comodule coalgebra

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

$$(\mathrm{\Delta }\otimes \mathrm{id})t(a)=\sum a_{(1)(1)}\otimes a_{(2)(1)}\otimes a_{(1)(2)}{a}_{(2)(2)},(\epsilon \otimes \mathrm{id})t(a)=\epsilon (a){1\mathrm{I}}_H,$$ (1)

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

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

Example 1

Let $H$ be a Hopf algebra. Then $H$ is itself a $H$-comodule coalgebra for the adjoint coaction $t\mathit{}\mathrm{(}h\mathrm{)}\mathrm{=}{h}_{\mathrm{(}\mathrm{2}\mathrm{)}}\mathrm{\otimes }S\mathit{}\mathrm{(}{h}_{\mathrm{(}\mathrm{1}\mathrm{)}}\mathrm{)}\mathit{}{h}_{\mathrm{(}\mathrm{3}\mathrm{)}}$.

Title	comodule coalgebra
Canonical name	ComoduleCoalgebra
Date of creation	2013-03-22 13:26:39
Last modified on	2013-03-22 13:26:39
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	7
Author	mhale (572)
Entry type	Definition
Classification	msc 16W30
Related topic	ModuleAlgebra
Related topic	ModuleCoalgebra
Related topic	ComoduleAlgebra