Quasitriangular Hopf algebra

Summary

A quasitriangular Hopf algebra is a Hopf algebra $H$ together with an invertible element $R \in H \otimes H$. This element, the universal R-matrix, conjugates the coproduct into the opposite coproduct. The element $R$ satisfies the Yang–Baxter equation. For 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 $V \otimes W \to W \otimes V$. These isomorphisms make the category of $H$-modules a braided monoidal category, which is why such algebras feed knot invariants and quantum groups.