Valued Graphs and the Representation Theory of Lie Algebras

TL;DR

This paper explores the representation theory of species and quivers, highlighting classification results, structural properties of tensor rings, and their connections to Lie algebras, with new insights into tensor algebra isomorphisms over perfect fields.

Contribution

It extends classification results of species, introduces new structural theorems about tensor rings, and links species representations to Lie algebra theory.

Findings

Classification of species of finite and tame representation type

Isomorphism conditions for tensor rings of species without oriented cycles

Morita equivalence of tensor algebras over algebraically closed fields

Abstract

Quivers (directed graphs) and species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel's extension of Gabriel's theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K-species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as "crushed" species, and we show that if K is a perfect field, then the tensor algebra of a K-species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.