comodule coalgebra

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

(\Delta\otimes\mathrm{id})t(a)=\sum a_{(1)(1)}\otimes a_{(2)(1)}\otimes a_{(1)% (2)}a_{(2)(2)},\quad(\varepsilon\otimes\mathrm{id})t(a)=\varepsilon(a)\mathord% {\mathrm{1\!\!\!\>I}}_{H},

(1)

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

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

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