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Introduction to Quivers

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 240)

Abstract

Quivers are directed graphs which are commonly used in fields such as representation theory and noncommutative geometry. This paper is meant to provide a short introduction for quivers and algebras produced from those quivers, called path algebras. We first look at basic definitions of quivers Q and path algebras kQ. We also cover some algebraic properties of path algebras in order to have a better understanding of the category of finite representations of a quiver Q. In fact, such category is equivalent the category of finitely generated left kQ-module corresponding to the quiver Q. As an example, we briefly describe how to obtain a representation of Q from a left kQ-module. At the end, we take a look at a bounded quiver Q (a.k.a. a quiver Q with a set of relations R) and its path algebra kQ / I where I is a two sided ideal generated by R. We use the Beilinson quiver for \(\mathscr {P}^2\) with relations as an example to illustrate the bounded quiver and its corresponding path algebra.

Keywords

Path Algebra Noncommutative Geometry Finite-dimensional Representation Algebraic Properties Indecomposable Projective Modules 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. Brion, Representations of Quivers, 2008, Available at: https://www-fourier.ujf-grenoble.fr/~mbrion/notes_quivers_rev.pdf
  2. 2.
    A. Craw, Explicit Methods for Derived Categories of Sheaves, 2007, available at: http://www.math.utah.edu/dc/tilting.pdf
  3. 3.
    A. Craw, Quiver Representations in Toric Geometry, 2008, available at: https://arxiv.org/pdf/0807.2191v1.pdf
  4. 4.
    T. Leinster, The bijection between projective indecomposable and simple modules, 2014, available at: https://arxiv.org/pdf/1410.3671.pdf

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of AlbertaEdmontonCanada

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