Isoclinism in regular Hom-Lie Yamaguti algebras

TL;DR

This paper develops the theory of isoclinism for regular Hom-Lie Yamaguti algebras, showing that in finite dimensions, isoclinism implies isomorphism, thus revealing a rigidity property in their classification.

Contribution

It introduces the notion of factor sets for analyzing isoclinism families and proves that isoclinism implies isomorphism for finite-dimensional regular Hom-Lie Yamaguti algebras.

Findings

Isoclinism implies isomorphism in finite-dimensional cases.

Factor sets are effective in analyzing algebraic structures.

Rigidity property generalizes classification results for related algebras.

Abstract

In this paper, we develop the theory of \emph{isoclinism} for regular Hom-Lie Yamaguti algebras, a class that unifies several generalizations of Lie algebras. Although isomorphism implies isoclinism by definition, the converse is not true in general. We introduce the notion of a \emph{factor set} and use it to analyze the structure of isoclinism families. Our main result establishes that for finite-dimensional regular Hom-Lie Yamaguti algebras of the same dimension, isoclinism implies isomorphism. This generalizes recent classification theorems for Lie-Yamaguti algebras and Hom-Lie superalgebras, highlighting a strong rigidity property in the finite-dimensional setting. The proof relies on the existence of stem algebras and a decomposition theorem within isoclin families.