Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras

Tomasz Ciborski

TL;DR

The work analyzes entropy and polynomial entropy of derived autoequivalences for derived discrete algebras, classifying autoequivalences via explicit generators (twists T_X, T_Y and shifts) in both finite and infinite global dimension. It delivers concrete, closed-form formulas for the entropy and polynomial entropy as functions of the generator data and the invariants p,q,r, revealing linear growth rates and vanishing polynomial entropy in most cases. By connecting these entropy notions to the structure of derived/discrete algebras and their Serre/twist structures, the paper links algebraic autoequivalences to dynamical-type invariants, with implications for mass growth under stable conditions. The results provide a precise, computable description of how derived autoequivalences of Λ(p,q,r) act on the bounded derived category and on perfect complexes, including explicit dependence on the underlying quiver data.

Abstract

The aim of this paper is to calculate entropy in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich and polynomial entropy as defined by Fan-Fu-Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field.

Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras

TL;DR

The work analyzes entropy and polynomial entropy of derived autoequivalences for derived discrete algebras, classifying autoequivalences via explicit generators (twists T_X, T_Y and shifts) in both finite and infinite global dimension. It delivers concrete, closed-form formulas for the entropy and polynomial entropy as functions of the generator data and the invariants p,q,r, revealing linear growth rates and vanishing polynomial entropy in most cases. By connecting these entropy notions to the structure of derived/discrete algebras and their Serre/twist structures, the paper links algebraic autoequivalences to dynamical-type invariants, with implications for mass growth under stable conditions. The results provide a precise, computable description of how derived autoequivalences of Λ(p,q,r) act on the bounded derived category and on perfect complexes, including explicit dependence on the underlying quiver data.

Abstract

The aim of this paper is to calculate entropy in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich and polynomial entropy as defined by Fan-Fu-Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field.

Paper Structure

This paper contains 12 sections, 45 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categorical complexity and entropy
  4. Polynomial entropy
  5. Example: shift functors
  6. Twist functors
  7. Derived autoequivalences of discrete algebras
  8. Finite global dimension
  9. Twist functors
  10. General functor
  11. Polynomial entropy
  12. Infinite global dimension

Key Result

Theorem 1.1

Let $\Lambda$ be a derived discrete $\mathbb{K}$-algebra, which is not piecewise hereditary of Dynkin type and of finite global dimension. Suppose that $\Lambda$ is derived equivalent to $\Lambda(p,q,r)$ where $q\geq 0$ and $1\leq r<p$. If $F$ is an autoequivalence of $\mathbf{D}^\mathrm{b}(\mathop{

Figures (1)

  • Figure 1: The gentle bound quiver of $\Lambda(p,q,r)$. Here, the dotted arcs represent the zero-relations.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: haiden
  • Lemma 2.2
  • Corollary 2.3
  • Definition 2.4: haiden
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 73 more