Symmetric polynomials in Leibniz algebras and their inner automorphisms
Abstract
Let Ln be the free metabelian Leibniz algebra generated by the set Xn=\x1,…,xn\ 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(Xn)∈ Ln symmetric if s(xσ(1),…,xσ(n))=s(x1,…,xn) for each permutation σ of \1,…,n\. The set LnSn of symmetric polynomials of Ln is the algebra of invariants of the symmetric group Sn. Let K[Xn] be the usual polynomial algebra with indeterminates from Xn. The description of the algebra K[Xn]Sn is well known, and the algebra (Ln')Sn in the commutator ideal Ln' is a right K[Xn]Sn-module. We give explicit forms of elements of the K[Xn]Sn-module (Ln')Sn. Additionally, we determine the description of the group Inn(LnSn) of inner automorphisms of the algebra LnSn. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank n.