Symmetric polynomials in Leibniz algebras and their inner automorphisms

Abstract

Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\dots ,x_n\}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call an element $s(X_n)\in L_n$ symmetric if $s(x_{\sigma (1)},\dots ,x_{\sigma (n)})=s(x_1,\dots ,x_n)$ for each permutation $\sigma$ of $\{1,\dots ,n\}$. The set $L_n^{S_n}$ of symmetric polynomials of $L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$ be the usual polynomial algebra with indeterminates from $X_n$. The description of the algebra $K[X_n{]}^{S_n}$ is well known, and the algebra $(L_n^{\prime }{)}^{S_n}$ in the commutator ideal $L_n^{\prime }$ is a right $K[X_n{]}^{S_n}$-module. We give explicit forms of elements of the $K[X_n{]}^{S_n}$-module $(L_n^{\prime }{)}^{S_n}$. Additionally, we determine the description of the group $\mathrm{Inn}(L_n^{S_n})$ of inner automorphisms of the algebra $L_n^{S_n}$. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank $n$.

7 pages