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＜E-Book＞
A Course on Hopf Algebras / by Rinat Kashaev
(Universitext. ISSN:21916675)

Edition	1st ed. 2023.
Publisher	Cham : Springer International Publishing : Imprint: Springer
Year	2023
Language	English
Size	XV, 165 p : online resource
Authors	*Kashaev, Rinat author SpringerLink (Online service)
Subjects	LCSH:Associative rings LCSH:Associative algebras LCSH:Manifolds (Mathematics) LCSH:Algebras, Linear LCSH:Topological groups LCSH:Lie groups LCSH:Mathematical physics LCSH:Algebra, Homological FREE:Associative Rings and Algebras FREE:Manifolds and Cell Complexes FREE:Linear Algebra FREE:Topological Groups and Lie Groups FREE:Mathematical Physics FREE:Category Theory, Homological Algebra
Notes	This textbook provides a concise, visual introduction to Hopf algebras and their application to knot theory, most notably the construction of solutions of the Yang–Baxter equations. 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 HTTP:URL=https://doi.org/10.1007/978-3-031-26306-4

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