Frobenius-Perron Theory of Modified ADE Bound Quiver Algebras

TL;DR

This paper applies Frobenius-Perron theory to modified ADE quiver algebras, revealing that the Frobenius-Perron dimension equals the maximum number of loops at a vertex, and introduces a method to compute this dimension.

Contribution

It introduces a new approach to calculate Frobenius-Perron dimensions for radical square zero bound quiver algebras, including cases with irrational dimensions.

Findings

Frobenius-Perron dimension equals maximum loops at a vertex.

A general method for calculating Frobenius-Perron dimension of radical square zero algebras.

Existence of categories with arbitrarily large irrational Frobenius-Perron dimensions.

Abstract

The Frobenius-Perron dimension for an abelian category was recently introduced. We apply this theory to the category of representations of the finite-dimensional radical squared zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius-Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius-Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius-Perron dimensions.