comodule algebra

Let $H$ be a bialgebra.A right $H$-comodule algebra is a unital algebra $A$ which is a right $H$-comodulesatisfying

$$t(ab)=t(a)t(b)=\sum a_{(1)}{b}_{(1)}\otimes a_{(2)}{b}_{(2)},t({1\mathrm{I}}_A)={1\mathrm{I}}_A\otimes {1\mathrm{I}}_H,$$(1)

for all $h\in H$ and $a,b\in A$.

There is a dual notion of a $H$-module coalgebra.

Example 1

Let $H$ be a bialgebra.Then $H$ is itself a $H$-comodule algebra for the right regular coaction$t\mathit{}\mathrm{(}h\mathrm{)}\mathrm{=}\mathrm{\Delta }\mathit{}\mathrm{(}h\mathrm{)}$.