A Basis for Free Assosymmetric Algebras

Abstract

Assosymmetric algebras are nonassociative algebras, where (xy)z−x(yz) remains invariant under each permutation ofx,y,z. In general, the free nonassociative algebra in a variety is difficult to describe. We show this is not the case for free assosymmetric algebras having characteristic ≠2,3. We exhibit a natural basis, describe how basis elements are multiplied and show how arbitrary elements can be expressed relative to this basis.