Irreducible and permutative representations of ultragraph Leavitt path algebras

Daniel Gon\c{c}alves; Danilo Royer

arxiv: 1906.11602 · v1 · pith:3MLEFWDKnew · submitted 2019-06-27 · 🧮 math.RA · math.OA

Irreducible and permutative representations of ultragraph Leavitt path algebras

Daniel Gon\c{c}alves , Danilo Royer This is my paper

Pith reviewed 2026-05-25 14:00 UTC · model grok-4.3

classification 🧮 math.RA math.OA

keywords ultragraph Leavitt path algebrasirreducible representationspermutative representationsbranching systemsChen's constructionfaithful representations

0 comments

The pith

Perfect branching systems on ultragraphs produce exactly the perfect permutative irreducible representations of the associated Leavitt path algebra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Chen's construction of irreducible representations from ordinary Leavitt path algebras to the more general ultragraph setting. It proves that every perfect permutative irreducible representation arises from a perfect branching system and gives a full characterization of those representations. The same extension also yields a sharpened criterion for when the resulting representations are faithful. A reader would care because the combinatorial branching systems replace abstract algebraic conditions with explicit graph-like data that can be checked directly on the ultragraph.

Core claim

We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this we extend to ultragraph Leavitt path algebras Chen's construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching systems and characterize irreducible representations associated to perfect branching systems. Along the way we improve the characterization of faithfulness of Chen's irreducible representations.

What carries the argument

Perfect branching systems on ultragraphs, combinatorial objects that extend Chen's branching systems and generate representations while preserving the defining relations of the ultragraph Leavitt path algebra.

If this is right

Every perfect permutative irreducible representation arises from some perfect branching system.
Irreducible representations coming from perfect branching systems are fully described.
Faithfulness of the extended Chen-type representations admits an improved combinatorial criterion.
The same branching-system data determines both the representation and its faithfulness properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method may classify representations for other generalizations of graph algebras whose relations are combinatorial.
Branching systems could supply new invariants that distinguish non-isomorphic ultragraph algebras.
One could test whether every irreducible representation (not just the permutative ones) admits a branching-system description.

Load-bearing premise

The combinatorial definition of branching systems on ultragraphs extends Chen's construction while preserving the algebraic relations that define the Leavitt path algebra.

What would settle it

An explicit perfect permutative irreducible representation of some ultragraph Leavitt path algebra that cannot be realized by any perfect branching system on that ultragraph, or a perfect branching system whose induced map fails to be a representation.

read the original abstract

We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this we extend to ultragraph Leavitt path algebras Chen's construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way we improve the characterization of faithfulness of Chen's irreducible representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Chen's branching system construction to ultragraph Leavitt path algebras and gives a characterization of the perfect permutative irreducible representations.

read the letter

This paper extends Chen's construction of irreducible representations from ordinary Leavitt path algebras to the ultragraph case and characterizes the perfect, permutative, irreducible ones as coming from perfect branching systems. It also improves the faithfulness criteria for these representations as a side result. The extension is the core new content, since ultragraphs have range maps that land on sets of vertices rather than single vertices, which requires adjusting the combinatorial data while keeping the algebra relations intact. The authors define the appropriate branching systems on ultragraphs and prove both directions of the correspondence for the perfect case. They also tighten the conditions that make the representations faithful. The work stays within the existing program on graph algebras and does not introduce new foundational ideas, but the adaptation looks clean and consistent with the cited literature. The main technical point is that the extended branching systems must induce well-defined homomorphisms, which means they have to satisfy the projection relation that sums the range projections over edges whose ranges cover a given vertex. The paper claims to handle this directly in the construction, and nothing in the abstract suggests an obvious gap there. This is specialized work aimed at people already working on Leavitt path algebras and their representations. Readers outside that small circle will find little to use. The proofs are not visible in the abstract, but the claims are stated clearly enough that a referee could check them. I would send it to peer review because the extension is a concrete, verifiable advance within the subfield even if its reach is narrow.

Referee Report

1 major / 0 minor

Summary. The paper claims a complete characterization of perfect, permutative, irreducible representations of ultragraph Leavitt path algebras. It extends Chen's construction of irreducible representations from (ordinary) Leavitt path algebras to the ultragraph setting, shows that the representations arise from perfect branching systems, and improves the faithfulness criterion for Chen's representations along the way.

