Generators of symmetric polynomials in free metabelian Leibniz algebras

Let K be a field of characteristic zero, X-n = {x(1),..., x(n)} and R-n = {r(1),..., r(n)} be two sets of variables, Ln be the free metabelian Leibniz algebra generated by X-n, and K[R-n] be the commutative polynomial algebra generated by R-n over the base field K. Polynomials p(X-n) is an element of L-n and q(R-n) is an element of K[R-n] are called symmetric if they satisfy p(x(pi(1)),..., x(pi(n))) = p(X-n) and q(r(pi(1)),..., r(pi(n))) = q(R-n), respectively, for all pi is an element of S-n. The sets L-n(Sn) and K[R-n](Sn) of symmetric polynomials are the S-n-invariant subalgebras of L-n and K[R-n], respectively. The Leibniz subalgebra (L-n ')(Sn) = L-n(Sn) boolean AND L-n ' in the commutator ideal L-n ' of Ln is a right K[R-n](Sn)-module by the adjoint action. In this study, we provide a finite generating set for the right K[R-n](Sn)-module (L-n ')(Sn). In particular, we give free generating sets for (L-n ')(S2) and (L-3 ')(S3) as K[R-2](S2)-module and K[R-3](S3)-module, respectively.