Abstract
In this paper, we classify all Leavitt path algebras which have the property that every Lie ideal is an ideal. As an application, we show that Leavitt path algebras with this property provide a class of locally finite, infinite-dimensional Lie algebras whose locally solvable radical is completely determined. This particularly gives us a new class of semisimple Lie algebras over a field of prime characteristic.
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References
Abrams, G., Ara, P., Siles Molina, M.: Leavitt Path Algebras, Lecture Notes in Mathematics Series, vol. 2191, Springer-Verlag Inc., 2017
Abrams, G., Aranda Pino, G.: The Leavitt path algebra of a graph. J. Algebra 293, 319–334 (2005)
Abrams, G., Aranda Pino, G., Siles Molina, M.: Locally finite Leavitt path algebras. Israel J. Math. 165, 329–348 (2008)
Abrams, G., Mesyan, Z.: Simple Lie algebras arising from Leavitt path algebras. J. Pure Appl. Algebra 216(10), 2302–2313 (2012)
Alahmadi, A., Alsulami, H.: Simplicity of the Lie algebra of skew symmetric elements of a Leavitt path algebra. Comm. Algebra 44, 3182–3190 (2016)
Alahmedi, A., Alsulami, H.: On the simplicity of the Lie algebra of a Leavitt path algebra. Comm. Algebra 44(9), 4114–4120 (2016)
Ara, P., Moreno, M.A., Pardo, E.: Nonstable \(K\)-theory for graph algebras. Algebr. Represent. Theory 10, 157–178 (2007)
Aranda Pino, G., Crow, K.: The center of a Leavitt path algebras. Rev. Math. Iberoam 27(2), 621–644 (2011)
Mesyan, Z.: Commutator Leavitt path algebras. Algebr. Represent. Theory 16(5), 1207–1232 (2013)
Nam, T.G., Zhang, Z.: Lie solvable Leavitt path algebras. J. Algebra Appl. 21(10), Paper No. 2250203, pp. 12 (2022)
Penkov, I., Hoyt, C.: Classical Lie algebras at infinity. Springer Monographs in Mathematics, Springer, Cham (2022)
Vaš, L.: Every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. Bull. Aust. Math. Soc. 105(2), 248–256 (2022)
Acknowledgements
This paper was carried out when the author was working as a postdoctoral researcher at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to express his warmest thanks to the VIASM for fruitful research environment and hospitality. The author would like to thank anonymous referees for carefully reading the manuscript and providing excellent suggestions for improvement.
Funding
This research is funded by the Vietnam Ministry of Education and Training under grant number B2024-CTT-02.
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Khánh, H.V. Leavitt Path Algebras in Which Every Lie Ideal is an Ideal and Applications. Transformation Groups (2024). https://doi.org/10.1007/s00031-024-09848-1
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DOI: https://doi.org/10.1007/s00031-024-09848-1