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.

Example 1

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

Title module algebra ModuleAlgebra 2013-03-22 13:26:31 2013-03-22 13:26:31 mhale (572) mhale (572) 8 mhale (572) Definition msc 16W30 ComoduleCoalgebra ModuleCoalgebra ComoduleAlgebra