The Nowicki conjecture for relatively free algebras

A linear locally nilpotent derivation of the polynomial algebra in m variables over a field K of characteristic 0 is called a Weitzenböck derivation. It is well known from the classical theorem of Weitzenböck that the algebra of constants of a Weitzenböck derivation δ is finitely generated. Assume that δ acts on the polynomial algebra in 2d variables as follows: , , . The Nowicki conjecture states that the algebra is generated by , and , . The conjecture was proved by several authors based on different techniques. We apply the same idea to two relatively free algebras of rank 2d. We give the infinite set of generators of the algebra of constants in the free metabelian associative algebras , and finite set of generators in the free algebra in the variety determined by the identities of the infinite dimensional Grassmann algebra.