Symmetric polynomials in Leibniz algebras and their inner automorphisms

Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n=\{x_1,\ldots,x_n\}$ 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 $s(x_{σ(1)},\ldots,x_{σ(n)})=s(x_1,\ldots,x_n)$ for each permutation $σ$ of $\{1,\ldots,n\}$. 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 $(L_n')^{S_n}$ in the commutator ideal $L_n'$ is a right $K[X_n]^{S_n}$-module. We give explicit forms of elements of the $K[X_n]^{S_n}$-module $(L_n')^{S_n}$. Additionally, we determine the description of the group ${\rm Inn}(L_{n}^{S_n})$ 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 $n$.

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