Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Valued Graphs and the Representation Theory of Lie Algebras   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.

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The categories of this paper are: funct-an
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
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The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
23
The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci