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  "body": "\n## Summary\n\nThe quantum enveloping algebra $U_q(\\mathfrak{g})$ is also called the quantized universal enveloping algebra or Drinfeld–Jimbo quantum group.\nIt is a one-parameter deformation of the [[universal enveloping algebra|SCZ8PRVY00]] of a semisimple Lie algebra $\\mathfrak{g}$.\nIt 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 $\\mathfrak{g}$.\nFor $q$ not a root of unity its finite-dimensional representations closely mirror those of $\\mathfrak{g}$.\nIt is the main example of a [[quantum group|AP2ZDXANKC]], and suitable versions of it are [[quasitriangular|NCRPWS9W5R]].\n\n## Explanations\n\n### L1 human\n\nA Lie algebra is described by generators and relations, and its enveloping algebra turns these into an associative algebra.\nThe quantum enveloping algebra keeps the same generators but bends the relations by a number $q$.\nFor $\\mathfrak{sl}_2$ the classical relation $[E, F] = H$ becomes $EF - FE = (K - K^{-1})/(q - q^{-1})$, where $K$ plays the role of $q^H$.\nWhen $q$ tends to $1$, the right side tends formally to $H$, and the classical algebra comes back.\nThe deformation keeps a coproduct, so tensor products of representations still make sense, but they are no longer symmetric in the two factors.\n",
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