Algebras and Hopf Algebras in Braided Categories

TL;DR

This paper introduces the theory of algebras and Hopf algebras within braided categories, generalizing classical structures and exploring their properties, with implications for quantum groups and related algebraic systems.

Contribution

It provides foundational concepts, properties, and a diagrammatic proof for braided Hopf algebras, extending classical algebraic theories into braided categorical contexts.

Findings

Recalled basic facts about braided categories

Studied modules and comodules of Hopf algebras in braided categories

Presented a diagrammatic proof of the reconstruction theorem for braided groups

Abstract

This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied,the notion of `braided -commutative' or `braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group aut(C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.