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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.

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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:

  1. $C$ is a right cosemisimple coalgebra
  2. $C$ is a left cosemisimple coalgebra
  3. $C=C_0$
  4. 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.

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