Freeness and divisibility for right $H$-simple left $H$-comodule algebras over a pointed Hopf algebra $H$

Abstract

Let $H$ be a pointed Hopf algebra and let $A$ be a right $H$-simple left $H$-comodule algebra. We show that every relative $(H,A)$-Hopf module is free as an $A$-module and give a characterization of when the category of relative $(H,A)$-Hopf modules is semisimple. We also give an embedding-type structure theorem for $A$ when $H$ and $A$ are $\mathbb{N}_0$-graded. As a consequence, we show that if $H$ is finite-dimensional and $A^{\mathrm{co} H}=\Bbbk$, then $\dim A$ divides $\dim H$.