# 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.

###### Example 1

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

Title comodule coalgebra ComoduleCoalgebra 2013-03-22 13:26:39 2013-03-22 13:26:39 mhale (572) mhale (572) 7 mhale (572) Definition msc 16W30 ModuleAlgebra ModuleCoalgebra ComoduleAlgebra