nLab coquasitriangular bialgebra

Coquasitriangularity is dual property to quasitriangularity.

A $k$-bialgebra (or, in particular, Hopf algebra) $(H, m,\eta,\Delta,\epsilon)$ is coquasitriangular (or dual quasitriangular) if it is equipped with a $k$-linear map $R:H\otimes H\to k$ which is invertible in convolution algebra $Hom_k(H\otimes H,k)$ (with respect to the convolution-unit $\epsilon\otimes\epsilon$) with a convolution inverse denoted $\bar{R}$ such that the opposite multiplication $m_{H_{op}} := m\circ \tau$ is given by

$$m_{H_{op}} = R\star m\star \bar{R}$$

and the following two identities hold when applied on $H\otimes H\otimes H$:

$$R(m\otimes id) = R_{13} R_{23}$$

$$R(id\otimes m) = R_{12} R_{23}$$

with the subscript notation as explained in the $n$lab entry quasitriangular Hopf algebra. The main examples come from quantized function algebras (that is, roughly, dual of quantized enveloping algebras).