The Nowicki conjecture for free metabelian Lie algebras.

Let K [ X d ] = K [ x 1 , ... , x d ] be the polynomial algebra in d variables over a field K of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations δ (known as Weitzenböck derivations), the algebra of constants K [ X d ] δ is finitely generated. When the Weitzenböck derivation δ acts on the polynomial algebra K [ X d , Y d ] in 2 d variables by δ (y i) = x i , δ (x i) = 0 , i = 1 , ... , d , Nowicki conjectured that K [ X d , Y d ] δ is generated by X d and x i y j − y i x j for all 1 ≤ i < j ≤ d. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free d -generated metabelian Lie algebra F d , with few trivial exceptions, the algebra F d δ is not finitely generated. However, the vector subspace (F d ′) δ of the commutator ideal F d ′ of F d is finitely generated as a K [ X d ] δ -module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the K [ X d , Y d ] δ -module (F 2 d ′) δ . [ABSTRACT FROM AUTHOR]