Categorical dynamics on stable module categories
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 $.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.09964
- arXiv:
- arXiv:2212.09964
- Bibcode:
- 2022arXiv221209964Y
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Category Theory;
- Mathematics - Dynamical Systems;
- Mathematics - Representation Theory;
- 55P42;
- 18G65;
- 16T05
- E-Print:
- 43 pages, comments welcome