On equivalence classes of homotopes of algebras and trilinear forms

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.