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The groupoid approach to Leavitt path algebras

Simon Rigby (UGent)
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Abstract
When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph’s boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.
Keywords
ample groupoids, Leavitt path algebras, Steinberg algebras

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Please use this url to cite or link to this publication:

MLA
Rigby, Simon. “The Groupoid Approach to Leavitt Path Algebras.” Leavitt Path Algebras and Classical K-Theory, edited by A. A. Ambily et al., Springer, 2020, pp. 21–72.
APA
Rigby, S. (2020). The groupoid approach to Leavitt path algebras. In A. A. Ambily, R. Hazrat, & B. Sury (Eds.), Leavitt Path Algebras and Classical K-Theory (pp. 21–72). Singapore: Springer.
Chicago author-date
Rigby, Simon. 2020. “The Groupoid Approach to Leavitt Path Algebras.” In Leavitt Path Algebras and Classical K-Theory, edited by A. A. Ambily, R. Hazrat, and B. Sury, 21–72. Singapore: Springer.
Chicago author-date (all authors)
Rigby, Simon. 2020. “The Groupoid Approach to Leavitt Path Algebras.” In Leavitt Path Algebras and Classical K-Theory, ed by. A. A. Ambily, R. Hazrat, and B. Sury, 21–72. Singapore: Springer.
Vancouver
1.
Rigby S. The groupoid approach to Leavitt path algebras. In: Ambily AA, Hazrat R, Sury B, editors. Leavitt Path Algebras and Classical K-Theory. Singapore: Springer; 2020. p. 21–72.
IEEE
[1]
S. Rigby, “The groupoid approach to Leavitt path algebras,” in Leavitt Path Algebras and Classical K-Theory, A. A. Ambily, R. Hazrat, and B. Sury, Eds. Singapore: Springer, 2020, pp. 21–72.
@incollection{8616832,
  abstract     = {When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph’s boundary path groupoid. This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.},
  author       = {Rigby, Simon},
  booktitle    = {Leavitt Path Algebras and Classical K-Theory},
  editor       = {Ambily, A. A. and Hazrat, R. and Sury, B.},
  isbn         = {978-981-15-1610-8},
  keywords     = {ample groupoids,Leavitt path algebras,Steinberg algebras},
  language     = {eng},
  pages        = {21--72},
  publisher    = {Springer},
  series       = {Indian Statistical Institute Series},
  title        = {The groupoid approach to Leavitt path algebras},
  url          = {http://dx.doi.org/10.1007/978-981-15-1611-5},
  year         = {2020},
}

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