Entropy of monomial algebras and derived categories

TL;DR

This paper investigates the entropy of monomial algebras and their derived categories, establishing connections between categorical entropy, algebra entropy, and graph entropy, especially for path algebras of quivers.

Contribution

It computes the categorical entropy of the Serre twist functor for monomial algebras and relates it to algebra and graph entropies, revealing new links between these concepts.

Findings

Categorical entropy equals the logarithm of the algebra's entropy.

For path algebras of quivers, entropy equals the logarithm of the spectral radius.

Established relationships between algebra, categorical, and graph entropies.

Abstract

Let A be a finitely presented associative monomial algebra. We study the category qgr(A) which is a quotient of the category of graded finitely presented A-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category. We calculate the categorical entropy of the Serre twist functor this derived category and show that it is equal to logarithm of the entropy of the algebra A itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of A and the entropy of the path algebra of the graph. If A is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver's adjacency matrix.