On module categories over finite-dimensional Hopf algebras

TL;DR

The paper классиfies indecomposable exact module categories over the finite tensor category ${ m Rep}\,H$ of a finite-dimensional Hopf algebra $H$ by $H$-indecomposable left $H$-comodule algebras, up to equivariant Morita equivalence. It develops the Yan–Zhu stabilizer construction, showing it coincides with the internal Hom and yields a duality framework that clarifies the correspondence between module categories over ${ m Rep}\,H$ and ${ m Rep}igl(H^*igr)$. A dimension formula for stabilizers and explicit examples (notably coideal subalgebras and Hopf-Galois extensions) are provided, along with a detailed exploitation of stabilizers to classify exact module categories and describe the dual module category. The results offer an alternative, comodule-algebra–based route to the Etingof–Ostrik classification and illuminate the interplay between stabilizers, internal End, and Morita theory in the Hopf-algebra setting.

Abstract

We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module categories is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We also describe the correspondence of module categories between Rep H and Rep (H^*).

Theorems & Definitions (101)

• Lemma 1.1
• Definition 1.2
• Lemma 1.3
• proof
• Theorem 1.4
• proof
• Theorem 1.5
• proof
• Proposition 1.6
• proof