Symmetric polynomials in the free metabelian associative algebra of rank 2 Dedicated to the 70th anniversary of Vesselin Drensky

Let F be the free metabelian associative algebra generated by x and y over a field of characteristic zero. We call a polynomial f is an element of F symmetric, if f(x, y) = f(y, x). The set of all symmetric polynomials coincides with the algebra F-S2 of invariants of the symmetric group S-2. In this paper, we give the full description of the algebra F-S2.