module coalgebra

Let $H$ be a bialgebra.A left $H$-module coalgebra is a coalgebra $A$ which is a left $H$-modulewith action $h▷a$ satisfying

$$\mathrm{\Delta }(h▷a)=\sum (h_{(1)}▷a_{(1)})\otimes (h_{(2)}▷a_{(2)}),\epsilon (h▷a)=\epsilon (h)\epsilon (a),$$(1)

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

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

Example 1

Let $H$ be a bialgebra.Then $H$ is itself a $H$-module coalgebra for the left regular action$g\mathrm{▷}h\mathrm{=}g\mathit{}h$.