---
tag: FTQDGFRRS0
kind: definition
name: Quantum group
msc: [16T20, 17B37, 81R50]
defines: AP2ZDXANKC
uses: [7XFYP0N84H, FHQKT6HPW2]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_group
---

## Definition

The term *quantum group* has no single agreed definition.
In the broad sense of Drinfeld, a quantum group is a [[Hopf algebra|7XFYP0N84H]] that is in general neither commutative nor cocommutative.
The base ring is a field or a ring of formal power series in a parameter $h$.
In the narrow sense, it is one of the following families, each depending on a parameter $q$ and recovering a classical object at $q = 1$:

1. the quantized enveloping algebra $U_q(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$, a deformation of the universal enveloping algebra;
2. the quantized coordinate algebra $\mathcal{O}_q(G)$ of a semisimple algebraic group $G$, a deformation of the coordinate ring;
3. a [[compact quantum group|FHQKT6HPW2]] in the sense of Woronowicz, a C*-algebraic analogue of the algebra of continuous functions on a compact group.

## Remarks

Example: let $k$ be a field and $q \in k$ nonzero.
The algebra $\mathcal{O}_q(SL_2)$ is generated by $a, b, c, d$ with the relations
$$ab = q\,ba, \quad ac = q\,ca, \quad bd = q\,db, \quad cd = q\,dc, \quad bc = cb,$$
$$ad - da = (q - q^{-1})\,bc, \qquad ad - q\,bc = 1.$$
Its coproduct is matrix multiplication of the generator matrix: $\Delta(a) = a \otimes a + b \otimes c$, $\Delta(b) = a \otimes b + b \otimes d$, $\Delta(c) = c \otimes a + d \otimes c$, $\Delta(d) = c \otimes b + d \otimes d$.
At $q = 1$ the relations say that $a, b, c, d$ commute and $ad - bc = 1$, which is the coordinate ring of $SL_2$.
Non-example in the broad sense: the group algebra $k[G]$ of a group is a Hopf algebra, but it is cocommutative.
So it is a classical object rather than a genuine quantum group.