= 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>:
$$
\begin{aligned}
r(ab,c)&=\sum r(a,c_{(1)})r(b,c_{(2)}),\\
r(a,bc)&=\sum r(a_{(1)},c)r(a_{(2)},b),\\
\sum r(a_{(1)},b_{(1)})a_{(2)}b_{(2)}
&=\sum b_{(1)}a_{(1)}r(a_{(2)},b_{(2)}),\\
r(1,a)&=r(a,1)=\varepsilon(a).
\end{aligned}
$$
Convolution invertibility means there is $\bar r$ with $\sum r(a_{(1)},b_{(1)})\bar r(a_{(2)},b_{(2)})=\varepsilon(a)\varepsilon(b)$, and likewise with $r,\bar r$ interchanged. This is a <coquasitriangular structure>. On right <comodules> it induces the <braiding>
$$
c(v\otimes w)=\sum w_{(0)}\otimes v_{(0)}r(v_{(1)},w_{(1)}).
$$
The third identity makes this a <comodule> map, the first two give its two hexagon identities, and the convolution inverse gives the inverse <braiding>.
Back to article page
