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.
Author(s): Rinat Kashaev
Series: Universitext
Publisher: Springer
Year: 2023
Language: English
Pages: 172
City: Cham
Preface
Notation and Conventions
Contents
1 Groups and Hopf Algebras
1.1 Monoidal Categories
1.1.1 Monoidal Categories
1.1.2 Braided Monoidal Categories
1.1.3 The Graphical Notation of String Diagrams
1.2 Groups in Terms of Structural Maps
1.2.1 The Structural Maps of a Group in Graphical Notation
1.3 Monoids and Comonoids
1.4 Hopf Algebras
1.5 Group Algebras as Hopf Algebras
1.6 Algebras
1.6.1 Iterated Products
1.6.2 Modules
1.7 Coalgebras
1.7.1 Iterated Coproducts
1.7.2 Sweedler's Sigma Notation for the Iterated Coproducts
1.7.3 The Fundamental Theorem of Coalgebras
1.7.4 Comodules
1.8 Convolution Algebras
1.9 Some Properties of Hopf Algebras
1.10 Bialgebras
2 Constructions of Algebras, Coalgebras, Bialgebras, and Hopf Algebras
2.1 Construction of Algebras
2.1.1 The Tensor Algebra
2.1.2 The Universal Property of the Tensor Algebra
2.1.3 Presentations of Algebras
2.2 Construction of Coalgebras
2.2.1 Dual Coalgebras
2.2.2 Quotient Coalgebras
2.2.3 Direct Sum Coalgebras
2.3 Construction of Bialgebras
2.3.1 Presentations of Bialgebras
2.4 Construction of Hopf Algebras
2.4.1 Free Hopf Algebras on Coalgebras
2.4.2 Presentations of Hopf Algebras
3 The Restricted Dual of an Algebra
3.1 The Restricted Dual and Finite Dimensional Representations
3.1.1 An Algebra with Trivial Restricted Dual
3.1.2 An Infinite Dimensional Algebra A with Ao=A*
3.2 The Restricted Dual of the Tensor Product of Two Algebras
3.3 The Restricted Dual of a Hopf Algebra
4 The Restricted Dual of Hopf Algebras: Examples of Calculations
4.1 The Hopf Algebra C[x]
4.2 The Group Algebra C[Z]
4.3 The Hopf Algebra J.12em.1emdotteddotteddotted.76dotted.6h
4.4 The Quantum Group Bq
5 The Quantum Double
5.1 Bialgebras Twisted by Cocycles
5.1.1 Dual Pairings
5.2 Cobraided Bialgebras
5.2.1 The Quantum Double
5.3 The Quantum Double D(Bq)
5.3.1 Irreducible Representations of D(Bq)
5.3.2 Quantum Group Uq(sl2)
5.4 The Hopf Algebra D(B1)
5.4.1 The Restricted Dual Hopf Algebra B1o,op
5.4.2 The Quantum Double D(B1)
5.4.3 The Center of D(B1)
5.5 Solutions of the Yang–Baxter Equation
6 Applications in Knot Theory
6.1 Polygonal Links and Diagrams
6.1.1 Polygonal Knots and Links
6.1.2 Link Diagrams
6.1.3 Oriented Links and Diagrams
6.2 Long Knots
6.3 Invariants of Long Knots from Rigid r-Matrices
6.4 Rigid r-Matrices from Racks
6.4.1 Categories of Spans and Relations
6.4.2 Racks and Rigid r-Matrices in the Category of Spans
6.4.3 Racks Associated to Pointed Groups
6.5 The Alexander Polynomial as a Universal Invariant
6.5.1 Universal Knot Invariants from Hopf Algebras
6.5.2 The Universal Invariant Associted to B1
6.5.3 Schrödinger's Coherent States
6.5.4 A Dense Subspace of Hn
6.5.5 Gaussian Integration Formula
6.5.6 Representations of D(B1) in A1[[.12em.1emdotteddotteddotted.76dotted.6h]]
6.5.7 The Diagrammatic Rules for the Reshetikhin–Turaev Functor
6.6 The Alexander Polynomial from the Burau Representation
References
Index
