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By Ross Street
Publisher: Cambridge University Press
Print Publication Year:2007
Online Publication Date:December 2009
Online ISBN:9780511618505
Paperback ISBN:9780521695244
Book DOI: http://dx.doi.org/10.1017/CBO9780511618505
Subjects: Logic, categories and Sets , Theoretical Physics and Mathematical Physics
Algebra 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.
Reviews:
pp. i-vi
pp. vii-viii
pp. ix-xviii
1 - Revision of basic structures: Read PDF
pp. 1-4
2 - Duality between geometry and algebra: Read PDF
pp. 5-8
3 - The quantum general linear group: Read PDF
pp. 9-12
4 - Modules and tensor products: Read PDF
pp. 13-20
pp. 21-26
pp. 27-36
7 - Coalgebras and bialgebras: Read PDF
pp. 37-46
8 - Dual coalgebras of algebras: Read PDF
pp. 47-50
pp. 51-58
10 - Representations of quantum groups: Read PDF
pp. 59-66
11 - Tensor categories: Read PDF
pp. 67-76
12 - Internal homs and duals: Read PDF
pp. 77-84
13 - Tensor functors and Yang–Baxter operators: Read PDF
pp. 85-92
14 - A tortile Yang–Baxter operator for each finite–dimensional vector space: Read PDF
pp. 93-96
15 - Monoids in tensor categories: Read PDF
pp. 97-108
16 - Tannaka duality: Read PDF
pp. 109-116
17 - Adjoining an antipode to a bialgebra: Read PDF
pp. 117-118
18 - The quantum general linear group again: Read PDF
pp. 119-120
19 - Solutions to Exercises: Read PDF
pp. 121-132
pp. 133-134
pp. 135-141