Algebraic entropy and a complete classification of path algebras over finite graphs by growth

TL;DR

This paper introduces algebraic entropy for path algebras and provides a comprehensive classification of such algebras over finite graphs based on growth measures, highlighting the role of cycles.

Contribution

It establishes the algebraic entropy as a new invariant for classifying path algebras and compares it with Gelfand-Kirillov dimension, showing their dependence on graph cycles.

Findings

Algebraic entropy is conserved under Morita equivalence.

Complete classification of path algebras by dimension, Gelfand-Kirillov dimension, and entropy.

Examples illustrating entropy in path and Leavitt path algebras.

Abstract

The Gelfand-Kirillov dimension is a well established quantity to classify the growth of infinite dimensional algebras. In this article we introduce the algebraic entropy for path algebras. For the path algebras, Leavitt path algebras and the path algebra of the extended (double) graph, we compare the Gelfand-Kirillov dimension and the entropy. We give a complete classification of path algebras over finite graphs by dimension, Gelfand-Kirillov dimension and algebraic entropy. We show indeed how these three quantities are dependent on cycles inside the graph. Moreover we show that the algebraic entropy is conserved under Morita equivalence. In addition we give several examples of the entropy in path algebras and Leavitt path algebras.