Symmetric polynomials in the free metabelian Lie algebras

Abstract

Let $K[X_n]$ be the commutative polynomial algebra in the variables $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. A theorem from undergraduate course of algebra states that the algebra $K[X_n]^{S_n}$ of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over $K$. In the present paper we study a noncommutative and nonassociative analogue of the algebra $K[X_n]^{S_n}$ replacing $K[X_n]$ with the free metabelian Lie algebra $F_n$ of rank $n\geq 2$ over $K$. It is known that the algebra $F_n^{S_n}$ is not finitely generated but its ideal $(F_n')^{S_n}$ consisting of the elements of $F_n^{S_n}$ in the commutator ideal $F_n'$ of $F_n$ is a finitely generated $K[X_n]^{S_n}$-module. In our main result we describe the generators of the $K[X_n]^{S_n}$-module $(F_n')^{S_n}$ which gives the complete description of the algebra $F_n^{S_n}$.