Infinite dimensional representations of finite dimensional algebras and amenability

TL;DR

This paper introduces a new framework for understanding finite dimensional algebra representations using graph limit-inspired rank spectra, classifies algebras into types, and explores implications for module parameter testing.

Contribution

It develops the concept of the rank spectrum for finite dimensional algebras, classifies algebras into finite, amenable, and non-amenable types, and connects limit representations to von Neumann continuous rings.

Findings

String algebras are of amenable representation type.

Wild Kronecker algebras are not of amenable type.

Reasonable parameters for string algebras are testable.

Abstract

We present a novel approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra $R$ over a finite field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on $R$-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930's. As an application in algorithm theory, we introduce and study the notion of testing of parameters of modules over finite dimensional algebras, that is analogous to the property testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.