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 $$A=\oplus _{n\ge 0} A(n)$$ A = ⊕ n ≥ 0 A ( n ) and $$H_0$$ H 0 -equivariant Morita equivalence $$A(0)\simeq _{H_0} X$$ A ( 0 ) ≃ H 0 X , there exists a Loewy-graded H -comodule algebra B (isomorphic to X in degree zero) realizing an H -equivariant Morita equivalence $$A\simeq _H B$$ A ≃ H B . In addition, if every exact $$H_0$$ H 0 -comodule algebra is $$H_0$$ H 0 -equivariant Morita equivalent to a coideal subalgebra of $$H_0$$ 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 $$H_0={\mathcal{K}\mathcal{P}}$$ H 0 = K P , the Kac-Paljutkin Hopf algebra.