$H$-Equivariant Morita equivalences of Loewy-graded comodule algebras
Jacob Van Grinsven
TL;DR
The article develops a framework for $H$‑equivariant Morita theory of Loewy‑graded AM‑exact $H$‑comodule algebras when $H$ is coradically graded with coradical $H_0$. It shows how Morita equivalences at degree zero, given by $H_0$‑equivariant Morita data, lift to the full Loewy‑graded setting via endomorphism constructions and ambient crossed products $ ext{B}(V) obreak times A(0)$, yielding $H$‑equivariant Morita equivalences to coideal subalgebras of $H$. A key technical result is that every Loewy‑graded AM‑exacting $A$ embeds into $Hoxtimes A(0)$, enabling a maximal AM‑exacting subalgebra and a reduction to coideal subalgebras under suitable hypotheses; these ideas culminate in a general Morita lifting principle and a concrete KP case. In the ${ m KP}$ setting, the paper classifies AM‑exacting ${ m KP}$‑comodule algebras as either coideal subalgebras or twisted group algebras, and proves that all indecomposable exact ${ m KP}$‑module categories arise from homogeneous coideal subalgebras, enriching the classification of module categories for coradically graded Hopf 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.