# Higher polynomial identities for mutations of associative algebras

## TL;DR

This paper investigates polynomial identities in the mutation algebra derived from associative algebras, simplifying known identities in degree 4, and exploring new identities in degrees 5 and 6.

## Contribution

It provides a simplified set of identities for degree 4 and identifies new identities emerging in degrees 5 and 6 for mutation products.

## Key findings

- Only two identities generate degree 4 identities.
- Adding one identity suffices for degree 5.
- New identities appear in degree 6.

## Abstract

We study polynomial identities satisfied by the mutation product $xpy - yqx$ 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 number of new identities.