Weitzenböck derivations of free metabelian Lie algebras

Abstract

A nonzero locally nilpotent linear derivation δ of the polynomial algebra K[Xd]=K[x1,…,xd] in several variables over a field K of characteristic 0 is called a Weitzenböck derivation. The classical theorem of Weitzenböck states that the algebra of constants K[Xd]δ (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 δ 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 (Ld/Ld″)δ of the free metabelian Lie algebra Ld/Ld″ with d generators. We show that the vector space of the constants (Ld/Ld″)δ in the commutator ideal Ld′/Ld″ is a finitely generated K[Xd]δ-module. For small d, we calculate the Hilbert series of (Ld/Ld″)δ and find the generators of the K[Xd]δ-module (Ld/Ld″)δ. This gives also an (infinite) set of generators of the algebra (Ld/Ld″)δ.