Work

Entropy of Monomial Algebras and Derived Categories

Year: 2021

Type: article

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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnab312

Authors: Li Lu, Dmitri Piontkovski

Institutions: Chongqing University, National Research University Higher School of Economics

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Russian Science Foundation

Award: G8227239686