When Hom-Lie structures form a Jordan algebra

TL;DR

This paper investigates conditions under which Hom-Lie structures on a Lie algebra form a Jordan algebra, revealing unexpected links to the Yang-Baxter equation and Lie algebra decompositions.

Contribution

It introduces new criteria for Hom-Lie structures to form Jordan algebras and explores their connections to the Yang-Baxter equation and algebra decomposition.

Findings

Hom-Lie structures can form Jordan algebras under specific conditions

Connections established between Hom-Lie structures and the Yang-Baxter equation

Insights into Lie algebra decomposition related to these structures

Abstract

We are concerned with the question when Hom-Lie structures on a Lie algebra are closed with respect to the Jordan product. Somewhat unexpectedly, this leads us to certain questions connected with the Yang-Baxter equation, and with decomposition of a Lie algebra into the sum of subalgebras with given properties.