Proceedings of the International Congress of Mathematicians Berkeley, California, USA, 1986

Simple Lie Algebras over Fields of Prime Characteristic ROBERT LEE WILSON Dedicated

to the memory

of Boris

Weisfeiler

The classification problem for finite-dimensional simple Lie algebras over a field F of prime characteristic p has attracted attention for about fifty years. Although it is still open, much progress has been made towards its solution. In particular, the known simple Lie algebras have a natural description (as discussed in Kostrikin's talk at the 1970 International Congress of Mathematicians [25]), the classification has been completed recently in several important special cases, and there are good techniques available for attacking the general problem. We will assume throughout this article that F is algebraically closed of characteristic p > 7. For p = 2,3,5 counterexamples exist to some of the results stated here, and for p = 7 some of the proofs fail. For an important class of Lie algebras over F, the restricted Lie algebras, the classification is now complete (see Theorem 3.3). We will (in §1) define these algebras and discuss some of the technical properties which make their study easier than that of general algebras. We will then describe the known finitedimensional simple Lie algebras over F and discuss some of their structural properties (§2) and state three recent classification results (§3). We conclude (§4) by describing the main techniques used to establish these results: filtration theory and local analysis. 1. Restricted algebras. We will devote particular attention to restricted Lie algebras (= Lie p-algebras) over F (defined by Jacobson [14]). A Lie algebra L over F is restricted if it has a mapping x \-> x^ satisfying (1.1) (adz)? = ad(aM) for all x G L, (1.2) (arc) M = apxW for all a G F, xG L, and (1.3) (x + 2/)'pl = x'p] + 2/'p] + X^=i si{x)V) where isi(x,y) is the coefficient o f f " 1 in (a>d(tx + y)Y~1x. If A is an associative algebra over F, the ordinary pth power map gives the Lie algebra A~ (with product [x, y] = xy — yx) the structure of a restricted Lie Supported, in part, by NSF Grant #DMS-8603151. © 1987 International Congress of Mathematicians 1986

407

408

R. L. WILSON

algebra. Then any Lie subalgebra of A~ closed under taking pth powers is again a restricted Lie algebra. Applying this remark to Deri? C(End B)~ (where B is any algebra over F) shows that any derivation algebra is restricted. Restricted Lie algebras are more tractable than arbitrary Lie algebras over F for a number of technical reasons. We note some of these. If L is a restricted Lie algebra, an element x G L is said to be semisimple if x G span{a;tpl,a;'p ' , . . . } and is said to be nilpotent if x' p n ' = 0 for some n. A torus T is a subalgebra (necessarily abelian) such that every x G T is semisimple. A subalgebra I is said to be nil if every a; G J is nilpotent. The following theorems are due to Seligman [32]. THEOREM 1.1 (Jordan-Chevalley-Seligman Decomposition). Let L be a restricted Lie algebra over F and x G L. Then there exist unique elements x3,xn G L satisfying: ^aj x

==

Xg -j- Xfi,

(b) x3 is semisimple and xn is nilpotent, (C) [Xg,Xn] = 0. THEOREM 1.2. Let L be a restricted Lie algebra over F. Then H is a Cartan subalgebra of L if and only if H = CL(T) (= {x G L\[x,T] = (0)}) where T is a maximal torus in L. The following lemma (proved by applying the Engel-Jacobson theorem [16, Theorem 2.1] to \JlET* J 7 and applying Theorem 1.1) will be useful. LEMMA 1.3. Let (0) ^ I be a restricted ideal in a restricted semisimple Lie algebra L. Let T be a maximal torus in L. Then InT ^ (0). 2. Description of the known simple Lie algebras over F. The known finite-dimensional simple Lie algebras over F are of two types: classical type (analogues over F of finite-dimensional simple Lie algebras over C) and Cartan type (finite-dimensional analogues over F of the infinite Lie algebras of Cartan [9, 24, 34] over C). The classical algebras may be constructed as follows: Let L be a finitedimensional simple Lie algebra over C and let Lz be a Chevalley lattice in L. Let LF = Lz ®z F. Then Lp /center Lp is simple ( and dim center Lp < 1). Such algebras over F are called classical. Note that the algebras of types E-G are considered to be classical according to this definition. If L is a classical simple Lie algebra over F then L is restricted and (2.1) L contains a Cartan subalgebra H which is a torus. Letting T denote the set of roots of L with respect to H (so L = H + S 7 e r ^"P L 7 = {x G L\[h, x] = i(h)x for all h G H}) we have: (2.2) dim[L 7 ,L_ 7 ] < 1 for all 7 G T; (2.3) if a, ß G Y then there is some k G Z such that ß + ha i|0 < i < p — 1} and that e^ H-> (x\ + iyt1Di gives an isomorphism of W(l : 1) onto Der S i . This construction of W(l : 1) makes the following points apparent. (2.6) s p a n a i . D i | i > 1} (= W(l : l)o) is a subalgebra of codimension one in W(l : 1). This subalgebra may be characterized as the unique compositionally classical subalgebra of maximal dimension in W(l : 1). It is therefore invariant under A u W ( l : 1). (2.7) x\D\ and (x\ + 1)D\ span maximal tori (which are also Cartan subalgebras) in W(l : 1). Since ziZ>i G W(\ : 1) 0 , (zi + l)£>i £ W(l : 1) 0 we see that these Cartan subalgebras are not conjugate under Aut W(l : 1). Thus (2.4) fails for W(l : 1). (2.8) If Ex = £?=o( a d f l 0 , 'AI then E^faDt) = (a;i + l)Z?i. This, of course, implies that EDl £ Aut W(l : 1). Call Ex a Winter exponential [42]. Jacobson [15] studied the algebras W(m : 1) = Der Bm,

