A cosemisimple Hopf algebra is usually defined to a Hopf algebra which is the sum of its cosimple subcoalgebras. Does this definition assume that each cosimple subcoalgebra appears only once in the sum, or is this a consequence?


Also, assuming this is true, then the decomposition must be unique by a cosimplicity argument, i.e. given two decompositions $H = \bigoplus\_i H\_i = \bigoplus H'\_i$, with each $H\_i$ and $H'\_i$ cosemisimple, then for each $i$ there exists a $j$ such that $H\_i = H'\_j$.



Definition of a cosemisimple Hopf algebra
For the finite dimensional case, cosemisimple is the same as the dual being semisimple. So you can apply your knowledge of semisimple algebras. So the answer to your first question is no, you're allowed to have repeats just like semisimple algebras can have the same matrix factor repeated.


For uniqueness of decomposition, look at H as a comodule over itself, and use isotypic decomposition. This gives exactly the uniqueness statement you want.


