The Hom-Yang-Baxter equation and Hom-Lie algebras

TL;DR

This paper explores the Hom-Yang-Baxter equation, a twisted version of the Yang-Baxter equation inspired by Hom-Lie algebras, constructing new solutions and relating them to quantum groups and knot invariants.

Contribution

It introduces new classes of solutions to the Hom-Yang-Baxter equation and connects these solutions to quantum algebra structures and knot theory.

Findings

Constructed several new solutions of the HYBE.

Linked solutions of the HYBE to quantum enveloping algebras and knot invariants.

Computed all Lie algebra endomorphisms on specific algebras.

Abstract

Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by the author in an earlier paper. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway polynomial, and Yetter-Drinfel'd modules. We also construct a new infinite sequence of solutions of the HYBE from a given one. Along the way, we compute all the Lie algebra endomorphisms on the (1+1)-Poincare algebra and sl(2).