Symmetric polynomials in the free metabelian Lie algebras

TL;DR

This paper investigates the structure of symmetric invariants in free metabelian Lie algebras, providing a complete description of their generators and extending classical symmetric polynomial results to a noncommutative setting.

Contribution

It describes the generators of the symmetric invariants in the free metabelian Lie algebra, extending classical polynomial symmetric functions to a noncommutative algebraic context.

Findings

The algebra of symmetric invariants in the free metabelian Lie algebra is not finitely generated.

The ideal of symmetric invariants in the commutator ideal is finitely generated as a module.

The paper provides explicit generators for this module.

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 $F_n^{S_n}$.