Inner isotopes associated with automorphisms of commutative associative algebras

TL;DR

This paper explores how inner isotopies can be used to construct and analyze new classes of commutative nonassociative algebras, revealing their structure, idempotents, and spectra.

Contribution

It introduces a novel class of commutative nonassociative algebras obtained via inner isotopy, establishing their classification and structural properties.

Findings

Algebras are classified by integer partitions of their dimensions.

Constructed algebras are generic, with some being axial and metrized.

Complete description of idempotents and spectra of these algebras.

Abstract

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.