---
tag: 1V8H34F6N9
kind: concept
name: Quantum enveloping algebra
msc: [17B37, 16T20]
specializes: [AP2ZDXANKC]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_group
    locator: Drinfeld–Jimbo type quantum groups
---

## Summary

The quantum enveloping algebra $U_q(\mathfrak{g})$ is also called the quantized universal enveloping algebra or Drinfeld–Jimbo quantum group.
It is a one-parameter deformation of the [[universal enveloping algebra|SCZ8PRVY00]] of a semisimple Lie algebra $\mathfrak{g}$.
It is a [[Hopf algebra|7XFYP0N84H]] given by generators $E_i, F_i, K_i^{\pm 1}$ and relations read off from the Cartan matrix of $\mathfrak{g}$.
For $q$ not a root of unity its finite-dimensional representations closely mirror those of $\mathfrak{g}$.
It is the main example of a [[quantum group|AP2ZDXANKC]], and suitable versions of it are [[quasitriangular|NCRPWS9W5R]].

## Explanations

### L1 human

A Lie algebra is described by generators and relations, and its enveloping algebra turns these into an associative algebra.
The quantum enveloping algebra keeps the same generators but bends the relations by a number $q$.
For $\mathfrak{sl}_2$ the classical relation $[E, F] = H$ becomes $EF - FE = (K - K^{-1})/(q - q^{-1})$, where $K$ plays the role of $q^H$.
When $q$ tends to $1$, the right side tends formally to $H$, and the classical algebra comes back.
The deformation keeps a coproduct, so tensor products of representations still make sense, but they are no longer symmetric in the two factors.