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A brief introduction to quantum groups

Pavel Etingof, Mykola Semenyakin

Abstract

These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math Science Literature Lecture Series in May 2020. We attempt to give a bird's-eye view of basic aspects of the theory of quantum groups.

A brief introduction to quantum groups

Abstract

These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math Science Literature Lecture Series in May 2020. We attempt to give a bird's-eye view of basic aspects of the theory of quantum groups.

Paper Structure

This paper contains 37 sections, 38 theorems, 192 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hopf algebras and monoidal categories
  3. The definition of a Hopf algebra
  4. Properties and examples of Hopf algebras
  5. Monoidal categories
  6. Duals and rigid categories
  7. Monoidal functors
  8. Strict monoidal categories
  9. Problems
  10. Braided monoidal categories and quasitriangular Hopf algebras
  11. Braided monoidal categories and the Drinfeld center
  12. Yetter-Drinfeld modules and the quantum double
  13. Group actions on monoidal categories, equivariantization, and the Drinfeld center of ${\rm Vec}(G)$
  14. Tannakian reconstruction and quasitriangular Hopf algebras
  15. $U_q(\mathfrak{sl}_2)$ at a root of unity
  16. ...and 22 more sections

Key Result

Proposition 2.2

(i) If $H$ is a finite dimensional Hopf algebra then so is $H^*$, with the operations of $H^*$ being dual to the operations of $H$.Here we should view the unit of $H$ as a linear map $\iota: \Bbb C\to H$ such that $\iota(1)=1$. Then the dual of $\varepsilon_H$ is $\iota_{H^*}$. (ii) $\varepsilon$ is

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (97)

  • Definition 2.1
  • Proposition 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.11
  • ...and 87 more