Crossed modules and quantum groups in braided categories I

TL;DR

This paper explores the structure of crossed modules over Hopf algebras in braided categories, establishing their properties, connections with quantum braided groups, and extending classical theorems to the braided setting.

Contribution

It introduces a braided variant of Radford-Majid's theorem and constructs the correct cross product for quantum braided groups, expanding the theory of Hopf algebras in braided categories.

Findings

Category of crossed modules is braided and concrete.

Full subcategory of modules over quantum braided groups identified.

Braided Radford-Majid theorem extended to quantum braided groups.

Abstract

Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is concrete realization of general categorical construction. For quantum braided group $(H,{\cal R})$ corresponding braided category ${\cal C}^{\cal R}_H$ of modules is identifyed with full subcategory in ${\cal C}_H^H$. Connection with crossproducts is discussed. Correct cross product in the class of quantum braided groups is built. Radford's--Majid's theorem gives equivalent condition for usual Hopf algebra to be crossproduct. Braided variant and analog of this theorem for quantum braided qroups are obtained.