Algebras and Hopf Algebras in Braided Categories
نویسنده
چکیده
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, as well 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. One of the main motivations of the theory of Hopf algebras is that they provide a generalization of groups. Hopf algebras of functions on groups provide examples of commutative Hopf algebras, but it turns out that many group-theoretical constructions work just as well when the Hopf algebra is allowed to be non-commutative. This is the philosophy associated to some kind of non-commutative (or so-called quantum) algebraic geometry. In a Hopf algebra context one can say the same thing in a dual way: group algebras and enveloping algebras are cocommutative but many constructions are not tied to this. This point of view has been highly successful in recent years, especially in regard to the quasitriangular Hopf algebras of Drinfeld[11]. These are non-cocommutative but the non-cocommutativity is controlled by a quasitriangular structure R. Such objects are commonly called quantum groups. Coming out of physics, notably associated to solutions of the Quantum YangBaxter Equations (QYBE) is a rich supply of quantum groups. Here we want to describe some kind of rival or variant of these quantum groups, which we call braided groups[44]–[52]. These are motivated by an earlier revolution that was very popular some decades ago in mathematics and physics, namely the theory of super or 1991 Mathematics Subject Classification 18D10, 18D35, 16W30, 57M25, 81R50, 17B37 This paper is in final form and no version of it will be submitted for publication elsewhere SERC Fellow and Fellow of Pembroke College, Cambridge
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