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Record Nr. |
UNINA9910698648803321 |
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Autore |
Kashaev Rinat |
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Titolo |
A Course on Hopf Algebras / / by Rinat Kashaev |
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Pubbl/distr/stampa |
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Cham : , : Springer International Publishing : , : Imprint : Springer, , 2023 |
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ISBN |
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9783031263064 |
9783031263057 |
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Edizione |
[1st ed. 2023.] |
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Descrizione fisica |
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1 online resource (173 pages) |
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Collana |
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Universitext, , 2191-6675 |
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Disciplina |
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Soggetti |
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Associative rings |
Associative algebras |
Manifolds (Mathematics) |
Algebras, Linear |
Topological groups |
Lie groups |
Mathematical physics |
Algebra, Homological |
Associative Rings and Algebras |
Manifolds and Cell Complexes |
Linear Algebra |
Topological Groups and Lie Groups |
Mathematical Physics |
Category Theory, Homological Algebra |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Sommario/riassunto |
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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 |
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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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