On equivalence classes of homotopes of algebras and trilinear forms

TL;DR

This paper investigates the diversity of homotopes and trilinear forms in finite-dimensional algebras, revealing that generic cases possess infinitely many non-isotopic homotopes, with special classes exhibiting similar complexity.

Contribution

It demonstrates that generic finite-dimensional algebras and trilinear forms have infinitely many non-isotopic homotopes, extending understanding of algebraic mutation classes.

Findings

Generic finite-dimensional algebras of dimension >3 have infinitely many non-isotopic homotopes.

Similar results hold for generic trilinear forms.

Certain classes of homotopes, like Δ-homotopes, also exhibit infinite diversity.

Abstract

A homotope, or a mutation, of a $k$-algebra is a new algebra with the same underlying space, but with the multiplication law dependent on the multiplication law of the original algebra. In this paper, we show that a generic finite-dimensional algebra of dimension greater than 3 has infinitely many non-isotopic homotopes, and that, more generally, a similar result is true for generic trilinear forms. We also study a particular class of homotopes called $\Delta$-homotopes, where $\Delta$ is an element of the algebra, and show that there are algebras with infinitely many non-isomorphic homotopes even under some additional assumptions, such as the associativity of the algebra or $\Delta$ being well-tempered.