Weitzenböck derivations of free metabelian associative algebras

Abstract

By the classical theorem of Weitzenböck the algebra of constants [Formula: see text] of a nonzero locally nilpotent linear derivation [Formula: see text] of the polynomial algebra [Formula: see text] in several variables over a field [Formula: see text] of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants [Formula: see text] of a locally nilpotent linear derivation [Formula: see text] of a finitely generated relatively free algebra [Formula: see text] in a variety [Formula: see text] of unitary associative algebras over [Formula: see text]. It is known that [Formula: see text] is finitely generated if and only if [Formula: see text] satisfies a polynomial identity which does not hold for the algebra [Formula: see text] of [Formula: see text] upper triangular matrices. Hence the free metabelian associative algebra [Formula: see text] is a crucial object to study. We show that the vector space of the constants [Formula: see text] in the commutator ideal [Formula: see text] is a finitely generated [Formula: see text]-module, where [Formula: see text] acts on [Formula: see text] and [Formula: see text] in the same way as on [Formula: see text]. For small [Formula: see text], we calculate the Hilbert series of [Formula: see text] and find the generators of the [Formula: see text]-module [Formula: see text]. This gives also an (infinite) set of generators of the algebra [Formula: see text].