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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 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.

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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".

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    $\begingroup$ Insofar as the typical reader of the site that comes across this question may have no easy access to the indicated text, this doesn't do much to answer the question. $\endgroup$ – zibadawa timmy Apr 27 '16 at 9:42
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    $\begingroup$ Since what the OP asked is rather standard, I preferred to give him a reference where finding the material he need. In general, I think that this is just the policy adopted on MathOverflow for this kind of questions. $\endgroup$ – Salvatore Siciliano Apr 27 '16 at 10:58
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    $\begingroup$ Well, I won't claim to be an expert on MO policy, though from what I've seen usually a brief summary of the answer is given, and the reference serves for the proof, and possibly a more general statement. Otherwise, fair enough. $\endgroup$ – zibadawa timmy Apr 27 '16 at 11:39
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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, 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.

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