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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.
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.
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
| Erscheinungsjahr: | 2023 |
|---|---|
| Fachbereich: | Arithmetik & Algebra |
| Genre: | Mathematik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Taschenbuch |
| Reihe: | Universitext |
| Inhalt: |
xv
165 S. |
| ISBN-13: | 9783031263057 |
| ISBN-10: | 3031263057 |
| Sprache: | Englisch |
| Ausstattung / Beilage: | Paperback |
| Einband: | Kartoniert / Broschiert |
| Autor: | Kashaev, Rinat |
| Auflage: | 1st ed. 2023 |
| Hersteller: |
Springer Nature Switzerland
Springer International Publishing Springer International Publishing AG Universitext |
| Maße: | 235 x 155 x 11 mm |
| Von/Mit: | Rinat Kashaev |
| Erscheinungsdatum: | 15.04.2023 |
| Gewicht: | 0,289 kg |
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
| Erscheinungsjahr: | 2023 |
|---|---|
| Fachbereich: | Arithmetik & Algebra |
| Genre: | Mathematik |
| Rubrik: | Naturwissenschaften & Technik |
| Medium: | Taschenbuch |
| Reihe: | Universitext |
| Inhalt: |
xv
165 S. |
| ISBN-13: | 9783031263057 |
| ISBN-10: | 3031263057 |
| Sprache: | Englisch |
| Ausstattung / Beilage: | Paperback |
| Einband: | Kartoniert / Broschiert |
| Autor: | Kashaev, Rinat |
| Auflage: | 1st ed. 2023 |
| Hersteller: |
Springer Nature Switzerland
Springer International Publishing Springer International Publishing AG Universitext |
| Maße: | 235 x 155 x 11 mm |
| Von/Mit: | Rinat Kashaev |
| Erscheinungsdatum: | 15.04.2023 |
| Gewicht: | 0,289 kg |
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