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SUBSETS OF VERTICES GIVE MORITA EQUIVALENCES OF LEAVITT PATH ALGEBRAS

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  30 March 2017

LISA ORLOFF CLARK Open the ORCID record for LISA ORLOFF CLARK [Opens in a new window] ,
ASTRID AN HUEF Open the ORCID record for ASTRID AN HUEF [Opens in a new window]  and
PAREORANGA LUITEN-APIRANA
LISA ORLOFF CLARK
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email lclark@maths.otago.ac.nz
ASTRID AN HUEF*
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email astrid@maths.otago.ac.nz
PAREORANGA LUITEN-APIRANA
Affiliation:
Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand email pareorangaluitenapirana@gmail.com
*
✉astrid@maths.otago.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

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We show that every subset of vertices of a directed graph $E$ gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of $E$ can be contracted to a new graph $G$ such that the Leavitt path algebras of $E$ and $G$ are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.

Keywords

directed graphLeavitt path algebraMorita contextMorita equivalencegraph algebra

MSC classification

Primary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 212 - 222
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

This research has been supported by a University of Otago Research Grant.

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