Entropy of Monomial Algebras and Derived Categories

Abstract

Abstract Let $A$ be a finitely presented associative monomial algebra. We study the category $\textsf{qgr}(A)$, 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 $\textbf{D}^b(\textsf{qgr}(A))$. We calculate the categorical entropy of the Serre twist functor on $\textbf{D}^b(\textsf{qgr}(A))$ and show that it is equal to the (natural) 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.