Cosemisimple Hopf algebra and Krull-Schmidt 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 Krull-Schmidt, by which I mean will the type and multiplicity of the irreducible comodules appearing 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.
 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, 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. 
