Quantum Group and Crystal Bases (updating...)
Introduction of the course
We follows the textbook Introduction to Quantum Groups and Crystal Bases written by Jin Hong and Seok-Jin Kang. I show my great gratitude to professor Huang Min for his rigorous and crystal clear introduction in this domain full of beauty.
Chapiter 1
In this Chapter, we review some important conclusions of Lie algebras and introduce Hopf algebras.
Here is my quick review on root system and the classification of semisimple Lie algebras: pdf
Chapiter 2
In this Chapter, we review those important conclusions of Kac–Moody algebras.
Chapiter 3
In this Chapter, we introduce Quantum groups as quantumed Kac–Moody algebras, how to recover the original Kac–Moody algebra from a given Quantum group via $\mathbb{A}_1$ forms and taking classical limit. We also introduce the corresponding category $\mathcal{O}^{q}_{int}$ of Quantum groups.
For lemma 3.2.5 and proposition 3.3.6, I viewed them in a different way to supplement the material in the textbook. Click the Bilibili Links to see my explanation.