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SIMPLICITY AND CHAIN CONDITIONS FOR ULTRAGRAPH LEAVITT PATH ALGEBRAS VIA PARTIAL SKEW GROUP RING THEORY

Part of: Chain conditions, growth conditions, and other forms of finiteness Rings and algebras arising under various constructions

Published online by Cambridge University Press:  18 July 2019

DANIEL GONÇALVES Open the ORCID record for DANIEL GONÇALVES [Opens in a new window]  and
DANILO ROYER Open the ORCID record for DANILO ROYER [Opens in a new window]
DANIEL GONÇALVES*
Affiliation:
Department of Mathematics, Federal University of Santa Catarina, 88040900, Brazil
DANILO ROYER
Affiliation:
Department of Mathematics, Federal University of Santa Catarina, 88040900, Brazil email: daniloroyer@gmail.com
*
email: daemig@gmail.com
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Abstract

We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.

Keywords

ultragraph Leavitt path algebraspartial skew group ringsimplicityartinian ringnoetherian ring

MSC classification

Primary: 16S35: Twisted and skew group rings, crossed products
Secondary: 16S99: None of the above, but in this section 16P20: Artinian rings and modules
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 3 , December 2020 , pp. 299 - 319
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

Partially supported by CNPq and Capes-PrInt Brazil.

References

Abrams, G., ‘Leavitt path algebras: the first decade’, Bull. Math. Sci. 5 (2015), 59–120.Google Scholar
Abrams, G., Ara, P. and Molina, M. S., Leavitt Path Algebras, Lecture Notes in Mathematics (Springer, London, 2017).Google Scholar
Castro, G. G. and Gonçalves, D., ‘KMS and ground states on ultragraph C∗ -algebras’, Integral Equations Operator Theory 90 (2018), 63.Google Scholar
Dokuchaev, M., ‘Recent developments around partial actions’, São Paulo J. Math. Sci. 13(1) (2018), 195–247.Google Scholar
Gonçalves, D., ‘Simplicity of partial skew group rings of abelian groups’, Canad. Math. Bull. 57(3) (2014), 511–519.Google Scholar
Gonçalves, D., Li, H. and Royer, D., ‘Branching systems and general Cuntz–Krieger uniqueness theorem for ultragraph C∗ -algebras’, Internat. J. Math. 27(10) (2016), 1650083.Google Scholar
Gonçalves, D., Öinert, J. and Royer, D., ‘Simplicity of partial skew group rings with applications to Leavitt path algebras and topological dynamics’, J. Algebra 420 (2014), 201–216.Google Scholar
Gonçalves, D. and Royer, D., ‘Leavitt path algebras as partial skew group rings’, Comm. Algebra 42 (2014), 3578–3592.Google Scholar
Gonçalves, D. and Royer, D., ‘Ultragraphs and shift spaces over infinite alphabets’, Bull. Sci. Math. 141 (2017), 25–45.Google Scholar
Gonçalves, D. and Royer, D., ‘Infinite alphabet edge shift spaces via ultragraphs and their C∗ -algebras’, Int. Math. Res. Not. 2019 (2019), 2177–2203.Google Scholar
Gonçalves, D. and Sobottka, M., ‘Continuous shift commuting maps between ultragraph shift spaces’, Discrete Contin. Dyn. Syst. 39 (2019), 1033–1048.Google Scholar
Imanfar, M., Pourabbas, A. and Larki, H., ‘The Leavitt path algebras of ultragraphs’, Preprint, 2017, arXiv:1701.00323v3.Google Scholar
Katsura, T., Muhly, P. S., Sims, A. and Tomforde, M., ‘Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence’, J. reine angew. Math. 640 (2010), 135–165.Google Scholar
Nystedt, P., Öinert, J. and Pinedo, H., ‘Artinian and noetherian partial skew groupoid rings’, J. Algebra 503 (2018), 433–452.Google Scholar
Ott, W., Tomforde, M. and Willis, P. N., ‘One-sided shift spaces over infinite alphabets’, New York J. Math. NYJM Monographs 5 (2014).Google Scholar
Tomforde, M., ‘Simplicity of ultragraph algebras’, Indiana Univ. Math. J. 4 (2003), 901–925.Google Scholar
Tomforde, M., ‘A unified approach to Exel–Laca algebras and C∗ -algebras associated to graphs’, J. Operator Theory 50 (2003), 345–368.Google Scholar