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.
