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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 <fc V U {0}; (2.4) if Hi, H2 are Cartan subalgebras of L then there exists o G AutL such that oHx = H2. In fact, (2-l)-(2.3) characterize the classical simple algebras. SIMPLE LIE ALGEBRAS 409 THEOREM 2.1 (MILLS-SELIGMAN [29]). Let L be a finite-dimensional simple Lie algebra over F satisfying (2.1)-(2.3). Then L is classical. We will say that a Lie algebra L over F is compositionally classical if all of its composition factors are classical or abelian, Before giving the general definition of the algebras of Cartan type we will give a brief historical survey of the nonclassical simple Lie algebras. The first nonclassical simple Lie algebra over F was discovered by Witt (see [44, 11]). We denote it by W(l : 1). It is defined as the algebra with basis {ei\i G Z/(p)} and multiplication (2-5) [eutj] = C/-*><+r This algebra is restricted. Zassenhaus [44] obtained nonrestricted generalizations by letting G be a finite additive subgroup of F and defining multiplication on the algebra with basis {ei\i G G} by (2.5). Further constructions of simple Lie algebras in this spirit (by giving an explicit multiplication table for a basis indexed, at least in part, by an additive group) were given by Kaplansky [22], Albert and Frank [1], and Block [4]. The Witt algebra W(l : 1) can be constructed in another way (due, independently, to Jacobson and Witt). Let Bi = F[xi]/(x^). Then letting D\ denote the derivation of B\ induced by djdx\, we see that Der£?i has basis {^Z>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 <m} = g\(W(m : l)[-i]). W(m : 1) is simple of dimension mpm, 410 R. L. WILSON Albert and Frank [1] and Frank [12, 13] discovered restricted graded simple subalgebras M of W(m : 1) obtained by taking M = L^ where L is defined by L[_i] = W(m : l)[-ij; £[o] — sl(ra) (Frank [12]), sp(ra) for m even (Albert and Frank [1]), or W(r : 1) + Br acting on Br for m = pr (Frank [13]) and L[i\ = {xG W(m : l) w |(adl,[_i])*z Ç L [0 ]} for i > 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<°0 = Y<U*, z'ez £(«.« = £ Lia+jß> L[a] = L<a>/solvL<a>, L[a,ß) = £(".«/solvL<°'« for all roots a, ß. In the proof of Theorem 3.2, Benkart and Osborn observe that L^ contains a one-dimensional Cartan subalgebra F h with all the eigenvalues of ad h in the prime field. Algebras satisfying this condition have been classified by Yermolaev [43]. If such an algebra A is not solvable it has an abelian radical and Aj solv A = si(2) or W(l : 1). This is enough information to develop a representation theory of these algebras. Benkart and Osborn do this and apply it to L^ acting on Y, Lß+ia for any root ß. Information obtained in this way, together with use of filtration theory, allows them to show that the hypotheses of Block's theorem [5] are satisified and so complete the proof. In the proof of Theorem 3.3, the techniques of local analysis and filtration theory are combined. Local analysis is used to define an appropriate LQ for use in Theorem 4.1. Let A be any restricted Lie algebra and let T be a maximal torus in A. We say that T is standard in A if GA(T) = T + I where J is a nil ideal in CA(T). By [41] any maximal torus in a restricted simple Lie algebra L is standard. Suppose L is restricted simple and T is a torus of maximal dimension in L. Then M = L[a] is restricted semisimple, has no tori of dimension > 1 (as T has maximal dimension), and all one-dimensional tori in M are standard (as all maximal tori in L are standard). An algebra M satisfying these conditions is of one of four types: M = (0), M = sl(2), M = W(1: 1), H(2 : 1)<2) C M C H(2 : 1). It is easily seen that in any case M contains a unique compositionally classical subalgebra of maximal dimension (see (2.6) for the case W(l : 1)) and consequently L^ contains a unique compositionally classical subalgebra (which we denote Q^) of maximal dimension. If T Ç Q( a ) we say the root a is proper. We