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  "body": "\n## Definition\n\nLet $\\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan matrix $(a_{ij})_{1 \\le i, j \\le n}$.\nLet $d_1, \\ldots, d_n$ be positive integers with $d_i a_{ij} = d_j a_{ji}$ for all $i, j$.\nLet $q \\in \\C$ be nonzero and not a [[root of unity|0S61N75H45]], and put $q_i = q^{d_i}$.\nFor 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 $\\left[{m \\atop r}\\right]_q = \\frac{[m]_q!}{[r]_q!\\,[m-r]_q!}$.\nThe *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:\n\n1. $K_i K_i^{-1} = K_i^{-1} K_i = 1$ and $K_i K_j = K_j K_i$.\n2. $K_i E_j K_i^{-1} = q_i^{a_{ij}} E_j$ and $K_i F_j K_i^{-1} = q_i^{-a_{ij}} F_j$.\n3. $E_i F_j - F_j E_i = \\delta_{ij} \\dfrac{K_i - K_i^{-1}}{q_i - q_i^{-1}}$.\n4. 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 $F$.\n\nIt 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 $\\Delta(F_i) = F_i \\otimes K_i^{-1} + 1 \\otimes F_i$.\n\n## Remarks\n\nExample: for $\\mathfrak{g} = \\mathfrak{sl}_2$ the Cartan matrix is $(2)$, $d_1 = 1$, and there is no relation 4.\nSo $U_q(\\mathfrak{sl}_2)$ is generated by $E, F, K^{\\pm 1}$ with $K E K^{-1} = q^2 E$, $K F K^{-1} = q^{-2} F$ and $EF - FE = (K - K^{-1})/(q - q^{-1})$.\nNon-example: the value $q = 1$ is excluded, since the denominator $q_i - q_i^{-1}$ in relation 3 vanishes.\nThe 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 $q = e^h$.\nSign and normalization conventions differ between textbooks; the coproduct is sometimes replaced by its opposite.\n",
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