nLab
coquasitriangular bialgebra

Coquasitriangularity is dual property to quasitriangularity.

A $k$-bialgebra (or, in particular, Hopf algebra) $(H,m,\eta ,\Delta ,ϵ)$ 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 ${\mathrm{Hom}}_k(H\otimes H,k)$ (with respect to the convolution-unit $ϵ\otimes ϵ$) with a convolution inverse denoted $\overline{R}$ such that the opposite multiplication $m_{H_{\mathrm{op}}}:=m\circ \tau$ is given by

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

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

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

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