The Representation Theory of Dynkin Quivers. Three Contributions

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.