Hom-quantum groups I: quasi-triangular Hom-bialgebras

TL;DR

This paper introduces a new class of algebraic structures called quasi-triangular Hom-bialgebras, which generalize quantum groups by incorporating non-associativity controlled by a twisting map, and explores their solutions to a non-associative quantum Yang-Baxter equation.

Contribution

It defines quasi-triangular Hom-bialgebras, constructs them from classical quasi-triangular bialgebras, and introduces the quantum Hom-Yang-Baxter equation with solutions derived from their modules.

Findings

Defined quasi-triangular Hom-bialgebras.

Constructed examples from classical quantum groups.

Established solutions to the quantum Hom-Yang-Baxter equation.

Abstract

We introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained from modules of suitable quasi-triangular Hom-bialgebras.