THE NOWICKI CONJECTURE FOR BICOMMUTATIVE ALGEBRAS

Abstract

Let K be a field of characteristic zero, and K[Xn,Yn ] be the commutative associative unitary polynomial algebra of rank 2n generated by the set Xn∪Yn={x1,…,xn,y1,…,yn }. It is well known that the algebra K[Xn,Yn ]^δ of constants of the locally nilpotent linear derivation δ of K[Xn,Yn ] sending yi to xi, and xi to 0, is generated by x1,…,xn and the determinants of the form xi yj-xj yi; that was first conjectured by Nowicki in 1994, and later proved by several authors. Bicommutative algebras are nonassociative noncommutative algebras satisfying the identities (xy)z=(xz)y and x(yz)=y(xz). In this study, we work in the 2n generated free bicommutative algebra as a noncommutative nonassociative analogue of the Nowicki conjecture, and find the generators of the algebra of constants in this algebra.