Entropies of Serre functors for higher hereditary algebras

TL;DR

This paper computes various entropy invariants of Serre functors for higher hereditary algebras, revealing their dependence on algebraic properties like Calabi-Yau dimension and Coxeter matrix spectral radius.

Contribution

It provides explicit formulas for entropy invariants of Serre functors in higher hereditary algebras, confirming conjectured relations between entropy and Hochschild (co)homology entropy.

Findings

Entropy equals Calabi-Yau dimension for higher representation-finite algebras.

Entropy relates to global dimension and Coxeter matrix spectral radius for higher representation-infinite algebras.

Confirmed positive answer to Kikuta and Ouchi's question on entropy relations.

Abstract

For a higher hereditary algebra, we calculate its upper (lower) Serre dimension, the entropy and polynomial entropy of Serre functor, and the Hochschild (co)homology entropy of Serre quasi-functor. These invariants are given by its Calabi-Yau dimension for a higher representation-finite algebra, and by its global dimension and the spectral radius and polynomial growth rate of its Coxeter matrix for a higher representation-infinite algebra. For this, we prove the Yomdin type inequality on Hochschild homology entropy for a finite dimensional elementary algebra of finite global dimension. Our calculations imply that the Kikuta and Ouchi's question on relations between entropy and Hochschild (co)homology entropy has positive answer, and the Gromov-Yomdin type equalities on entropy and Hochschild (co)homology entropy hold, for the Serre functor on perfect derived category and the Serre quasi-functor on perfect dg module category of an indecomposable elementary higher hereditary algebra.