Approach to artinian algebras via natural quivers

TL;DR

This paper explores the relationships among various combinatorial objects associated with Artinian algebras, such as diagrams and quivers, to better understand and characterize these algebras.

Contribution

It investigates the relations among diagrams, natural quivers, and generalized species for Artinian algebras, providing new insights into their structure and classification.

Findings

The diagram $D_A$ and ext-quiver $ ext{Gamma}_A$ coincide when $A$ is splitting.

Relations among combinatorial objects are established, aiding algebra characterization.

Provides criteria for algebra classification based on these objects.

Abstract

Given an Artinian algebra $A$ over a field $k$, there are several combinatorial objects associated to $A$. They are the diagram $D_A$ as defined in [DK], the natural quiver $\Delta_A$ defined in \cite{Li} (cf. Section 2), and a generalized version of $k$-species $(A/r, r/r^2)$ with $r$ being the Jacobson radical of $A$. When $A$ is splitting over the field $k$, the diagram $D_A$ and the well-known ext-quiver $\Gamma_A$ are the same. The main objective of this paper is to investigate the relations among these combinatorial objects and in turn to use these relations to give a characterization of the algebra $A$.