Significance. If the extension is shown to yield well-defined algebra homomorphisms, the result supplies a combinatorial parametrization of a natural class of representations for a strictly larger family of algebras than ordinary graphs. This would be a useful addition to the literature on Leavitt path algebras and their generalizations, especially for questions about irreducibility and faithfulness.

major comments (1)

[Abstract / extension of Chen's construction] The central claim requires that the extended branching-system construction induces a well-defined homomorphism, i.e., that the operators satisfy all ultragraph Cuntz-Krieger relations (in particular the relation expressing a vertex projection as the sum of s_e s_e^* over edges whose range sets cover the vertex). The abstract states that the extension is performed and that the representations arise from perfect branching systems, but supplies no explicit verification that the set-valued range maps are handled so that the projection relation holds; without this step the characterization in both directions collapses.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit verification that the extended construction yields a well-defined homomorphism. We address the point directly below.

read point-by-point responses

Referee: [Abstract / extension of Chen's construction] The central claim requires that the extended branching-system construction induces a well-defined homomorphism, i.e., that the operators satisfy all ultragraph Cuntz-Krieger relations (in particular the relation expressing a vertex projection as the sum of s_e s_e^* over edges whose range sets cover the vertex). The abstract states that the extension is performed and that the representations arise from perfect branching systems, but supplies no explicit verification that the set-valued range maps are handled so that the projection relation holds; without this step the characterization in both directions collapses.

Authors: We agree that an explicit, self-contained verification of all ultragraph Cuntz-Krieger relations is required for the homomorphism property to be fully rigorous, especially the vertex-projection relation when range maps are set-valued. While the construction and the statement that it produces a representation appear in Section 3 and Theorem 3.5, the detailed check for the projection relation is not written out at the level of individual summands. In the revised manuscript we will insert a new lemma (placed immediately after the definition of the operators) that verifies each relation in turn, with a separate paragraph handling the covering condition for vertices via the perfect-branching-system axioms. This addition will make the argument complete in both directions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends external prior work independently

full rationale

The paper's central claim extends Chen's construction (external prior literature) to ultragraphs and characterizes representations via branching systems. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the provided abstract or description. The extension step is presented as preserving algebraic relations without reducing the new results to the inputs by construction. This is the common case of an independent combinatorial extension.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axiomatic definition of Leavitt path algebras and the combinatorial notion of branching systems; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard algebraic relations defining Leavitt path algebras from graphs or ultragraphs hold.
Invoked when extending Chen's construction; background from prior literature.

pith-pipeline@v0.9.0 · 5588 in / 1150 out tokens · 26402 ms · 2026-05-25T14:00:38.421950+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 4 internal anchors

[1]

Abrams, F

G. Abrams, F. Mantese, A. Tonolo, Extensions of simple modules over Leavitt path algebras , J. Algebra 431, 78-106, 2015

work page 2015
[2]

Abrams, K.M

G. Abrams, K.M. Rangaswamy, and M. Siles Molina, The socle series of a Leavitt path algebra , Israel J. Math. 184, 413-435, 2011

work page 2011
[3]

Ara and K.M

P. Ara and K.M. Rangaswamy, Finitely presented simple modules over Leavitt path algebras , J. Algebra 417, 333-352, 2015

work page 2015
[4]

Ara, K.M

P. Ara, K.M. Rangaswamy, Leavitt path algebras with at most countably many irreducible representations , Rev. Mat. Iberoam. 31 (4), 1263-1276, 2015. 26

work page 2015
[5]

Aranda Pino, D

G. Aranda Pino, D. Mart ´ ın Barquero, C. Mart ´ ın Gonz´ alez and M. Siles Molina, The socle of a Leavitt path algebra , J. Pure Appl. Algebra 212 (3), 500-509, 2008

work page 2008
[6]

ArandaPino, D

G. ArandaPino, D. Mart ´ ın Barquero, C. Mart ´ ın Gonz´ alez, M. S iles Molina, Socle theory for Leavitt path algebras of arbitrary graphs , Rev. Mat. Iberoam. 26 (2), 611-638, 2010

work page 2010
[7]

Bratteli and P

O. Bratteli and P. E. T. Jorgensen, Iterated function systems and permu- tation representations of the Cuntz algebra , Memoirs Amer. Math. Soc. 139 no.663, 1999

work page 1999
[8]

C. G. Canto, D. Gon¸ calves, Representations of relative Cohn path alge- bras, arXiv:1806.03077 [math.RA], 2018

work page arXiv 2018
[9]

G. G. Castro, D. Gon¸ calves, KMS and ground states on ultragraph C*- algebras, Integr. Equ. Oper. Theory 90:63, doi 10.1007/s00020-018-24 90- 2, 2018

work page doi:10.1007/s00020-018-24 2018
[10]

Chen, Irreducible representations of Leavitt path algebras , Forum Math

X.W. Chen, Irreducible representations of Leavitt path algebras , Forum Math. 27 (1), 549-57, 2015

work page 2015
[11]

Correia Ramos, N

C. Correia Ramos, N. Martins, P. R. Pinto, On graph algebras from interval maps , Ann. Funct. Anal. 10 (2), 203-217, 2019

work page 2019
[12]

Farsi, E

C. Farsi, E. Gillaspy, S. Kang, J. Packer, Separable representations, KMS states, and wavelets for higher-rank graphs , J. Math. Anal. Appl. 434, 241-270, 2016

work page 2016
[13]

Farsi, E

C. Farsi, E. Gillaspy, S. Kang, J. Packer, Wavelets and graph C*- algebras, Appl. Numer. Harmon. Anal. 5, 35-86, 2017

work page 2017
[14]

Farsi, E

C. Farsi, E. Gillaspy, P. Jorgensen, S. Kang, J. Packer, Monic represen- tations of ﬁnite higher-rank graphs , to appear at Ergodic Theory Dynam. Systems, doi.org/10.1017/etds.2018.79, 2019. 27

work page doi:10.1017/etds.2018.79 2018
[15]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Branching systems and general Cuntz- Krieger uniqueness theorem for ultragraph C*-algebras, Internat. J. Math. 27 (10), 1650083 (26 pg), 2016

work page 2016
[16]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Faithful representations of graph algebras via branching systems , Canad. Math. Bull. 59, 95-103, 2016

work page 2016
[17]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Branching systems for higher rank graph C*-algebras, Glasg. Math. J., doi:10.1017/S0017089518000058, 2018

work page doi:10.1017/s0017089518000058 2018
[18]

Gon¸ calves, D

D. Gon¸ calves, D. Royer,Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for inﬁnite matrices , J. Math. Anal. Appl. 351, 811-818, 2009

work page 2009
[19]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, On the representations of Leavitt path algebras , J. Algebra 333, 258-272, 2011

work page 2011
[20]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Unitary equivalence of representations of alge- bras associated with graphs, and branching systems , Funct. Anal. Appl. 45, 45-59, 2011

work page 2011
[21]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Graph C∗-algebras, branching systems and the Perron-Frobenius operator, J. Math. Anal. Appl. 391, 457-465, 2012

work page 2012
[22]

Gon¸ calves, D

D. Gon¸ calves, D. Royer,Leavitt path algebras as partial skew group rings , Comm. Algebra 42, 127-143, 2014

work page 2014
[23]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Branching systems and representations of Cohn–Leavitt path algebras of separated graphs , J. Algebra 422, 413-426, 2015

work page 2015
[24]

Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory

D. Gon¸ calves, D. Royer, Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory , to appear at J. Aust. Math. Soc., arXiv:1706.03628

work page internal anchor Pith review Pith/arXiv arXiv
[25]

Representations and the reduction theorem for ultragraph Leavitt path algebras

D. Gon¸ calves, D. Royer Representations and the reduction theorem for ultragraph Leavitt path algebras , arXiv:1902.00013 [math.RA], 2019. 28

work page internal anchor Pith review Pith/arXiv arXiv 1902
[26]

Gon¸ calves and D

D. Gon¸ calves and D. Royer, Inﬁnite alphabet edge shift spaces via ultra- graphs and their C*-algebras , Int. Math. Res. Not., v. 2019, 2177-2203, 2019

work page 2019
[27]

Gon¸ calves and D

D. Gon¸ calves and D. Royer, Ultragraphs and shift spaces over inﬁnite alphabets, Bull. Sci. Math., 141 (1), 25-45, 2017

work page 2017
[28]

Gon¸ calves and M

D. Gon¸ calves and M. Sobottka, Continuous shift commuting maps be- tween ultragraph shift spaces , Discrete Contin. Dyn. Syst., v.39, 1033- 1048, 2019

work page 2019
[29]

Li-Yorke chaos for ultragraph shift spaces

D. Gon¸ calves, B. B. Uggioni, Li-Yorke chaos for ultragraph shift spaces , to appear at Groups Geom. Dyn., arXiv:1806.07927 [math.DS]

work page internal anchor Pith review Pith/arXiv arXiv
[30]

Ultragraph shift spaces and chaos

D. Gon¸ calves, B. B. Uggioni, Ultragraph shift spaces and chaos , arXiv:1902.05784 [math.DS], 2019

work page internal anchor Pith review Pith/arXiv arXiv 1902
[31]

Hazrat, K.M

R. Hazrat, K.M. Rangaswamy, On graded irreducible representations of Leavitt path algebras , J. Algebra 450, 458-486, 2016

work page 2016
[32]

Imanfar, A

M. Imanfar, A. Pourabbas and H. Larki, The leavitt path algebras of ultragraphs, arXiv:1701.00323v3 [math.RA], 2017

work page arXiv 2017
[33]

Katsura, P

T. Katsura, P. S. Muhly, A. Sims and M. Tomforde, Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equiv- alence, J. Reine Angew. Math. 640, 135-165, 2010

work page 2010
[34]

T, Katsura, A. Sims, M. Tomforde, Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras , J. Funct. Anal. 257 (5), 1589-1620, 2009

work page 2009
[35]

Kawamura, Y

K. Kawamura, Y. Hayashi, D. Lascu, Continued fraction expansions and permutative representations of the Cuntz algebra O∞ , J. Number Theory 129 (12), 3069-3080, 2009

work page 2009
[36]

Larki, Primitive ideals and pure inﬁniteness of ultragraph C ∗- algebras, J

H. Larki, Primitive ideals and pure inﬁniteness of ultragraph C ∗- algebras, J. Korean Math. Soc. 56 (1), 1-23, 2019. 29

work page 2019
[37]

Rangaswamy, On simple modules over Leavitt path algebras , J

K.M. Rangaswamy, On simple modules over Leavitt path algebras , J. Algebra 423, 239-258, 2015

work page 2015
[38]

Rangaswamy, Leavitt path algebras with ﬁnitely presented irre- ducible representations, J

K.M. Rangaswamy, Leavitt path algebras with ﬁnitely presented irre- ducible representations, J. Algebra 447, 624-648, 2016

work page 2016
[39]

Tomforde, A uniﬁed approach to Exel-Laca algebras and C ∗-algebras associated to graphs , J

M. Tomforde, A uniﬁed approach to Exel-Laca algebras and C ∗-algebras associated to graphs , J. Operator Theory 50, 345-368, 2003

work page 2003
[40]

Tomforde, Simplicity of ultragraph algebras , Indiana Univ

M. Tomforde, Simplicity of ultragraph algebras , Indiana Univ. Math. J. 52 (4), 901-925, 2003. Daniel Gon¸ calves, Departamento de Matem´ atica, Universidade Federal de Santa Catarina, Florian´ opolis, 88040-900, Brazil. Email: daemig@gmail.com Danilo Royer, Departamento de Matem´ atica, Universidade Federa l de Santa Catarina, Florian´ opolis, 88040-900,...

work page 2003

[1] [1]

Abrams, F

G. Abrams, F. Mantese, A. Tonolo, Extensions of simple modules over Leavitt path algebras , J. Algebra 431, 78-106, 2015

work page 2015

[2] [2]

Abrams, K.M

G. Abrams, K.M. Rangaswamy, and M. Siles Molina, The socle series of a Leavitt path algebra , Israel J. Math. 184, 413-435, 2011

work page 2011

[3] [3]

Ara and K.M

P. Ara and K.M. Rangaswamy, Finitely presented simple modules over Leavitt path algebras , J. Algebra 417, 333-352, 2015

work page 2015

[4] [4]

Ara, K.M

P. Ara, K.M. Rangaswamy, Leavitt path algebras with at most countably many irreducible representations , Rev. Mat. Iberoam. 31 (4), 1263-1276, 2015. 26

work page 2015

[5] [5]

Aranda Pino, D

G. Aranda Pino, D. Mart ´ ın Barquero, C. Mart ´ ın Gonz´ alez and M. Siles Molina, The socle of a Leavitt path algebra , J. Pure Appl. Algebra 212 (3), 500-509, 2008

work page 2008

[6] [6]

ArandaPino, D

G. ArandaPino, D. Mart ´ ın Barquero, C. Mart ´ ın Gonz´ alez, M. S iles Molina, Socle theory for Leavitt path algebras of arbitrary graphs , Rev. Mat. Iberoam. 26 (2), 611-638, 2010

work page 2010

[7] [7]

Bratteli and P

O. Bratteli and P. E. T. Jorgensen, Iterated function systems and permu- tation representations of the Cuntz algebra , Memoirs Amer. Math. Soc. 139 no.663, 1999

work page 1999

[8] [8]

C. G. Canto, D. Gon¸ calves, Representations of relative Cohn path alge- bras, arXiv:1806.03077 [math.RA], 2018

work page arXiv 2018

[9] [9]

G. G. Castro, D. Gon¸ calves, KMS and ground states on ultragraph C*- algebras, Integr. Equ. Oper. Theory 90:63, doi 10.1007/s00020-018-24 90- 2, 2018

work page doi:10.1007/s00020-018-24 2018

[10] [10]

Chen, Irreducible representations of Leavitt path algebras , Forum Math

X.W. Chen, Irreducible representations of Leavitt path algebras , Forum Math. 27 (1), 549-57, 2015

work page 2015

[11] [11]

Correia Ramos, N

C. Correia Ramos, N. Martins, P. R. Pinto, On graph algebras from interval maps , Ann. Funct. Anal. 10 (2), 203-217, 2019

work page 2019

[12] [12]

Farsi, E

C. Farsi, E. Gillaspy, S. Kang, J. Packer, Separable representations, KMS states, and wavelets for higher-rank graphs , J. Math. Anal. Appl. 434, 241-270, 2016

work page 2016

[13] [13]

Farsi, E

C. Farsi, E. Gillaspy, S. Kang, J. Packer, Wavelets and graph C*- algebras, Appl. Numer. Harmon. Anal. 5, 35-86, 2017

work page 2017

[14] [14]

Farsi, E

C. Farsi, E. Gillaspy, P. Jorgensen, S. Kang, J. Packer, Monic represen- tations of ﬁnite higher-rank graphs , to appear at Ergodic Theory Dynam. Systems, doi.org/10.1017/etds.2018.79, 2019. 27

work page doi:10.1017/etds.2018.79 2018

[15] [15]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Branching systems and general Cuntz- Krieger uniqueness theorem for ultragraph C*-algebras, Internat. J. Math. 27 (10), 1650083 (26 pg), 2016

work page 2016

[16] [16]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Faithful representations of graph algebras via branching systems , Canad. Math. Bull. 59, 95-103, 2016

work page 2016

[17] [17]

Gon¸ calves, H

D. Gon¸ calves, H. Li, D. Royer, Branching systems for higher rank graph C*-algebras, Glasg. Math. J., doi:10.1017/S0017089518000058, 2018

work page doi:10.1017/s0017089518000058 2018

[18] [18]

Gon¸ calves, D

D. Gon¸ calves, D. Royer,Perron-Frobenius operators and representations of the Cuntz-Krieger algebras for inﬁnite matrices , J. Math. Anal. Appl. 351, 811-818, 2009

work page 2009

[19] [19]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, On the representations of Leavitt path algebras , J. Algebra 333, 258-272, 2011

work page 2011

[20] [20]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Unitary equivalence of representations of alge- bras associated with graphs, and branching systems , Funct. Anal. Appl. 45, 45-59, 2011

work page 2011

[21] [21]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Graph C∗-algebras, branching systems and the Perron-Frobenius operator, J. Math. Anal. Appl. 391, 457-465, 2012

work page 2012

[22] [22]

Gon¸ calves, D

D. Gon¸ calves, D. Royer,Leavitt path algebras as partial skew group rings , Comm. Algebra 42, 127-143, 2014

work page 2014

[23] [23]

Gon¸ calves, D

D. Gon¸ calves, D. Royer, Branching systems and representations of Cohn–Leavitt path algebras of separated graphs , J. Algebra 422, 413-426, 2015

work page 2015

[24] [24]

Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory

D. Gon¸ calves, D. Royer, Simplicity and chain conditions for ultragraph Leavitt path algebras via partial skew group ring theory , to appear at J. Aust. Math. Soc., arXiv:1706.03628

work page internal anchor Pith review Pith/arXiv arXiv

[25] [25]

Representations and the reduction theorem for ultragraph Leavitt path algebras

D. Gon¸ calves, D. Royer Representations and the reduction theorem for ultragraph Leavitt path algebras , arXiv:1902.00013 [math.RA], 2019. 28

work page internal anchor Pith review Pith/arXiv arXiv 1902

[26] [26]

Gon¸ calves and D

D. Gon¸ calves and D. Royer, Inﬁnite alphabet edge shift spaces via ultra- graphs and their C*-algebras , Int. Math. Res. Not., v. 2019, 2177-2203, 2019

work page 2019

[27] [27]

Gon¸ calves and D

D. Gon¸ calves and D. Royer, Ultragraphs and shift spaces over inﬁnite alphabets, Bull. Sci. Math., 141 (1), 25-45, 2017

work page 2017

[28] [28]

Gon¸ calves and M

D. Gon¸ calves and M. Sobottka, Continuous shift commuting maps be- tween ultragraph shift spaces , Discrete Contin. Dyn. Syst., v.39, 1033- 1048, 2019

work page 2019

[29] [29]

Li-Yorke chaos for ultragraph shift spaces

D. Gon¸ calves, B. B. Uggioni, Li-Yorke chaos for ultragraph shift spaces , to appear at Groups Geom. Dyn., arXiv:1806.07927 [math.DS]

work page internal anchor Pith review Pith/arXiv arXiv

[30] [30]

Ultragraph shift spaces and chaos

D. Gon¸ calves, B. B. Uggioni, Ultragraph shift spaces and chaos , arXiv:1902.05784 [math.DS], 2019

work page internal anchor Pith review Pith/arXiv arXiv 1902

[31] [31]

Hazrat, K.M

R. Hazrat, K.M. Rangaswamy, On graded irreducible representations of Leavitt path algebras , J. Algebra 450, 458-486, 2016

work page 2016

[32] [32]

Imanfar, A

M. Imanfar, A. Pourabbas and H. Larki, The leavitt path algebras of ultragraphs, arXiv:1701.00323v3 [math.RA], 2017

work page arXiv 2017

[33] [33]

Katsura, P

T. Katsura, P. S. Muhly, A. Sims and M. Tomforde, Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equiv- alence, J. Reine Angew. Math. 640, 135-165, 2010

work page 2010

[34] [34]

T, Katsura, A. Sims, M. Tomforde, Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras , J. Funct. Anal. 257 (5), 1589-1620, 2009

work page 2009

[35] [35]

Kawamura, Y

K. Kawamura, Y. Hayashi, D. Lascu, Continued fraction expansions and permutative representations of the Cuntz algebra O∞ , J. Number Theory 129 (12), 3069-3080, 2009

work page 2009

[36] [36]

Larki, Primitive ideals and pure inﬁniteness of ultragraph C ∗- algebras, J

H. Larki, Primitive ideals and pure inﬁniteness of ultragraph C ∗- algebras, J. Korean Math. Soc. 56 (1), 1-23, 2019. 29

work page 2019

[37] [37]

Rangaswamy, On simple modules over Leavitt path algebras , J

K.M. Rangaswamy, On simple modules over Leavitt path algebras , J. Algebra 423, 239-258, 2015

work page 2015

[38] [38]

Rangaswamy, Leavitt path algebras with ﬁnitely presented irre- ducible representations, J

K.M. Rangaswamy, Leavitt path algebras with ﬁnitely presented irre- ducible representations, J. Algebra 447, 624-648, 2016

work page 2016

[39] [39]

Tomforde, A uniﬁed approach to Exel-Laca algebras and C ∗-algebras associated to graphs , J

M. Tomforde, A uniﬁed approach to Exel-Laca algebras and C ∗-algebras associated to graphs , J. Operator Theory 50, 345-368, 2003

work page 2003

[40] [40]

Tomforde, Simplicity of ultragraph algebras , Indiana Univ

M. Tomforde, Simplicity of ultragraph algebras , Indiana Univ. Math. J. 52 (4), 901-925, 2003. Daniel Gon¸ calves, Departamento de Matem´ atica, Universidade Federal de Santa Catarina, Florian´ opolis, 88040-900, Brazil. Email: daemig@gmail.com Danilo Royer, Departamento de Matem´ atica, Universidade Federa l de Santa Catarina, Florian´ opolis, 88040-900,...

work page 2003