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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Table of contents (6 chapters)
Keywords
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.
Authors and Affiliations
About the author
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