Symmetric polynomials in the free metabelian Poisson algebras

Let K be a field of characteristic zero and X-n = {x(1),..., xn} be a finite set of variables. Consider the free metabelian Poisson algebra Pn of rank n generated by X-n over K. An element in Pn is called symmetric if it is preserved under any change of variables, i.e. under the action of each permutation in Sn. In this study, we determine the algebra P-S(n) (n) of symmetric polynomials of P-n.