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 \(F_n^{S_n}\).