Category of bicomodules of a cosemisimple Hopf algebra

Question

A cosemisimple Hopf algebra $H$ is one which is equal to the direct sum of its subcoalgebras. As is well-known, this is equivalent to its category of $H$-comodules being semisimple. Is this also true for the category of bicomodules of the Hopf algebra? (See here for the definition of a bicomodule.)

Answer 1

The answer is yes, if we are talking about finite dimensional, Hopf algebras over a field:

$\bullet$ $H$ being cosemisimple (as a coalgebra) is equivalent to the dual hopf algebra $H^*$ being semisimple (as an algebra). But a semisimple Hopf algebra is also separable (see Ex.5.2.12). Now, $H^*$ being separable is equivalent (see th.6.1.2) to $H^*\otimes (H^*)^{op}$ being semisimple thus any (right) $H^*\otimes (H^*)^{op}$-module is semisimple. Since an $H^*\otimes (H^*)^{op}$-module is essentially the same thing as an $H^*$-bimodule (in the sense that the corresponding categories are isomorphic), we conclude that any $H^*$-bimodule is semisimple. This means that the category ${}_{H^*}\mathcal{M}_{H^*}$ of $H^*$-bimodules is semisimple.

Now, to answer your question, recall (see th.2.3.3) that the category $Rat({}_{H^*}\mathcal{M}_{H^*})$ of rational $H^*$-bimodules is isomorphic to the category ${}^{H}\mathcal{M}^H$ of $H$-bicomodules and that for a finite dimensional Hopf algebra $H$, all $H^*$-modules are rational, i.e. $$ {}^{H}\mathcal{M}^H\cong Rat({}_{H^*}\mathcal{M}_{H^*})={}_{H^*}\mathcal{M}_{H^*} $$ thus, the category of $H$-bicomodules ${}^{H}\mathcal{M}^H$ is isomorphic to the (semisimple) category ${}_{H^*}\mathcal{M}_{H^*}$ of $H^*$-bimodules. (In general $Rat({}_{H^*}\mathcal{M}_{H^*})$ is a full subcategory of ${}_{H^*}\mathcal{M}_{H^*}$).

$\bullet$ For the converse, the above argument can easily run all the way back, recalling that any separable algebra is -by definition- semisimple.

Thus, the above shows that, for a finite dimensional hopf algebra $H$:

$H$ being cosemisimple is equivalent to the category ${}^{H}\mathcal{M}^H$ of $H$-bicomodules being semisimple.