Inner automorphisms of Lie algebras of symmetric polynomials

TL;DR

This paper investigates the structure of automorphism groups of symmetric polynomial Lie algebras, specifically determining inner automorphisms for certain free and metabelian Lie algebras invariant under symmetric group actions.

Contribution

It provides explicit descriptions of the groups of inner automorphisms for symmetric polynomial Lie algebras, including free, metabelian, and nilpotent cases, expanding understanding of their symmetry properties.

Findings

Determined the groups of inner automorphisms for symmetric invariant Lie algebras.

Described automorphism groups for specific cases like L_2^{S_2} and F_2^{S_2}.

Extended the analysis to metabelian and nilpotent Lie algebras of symmetric polynomials.

Abstract

Let $L_{n}$ be the free Lie algebra, $F_{n}$ be the free metabelian Lie algebra, and $L_{n,c}$ be the free metabelian nilpotent of class $c$ Lie algebra of rank $n$ generated by $x_1,\ldots,x_n$ over a field $K$ of characteristic zero. We call a polynomial $p(X_n)$ symmetric in these Lie algebras if $p(x_1,\ldots,x_n)=p(x_{\pi(1)},\ldots,x_{\pi(n)})$ for each element $\pi$ of the symmetric group $S_n$. The sets $L_n^{S_n}$, $F_n^{S_n}$, and $L_{n,c}^{S_n}$ of symmetric polynomials coincide with the algebras of invariants of the group $S_n$ in $L_{n}$, $F_{n}$, and $L_{n,c}$, respectively. We determine the groups $\text{Inn}(F_{n}^{S_n})$ and $\text{Inn}(L_{n,c}^{S_n})$ of inner automorphisms of the algebras $F_{n}^{S_n}$ and $L_{n,c}^{S_n}$, respectively. In particular, we obtain the descriptions of the groups $\text{Aut}(L_{2}^{S_2})$, $\text{Aut}(F_{2}^{S_2})$, and $\text{Aut}(L_{2,c}^{S_2})$ of all automorphisms of the algebras $L_{2}^{S_2}$, $F_{2}^{S_2}$, and $L_{2,c}^{S_2}$, respectively.