The Nowicki conjecture for free metabelian Lie algebras

Drensky V., FINDIK Ş.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.19, no.5, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 19 Issue: 5
  • Publication Date: 2020
  • Doi Number: 10.1142/s0219498820500954


Let K[X-d] = K[x1, ..., x(d)] be the polynomial algebra in d variables over a field K of characteristic 0. The classical theorem of Weitzenbock from 1932 states that for linear locally nilpotent derivations delta (known as Weitzenbock derivations), the algebra of constants K[X-d]delta is finitely generated. When the Weitzenbock derivation delta acts on the polynomial algebra K(X-d, Y-d) in 2d variables by delta(y(i)) = x(i), delta(x(i)) = 0, i = 1, ..., d, Now-icki conjectured that K[X-d, Y-d](delta) 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 Weitzenbock derivations of the free d-generated metabelian Lie algebra F-d, with few trivial exceptions, the algebra Fd(delta) is not finitely generated. However, the vector subspace (F-d')delta of the commutator ideal F-d' of F-d is finitely generated as a K[X-d](delta) smodule. 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](delta) -module (F-2d')(delta).