25 Years of Quantum Groups: from Definition to Classification

Abstract

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

Keywords

• Lie bialgebra
• quantum group
• Yang-Baxter equation
• quantization
• classical twist
• quantum twist
• integrable model