Hom-Tensor Categories and the Hom-Yang-Baxter Equation

TL;DR

This paper develops the theory of hom-tensor and hom-braided categories, providing a categorical framework for modules over hom-bialgebras and quasitriangular hom-bialgebras, and relating these to the hom-Yang-Baxter equation.

Contribution

It introduces hom-tensor and hom-braided categories, extending categorical structures to hom-bialgebras and their modules, and connects these to the hom-Yang-Baxter equation.

Findings

Hom-tensor categories provide a suitable setting for modules over hom-bialgebras.

Hom-braided categories are appropriate for modules over quasitriangular hom-bialgebras.

The category of Yetter-Drinfeld modules forms a hom-braided category.

Abstract

We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom-tensor category (respectively a hom-braided category).