comodule coalgebra

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

$$(\mathrm{\Delta }\otimes \mathrm{id})t(a)=\sum a_{(1)(1)}\otimes a_{(2)(1)}\otimes a_{(1)(2)}{a}_{(2)(2)},(\epsilon \otimes \mathrm{id})t(a)=\epsilon (a){1\mathrm{I}}_H,$$(1)

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

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

Example 1

Let $H$ be a Hopf algebra.Then $H$ is itself a $H$-comodule coalgebra for the adjoint coaction$t\mathit{}\mathrm{(}h\mathrm{)}\mathrm{=}{h}_{\mathrm{(}\mathrm{2}\mathrm{)}}\mathrm{\otimes }S\mathit{}\mathrm{(}{h}_{\mathrm{(}\mathrm{1}\mathrm{)}}\mathrm{)}\mathit{}{h}_{\mathrm{(}\mathrm{3}\mathrm{)}}$.