Irreducible and permutative representations of ultragraph Leavitt path algebras

Royer, Danilo

doi:10.1515/forum-2019-0270

Cited by 13 publications

(13 citation statements)

References 42 publications

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“…This will be accomplished by defining a map from a "large" subset of X to the path space Λ ∞ . The ideas presented here generalize the constructions in [10,32] to the k-graph setting.…”

Section: The Composition πmentioning

confidence: 89%

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Farsi¹,

Gillaspy²

2021

Preprint

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The representations of a k-graph C * -algebra C * (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C * (Λ). Throughout the paper, we work in the setting of rowfinite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.

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“…This will be accomplished by defining a map from a "large" subset of X to the path space Λ ∞ . The ideas presented here generalize the constructions in [10,32] to the k-graph setting.…”

Section: The Composition πmentioning

confidence: 89%

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Farsi¹,

Gillaspy²

2021

Preprint

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“…Ánh and the third author, in [6], gave another way to describe Chen simple modules and compute their annihilators. Royer and the second author, in [23], extended Chen's construction of simple modules for graph Leavitt path algebras to ultragraph Leavitt path algebras. The aim of this section is to extend Ara and Ragaswamy's construction of Chen simple modules for graph Leavitt path algebras to ulragraph Leavitt path algebras and compute their annihilators (Theorem 5.4).…”

Section: Chen Simple Modules Of Ultragraph Leavitt Path Algebrasmentioning

confidence: 99%

“…For [23,Proposition 3.9] Royer and the second author defined the simple left L K (G)-module V [p] to be the K-vector space having [p] as a basis and with the scalar multiplication satisfying the following: for all A ∈ G 0 , e ∈ G 1 and α ∈ [p],…”

Section: Chen Simple Modules Of Ultragraph Leavitt Path Algebrasmentioning

confidence: 99%

“…We mention the following. Gonçalves and Royer [23] extended Chen's construction of irreducible representations of graph Leavitt path algebras to ultragraph Leavitt path algebras (see [10]); Gonçalves and Royer [21] realized ultragraph Leavitt path algebras as partial skew group rings. Using this realization they characterized Artinian ultragraph Leavitt path algebras and gave simplicity criteria for these algebras; and de Castro, Gonçalves and van Wyk [13] realized ultragraph Leavitt path algebras as Steinberg algebras, and applied this result to obtain generalized uniqueness theorems for ultragraph Leavitt path algebras.…”

Section: Introductionmentioning

confidence: 99%

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On the ideals of ultragraph Leavitt path algebras

Duyen¹,

Nam²

2021

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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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“…Ultragraph Leavitt path algebras have been widely studied, see [11,8,7,12,13,22,14]. Ultragraph Leavitt path algebra L R (G) was introduced in [15] as the algebraic version of ultragraph C * -algebra C * (G) [24].…”

Section: Introductionmentioning

confidence: 99%

Prime and Primitive Ideals of Ultragraph Leavitt Path Algebras

Pourabbas,

Imanfar,

Larki

2021

Preprint

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Irreducible and permutative representations of ultragraph Leavitt path algebras

Cited by 13 publications

References 42 publications

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

Irreducibility and monicity for representations of $k$-graph $C^*$-algebras

On the ideals of ultragraph Leavitt path algebras

Prime and Primitive Ideals of Ultragraph Leavitt Path Algebras

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