Bm = F[xu • • •, xm]/(xp1,...,

xpm).

W(m : 1) may be graded and filtered by setting deg a;« = — deg Di = 1 for 1 < i < m. Then W(m : 1) = Y, W(m : % '

W m

^

:

^Ml

=

W A , . • • ,Dm},

i>-l

W(m : l)[o] = span{z;Dj|l < i,j 1. Simple nonrestricted subalgebras of W(m : 1) were identified by Albert and Frank [1], Ree [30], Jennings and Ree [17], Strade [35]. All these families of nonclassical algebras are now known to be contained in the family of algebras of Cartan type. In 1966 Kostrikin and âafarevté [26] constructed restricted simple Lie algebras corresponding to the infinite Lie algebras of Cartan over C and conjectured that the algebras so constructed were precisely the known restricted simple Lie algebras over F. (Only the isomorphism of the algebras K(m : 1) of Cartan type with one of the families discovered by Frank [13] was in question. This was proved in [10].) They also conjectured that all restricted simple Lie algebras over F were of classical or Cartan type. The original Kostrikin-Safareviö construction has been generalized to give nonrestricted algebras by Kostrikin-Safareviö [27], Kac [19, 20, 21], Shen [33], Wilson [39, 40]. We describe the most general version [21, 40] here. Give the polynomial algebra F[X\,..., Xm] its usual coalgebra structure with each Xi primitive. Then the dual space a(m) = F[Xi,... ,Xm]* is an infinitedimensional commutative associative algebra consisting of all formal sums £ xa, where a ranges over all m-tuples of nonnegative integers, with multiplication determined by xax^ = ("+" )**+" where (I ) = ( $ ] ) • • • ( j g j )• This algebra is called the completed free divided power algebra. For n = (n±,..., nm), an ra-tuple of positive integers, we let a(ra : n) denote the span of the xa with a(i) < pn% for all i. Then a(m : n) is a subalgebra of a(m). Write 1 for the mtuple ( 1 , . . . , 1). Note that a(m : 1) = Bm. For each i we get a derivation Di of a(m) with Di(xa) = xa~£i, where £i(j) = 6ij (and x& = 0 if some ß(i) < 0). We will usually write Xi for x£i. The set {uiD\-\ \-UmDm\ui G a(m) (respectively, U{ G a(m : n)}) (where (uD)v = u(Dv)) is a subalgebra of Dera(ra), which is denoted W(m) (respectively W(m : n)). The algebra W(m : n) is simple, of dimension mpn, where n = n\ -\ h nm. It is restricted if and only if n = 1. Define differential forms OJS^H^K by LJS

=

d x \

A

• • • A

d x

m

,

r

UH = ^2 dxi A dxi+r

(m = 2r),

2= 1

r

uK = dz 2 r+i + J ^ Xi+rdxi - Xidxi+r z=l

(m = 2r + 1).

SIMPLE LIE ALGEBRAS

411

Elements of W(m) act on differential forms according to the rules D(df) = d(Df),

D(a Aß) = (Da) Aß + aA

(Dß),

m

D(fa)

= (Df)a + f(Da),

df = £ ( A / ) dx{ 2=1

for all / G a(m) and all differential forms a and ß. Define subalgebras H(m),K(m)CW(m) by S(rn) = {DG W(m)

| D(UJS)

= 0}

(m > 3),

= 0}

(m = 2r>

H(m)

= {DG W(m)

\ D(LJH)

K(m)

= {DG

\ D(UJK) G a(m)u)K}

W(m)

S(m),

2),

(m = 2r + l>

3).

