An introduction to quantized Lie groups and algebras

TL;DR

This paper provides a comprehensive introduction to quantum groups, covering their formal theory, relation to Lie structures, quantization methods, and applications to the Yang-Baxter equation and representation theory.

Contribution

It offers a detailed, self-contained overview of quantum groups, including explicit examples like quantized $sl_2$, and explores their applications in solving the Yang-Baxter equation.

Findings

Explicit quantization of $sl_2$ demonstrated

Universal $R$-matrix constructed for quantum $sl_2$

Finite-dimensional irreducible representations at roots of unity analyzed

Abstract

We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of $sl_2$ is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal $R$-matrix for the quantum $sl_2$ algebra. In the last section we deduce all finite dimensional irreducible representations for $q$ a root of unity. We also give their tensor product decomposition (fusion rules) which is relevant to conformal field theory.