# Critical groups for Hopf algebra modules

## TL;DR

This paper introduces the concept of critical groups for modules over finite-dimensional Hopf algebras, generalizing previous work on finite group representations, and provides formulas for their cardinalities.

## Contribution

It generalizes the notion of critical groups to Hopf algebra modules and derives explicit formulas, including for the regular representation.

## Key findings

- Formula for the cardinality of the critical group
- Complete description of the critical group for the regular representation
- Role of the gcd of dimensions of indecomposable projectives

## Abstract

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.