Quasitriangular Hopf algebra

Definition
A Hopf algebra $$H$$ with product $$\nabla$$, unit $$\eta$$, coproduct $$\Delta$$ and counit $$\epsilon$$, and where $$\tau: H \otimes H \to H \otimes H$$ is the unique linear map sending any $$x \otimes y$$ to $$y \otimes x$$.

is said to be quasitriangular if there exists $$R \in H \otimes H$$ such that:


 * $$\nabla(R \otimes \Delta(x)) = \nabla((\tau \circ \Delta)(x) \otimes R)$$
 * $$(\Delta \otimes id) R = R_{13}R_{23}$$
 * $$(id \otimes \Delta) R = R_{13}R_{12}$$

Here, $$R_{ij} = \phi_{ij}(R)$$ where:


 * $$\phi_{12}(a \otimes b) = a \otimes b \otimes \eta$$
 * $$\phi_{23}(a \otimes b) = \eta \otimes a \otimes b$$
 * $$\phi_{13}(a \otimes b) = a \otimes \eta \otimes b$$

The $$R$$ is often considered part of the quasitriangular structure and is called the R-matrix of the algebra.

Solution to Yang-Baxter equation
The R-matrix of a quasitriangular Hopf algebra is a solution to the Yang-Baxter equation.