Symmetric Polynomials in the Free Metabelian Lie Algebras

Let K[X-n] be the commutative polynomial algebra in the variables X-n = {x(1), ... , x(n)} over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra K[X-n](Sn) of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over K. In the present paper, we study a noncommutative and nonassociative analogue of the algebra K[X-n](Sn) replacing K[X-n] with the free metabelian Lie algebra F-n of rank n >= 2 over K. It is known that the algebra F-n(Sn) is not finitely generated, but its ideal (F'(n))(Sn) consisting of the elements of F-n(Sn) in the commutator ideal F'(n) of F-n is a finitely generated K[X-n](Sn)-module. In our main result, we describe the generators of the K[X-n](Sn)-module (F'(n))(Sn) which gives the complete description of the algebra F-n(Sn).