Higher Polynomial Identities for Mutations of Associative Algebras

Abstract

We study polynomial identities satisfied by the mutation product $$xpy - yqx$$ x p y - y q x on the underlying vector space of an associative algebra A, where p, q are fixed elements of A. We simplify known results for identities in degree 4, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new identity suffices; in degree 6, we demonstrate the existence of a significant number of new identities, which induce us to conjecture that the variety generated by mutation algebras of associative algebras is not finitely based.