Abstract
Quivers (directed graphs), 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. 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.
Highlights
I would like to thank Alistair Savage for introducing me to this topic and for his invaluable guidance and encouragement
In his paper [1], Gabriel outlined how one could classify all species of finite representation type over non-algebraically closed fields. It was Dlab and Ringel in 1976 who were able to generalize Gabriel’s theorem and show that a species is of finite representation type if and only if its underlying valued graph is a Dynkin diagram of finite type
We show that if Q is an Fq -species, the tensor algebra of Q is isomorphic to the fixed point algebra of the path algebra of a quiver under the Frobenius morphism
Summary
I would like to thank Alistair Savage for introducing me to this topic and for his invaluable guidance and encouragement. It was Dlab and Ringel in 1976 (see [3]) who were able to generalize Gabriel’s theorem and show that a species is of finite representation type if and only if its underlying valued graph is a Dynkin diagram of finite type They showed that, just as for quivers, there is a bijection between the isomorphism classes of the indecomposable representations and the positive roots of the corresponding Kac–Moody Lie algebra. The seventh and final section deals with the Ringel–Hall algebra of a species It is well-known that the generic composition algebra of a quiver is isomorphic to the positive part of the quantized enveloping algebra of the associated Kac–Moody Lie algebra. We assume throughout that all algebras (other than Lie algebras) are associative and unital
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