A note on the representation type of generalized path algebras

TL;DR

This paper extends Gabriel's representation-type analysis to generalized path algebras by realizing gp-algebras as bound path algebras $kQ/I$ and applying Dynkin/Euclidean classifications. It provides explicit necessary and sufficient conditions on the base quiver $\Gamma$ and the vertex algebras $A_i$ that determine when $k(\Gamma,\mathcal{A})$ is representation-finite or of strict tame type. The main results show that rep-finiteness occurs only for Dynkin $\Gamma$ with constrained $A_i$ (often $k$ or $k^2$ at endpoints), while strict tameness requires $\Gamma$ to be Dynkin or Euclidean with additional restrictions on $A_i$, thereby extending Gabriel’s theorem to the generalized setting. These criteria offer a concrete classification framework for the module categories of gp-algebras and illuminate the interplay between quiver structure and local vertex algebras in determining representation type.

Abstract

In (Ibáñez-Cobos et al., 2008), the authors describe the ordinary quiver of a given generalized path algebra, a concept introduced by Coelho and Liu in (Coelho, Liu, 2000). In this short note, we use this result to characterize which generalized path algebras have finite or tame representation type.

Theorems & Definitions (16)

• Theorem 2.1
• Remark 2.2
• Example 2.3
• Proposition 2.4
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• proof
• proof
• Theorem 4.1