# An in-Depth Look at Quotient Modules

## TL;DR

This paper explores the generalization of quotient modules from finite groups to Hopf algebras, analyzing their properties, integrals, and module-theoretic structures, with implications for semisimple cases and algebraic invariants.

## Contribution

It extends the concept of quotient modules to Hopf subalgebras, providing new results on integrals, Morita equivalence, and module structures, and organizing existing knowledge through various algebraic frameworks.

## Key findings

- Q has a nonzero integral iff the modular function restricts appropriately.
- Organized knowledge of Q using annihilator ideals, trace ideals, and Burnside ring formulas.
- In semisimple cases, related the depth of Q to McKay quiver and Green ring.

## Abstract

The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that $Q$ has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of $Q$ and its tensor powers in terms of annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and when considering semisimple Hopf algebras, the depth of $Q$ in terms of the McKay quiver and the Green ring.