Modulation and natural valued quiver of an algebra

TL;DR

This paper generalizes the concept of modulation to pseudo-modulation and introduces the natural valued quiver for artinian algebras, establishing relations with valued Ext-quivers and exploring their representation categories.

Contribution

It defines the natural valued quiver for algebras, relates it to valued Ext-quivers, and characterizes the representation categories of pseudo-modulations and pre-modulations.

Findings

Representation categories are equivalent to tensor and generalized path algebras.

Established isomorphism theorem for normal generalized path algebras.

Provided examples including group species, path algebras with loops, and differential pseudo-modulations.

Abstract

The concept of modulation is generalized to pseudo-modulation and its subclasses including pre-modulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called {\em natural valued quiver}, of an artinian algebra so as to correspond to its valued Ext-quiver when this algebra is not $k$-splitting over the field $k$. Moreover, we illustrate the relation between the valued Ext-quiver and the natural valued quiver. The interesting fact we find is that the representation categories of a pseudo-modulation and of a pre-modulation are equivalent respectively to that of a tensor algebra of $\mathcal A$-path type and of a generalized path algebra. Their examples are given respectively from two kinds of artinian hereditary algebras. Furthermore, the isomorphism theorem is given for normal generalized path algebras with finite (acyclic) quivers and normal pre-modulations. Four examples of pseudo-modulations are given: (i) group species in mutation theory as a semi-normal generalized modulation; (ii) viewing a path algebra with loops as a pre-modulation with valued quiver which has not loops; (iii) differential pseudo-modulation and its relation with differential tensor algebras; (iv) a pseudo-modulation is considered as a free graded category.