Weitzenboeck derivations of free metabelian Lie algebras

TL;DR

This paper investigates the algebra of constants under Weitzenboeck derivations in free metabelian Lie algebras, showing finite generation of certain modules and computing Hilbert series for small cases.

Contribution

It demonstrates that the vector space of constants in the commutator ideal is finitely generated over the polynomial algebra constants and provides explicit calculations for small numbers of generators.

Findings

The constants form a finitely generated module over polynomial algebra constants.

Explicit Hilbert series are computed for small d.

Generators of the module and Lie algebra of constants are identified.

Abstract

A nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K. Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants of the free metabelian Lie algebra L/L" with d generators. We show that the vector space of the constants in the commutator ideal L'/L" is a finitely generated module over the algebra of constants in K[X]. For small d, we calculate the Hilbert series of the algebra of constants in L/L" and find the generators of the module of the constants in L'/L". This gives also an (infinite) set of generators of the Lie algebra of constants in L/L".