Higher Polynomial Identities for Mutations of Associative Algebras

Murray R Bremner; Jose Brox; Juana Sánchez-Ortega

doi:10.1007/s00025-023-01986-4

Higher Polynomial Identities for Mutations of Associative Algebras

Results Math. 2023;78(6):237. doi: 10.1007/s00025-023-01986-4. Epub 2023 Sep 22.

Authors

Murray R Bremner¹, Jose Brox², Juana Sánchez-Ortega³

Affiliations

¹ Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada.
² Department of Mathematics, Centre for Mathematics of the University of Coimbra, 3004-504 Coimbra, Portugal.
³ Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa.

Abstract

We study polynomial identities satisfied by the mutation product $x p y - y q x$ 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.

Keywords: Jordan-admissible; Lie-admissible; Mutation algebras; algebraic operads; computer algebra; polynomial identities; theoretical particle-physics.