For any automorphism $ of W(m) and for X = S, H or K define X(m : n : $) = $X(m) H W(m : n). Then the algebras W(m : n) and, for appropriate m, n, and $ depending on X (see [21]), the algebras X(m : n : $)( 2 ) are simple. These are the algebras of Cartan type. Kac has shown [21] that the only restricted simple Lie algebras of Cartan type are those originally defined by Kostrikin and Safareviö: X(m:l)W, X = W,S,H,K. 3. Classification results. We now state three recent classification results. THEOREM 3.1 ( WEISFEILER [38] ) . Let L be finite-dimensional and simple over F and let LQ be a maximal subalgebra of L which is solvable. Then L = sl(2) or W(l : n) for some n. This generalizes earlier results of Schue [31] (where it is assumed that all proper subalgebras are solvable) and Kuznecov [28] (where it is assumed that LQ acts irreducibly on L/LQ). THEOREM 3.2 ( B E N K A R T - O S B O R N [2, 3]). Let L be finite-dimensional and simple over F and let L contain a one-dimensional Cartan subalgebra. Then L = sl(2) or L is an Albert-Zassenhaus algebra (i.e., some W(l : n) or an appropriate H(2 : n : $)W). This generalizes earlier work of Kaplansky [23] (where the theorem is proved for restricted L and structural results on nonrestricted algebras are obtained) and Block [5] (where the theorem is proved under the additional hypotheses that dim.L a = 1 and [La,L-a] ^ (0) for all roots a). THEOREM 3.3 (BLOCK-WILSON [7, 8]). Let L be a finite-dimensional restricted simple Lie algebra over F. Then L is of classical or Cartan type. This verifies the 1966 conjecture of Kostrikin and âafareviC. 4. Techniques of proof. We now discuss briefly two techniques which are used in the proofs of the theorems cited above: filtration theory and local analysis.

412

R. L. WILSON

Notice that the algebras of Cartan type all have natural filtrations by degree (in type K we take degZ2r+i = 2, in all other cases we take dega;^ = 1). This, together with the heavy use of filtration theory in Cartan's work, suggests that we should try to equip an arbitrary simple algebra L with a filtration and use this filtration in our analysis. Let L be simple and LQ be a maximal subalgebra. Define a filtration (Cartan [9], Weisfeiler [36]) as follows: Lo acts on L/L0. Let L _ i / L 0 be an irreducible Lo-submodule. Set L^+i = {x G Li \ [x,L-i] Ç Li} for i > 0 and Li-i = [Li, L-i] + Li for i < 0. Let G = YQ be the associated graded algebra. The aim of filtration theory is to determine first G and then L. The following "Recognition Theorem" which combines work of Kostrikin-Öafareviö [27], Kac [18, 21], and Wilson [40] shows that under appropriate hypotheses this can be done. THEOREM 4 . 1 . Let L be finite-dimensional simple over F, filtered as above. Assume: (a) Go — a direct sum of restricted ideals, each of which is either classical simple, gl(A), sl(fc), or pgl(&) where p divides k, or abelian. (b) x G Go implies ad(xp) and (adz) p agree on G-\. (c) xGGi, i < 0, [x, Gi] = (0) implies x = 0. Then L is classical or of Cartan type (with the usual filtration). A second major result in filtration theory (Weisfeiler [37]) gives the structure of G (without hypotheses on Lo)- Before stating this result we recall Block's characterization of semisimple Lie algebras over F (which is used in Weisfeiler's result and also in local analysis). THEOREM 4.2 (BLOCK [6]). / / L is a finite-dimensional semisimple Lie algebra over F then there exist integers m, n\,..., nm > 0 and simple Lie algebras Si,..., Sm such that m

/ m

\

i=l

\i=l

J

THEOREM 4.3 (WEISFEILER [37]). Let L be finite-dimensional simple over F, filtered as above. Then solvG Ç ^ < 0 G i and G/solvG contains a unique minimal ideal A = S®Bn, S simple, n > 0. Moreover, A is graded (with the grading determined by a grading on S if Ai ^ (0) and by a grading on Bn if Ai = (0)) and Ai = (G/solvG); for i < 0. In applications of this theorem one usually tries to show that n = 0 and that solvG = (0). If this can be done it is frequently possible to apply Theorem 4.1. Weisfeiler used these techniques to prove Theorem 3.1. The solvable maximal subalgebra L 0 is used to define the filtration. Then the solvable algebra Go acts irreducibly on G_i. Representation theory (in particular, the theory of induced modules) allows one to describe G_i well enough to show (after much work)

SIMPLE LIE ALGEBRAS

413

that if G-2 7^ (0) then dim G = oo. Thus L = L_i and Kuznecov's result [28] applies. The second major technique we will discuss, local analysis, involves studying subalgebras of L of toral rank 1 and 2 (the toral rank of algebra is the rank of the additive group generated by its roots) and using information obtained in this way to reconstruct L. We set L