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 L Sn n 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 (L ′ n) Sn in the commutator ideal L ′ n is a right K[Xn] Sn -module. We give explicit forms of elements of the K[Xn] Sn -module (L ′ n) Sn . Additionally, we determine the description of the group Inn(L Sn n ) of inner automorphisms of the algebra L Sn n . The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank n.