Modulation and natural valued quiver of an algebra

Abstract

We generalize the concept of modulation to pseudomodulation and its subclasses including premodulation, generalized modulation and regular modulation. The motivation is to define the valued analogue of natural quiver, called 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 pseudomodulation and of a premodulation are equivalent respectively to that of a tensor algebra of A-path type and of a generalized path algebra. Their examples are given from two kinds of artinian hereditary algebras. Furthermore, the isomorphism theorem is given for normal generalized path algebras with finite (acyclic) quivers and normal premodulations. We give four examples of pseudomodulations: first, group species in mutation theory as a seminormal generalized modulation; second, viewing a path algebra with loops as a premodulation with valued quiver that has no loops; third, differential pseudomodulation and its relation with differential tensor algebras; fourth, a pseudomodulation considered as a free graded category. © 2012 by Pacific Journal of Mathematics.