2022 Invariant ideals in Leavitt path algebras
Cristóbal Gil Canto, Dolores Martín Barquero, Cándido Martín González
Author Affiliations +
Publ. Mat. 66(2): 541-569 (2022). DOI: 10.5565/PUBLMAT6622203

Abstract

It is known that the ideals of a Leavitt path algebra LK(E) generated by Pl(E), by Pc(E), or by Pec(E) are invariant under isomorphism. Though the ideal generated by Pb(E) is not invariant we find its “natural” replacement (which is indeed invariant): the one generated by the vertices of Pbp (vertices with pure infinite bifurcations). We also give some procedures to construct invariant ideals from previous known invariant ideals. One of these procedures involves topology, so we introduce the DCC topology and relate it to annihilators in the algebraic counterpart of the work. To be more explicit: if H is a hereditary saturated subset of vertices providing an invariant ideal, its exterior ext(H) in the DCC topology of E0 generates a new invariant ideal. The other constructor of invariant ideals is more categorical in nature. Some hereditary sets can be seen as functors from graphs to sets (for instance Pl, etc.). Thus a second method emerges from the possibility of applying the induced functor to the quotient graph. The easiest example is the known socle chain Soc(1)()Soc(2)(), all of which are proved to be invariant. We generalize this idea to any hereditary and saturated invariant functor. Finally we investigate a kind of composition of hereditary and saturated functors which is associative.

Funding Statement

The authors are supported by the Junta de Andalucía through projects FQM-336 and UMA18-FEDERJA-119 and by the Spanish Ministerio de Ciencia e Innovación through project PID2019-104236GB-I00, all of them with FEDER funds.

Citation

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Cristóbal Gil Canto. Dolores Martín Barquero. Cándido Martín González. "Invariant ideals in Leavitt path algebras." Publ. Mat. 66 (2) 541 - 569, 2022. https://doi.org/10.5565/PUBLMAT6622203

Information

Received: 27 January 2021; Accepted: 23 June 2020; Published: 2022
First available in Project Euclid: 22 June 2022

MathSciNet: MR4443748
zbMATH: 1509.16004
Digital Object Identifier: 10.5565/PUBLMAT6622203

Subjects:
Primary: 16D25 , 16S88

Keywords: annihilator , DCC topology , hereditary and saturated point functors , invariant ideal , Leavitt path algebra , socle

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.66 • No. 2 • 2022
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