Classical Invariant Theory for Free Metabelian Lie Algebras

WOS: 000484013200011Let W-d = K-d be the d-dimensional vector space over a field K of characteristic 0 with the canonical action of the general linear group GL(d)(K) and let KXd be the vector space of the linear functions on W-d. One of the main topics of classical invariant theory is the study of the algebra of invariants K[Xd](SL2(K)) of the special linear group SL2(K), when KXd is a direct sum of SL2(K)-modules of binary forms. Noncommutative invariant theory deals with the algebra of invariants F-d(A)(G) of a group G < GL(d) (K) acting on the relatively free algebra F-d(A) of a variety of K-algebras A. Due to the noncommutativity it is more convenient to assume that F-d(A) is generated by W-d instead of by KXd, with the corresponding action of GL(d)(K). In this paper we consider the free metabelian Lie algebra F-d(A)(2) which is the relatively free algebra in the variety A(2) of metabelian (solvable of class 2) Lie algebras. We study the algebra F-d (A(2))(SL2(K)) and describe the cases when it is finitely generated. This happens if and only if as an SL2(K)-module W-d congruent to K-2 circle plus K circle plus ... circle plus K or W-d congruent to S-2(K-2) (and in the trivial case KWd congruent to K circle plus ... circle plus K ). Here SL2(K) acts canonically on K-2, trivially on K, and S-2 (K-2) is the symmetric square of K-2. For small d we give a list of generators even when F-d (A(2))(SL2(K)) is not finitely generated. The methods for establishing that the algebra F-d (A(2))(SL2(K)) is not finitely generated work also for other relatively free algebras F-d(A) and for other groups G.Bulgarian Science FundNational Science Fund of Bulgaria [I02/18]; Grant Ukraine of the Bulgarian Science Fund for Bilateral Scientific Cooperation between Bulgaria and Ukraine [01/0007]; Council of Higher Education (YOK) in TurkeyMinistry of National Education - TurkeyPartially supported by Grant I02/18 of the Bulgarian Science Fund. The research of the first named author was partially supported by Grant Ukraine 01/0007 of the Bulgarian Science Fund for Bilateral Scientific Cooperation between Bulgaria and Ukraine. The research of the second named author was partially supported by the Council of Higher Education (YOK) in Turkey