The natural quiver of an artinian algebra

TL;DR

This paper explores the natural quiver of an artinian algebra, establishing its relationship with the ordinary quiver, generalizing Gabriel's theorem for radical-graded cases, and introducing Gabriel-type algebras with unique generalized path algebra representations.

Contribution

It introduces the concept of Gabriel-type algebras, extends the generalized Gabriel theorem to radical-graded artinian algebras, and analyzes the relationship between natural and ordinary quivers.

Findings

Relationship between natural and ordinary quivers established

Generalized Gabriel theorem proven for radical-graded artinian algebras

Uniqueness of generalized path algebra and quiver up to isomorphism

Abstract

The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In \cite{Li}, the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying the generalized Gabriel theorem here and in \cite{Li}. For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed.