Frobenius-Perron theory of the bound quiver algebras containing loops

TL;DR

This paper investigates the Frobenius-Perron theory for bound quiver algebras with loops, providing methods to compute their dimensions and revealing relationships between loops and algebraic properties.

Contribution

It introduces a way to calculate Frobenius-Perron dimensions for algebras with loops and links these dimensions to the number of loops at vertices.

Findings

Frobenius-Perron dimension equals the maximum number of loops at a vertex in certain algebras.

A method to compute Frobenius-Perron dimensions for algebras with loops satisfying commutativity.

Existence of infinite dimensional algebras with Frobenius-Perron dimension equal to the maximum number of loops.

Abstract

The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, find a way to calculate the Frobenius-Perron dimension of these algebras when they satisfy the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximum number of loops at a vertex. Finally, we point out that there also exists infinite dimensional algebras whose Frobenius-Perron dimension is equal to the maximal number of loops by giving an example.