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.

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