Generators of symmetric polynomials in free metabelian Leibniz algebras

Abstract

Let [Formula: see text] be a field of characteristic zero, [Formula: see text] and [Formula: see text] be two sets of variables, [Formula: see text] be the free metabelian Leibniz algebra generated by [Formula: see text], and [Formula: see text] be the commutative polynomial algebra generated by [Formula: see text] over the base field [Formula: see text]. Polynomials [Formula: see text] and [Formula: see text] are called symmetric if they satisfy [Formula: see text] and [Formula: see text], respectively, for all [Formula: see text]. The sets [Formula: see text] and [Formula: see text] of symmetric polynomials are the [Formula: see text]-invariant subalgebras of [Formula: see text] and [Formula: see text], respectively. The Leibniz subalgebra [Formula: see text] in the commutator ideal [Formula: see text] of [Formula: see text] is a right [Formula: see text]-module by the adjoint action. In this study, we provide a finite generating set for the right [Formula: see text]-module [Formula: see text]. In particular, we give free generating sets for [Formula: see text] and [Formula: see text] as [Formula: see text]-module and [Formula: see text]-module, respectively.