Cohomology and 1-parameter formal deformations of Hom-Lie triple systems

TL;DR

This paper develops a cohomology theory for Hom-Lie triple systems, introduces their central extension theory, and establishes a link between deformations and cohomology, extending classical Lie theory.

Contribution

It generalizes Yamaguti cohomology to Hom-Lie triple systems and connects deformation theory with cohomology in this new context.

Findings

Cohomology groups classify central extensions.

Deformation theory is governed by the third cohomology group.

Established a one-to-one correspondence between extensions and cohomology classes.

Abstract

In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems, and prove that it is governed by the cohomology group.