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Identities in the Enveloping Algebras for Modular Lie Superalgebras

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Identities in the Enveloping Algebras for Modular Lie Superalgebras View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 145, 1-21 (1992) Identities in the Enveloping Algebras for Modular Lie Superalgebras V. M. PETROGRADSKI Departmew of Mechanics and Malhemarics (Algebra), Moscow State University, Moscow I 17234, USSR Communicaled by Susan Montgomery Received November 1. 1989 INTRODUCTION The universal enveloping algebra of a Lie algebra in characteristic zero satisfies a nontrivial identity if and only if the Lie algebra is abelian [13]. Necessary and sufficient conditions for a universal enveloping algebra in positive characteristic to satisfy a nontrivial identity have been found in [S]. In [S] the analogous problem for the universal enveloping algebra of a Lie superalgebra in characteristic zero has been settled. These results may also be found in the monograph [l]. The present author has also found necessary and sufficient conditions for the restricted envelope of a Lie p-algebra to be a PI-algebra [ 18, 191. Independently, this result has been obtained by D. S. Passman using somewhat different methods [16, 171. Our main results are Theorems 2.1 and 2.4 which give necessary and sufficient conditions for the restricted enveloping algebra of a restricted Lie superalgebra to satisfy a nontrivial identity. These theorems generalize previously mentioned results of [IS]. Corollary 2.5 also generalizes a result of Ju. A. Bahturin [6]. In Section 6 the methods of this paper are used to specify those Lie superalgebras in both zero and positive characteristics such that the dimen- sions of all their irreducible representations are bounded by a finite constant (Theorems 6.2 and 6.3). The first of these results was obtained by Ju. A. Bahturin [ 10, 111. From Theorem 2.1 we see that the properties of the restricted envelope of a Lie p-superalgebra resemble those of a group ring in positive charac- teristic. Indeed, necessary and sufficient conditions for a modular group ring to satisfy nontrivial identities [ 151 are close analogs of Theorem 2.1. GQ21-8693192$3.00 Copyright 0 1992 by Academic Press, Inc. All rights 01 reproduction m any form reserved V. M. PETROGRADSKI 1. RESTRICTED LIE SUPERALGEBRAS The main field k is of positive characteristic p > 2. In case p = 3 we add to the axioms of a Lie superalgebra L = L,@ L, an identity [[y, y], y] = 0, ,v E L,. (If p > 3 then it follows from Jacobi’s identity.) This is necessary to get an embedding of L into its universal enveloping algebra; see 1.2, 1.3. We know from [ 141 the notion of a Lie p-superalgebra. Let us recall it supposing the notion of a Lie p-algebra is well known [2]. DEFINITION 1.1. A Lie superalgebra L = L, @ L, over a field k of positive characteristic p is called restricted (or a p-superalgebra) if (1) Lie algebra L, is restricted (i.e., has a unary operation CPl:&i+~o; L, 3 X H X[P’ E L, which satisfies some special conditions [2]). (2) The action of p-algebra L, on module L, is restricted. By U(L) we shall denote, for a Lie superalgebra L = L, @ L, , its univer- sal enveloping algebra. (It exists if char k # 2 [ 11.) DEFINITION 1.2. Let us define for a Lie p-superalgebra over a field k with char k = p # 2 a restricted enveloping algebra by u(L) = U(L)/Id(xP - xcp’ 1x E L,). Suppose that L,= (e,IccEZ), L, = (fDIBEJ) are ordered bases; then by analogy with [2] we have. LEMMA 1.3, The restricted enveloping algebra has the standard basis The degree of a basis monomial v written above is the number deg(v) = n, + . + n, + t; the degree of any element w E u(L) is maximum of degrees of basis monomials which enter its decomposition, DEFINITION 1.4. For any m = 0, 1, 2, .. define u,=u,(L)= {wEu(L)Ideg(w)dm}. By standard methods we have MODULAR LIE SUPERALGEBRAS 3 LEMMA 1.5. (1) The degree of an element defined above does not depend on the ordered basis of L, and L, (2) The set (u,(m =O, 1,2, .. } forms a filtration. Its associated graded algebra is isomorphic to the tensor product gr {u,,, } g A, Q A,, where A, = k[X, 1u E II/(X: 1c1 E I) is a ring of truncated polynomials, the number of variables being equal to the cardinality of I; A, = A( Y, 1fi E J) is a Grassmann algebra, the number of variables being equal to the cardinality of J. If M is a homogeneous subalgebra in a Lie superalgebra L = Lo @ L, then M= M, 0 M, is the decomposition into its homogeneous com- ponents. DEFINITION 1.6. Let A4 be a homogeneous subalgebra in a restricted Lie superalgebra L. (1) Subalgebra A4 is called restricted if M0 is a restricted Lie sub- algebra in Lie p-algebra L,. (2) We call the minimal restricted subalgebra of L containing it4 the p-hull M, of M. Evidently, M,= (MO)p@M,, where (MO)p is the p-hull of the restricted Lie subalgebra M0 c L,. LEMMA 1.7. Let L be a restricted superalgebra and Mc L be a homogeneous restricted subalgebra of finite codimension. Then there exists a homogeneous restricted ideal fin L of finite codimension in L which is contained in M. Proof The same as in [12, Lemma lo] where this fact has been established for restricted Lie algebras. 1 2. MAIN RESULT THEOREM 2.1. Let L = L, @ L, be a restricted Lie superalgebra over a field k of positive characteristic p # 2. Then the restricted enveloping algebra u(L) satisfies a nontrivial identity tf and only tf there exist homogeneous restricted ideals Q s R s L such that (1) dimL/R<co,dimQ<co. (2) R’cQ, Q2=0. (3) Zn Q = Q,@ Q, the restricted Lie subalgebra Q,E L, has a nilpotent p-map. 4 V. M. PETROGRADSKI Let us prove sufficiency conditions of the theorem without having supposed Q and R to be ideals. PROPOSITION 2.2. Suppose that there exist restricted homogeneous sub- algebras Q E R G L satisfying conditions (l)-(3) of Theorem 2.1 and that dim L/R = t, dim Q = m. Then u(L) satisfies a nontrivial identity of degree 108~2(m + r) Proof Consider canonical embeddings u(Q) c u(R) c u(L). The augmentation ideal Z= u,(Q) of u(Q) is nifpotent: Idim” = 0, where dim u(Q) =pdim Qopm QI Gpdim Q =pm. The ideal J in u(R) generated by Q is also nilpotent: J= u(R) Qu(R) = u(R) Q, Jpm= u(R) Qpm= 0. We have u(R)/J= u(R/Q) (by analogy with [2]). By the hypothesis R/Q is a commutative superalgebra, hence u(R/Q) satisfies an identity of the form [[X, Y], Z] = 0. Therefore u(R) satisfies [[X, Y], Z]“” = 0. By identifying any XE u(L) with a right multiplication we obtain an embedding of u(L) into the endomorphism ring of a free module uCRJu(L) of finite rank p dimLolRo2dim &/RI dpl. Thus we have u(L)~End,,,,zt(L)~M,,(u(R))rM,,(k)@u(R). By the proof of [ 1, 6.1.111, if algebras A, B satisfy nontrivial identities of degrees ql, q2, respectively, then, for n! > (q1q2)2”, there exists a nontrivial identity of degree n in A 0 B. Now the matrix ring M,,(k) satisfies standard identity SzP,. Since n! > (n/2)” > (n/3)“, we obtain an inequality of the form (n/3)” > (6~~+l)*~. Hence by n = 108p2@‘+” the result follows. i Remark 2.3. Let Q c R z L be ideals satisfying the hypotheses of 2.2. Then D = Qu(L) is an ideal in u(L) with Dpm= 0. By analogy, we obtain u(L)/D z u(L/Q) E M,,(u(R/Q)). Now u(R/Q) satisfies [X, Y]’ E 0 and, by [20], the latter matrix ring satisfies a power of standard identity (S,!)” E 0 for N= N(t) E N. Thus u(L) satisfies (S2,,,)NP”=0, the degree of this identity being 2Np”‘+‘. THEOREM 2.4. Let L be a restricted superalgebra over a field k, with char k =p > 2. Suppose that u(L) satisfies a nontrivial identity of degree d. Then there exist homogeneous restricted subalgebras Q c R c L such that (1) dim L/R< co, dim Q< co. (2) R2sQ, Q’=O. (3) The p-algebra Q, of Q = Q0 @ QI has nilpotent p-mapping. (4) Moreover, if k is a perfect field then we have estimates of the form dim LfR d 242d4,dim Q d 227d4. MODULAR LIESUPERALGEBRAS Proof Contained in Sections 335. Proof The proof of necessity of conditions of Theorem 2.1 immediately follows from 2.4. Indeed, by our assumption we have subalgebras Q c R E L satisfying conditions (l)-(3) and we must construct a chain of ideals. By 1.7 we get an ideal i? G L with Z?c R, dim L/R < cc and consider Q = (Z?2)p~ Q. Then we get the chain Q c i?c L, which completes the proof. 1 From Theorem 2.1 we have: COROLLARY 2.5. Let L = L,@ L, be a superalgebra over field k with char k = p > 2. Then its universal enveloping algebra U(L) satisfies a non- trivial identity tf and only tf there exist homogeneous subalgebras B G A c L such that (1) dim L/A< co, dim B< co.
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