---
tag: JGTGXJB7V5
kind: definition
name: Quantum enveloping algebra
msc: [17B37, 16T20]
defines: 1V8H34F6N9
uses: [7XFYP0N84H, SCZ8PRVY00, 0S61N75H45]
sources:
  - url: https://en.wikipedia.org/wiki/Quantum_group
    locator: Drinfeld–Jimbo type quantum groups
---

## Definition

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan matrix $(a_{ij})_{1 \le i, j \le n}$.
Let $d_1, \ldots, d_n$ be positive integers with $d_i a_{ij} = d_j a_{ji}$ for all $i, j$.
Let $q \in \C$ be nonzero and not a [[root of unity|0S61N75H45]], and put $q_i = q^{d_i}$.
For an integer $m \ge 0$ put $[m]_{q} = \frac{q^m - q^{-m}}{q - q^{-1}}$, $[m]_q! = [1]_q [2]_q \cdots [m]_q$, and $\left[{m \atop r}\right]_q = \frac{[m]_q!}{[r]_q!\,[m-r]_q!}$.
The *quantum enveloping algebra* $U_q(\mathfrak{g})$ is the $\C$-algebra with generators $E_i, F_i, K_i, K_i^{-1}$ for $1 \le i \le n$ and relations:

1. $K_i K_i^{-1} = K_i^{-1} K_i = 1$ and $K_i K_j = K_j K_i$.
2. $K_i E_j K_i^{-1} = q_i^{a_{ij}} E_j$ and $K_i F_j K_i^{-1} = q_i^{-a_{ij}} F_j$.
3. $E_i F_j - F_j E_i = \delta_{ij} \dfrac{K_i - K_i^{-1}}{q_i - q_i^{-1}}$.
4. For $i \ne j$, with $m = 1 - a_{ij}$: $\sum_{r=0}^{m} (-1)^r \left[{m \atop r}\right]_{q_i} E_i^{m-r} E_j E_i^{r} = 0$, and the same relation with every $E$ replaced by $F$.

It is a [[Hopf algebra|7XFYP0N84H]] with $\Delta(K_i) = K_i \otimes K_i$, $\Delta(E_i) = E_i \otimes 1 + K_i \otimes E_i$ and $\Delta(F_i) = F_i \otimes K_i^{-1} + 1 \otimes F_i$.

## Remarks

Example: for $\mathfrak{g} = \mathfrak{sl}_2$ the Cartan matrix is $(2)$, $d_1 = 1$, and there is no relation 4.
So $U_q(\mathfrak{sl}_2)$ is generated by $E, F, K^{\pm 1}$ with $K E K^{-1} = q^2 E$, $K F K^{-1} = q^{-2} F$ and $EF - FE = (K - K^{-1})/(q - q^{-1})$.
Non-example: the value $q = 1$ is excluded, since the denominator $q_i - q_i^{-1}$ in relation 3 vanishes.
The classical [[universal enveloping algebra|SCZ8PRVY00]] $U(\mathfrak{g})$ is recovered only through a limit, or through a version over formal power series in $h$ with $q = e^h$.
Sign and normalization conventions differ between textbooks; the coproduct is sometimes replaced by its opposite.