Symmetric polynomials in the free metabelian Poisson algebras

Abstract

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] be a finite set of variables. Consider the free metabelian Poisson algebra [Formula: see text] of rank [Formula: see text] generated by [Formula: see text] over [Formula: see text]. An element in [Formula: see text] is called symmetric if it is preserved under any change of variables, i.e. under the action of each permutation in [Formula: see text]. In this study, we determine the algebra [Formula: see text] of symmetric polynomials of [Formula: see text].