Categorical dynamics on stable module categories

TL;DR

This paper investigates the categorical entropy of a grading-twisting functor on stable module categories of finite graded Hopf algebras, providing bounds and explicit calculations for specific examples over .

Contribution

It introduces bounds for the categorical polynomial entropy of the twist functor and computes this entropy for certain finite graded Hopf algebras over .

Findings

Categorical entropy of the twist functor is zero.

Lower bounds relate entropy to the Krull dimension of cohomology.

Explicit entropy calculations are provided for examples over .

Abstract

Let $ A $ be a finite connected graded cocommutative Hopf algebra over a field $ k $. There is an endofunctor $ \mathsf{tw} $ on the stable module category $ \mathrm{StMod}_A $ of $ A $ which twists the grading by $ 1 $. We show the categorical entropy of $ \mathsf{tw} $ is zero. We provide a lower bound for the categorical polynomial entropy of $ \mathsf{tw} $ in terms of the Krull dimension of the cohomology of $ A $, and an upper bound in terms of the existence of finite resolutions of $ A $-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of the twist functor for examples of finite graded Hopf algebras over $ \mathbb{F}_2 $.