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  "body": "\n## Summary\n\nQuantum group is a common name for several families of [[Hopf algebras|7XFYP0N84H]] that deform classical objects attached to Lie groups and Lie algebras.\nThe main families are the [[quantum enveloping algebras|1V8H34F6N9]] $U_q(\\mathfrak{g})$ of Drinfeld and Jimbo, their duals the quantized coordinate algebras $\\mathcal{O}_q(G)$, and Woronowicz's [[compact quantum groups|FHQKT6HPW2]].\nThey are neither commutative nor cocommutative in general, and many carry a [[quasitriangular|NCRPWS9W5R]] structure.\nThis structure gives solutions of the [[Yang–Baxter equation|663BP8ZHS5]] and invariants of knots and links.\n\n## Explanations\n\n### L1 human\n\nA group can be studied through the algebra of functions on it.\nThat algebra is commutative.\nA quantum group is what one gets by allowing this algebra of functions to become noncommutative while keeping a rule for \"multiplying points\", the coproduct.\nA typical example depends on a parameter $q$: at $q = 1$ it is the coordinate ring of $SL_2$, and for other $q$ the coordinates $a, b$ satisfy $ab = q\\, ba$ instead of $ab = ba$.\nThere is no actual group behind it, only the algebra, but much of the theory of groups and their representations survives.\n",
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