# 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.

###### Example 1

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

Title module coalgebra ModuleCoalgebra 2013-03-22 13:26:37 2013-03-22 13:26:37 mhale (572) mhale (572) 9 mhale (572) Definition msc 16W30 ComoduleAlgebra ModuleAlgebra ComoduleCoalgebra