A functorial approach to Gabriel quiver constructions - Quirino, Samuel Amador dos Santos (2023)
The aim of this work is to establish the Gabriel quiver constructions via functors. By Gabriel quiver constructions we mean the Gabriels theorem which states that every pointed finite dimensional algebra is a quotient of the path algebra of its Gabriel quiver by an admissible ideal. In order to accomplish this, we consider the category of pointed coalgebras and the category of k-quivers, than we construct a pair of covariant functors between both categories, which translates the path coalgebra of a quiver and the Gabriel quiver of a pointed coalgebra, and show that these functors induce an adjoint pair when considering the quotient category of pointed coalgebras by an equivalence relation on coalgebra homomorphisms. The unit of the adjunction shows that every pointed coalgebra is an admissible subcoalgebra of the path coalgebra of its Gabriel quiver. By duality, we obtain a pair of contravariant functors from the category o k-quivers and the quotient category of pointed pseudocompact algebras by an equivalence relation on continuous algebra homomorphisms, which are adjoint on the left, and conclude that every pointed pseudocompact algebra is the quotient of the complete path algebra of its Gabriel quiver by an admissible ideal. We generalize these results for basic coalgebras with separable coradical and the concept of k-species for coalgebras. In parallel, we prove that the algebra of invariants of a complete path algebra under the action of a homogeneous group of continuous algebra automorphisms is a complete path algebra and preserves finite or tame representation type of the quiver.