Comodules of Cosemisimple Hopf Algebras A cosemisimple Hopf algebra is one which is the sum of its cosimple sub-cobalgebras. Is it clear that a comodule of a cosemisimple Hopf algebra always decomposes into irreducible parts? Moreover, will this decomposition obey the property that the type and multiplicity of the irreducible comodules appear be the same in any decomposition.
I am sure that this should be the case but I can't see how to prove it. One thing that confuses me is the prospect of infinite multiplicity in the case of an infinite dimensional comodule.
Note: I previously asked this question on stackexchange, but moved it here after no response.
 A: I suggest you to have look, e.g., at Chapter XIV in the book "M.E. Sweedler: Hopf algebras", or even at Section 3.1 in the book "S. Dascalescu, C. Nastasescu, and S. Raianu: Hopf Algebras. An Introduction". 
A: Regarding your first question, I think the following definition and theorem settles the answer to the affirmative: 
Definition: A coalgebra $C$ is called right cosemisimple (or right completely reducible coalgebra) if the Category $M^C$ is a semisimple Category i.e. if every right $C$-comodule is cosemisimple. 
Similarly, left cosemisimple coalgebras are defined by the semisimplicity of the Category of left comodules. 
Theorem: Let $C$ be a coalgebra. The following assertions are equivalent:


*

*$C$ is a right cosemisimple coalgebra

*$C$ is a left cosemisimple coalgebra

*$C=C_0$

*Every left (right) rational $C^*$-module is semisimple


where $C_0=Corad(C)$ i.e. the coradical of $C$, which is the sum of all its simple subcoalgebras. 
For a proof of the above you can see for example Hopf algebras-an introduction, Dascalescu-Nastasescu-Raianu, Ch.3, p.118-119, Theorem 3.1.5. (However this is more or less standard material, you can also find it in Sweedler's book on Hopf algebras, S.Montgomery's book etc). 
Now, regarding your second question, I think you can use the above theorem together with the correspondence between $C$-comodules and rational $C^*$-modules to investigate the situation closer. 
