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  "body": "\n## Definition\n\nThe term *quantum group* has no single agreed definition.\nIn the broad sense of Drinfeld, a quantum group is a [[Hopf algebra|7XFYP0N84H]] that is in general neither commutative nor cocommutative.\nThe base ring is a field or a ring of formal power series in a parameter $h$.\nIn the narrow sense, it is one of the following families, each depending on a parameter $q$ and recovering a classical object at $q = 1$:\n\n1. the quantized enveloping algebra $U_q(\\mathfrak{g})$ of a semisimple Lie algebra $\\mathfrak{g}$, a deformation of the universal enveloping algebra;\n2. the quantized coordinate algebra $\\mathcal{O}_q(G)$ of a semisimple algebraic group $G$, a deformation of the coordinate ring;\n3. a [[compact quantum group|FHQKT6HPW2]] in the sense of Woronowicz, a C*-algebraic analogue of the algebra of continuous functions on a compact group.\n\n## Remarks\n\nExample: let $k$ be a field and $q \\in k$ nonzero.\nThe algebra $\\mathcal{O}_q(SL_2)$ is generated by $a, b, c, d$ with the relations\n$$ab = q\\,ba, \\quad ac = q\\,ca, \\quad bd = q\\,db, \\quad cd = q\\,dc, \\quad bc = cb,$$\n$$ad - da = (q - q^{-1})\\,bc, \\qquad ad - q\\,bc = 1.$$\nIts coproduct is matrix multiplication of the generator matrix: $\\Delta(a) = a \\otimes a + b \\otimes c$, $\\Delta(b) = a \\otimes b + b \\otimes d$, $\\Delta(c) = c \\otimes a + d \\otimes c$, $\\Delta(d) = c \\otimes b + d \\otimes d$.\nAt $q = 1$ the relations say that $a, b, c, d$ commute and $ad - bc = 1$, which is the coordinate ring of $SL_2$.\nNon-example in the broad sense: the group algebra $k[G]$ of a group is a Hopf algebra, but it is cocommutative.\nSo it is a classical object rather than a genuine quantum group.\n",
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