Weitzenboeck derivations of free metabelian associative algebras

TL;DR

This paper investigates the algebra of constants of Weitzenboeck derivations in free metabelian associative algebras, showing finite generation properties and providing explicit generators and Hilbert series for small cases.

Contribution

It extends classical results to noncommutative free metabelian associative algebras, identifying conditions for finite generation and explicitly constructing generators.

Findings

The algebra of constants in the commutator ideal is finitely generated as a module.

Explicit generators and Hilbert series are computed for small dimensions.

An infinite set of generators for the algebra of constants in F(M) is obtained.

Abstract

By the classical theorem of Weitzenboeck the algebra of constants (i.e., the kernel) of a nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants of a locally nilpotent linear derivation of a d-generated relatively free algebra F(V) in a variety V of unitary associative algebras over K. It is known that the algebra of constants of F(V) is finitely generated if and only if V satisfies a polynomial identity which does not hold for the algebra of 2 x 2 upper triangular matrices. Hence the free metabelian associative algebra F(M) is a crucial object to study. We show that the vector space of the constants in the commutator ideal F'(M) is a finitely generated module of the algebra of constants of the polynomial algebra K[U,V] in 2d variables, where the derivation acts on U and V in the same way as on X. For small d, we calculate the Hilbert series of the constants in F'(M) and find the generators of the related module. This gives also an (infinite) set of generators of the algebra of constants in F(M).