Chapter 4. Identities in enveloping algebras

Abstract

In this section we present the main results of this chapter on identities in enveloping algebras. Our goal is to find necessary and sufficient conditions for a colour Lie superalgebra under which its restricted or universal enveloping algebra satisfies a non-trivial associative identity.1.1. Theorem. Let G be a finite group and let L = © 9e G^9 be a restricted colour Lie superalgebra over a field of positive characteristic p > 2. Then the restricted enveloping algebra u(L) satisfies a non-trivial identity if, and only if, there exist restricted homogeneous ideals Q <= R c: L such that 1) dimL/R < oo, dim Q < oo; 2) R 2 c β, 2 = 0; 3) Q has a nilpotent p-map (i.e. for each even homogeneous χ e G g ,g e G + we have x [/7 " ] = 0 for some integer n).This theorem will also be proved in the following modification. We may assume that Q, R are not ideals, but subalgebras. In this case bounds on the dimensions in Claim 1 can be established, provided that u(L) satisfies a polynomial identity of given degree. Sufficiency will also be proved without the assumption that Q is abelian; and some bounds on the degree of the identity will be given as functions of the dimensions in Claim 1. As a corollary we will get a theorem on identities in the universal enveloping algebra U(L) in the positive characteristic case, where the colour Lie superalgebra L need not, in general, be restricted.In the case of characteristic zero we will prove the following result. 1.2. Theorem. Let G be a finite group and let L = ©^ec-^» be a colour Lie superalgebra over a field of characteristic zero. Then a universal enveloping algebra U(L) satisfies a non-trivial identity if, and only if, there exists a homogeneous L+-submodule M c L_ such that 1) dimL_/M < oo, dim[L + ,M] < oo; 2) L+ is abelian and [M,M] = 0.This example gives a negative answer to a question raised by V.K. Kharchenko [Dnestrovskaia tetrad, Problem 2.118]: is it true that the existence of a non-trivial Lie identity in a p-algebra L implies the existence of a non-trivial associative identity in u(L)? In the example Γ is a 2-step nilpotent Lie p-algebra.Remark. In the case of an algebraically closed field, there are, up to the choice of bases, two options for the p-map: z [p] = z and z [p] = 0. Let us remark that in the second case u(F) does satisfy an identity [X, Y~\ p = 0. To see this, one should only remark that all non-trivial terms in the commutator of two elements a, b ε Μ (Γ) contain, as a factor, a central element ζ which is nilpotent, i.e. z lpl = 0. Theorem. Let A, B be associative algebras over a field K satisfying non-trivial identities of degrees d l5 d 2 , respectively. Then A ® K B also is a Pi-algebra satisfying a non-trivial identity of degree n = 3(d 1 -I) 2 (d 2 -I) 2 ·Proof. See [Bahturin, 1987a]; the bound for the degree of the identity may be derived easily by modifying arguments there. Π [Procesi, Small, 1968]. Let A satisfy some power of a standard identity (5 2m )' = 0. Then M n (A] satisfies also some power of a standard identity eN. Π 1.6. Abelian colour Li...