Bialgebra scalar cocycle ($\gamma :H\otimes H\to k$)

A normalized scalar map $\gamma :H\otimes H\to k$ satisfying $\sum \gamma (a_{(1)},b_{(1)})\gamma (a_{(2)}{b}_{(2)},c)=\sum \gamma (b_{(1)},c_{(1)})\gamma (a,b_{(2)}{c}_{(2)})$, and $\gamma (1,a)=\epsilon (a)=\gamma (a,1)$. Some conventions require convolution invertibility; that extra hypothesis distinguishes strong from lax tensor transformations.

With the convention $m\ast \gamma =\gamma \ast n$ and invertible bialgebra scalar cocycle $\gamma$ for $n$, the same coalgebra has $n=\bar{\gamma }\ast m\ast \gamma$, or equivalently $m=\gamma \ast n\ast \bar{\gamma }$. Thus $n(a,b)=\sum \bar{\gamma }(a_{(1)},b_{(1)})m(a_{(2)},b_{(2)})\gamma (a_{(3)},b_{(3)})$. Specifying the equation prevents opposite twist conventions from being conflated.