have seen (in (2.7)) that a root a can be either proper or improper. Use of Winter exponentials (see (2.8)) shows that if en G T* is improper we may switch to a new maximal torus T" such that the corresponding root a1 G T"* is proper. (The rank one case can be completely analyzed for the nonrestricted case as well, but the list of possibilities is longer.) Now consider the possible rank two algebras. We take T as above and further require that the number of proper roots with respect to T is maximal (so is at least 1). Then M = L[a,ß] is restricted semisimple, has no tori of dimension 414 R. L. WILSON > 2, and all two-dimensional tori in M are standard. Using Theorem 4.2 and the restrictedness of M (in particular, using Lemma 1.3) we see that either toral rank M < 1 or one of the following occurs: (a) Si e S2 Ç M C (DerSi)^) 0 (DerS 2 )W where Si, S2 = sl(2), W(l : 1), or#(2:l)(2). (b) S®Bn CMC Der(S<g)£n), n > 0, S = sl(2), W(l : 1), or an appropriate i/(2:l:$)(2). (c) S Ç M Ç Der S where S denotes the restricted subalgebra of Der S generated by S, S is simple, and S + I ^ M. Here S = W(l : 2) or an appropriate H(2 : 1 : $)( 2 ). ( d ) S Ç M Ç Der S where S is simple and S + J = M. (The corresponding classification problem in the nonrestricted case is unsolved.) To complete the analysis of case (d) we let R(M)a = {x G Ma \ [x,M-a] Ç 1} and let R(M) = GM(T) + £ Ä ( M ) a . We let M 0 2 R(M) be a maximal subalgebra. Analysis of the rank one cases shows dim((M/Mo) 7 ) < 7 for all 7 G T* (this uses the hypothesis that rank M = 2), and the fact that there is at least one proper root with respect to T shows that (M/MQ)1 ^ (0) for 2 at most p — p + 6 values of 7. Then (filtering M as above and letting G be the associated graded algebra) the Go-module G_i has the same properties. With this information we are able to show that Theorem 4.1 applies and hence that the simple algebra S in (d) is one of A2, C2,G2, W(2 : 1), 5(3 : l j W , H(4:1)W, Jf(3:l). Finally, consider a restricted simple algebra of arbitrary rank. Considering the subalgebra (X(m : n : 4>)(2))o of X(m : n : 4>)(2), we observe that it contains a maximal torus T with respect to which all roots are proper and that, with respect to this torus, we have (X(m : n : $)( 2 ))o = YQ^This leads us to make the following claims: (4.1) There is a torus T of maximal dimension in L with respect to which all roots are proper. (4.2) YQ{a) i s a subalgebra (with Q^ taken with respect to T). (4.3) If L = E Q ( a ) then L is classical. (4.4) If Lo is a maximal subalgebra of L containing Y Q^ a n d if H = {7 G r I L1 ^ L0tl} then for any a,ß G T we have \(ß + Za) H H| < 2. Using (4.4) we can show that Lo satisfies the hypotheses of Theorem 4.1. Properties (4.1)-(4.4) can all be checked in the rank two case. For example, for (4.1) it is sufficient to show that if a is improper and ß proper, and if a1 ,ß' are corresponding roots under a Winter exponential with ol proper then ßf is proper. If this property holds in every L[a, ß] then it holds in L. Unfortunately, these rank two properties do not hold in all of the semisimple algebras listed in (a)-(d). Thus we must show that some of these semisimple algebras cannot occur as L[a,ß] for L simple. We give an example of how this can be done. Suppose L[a, ß] ^ sl(2)®Bi+FDi+F(xiDi) and that T = span{ft<g)l, ziL>i} (where h is a nonzero diagonal matrix in sl(2)). Let a and u G La, v G L _ a SIMPLE LIE ALGEBRAS 415 be such that ü = u + solvi/"1'9) = D\, v = h ® xi. Then [ü,v] = h <8> 1 and so [u,v] £ I. However, a([u,L-a]) = (0). Let r ' = {7 G T \ i([u,v]) ^ 0}. Then ]TLer' ^1 + X^ser' [^75 Lg] is a nonzero ideal in L. Since L is simple this ideal equals L and so GL(T) — ^ 7 e r / [ L 7 , L _ 7 ] . (This is an argument due to Schue [31].) Thus there exists 7 G Y satisfying q([u, v\) ^ 0, a([L 7 ,L_ 7 ]) ^ (0). From this it is easily seen that if J is any nonzero ideal in Af = L[a,7] then M = J + I. 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