Quantum group

Summary

Quantum group is a common name for several families of Hopf algebras that deform classical objects attached to Lie groups and Lie algebras. The main families are the quantum enveloping algebras $U_q(\mathfrak{g})$ of Drinfeld and Jimbo, their duals the quantized coordinate algebras $\mathcal{O}_q(G)$, and Woronowicz's compact quantum groups. They are neither commutative nor cocommutative in general, and many carry a quasitriangular structure. This structure gives solutions of the Yang–Baxter equation and invariants of knots and links.