---
tag: AP2ZDXANKC
kind: concept
name: Quantum group
msc: [16T20, 17B37, 81R50]
specializes: [7XFYP0N84H]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_group
same_as:
  wikidata: Q2122223
  wikipedia: Quantum group
---

## Summary

Quantum group is a common name for several families of [[Hopf algebras|7XFYP0N84H]] that deform classical objects attached to Lie groups and Lie algebras.
The main families are the [[quantum enveloping algebras|1V8H34F6N9]] $U_q(\mathfrak{g})$ of Drinfeld and Jimbo, their duals the quantized coordinate algebras $\mathcal{O}_q(G)$, and Woronowicz's [[compact quantum groups|FHQKT6HPW2]].
They are neither commutative nor cocommutative in general, and many carry a [[quasitriangular|NCRPWS9W5R]] structure.
This structure gives solutions of the [[Yang–Baxter equation|663BP8ZHS5]] and invariants of knots and links.

## Explanations

### L1 human

A group can be studied through the algebra of functions on it.
That algebra is commutative.
A quantum group is what one gets by allowing this algebra of functions to become noncommutative while keeping a rule for "multiplying points", the coproduct.
A typical example depends on a parameter $q$: at $q = 1$ it is the coordinate ring of $SL_2$, and for other $q$ the coordinates $a, b$ satisfy $ab = q\, ba$ instead of $ab = ba$.
There is no actual group behind it, only the algebra, but much of the theory of groups and their representations survives.