Algebras and Hopf Algebras in Braided Categories

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.

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Cited by in corpus (15)

• Tannaka-Krein duality for Hopf algebroids
• The Nichols algebra of screenings
• SL(2,Z)-action for ribbon quasi-Hopf algebras
• q-Deformed quantum Lie algebras
• Weak bimonads and weak Hopf monads
• Representation Theory of Lattice Current Algebras
• Moduli spaces of noncommutative instantons: gauging away noncommutative parameters
• Braided Hopf algebras obtained from coquasitriangular Hopf algebras
• Integrable perturbations of conformal field theories and Yetter-Drinfeld modules
• Braided Oscillators
• Duality Theorems for Infinite Braided Hopf Algebras
• ${\mathbb{Z}}_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms
• Modified toric code models with flux attachment from Hopf algebra gauge theory
• A monoidal structure on the category of relative Hopf modules
• On braided Poisson and quantum inhomogeneous groups