Quasitriangular Hopf algebra

This article defines a property that can be evaluated for any Hopf algebra

This article or section of article is sourced from:Wikipedia

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.

Facts

Solution to Yang-Baxter equation

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

External links

• Wikipedia page