$H$-Equivariant Morita equivalences of Loewy-graded comodule algebras

Abstract: Let $H$ be a coradically graded Hopf algebra. For every Loewy-graded exact $H$-comodule algebra $A=\oplus_{n\geq 0} A(n)$ and $H_0$-equivariant Morita equivalence $A(0)\simeq_{H_0} X$, there exists a Loewy-graded $H$-comodule algebra $B$ (isomorphic to $X$ in degree zero) realizing an $H$-equivariant Morita equivalence $A\simeq_H B$. In addition, if every exact $H_0$-comodule algebra is $H_0$-equivariant Morita equivalent to a coideal subalgebra of $H_0$, then every Loewy-graded exact $H$-comodule algebra is $H$-equivariant Morita equivalent to a coideal subalgebra of $H$. We also discuss Loewy-graded $H$-comodule algebras with $H_0=\mathcal{KP}$, the Kac-Paljutkin Hopf algebra.