Classical Invariant Theory for Free Metabelian Lie Algebras

Let 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 KX d 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[X d ] SL2(K) of the special linear group SL 2 (K) , when KX d is a direct sum of SL 2 (K) -modules of binary forms.Noncommutative invariant theory deals with the algebra of invariants F d (V) G of a group G < GL d (K) acting on the relatively free algebra F d (V) of a variety of K -algebras V. Due to the noncommutativity it is more convenient to assume that F d (V) is generated by W d instead of by KX d , 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 SL 2 (K) -moduleHere SL 2 (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 (V) and for other groups G .