Introduction to Quantum Groups and Crystal Bases

Introduction to Quantum Groups and Crystal Bases, Volume 1, provides a foundational exploration of the algebraic structures known as quantum groups. Originally introduced by V.G. Drinfel'd and M. Jimbo through their research on the quantum Yang-Baxter equation, these Hopf algebras serve as deformations of universal enveloping algebras of Kac-Moody algebras. This mathematics text examines how these structures underpin diverse fields, including statistical mechanics, topological invariant theory, and geometric representation theory.

The book focuses on the theory of crystal bases, also called canonical bases, which were developed by M. Kashiwara and G. Lusztig. These bases offer significant combinatorial and geometric tools for analyzing the representations of quantum groups. Jin Hong presents an elementary introduction to these complex topics, emphasizing the combinatorial aspects of the theory. This volume serves as a specialized resource for students and researchers studying advanced mathematics and mathematical physics.