# comodule algebra

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

 $t(ab)=t(a)t(b)=\sum a_{(1)}b_{(1)}\otimes a_{(2)}b_{(2)},\quad t(\mathord{% \mathrm{1\!\!\!\>I}}_{A})=\mathord{\mathrm{1\!\!\!\>I}}_{A}\otimes\mathord{% \mathrm{1\!\!\!\>I}}_{H},$ (1)

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

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

###### Example 1

Let $H$ be a bialgebra. Then $H$ is itself a $H$-comodule algebra for the right regular coaction $t(h)=\Delta(h)$.

Title comodule algebra ComoduleAlgebra 2013-03-22 13:26:34 2013-03-22 13:26:34 mhale (572) mhale (572) 8 mhale (572) Definition msc 16W30 ModuleCoalgebra ModuleAlgebra ComoduleCoalgebra