Quantum Groups: A Path to Current AlgebraAlgebra has moved well beyond the topics discussed in standard undergraduate texts on 'modern algebra'. Those books typically dealt with algebraic structures such as groups, rings and fields: still very important concepts! However Quantum Groups: A Path to Current Algebra is written for the reader at ease with at least one such structure and keen to learn algebraic concepts and techniques. A key to understanding these new developments is categorical duality. A quantum group is a vector space with structure. Part of the structure is standard: a multiplication making it an 'algebra'. Another part is not in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a 'coalgebra'. While coalgebras, bialgebras and Hopf algebras have been around for half a century, the term 'quantum group', along with revolutionary new examples, was launched by Drinfel'd in 1986. |
Contents
Revision of basic structures | 1 |
Duality between geometry and algebra | 5 |
The quantum general linear group | 9 |
Modules and tensor products | 13 |
Cauchy modules | 21 |
Algebras | 27 |
Coalgebras and bialgebras | 37 |
Dual coalgebras of algebras | 47 |
Internal homs and duals | 77 |
Tensor functors and YangBaxter operators | 85 |
A tortile YangBaxter operator for each finitedimensional vector space | 93 |
Monoids in tensor categories | 97 |
Tannaka duality | 109 |
Adjoining an antipode to a bialgebra | 117 |
The quantum general linear group again | 119 |
Solutions to Exercises | 121 |
Hopf algebras | 51 |
Representations of quantum groups | 59 |
Tensor categories | 67 |
| 133 | |
| 135 | |
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Common terms and phrases
A A A A-point abelian group algebra H algebra morphism algebra structure antipode axioms bialgebra bijection bimonoid braided tensor category C-comodule called canonical isomorphisms cartesian product Cat/V Cauchy R-module Chapter coalgebra cocommutative commutative ring composite comultiplication CORC counit defined denote dualizing equations Example Exercise finite following diagram free module function f functor F given GL(n Hom(M homomorphism Homp HomR Hopf algebra internal hom left dual left R-module Lie algebra linear function M₁ ModŔ module morphism Mody Mon(V monoid monoid arrows monoid morphism MORM morphism f natural transformation object obtain pair precisely Proof Proposition quantum group R-algebra R-coalgebra R-linear R-module R(G)-module ring morphism satisfying scalar multiplication submodule subset Suppose surjective tensor category tensor product Theorem 5.2 twist uniquely determined vector space weak tensor functor write YB-operator μεμ