The Representation Theory of Dynkin Quivers. Three Contributions

TL;DR

This paper presents three new contributions to the representation theory of Dynkin and Euclidean quivers, including inductive procedures and uniform constructions for maximal indecomposable representations.

Contribution

It introduces an inductive approach for D-type quivers and a unified method for constructing maximal indecomposable representations in E-type and Euclidean quivers.

Findings

Category of D_n contains a subcategory equivalent to D_{n-1} representations

Provides a uniform construction for maximal indecomposable representations in E types

Offers an inductive procedure starting from D_3 = A_3

Abstract

These notes provide three contributions to the (well-established) representation theory of Dynkin and Euclidean quivers. They should be helpful as part of a direct approach to study representations of quivers, and they may shed some new light on properties of Dynkin and Euclidean quivers. Part 1 deals with the case A (see arXiv:1304.5720). Part 2 concerns the case D. We show that the category of representation of D_n contains a full subcategory which is equivalent to the category of representations of a quiver of type D_{n-1} such that the remaining indecomposable representations are thin. This suggests an inductive procedure to deal with the D-cases, starting with D_3 = A_3. Part 3 deals with the cases E. It provides a uniform way to construct the maximal indecomposable representation for these Dynkin quivers as well as a unified method to deal with the corresponding Euclidean quivers. The level of the presentation varies considerably and increases throughout the discussion. Whereas Part 1 is completely elementary, just based on some results in linear algebra, the further text is less self-contained and uses one-point extensions, simplification, perpendicular categories, Auslander-Reiten quivers and hammocks.