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 $g\\triangleright h=\\sum g_{(1)}hS(g_{(2)})$ .