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The Groupoid Approach to Leavitt Path Algebras

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Leavitt Path Algebras and Classical K-Theory

Part of the book series: Indian Statistical Institute Series ((INSIS))

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.

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Notes

  1. 1.

    Note the subtle difference in notation: we were using Z for basic open sets in \(\partial E\) and now we are using \({\mathcal {Z}}\) for basic open sets in \(\mathcal {G}_E\).

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Acknowledgements

I thank Juana Sánchez Ortega, for her valuable advice and guidance throughout my Masters degree. I also thank Giang Nam Tran for finding an important error in an earlier version of the paper and suggesting a way to fix it. Finally, I thank the two examiners of my Masters thesis, Pere Ara and Aidan Sims, who wrote very insightful comments that led to an improvement of this work.

 I acknowledge the support of the National Research Foundation of South Africa.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Cape Town, South Africa

    Simon W. Rigby

  2. Department of Mathematics: Algebra and Geometry, Ghent University, Ghent, Belgium

    Simon W. Rigby

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  1. Simon W. Rigby
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Correspondence to Simon W. Rigby .

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Editors and Affiliations

  1. Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India

    A. A. Ambily

  2. Centre for Research in Mathematics, Western Sydney University, Sydney, NSW, Australia

    Roozbeh Hazrat

  3. Statistics and Mathematics Unit, Indian Statistical Institute, Bengaluru, Karnataka, India

    B. Sury

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Rigby, S.W. (2020). The Groupoid Approach to Leavitt Path Algebras. In: Ambily, A., Hazrat, R., Sury, B. (eds) Leavitt Path Algebras and Classical K-Theory. Indian Statistical Institute Series. Springer, Singapore. https://doi.org/10.1007/978-981-15-1611-5_2

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