Weitzenbock derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation delta of the polynomial algebra K[X-d] = K[x(1), .. , x(d)] in several variables over a field K of characteristic 0 is called a Weitzenbock derivation. The classical theorem of Weitzenbock states that the algebra of constants K[X-d](delta) (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 8 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 (L-d/L-d '')(delta) of the free metabelian Lie algebra L-d/L-d '' with d generators. We show that the vector space of the constants K[X-d](delta) in the commutator ideal (L-d/L-d '')(delta) is a finitely generated K[X-d](delta)-module. For small d, we calculate the Hilbert series of (L-d/L-d '')(delta) and find the generators of the K[X-d](delta)-module (L-d/L-d '')(delta). This gives also an (infinite) set of generators of the algebra (L-d/L-d '')(delta). (C) 2013 Elsevier Inc. All rights reserved.