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On the Ideals of Ultragraph Leavitt Path Algebras

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Abstract

In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen simple module. Consequently, we prove Exel’s Effros-Hahn conjecture on primitive ideals in the ultragraph Leavitt path algebra setting (a conclusion that is also new in the context of Leavitt path algebras of graphs).

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Funding

Daniel Gonçalves was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq and Capes-PrInt, Brazil. The third author was supported by the International Center of Research and Postgraduate Training in Mathematics under grant ICRTM01-2021.04.

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Authors and Affiliations

  1. Faculty of Pedagogy, Hanoi Pedagogical University 2, Hanoi, Vietnam

    T. T. H. Duyen

  2. Departamento de Matemática, Universidade Federal de Santa Catarina, Florianópolis, 88040-900, Brazil

    D. Gonçalves

  3. Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    T. G. Nam

Authors
  1. T. T. H. Duyen
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  2. D. Gonçalves
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  3. T. G. Nam
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Correspondence to D. Gonçalves.

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Presented by: Kenneth Goodearl.

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Duyen, T.T.H., Gonçalves, D. & Nam, T.G. On the Ideals of Ultragraph Leavitt Path Algebras. Algebr Represent Theor 27, 77–113 (2024). https://doi.org/10.1007/s10468-023-10206-0

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