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.

Rinat Kashaev is a professor at the department of mathematics of the University of Geneva, where he has worked since 2002. His research covers topics in quantum topology, representation theory of Hopf algebras, integrable systems of low dimensional lattice statistical mechanics and quantum field theory, as well as Yang-Baxter and tetrahedron equations. He is the originator of the Volume Conjecture that relates quantum invariants of knots to the hyperbolic geometry of knot complements.

- 1. Groups and Hopf Algebras. - 2. Constructions of Algebras, Coalgebras, Bialgebras, and Hopf Algebras. - 3. The Restricted Dual of an Algebra. - 4. The Restricted Dual of Hopf Algebras: Examples of Calculations. - 5. The Quantum Double. - 6. Applications in Knot Theory.