quantumgroup

Quantum groups are Hopf algebras that deform classical algebraic structures associated with Lie groups and Lie algebras. The term commonly refers to quantized enveloping algebras U_q(g), which deform the universal enveloping algebra U(g) of a finite-dimensional semisimple Lie algebra g, and to dual quantized function algebras O_q(G) that deform coordinate rings of algebraic groups. Introduced independently by Drinfeld and Jimbo in the 1980s, quantum groups arose from solutions to the quantum Yang-Baxter equation and their role in integrable models.

Structurally, quantum groups are typically quasi-triangular Hopf algebras equipped with a universal R-matrix R that satisfies the Yang-Baxter equation. This R-matrix endows the category of representations with a braided tensor structure, permitting representations of braid groups and yielding knot and link invariants via constructions such as Reshetikhin–Turaev. In the limit q → 1, these algebras recover the classical enveloping algebras U(g); at roots of unity, they admit finite-dimensional quotients with rich representation theory, including the so-called small quantum groups.

Prominent examples include U_q(sl_2) and quantum affine algebras; dual objects include O_q(G) for groups such as SL_n. Quantum groups connect to many areas of mathematics and physics, with applications to integrable systems, exactly solvable models, and three-dimensional topological quantum field theories. They also play a central role in categorification programs and in the study of braided tensor categories arising from quantum group representations.