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 that this freeness characterizes the class of pointed Hopf algebras. We give a criterion for the category of relative $(H,A)$-Hopf modules to be semisimple. We also show that $A$ can be embedded into a left $H$-comodule algebra of a specific form when $H$ and $A$ are ${\mathbb{N}}_0$-graded. As a consequence, we prove that if $H$ is finite-dimensional and $A^{\mathrm{co}H}=\mathbb{k}$, then $A$ is finite-dimensional and $\dim A$ divides $\dim H$.

31 pages; added a section and an appendix