module algebra

Let $H$ be a bialgebra. A left $H$ -module algebra is a unital algebra $A$ which is a left $H$ -module with action $h\triangleright a$ satisfying

h\triangleright(ab)=\sum(h_{(1)}\triangleright a)(h_{(2)}\triangleright b),% \quad h\triangleright\mathord{\mathrm{1\!\!\!\>I}}_{A}=\varepsilon(h)\mathord{% \mathrm{1\!\!\!\>I}}_{A},

(1)

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

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

Let $H$ be a Hopf algebra. Then $H$ is itself a $H$ -module algebra for the adjoint action $g\triangleright h=\sum g_{(1)}hS(g_{(2)})$ .