Overview
- Includes advanced applications to knot theory, such as the construction of solutions to Yang–Baxter equations
- Uses string diagrams to facilitate understanding
- Assumes only minimal background for most of the book
Part of the book series: Universitext (UTX)
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About this book
Starting with a reformulation of the definition of a group in terms of structural maps as motivation for the definition of a Hopf algebra, the book introduces the related algebraic notions: algebras, coalgebras, bialgebras, convolution algebras, modules, comodules. Next, Drinfel’d’s quantum double construction is achieved through the important notion of the restricted (or finite) dual of a Hopf algebra, which allows one to work purely algebraically, without completions. As a result, in applications to knot theory, to any Hopf algebra with invertible antipode one can associate a universal invariant of long knots. These constructions are elucidated in detailed analyses of a few examples of Hopf algebras.
The presentation of the material is mostly based on multilinear algebra, with all definitions carefully formulated and proofs self-contained. The general theory is illustrated with concrete examples, and many technicalities are handled with the help of visual aids, namely string diagrams. As a result, most of this text is accessible with minimal prerequisites and can serve as the basis of introductory courses to beginning graduate students.
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Table of contents (6 chapters)
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Bibliographic Information
Book Title: A Course on Hopf Algebras
Authors: Rinat Kashaev
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-031-26306-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Softcover ISBN: 978-3-031-26305-7Published: 15 April 2023
eBook ISBN: 978-3-031-26306-4Published: 12 April 2023
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XV, 165
Topics: Associative Rings and Algebras, Topology, Linear Algebra, Topological Groups, Lie Groups, Mathematical Physics, Category Theory, Homological Algebra