module coalgebra

Let $H$ be a bialgebra. A left $H$ -module coalgebra is a coalgebra $A$ which is a left $H$ -module with action $h\triangleright a$ satisfying

\Delta(h\triangleright a)=\sum(h_{(1)}\triangleright a_{(1)})\otimes(h_{(2)}% \triangleright a_{(2)}),\quad\varepsilon(h\triangleright a)=\varepsilon(h)% \varepsilon(a),

(1)

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

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

Let $H$ be a bialgebra. Then $H$ is itself a $H$ -module coalgebra for the left regular action $g\triangleright h=gh$ .