Inner automorphisms of 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.