Abstract
Let n be a positive integer. For each \(0\le j \le n-1\) we let \(C_n^j\) denote the Cayley graph of the cyclic group \({\mathbb {Z}}_n\) with respect to the subset \(\{1,j\}\). Utilizing the Smith Normal Form process, we give an explicit description of the Grothendieck group of each of the Leavitt path algebras \(L_K(C_n^j)\) for any field K. Our general method significantly streamlines the approach that was used in a previous work to establish this description in the specific case \(j=2\). Along the way, we give necessary and sufficient conditions on the pairs (j, n) which yield that this group is infinite. We subsequently focus on the case \(j = 3\), where the structure of this group turns out to be related to a Fibonacci-like sequence, called the Narayana’s cows sequence.
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Acknowledgements
The authors would like to thank Aranda Pino and Iovanov for fruitful discussions during the preparation of this paper. Some of these results were anticipated and suggested by looking at output from the software package Magma. The authors are grateful to Viruel for his valuable help with this software. The first author was partially supported by a Simons Foundation Collaboration Grant #208941. The third author was partially supported by the Spanish MEC and Fondos FEDER through projects MTM2013-41208-P and MTM2016-76327-C3-1-P; by the Junta de Andalucía and Fondos FEDER, jointly, through project FQM-7156; and by the Grant “Ayudas para la realización de estancias en centros de investigación de calidad” of the “Plan Propio de Investigación y Transferencia” of the University of Málaga, Spain. Part of this work was carried out during a visit of the third author to the University of Colorado, Colorado Springs, USA. The third author thanks this host institution for its warm hospitality and support.
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Abrams, G., Erickson, S. & Gil Canto, C. Leavitt Path Algebras of Cayley Graphs \(C_n^j\). Mediterr. J. Math. 15, 197 (2018). https://doi.org/10.1007/s00009-018-1246-1
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DOI: https://doi.org/10.1007/s00009-018-1246-1