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  "body": "\n## Summary\n\nA quasitriangular Hopf algebra is a [[Hopf algebra|7XFYP0N84H]] $H$ together with an invertible element $R \\in H \\otimes H$.\nThis element, the universal R-matrix, conjugates the coproduct into the opposite coproduct.\nThe element $R$ satisfies the [[Yang–Baxter equation|663BP8ZHS5]].\nFor two modules $V, W$ over $H$, the map $v \\otimes w \\mapsto \\tau(R \\cdot (v \\otimes w))$, with $\\tau$ the flip, is an isomorphism $V \\otimes W \\to W \\otimes V$.\nThese isomorphisms make the category of $H$-modules a [[braided monoidal category|VT8SFWT0XW]], which is why such algebras feed knot invariants and [[quantum groups|AP2ZDXANKC]].\n\n## Explanations\n\n### L1 human\n\nFor a group, the representations $V \\otimes W$ and $W \\otimes V$ are the same up to swapping the factors.\nFor a general Hopf algebra the plain swap is no longer a map of modules.\nA quasitriangular structure supplies a corrected swap: first act by a fixed element $R$, then swap.\nIn the simplest case, the group algebra of a group with $R = 1 \\otimes 1$, the corrected swap is the ordinary swap.\nThe interesting examples, such as the Drinfeld double of a finite-dimensional Hopf algebra, can have a corrected swap that applied twice is not the identity.\n",
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