Название: A Course on Hopf Algebras
Автор: Rinat Kashaev
Издательство: Springer
Год: 2023
Страниц: 173
Язык: английский
Формат: pdf (true), epub
Размер: 14.2 MB

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 book is based on the one-semester introductory courses on Hopf algebras that the author taught on several occasions at the mathematical department of University of Geneva, and it is addressed mostly to those who learn the subject for the first time. Compared to the already existing vast literature on the subject, the distinguishing features of this book are as follows:

• We restrict ourselves to only minimal amount of material needed for applications in knot theory, namely, construction of solutions of the Yang–Baxter equations.
• The presentation is purely algebraic, essentially based on the multilinear algebra.
• When possible, the string diagram notation for monoidal categories is used which facilitates many proofs.
• The restricted (or finite) dual of a Hopf algebra plays the central role.
• The general theory is developed in a minimal way, while few simple examples are considered and worked out in detail.
• The universal R-matrix is given in dual terms as a bilinear form, so that for infinite dimensional Hopf algebras no completion is needed.

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