Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 2 / 3 / a / Solution

A cobraided bialgebra is a bialgebra $H$ equipped with a convolution-invertible bilinear form $r:H\otimes H\to k$ satisfying the following identities, in Sweedler notation:Convolution invertibility means there is $\bar{r}$ with $\sum r(a_{(1)},b_{(1)})\bar{r}(a_{(2)},b_{(2)})=\epsilon (a)\epsilon (b)$, and likewise with $r,\bar{r}$ interchanged. This is a coquasitriangular structure. On right comodules it induces the braidingThe third identity makes this a comodule map, the first two give its two hexagon identities, and the convolution inverse gives the inverse braiding.