) c L c gl(U) and let S(L)p,Lp denote

the p-envelopes in gl(U). Note that since Sx is not restrictable S(L)p =

(Sx)p ® C(L) = Der(W(l; 2)) © C(L). Then Lp = Der^l ; 2)) ® I where

I := {x e Lp|[x, Sx] = 0}. 7(1) c L centralizes Sx and as above we conclude

that 7 c C(L) and hence 7 is nilpotent. Application of (11.10) to L yields

l = CiLp).

(4) S(L)/C(L) = 77(2; n; 4>){ ' : A simple dimension argument yields

dimS(L) <dimgl(U) < (p2 - I)2 <p4 -p.

Therefore n = (1, 1) or n = (2, 1). D

It seems reasonable that also in (4) the central extension splits and L —►

Der(77(2 ; 1 ; cpy ) ® C(L) is true. A proof of that, however, seems to involve

an awful mass of computations and yields only comparatively little additional

information.

In cases (l)-(3) there exist faithful irreducible modules M of dimension 2,

M = W(l ; 1) of dimension p, M = A(l ; 2)/F of dimension p - 1 , respec-

tively. Here we consider C(L) acting as 7-"id on these modules if C(L) f^ 0.

References

1. G. Benkart and J. M. Osborn, Toral rank one Lie algebras, J. Algebra 115(1988), 238-250.

2. _, Simple Lie algebras of characteristic p with dependent roots, preprint.

3. R. E. Block, Trace forms on Lie algebras. Cañad. J. Math. 14 (1962), 553-564.

4. R. E. Block and H. Zassenhaus, The Lie algebras with a non-degenerate trace form, Illinois

J. Math. 8(1964), 543-549.

5. R. E. Block, Differentiably simple algebras. Bull. Amer. Math. Soc. 74 (1968), 1086-1090.

6. R. E. Block and R. L. Wilson, Classification of the restricted simple Lie algebras, J. Algebra

114 (1988), 115-259.

7. J. Dixmier, Algèbre enveloppantes, Gauthier-Villars, 1974.

8. A. S. Dzhumadil'daev, Central extensions of the Zassenhaus algebra and their irreducible

representations, Math. USSR Sb. 54 (1986), 457-474.

9. R. Farnsteiner and H. Strade, On the derived length of solvable Lie algebras. Math. Ann.

282 (1988), 503-511.

10. J. B. Jacobs, On classifying simple Lie algebras of prime characteristic by nilpotent elements,

J. Algebra 19 (1971), 31-50.


11. A. I. Kostrikin, Theorem on semisimple Lie- p-algebras, Mat. Zametki 2 (1967), 465-474.

12. H. Strade, Lie algebra representations of dimension p - 1 , Proc. Amer. Math. Soc. 41

(1973), 419-424.

13. _, Zur Darstellungstheorie von Lie-Algebren, Abh. Math. Sem. Univ. Hamburg 52 ( 1982),66-82.

14. _, Cyclically graded Lie algebras (in preparation).

15. H. Strade and R. Farnsteiner, Modular Lie algebras and their representations, Textbooks

and Monographs, vol. 116, Dekker, 1988.

16. R. L. Wilson, Simple Lie algebras of toral rank one, Trans. Amer. Math. Soc. 236 (1978),287-295.

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53705

Department of Mathematics, University of Hamburg, Bundesstrasse 55, 2 Hamburg

13, Federal Republic of Germany (Current address)


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