Extensions of hom-Lie color algebras

TL;DR

This paper explores the structure and classification of extensions of hom-Lie color algebras, providing geometric interpretations and cohomological criteria for their existence.

Contribution

It characterizes extensions of hom-Lie color algebras, introduces geometric interpretations, and identifies cohomological obstructions to their extension.

Findings

Characterization of hom-Lie algebra extensions

Geometric interpretation of extensions in differential geometry

Cohomological obstruction in third cohomology

Abstract

In this paper we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie algebra $\mathfrak{g}$ by another hom-Lie algebra $\mathfrak{h}$ and we discuss the case where $\mathfrak{h}$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i. e. we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology.