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 $$n\ge 2$$ 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 $$F_n^{S_n}$$ .