# Leavitt Path Algebras of Cayley Graphs \(C_n^j\)

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## 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.

## Keywords

Leavitt path algebra Cayley graph Narayana’s cows sequence## Mathematics Subject Classification

Primary 16S99 Secondary 11B39 05C99## Notes

### 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.

## References

- 1.Abrams, G., Ánh, P.N., Louly, A., Pardo, E.: The classification question for Leavitt path algebras. J. Algebra
**320**(5), 1983–2026 (2008)MathSciNetCrossRefGoogle Scholar - 2.Abrams, G., Ara, P., Siles Molina, M.: Leavitt path algebras, Lecture Notes in Mathematics Vol. 2191. Springer Verlag, London (2017)zbMATHGoogle Scholar
- 3.Abrams, G., Aranda Pino, G.: Purely infinite simple Leavitt path algebras. J. Pure Appl. Algebra
**207**(3), 553–563 (2006)MathSciNetCrossRefGoogle Scholar - 4.Abrams, G., Aranda Pino, G.: Leavitt path algebras of generalized Cayley graphs. Mediterr. J. Math.
**13**(1), 1–27 (2016)MathSciNetCrossRefGoogle Scholar - 5.Abrams, G., Erickson, S., Gil Canto, C.: Number-theoretic properties of the Haselgrove sequences
**(in preparation)**Google Scholar - 6.Abrams, G., Louly, A., Pardo, E., Smith, C.: Flow invariants in the classification of Leavitt path algebras. J. Algebra
**333**, 202–231 (2011)MathSciNetCrossRefGoogle Scholar - 7.Abrams, G., Schoonmaker, B.: Leavitt path algebras of Cayley graphs arising from cyclic groups. Am. Math. Soc. Contemp. Math. Ser.
**634**, 1–10 (2015)MathSciNetzbMATHGoogle Scholar - 8.Ara, P., Goodearl, K., Pardo, E.: \(K_0\) of purely infinite simple regular rings. K-Theory
**26**, 69–100 (2002)MathSciNetCrossRefGoogle Scholar - 9.Ara, P., Moreno, M.A., Pardo, E.: Non-stable \(K\)-theory for graph algebras. Algebras Represent. Theory
**10**(2), 157–178 (2007)MathSciNetCrossRefGoogle Scholar - 10.Haselgrove, C.B.: A note on Fermat’s last theorem and the Mersenne numbers. Eureka Archimedians’ J.
**11**, 19–22 (1949)Google Scholar - 11.Newman, M.: Integral Matrices, Monographs and Textbooks in Pure and Applied Mathematics, vol. 45. Academic Press, New York (1972)Google Scholar
- 12.Online Encyclopedia of Integer Sequences, http://oeis.org/