Yetter–Drinfeld categories associated with a weak braided Hopf algebra

J N Alonso Álvarez,C Soneira Calvo,R González Rodríguez,J M Fernández Vilaboa

doi:10.1007/s40065-012-0047-9

J N Alonso Álvarez, C Soneira Calvo + Show 2 more

Open Access

https://doi.org/10.1007/s40065-012-0047-9

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Abstract

In a previous paper, the authors introduced the monoidal category of left–left Yetter–Drinfeld modules over a weak braided Hopf algebra in a strict monoidal category. The main goal of this work is to define the categories of right–right, left–right and right–left Yetter–Drinfeld modules over a weak braided Hopf algebra and prove that there exists a categorical equivalence between all of them. We also establish the categorical equivalences by changing the weak braided Hopf algebra D by its (co)opposite. Finally, the general results are illustrated with an example coming from the projections of weak braided Hopf algebras.

Highlights

1 Introduction To study the projections of Hopf algebras, Radford [13] establishes conditions that led to the notion of Yetter–Drinfeld module introduced by Yetter [17] in order to explain the relationship between different theories in mathematics and physics, as low dimensional topology, knots and links, Hopf algebras, quantum integrable systems, and exactly solvable models in statistical mechanics
Every Yetter–Drinfeld module gives rise to a solution of the quantum Yang–Baxter equation, as was proved in [9], and if H is a finite Hopf algebra
Yetter–Drinfeld modules is isomorphic to the category of modules over the quantum double, which was originally conceived to find solutions of the

Summary

Introduction

To study the projections of Hopf algebras, Radford [13] establishes conditions that led to the notion of Yetter–Drinfeld module introduced by Yetter [17] in order to explain the relationship between different theories in mathematics and physics, as low dimensional topology, knots and links, Hopf algebras, quantum integrable systems, and exactly solvable models in statistical mechanics. In [4], the authors introduce the notion of weak operator and use it to establish a left–left Yetter–Drinfeld module theory in a general strict monoidal category. 2, the general framework is stated recalling the definitions of weak Yang–Baxter operator, weak braided bialgebra and weak braided Hopf algebra; as well as that of weak operator and its main properties, including the notion of compatibility for the (co)module structures that emerges naturally when dealing with the Yetter–Drinfeld categories. We prove that all of them are categorically equivalent and provide an example of application

Weak operators

The morphisms

Equivalences between Yetter–Drinfeld categories