The Nowicki Conjecture for free metabelian Lie algebras

Abstract

Let $K[X_d]=K[x_1,\dots ,x_d]$ 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 $\delta$ (known as Weitzenböck derivations) the algebra of constants $K[X_d{]}^{\delta }$ is finitely generated. When the Weitzenböck derivation $\delta$ acts on the polynomial algebra $K[X_d,Y_d]$ in $2d$ variables by $\delta (y_i)=x_i$, $\delta (x_i)=0$, $i=1,\dots ,d$, Nowicki conjectured that $K[X_d,Y_d{]}^{\delta }$ is generated by $X_d$ and $x_i{y}_j-y_i{x}_j$ for all $1\le i<j\le d$. 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 $F_d^{\delta }$ is not finitely generated. However, the vector subspace $(F_d^{\prime }{)}^{\delta }$ of the commutator ideal $F_d^{\prime }$ of $F_d$ is finitely generated as a $K[X_d{]}^{\delta }$-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 $K[X_d,Y_d{]}^{\delta }$-module $(F_{2d}^{\prime }{)}^{\delta }$.

8 pages

References in corpus (1)

• Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras