Modules for Leavitt Path Algebras of Bi-separated Graphs via Representation Graphs

Abstract

Leavitt path algebras of bi-separated graphs have been recently introduced by Mohan and Suhas. These algebras provide a common framework for studying various generalisations of Leavitt path algebras. In this paper, the author obtains modules for the Leavitt path algebra L(Ė) of a finitely bi-separated graph Ė = (E, C, D) by introducing the notion of a representation graph for Ė. Among these modules the author finds a class of simple modules. If the bi-separation on E is the Cuntz-Krieger bi-separation (and hence L(Ė) is isomorphic to the usual Leavitt path algebra L(E)), one recovers the celebrated Chen simple modules.

Keywords

Leavitt path algebra / Bi-separated graph / Simple modules / 16S88 / 16D70

References

[1] Abrams G, Ara P, Siles Molina M. Leavitt Path Algebras, 2017, London, Springer-Verlag2191

[2] Abrams G, Aranda Pino G. The Leavitt path algebra of a graph. J. Algebra, 2005, 293: 319-334

[3] Abrams, G. and Hazrat, R., Connections between abelian sandpile models and the K-theory of weighted Leavitt path algebras, Eur. J. Math., 9, 2023, Paper No. 21.

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

[5] Alahmadi A, Alsulami H, Jain S, Zelmanov E L. Leavitt path algebras of finite Gelfand-Kirillov dimension. J. Algebra Appl., 2012, 11(6): 1250225

[6] Alahmedi A, Alsulami H, Jain S, Zelmanov E I. Structure of Leavitt path algebras of polynomial growth. Proc. Natl. Acad. Sci. USA, 2013, 110(38): 15222-15224

[7] Ánh P N, Nam T G. Special irreducible representations of Leavitt path algebras. Adv. Math., 2021, 377: 107483

[8] Ara P, Brustenga M. Module theory over Leavitt path algebras and K-theory. J. Pure Appl. Algebra, 2010, 214: 1131-1151

[9] Ara P, Goodearl K. Leavitt path algebras of separated graphs. J. Reine Angew. Math., 2012, 669: 165-224

[10] Ara P, Moreno M A, Pardo E. Nonstable K-theory for Graph Algebras. Algebr. Represent. Theory, 2007, 10: 157-178

[11] Ara P, Rangaswamy K M. Finitely presented simple modules over Leavitt path algebras. J. Algebra, 2014, 417: 333-352

[12] Arnone G. Lifting morphisms between graded Grothendieck groups of Leavitt path algebras. J. Algebra, 2023, 631: 804-829

[13] Boava G, de Castro G G, Gonçalves D, van Wyk D W. Leavitt path algebras of labelled graphs. J. Algebra, 2023, 629: 265-306

[14] Bock W, Sebandal A, Vilela J P. A talented monoid view on Lie bracket algebras over Leavitt path algebras. J. Algebra Appl., 2023, 22: 2350170

[15] Chen X W. Irreducible representations of Leavitt path algebras. Forum Math., 2015, 27: 549-574

[16] Chirvasitu A. C* completions of Leavitt-path-algebra pullbacks. Appl. Categ. Structures, 2022, 30: 1123-1133

[17] Crisp T, MacDonald D. Flow equivalence of diagram categories and Leavitt path algebras. J. Pure Appl. Algebra, 2023, 227: 107232

[18] Deo T T, Chi V T. On some ideal structure of Leavitt path algebras with coefficients in integral domains. Int. Electron. J. Algebra, 2023, 33: 34-53

[19] Gonçalves D, Royer D. On the representations of Leavitt path algebras. J. Algebra, 2011, 333: 258-272

[20] Gonçalves D, Royer D. Branching systems and representations of Cohn-Leavitt path algebras of separated graphs. J. Algebra, 2015, 422: 413-426

[21] Gupta R S, Sen M K. k-Simplicity of Leavitt path algebras with coefficients in a k-Semifield. Discuss. Math. Gen. Algebra Appl., 2022, 42: 241-253

[22] Hazrat R. The graded structure of Leavitt path algebras. Israel J. Math., 2013, 195: 833-895

[23] Hazrat R, Nam T G. Realizing ultragraph Leavitt path algebras as Steinberg algebras. J. Pure Appl. Algebra, 2023, 227: 107275

[24] Hazrat R, Preusser R, Shchegolev A. Irreducible representations of Leavitt algebras. J. Algebra, 2022, 612: 147-207

[25] Higman G. The units of group-rings. Proc. London Math. Soc, 1940, 46(2): 231-248

[26] Kumjian A, Pask D. C*-algebras of directed graphs and group actions. Ergodic. Theory Dynam. Systems, 1999, 19: 1503-1519

[27] Ma Y, Bao Y-H, Li H. Realization for tensor products of Leavitt path algebras. Arch. Math., 2022, 119: 121-133

[28] Martínez L, Pinedo H, Soler Y. Ring theoretical properties of epsilon-strongly graded rings and Leavitt path algebras. Comm. Algebra, 2022, 50: 3201-3217

[29] Leavitt W G. Modules over rings of words. Proc. Amer. Math. Soc, 1956, 7: 188-193

[30] Leavitt W G. Modules without invariant basis number. Proc. Amer. Math. Soc., 1957, 8: 322-328

[31] Leavitt W G. The module type of a ring. Trans. Amer. Math. Soc, 1962, 103: 113-130

[32] Leavitt W G. The module type of homomorphic images. Duke Math. J., 1965, 32: 305-311

[33] Mohan R, Suhas B N. Cohn-Leavitt path algebras of bi-separated graphs. Comm. Algebra, 2021, 49: 1991-2021

[34] Preusser R. Leavitt path algebras of hypergraphs. Bull. Braz. Math. Soc. (N.S.), 2020, 51: 185-221

[35] Preusser, R., Modules for Leavitt path algebras via extended algebraic branching systems, Bull. Braz. Math. Soc. (N.S.), 54, 2023, Paper No. 18.

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

[37] Rangaswamy K M. Leavitt path algebras with finitely presented irreducible representations. J. Algebra, 2016, 447: 624-648

[38] Rangaswamy K M. A note on the Dixmier-Moeglin equivalence in Leavitt path algebras of arbitrary graphs over a field. Comm. Algebra, 2023, 51: 20-28

[39] Smith S P. Category equivalences involving graded modules over path algebras of quivers. Adv. Math., 2012, 230: 1780-1810

[40] Vaš L. Cancellation properties of graded and nonunital rings, Graded clean and graded exchange Leavitt path algebras. J. Algebra Appl., 2023, 22: 2350050

[41] Vaš L. Graded irreducible representations of Leavitt path algebras: A new type and complete classification. J. Pure Appl. Algebra, 2023, 227: 107213

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