Algebra & Number Theory

Graded Steinberg algebras and their representations

Pere Ara, Roozbeh Hazrat, Huanhuan Li, and Aidan Sims

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the category of left unital graded modules over the Steinberg algebra of a graded ample Hausdorff groupoid. In the first part of the paper, we show that this category is isomorphic to the category of unital left modules over the Steinberg algebra of the skew-product groupoid arising from the grading. To do this, we show that the Steinberg algebra of the skew product is graded isomorphic to a natural generalisation of the Cohen–Montgomery smash product of the Steinberg algebra of the underlying groupoid with the grading group. In the second part of the paper, we study the minimal (that is, irreducible) representations in the category of graded modules of a Steinberg algebra, and establish a connection between the annihilator ideals of these minimal representations, and effectiveness of the groupoid.

Specialising our results, we produce a representation of the monoid of graded finitely generated projective modules over a Leavitt path algebra. We deduce that the lattice of order-ideals in the K 0 -group of the Leavitt path algebra is isomorphic to the lattice of graded ideals of the algebra. We also investigate the graded monoid for Kumjian–Pask algebras of row-finite k -graphs with no sources. We prove that these algebras are graded von Neumann regular rings, and record some structural consequences of this.

Article information

Source
Algebra Number Theory, Volume 12, Number 1 (2018), 131-172.

Dates
Received: 16 April 2017
Revised: 6 November 2017
Accepted: 8 November 2017
First available in Project Euclid: 4 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1522807233

Digital Object Identifier
doi:10.2140/ant.2018.12.131

Mathematical Reviews number (MathSciNet)
MR3781435

Zentralblatt MATH identifier
06861738

Subjects
Primary: 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05]
Secondary: 16G30: Representations of orders, lattices, algebras over commutative rings [See also 16Hxx] 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx]

Keywords
Steinberg algebra Leavitt path algebra skew-product smash product graded irreducible representation annihilator ideal effective groupoid

Citation

Ara, Pere; Hazrat, Roozbeh; Li, Huanhuan; Sims, Aidan. Graded Steinberg algebras and their representations. Algebra Number Theory 12 (2018), no. 1, 131--172. doi:10.2140/ant.2018.12.131. https://projecteuclid.org/euclid.ant/1522807233


Export citation

References

  • G. Abrams, “Leavitt path algebras: the first decade”, Bull. Math. Sci. 5:1 (2015), 59–120.
  • G. Abrams and G. Aranda Pino, “The Leavitt path algebra of a graph”, J. Algebra 293:2 (2005), 319–334.
  • G. Abrams and G. Aranda Pino, “The Leavitt path algebras of arbitrary graphs”, Houston J. Math. 34:2 (2008), 423–442.
  • G. Abrams, F. Mantese, and A. Tonolo, “Extensions of simple modules over Leavitt path algebras”, J. Algebra 431 (2015), 78–106.
  • P. Ara and K. R. Goodearl, “Leavitt path algebras of separated graphs”, J. Reine Angew. Math. 669 (2012), 165–224.
  • P. Ara and K. R. Goodearl, “Tame and wild refinement monoids”, Semigroup Forum 91:1 (2015), 1–27.
  • P. Ara and E. Pardo, “Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras”, J. K-Theory 14:2 (2014), 203–245.
  • P. Ara and K. M. Rangaswamy, “Finitely presented simple modules over Leavitt path algebras”, J. Algebra 417 (2014), 333–352.
  • P. Ara and K. M. Rangaswamy, “Leavitt path algebras with at most countably many irreducible representations”, Rev. Mat. Iberoam. 31:4 (2015), 1263–1276.
  • P. Ara, M. A. Moreno, and E. Pardo, “Nonstable $K$-theory for graph algebras”, Algebr. Represent. Theory 10:2 (2007), 157–178.
  • G. Aranda Pino, J. Clark, A. an Huef, and I. Raeburn, “Kumjian–Pask algebras of higher-rank graphs”, Trans. Amer. Math. Soc. 365:7 (2013), 3613–3641.
  • M. Beattie, “A generalization of the smash product of a graded ring”, J. Pure Appl. Algebra 52:3 (1988), 219–226.
  • S. K. Berberian, Baer $*$-rings, Grundlehren der Math. Wissenschaften 195, Springer, 1972.
  • G. M. Bergman, “Modules over coproducts of rings”, Trans. Amer. Math. Soc. 200 (1974), 1–32.
  • J. Brown, L. O. Clark, C. Farthing, and A. Sims, “Simplicity of algebras associated to étale groupoids”, Semigroup Forum 88:2 (2014), 433–452.
  • X.-W. Chen, “Irreducible representations of Leavitt path algebras”, Forum Math. 27:1 (2015), 549–574.
  • L. O. Clark and C. Edie-Michell, “Uniqueness theorems for Steinberg algebras”, Algebr. Represent. Theory 18:4 (2015), 907–916.
  • L. O. Clark and Y. E. P. Pangalela, “Kumjian–Pask algebras of finitely aligned higher-rank graphs”, J. Algebra 482 (2017), 364–397.
  • L. O. Clark and A. Sims, “Equivalent groupoids have Morita equivalent Steinberg algebras”, J. Pure Appl. Algebra 219:6 (2015), 2062–2075.
  • L. O. Clark, C. Farthing, A. Sims, and M. Tomforde, “A groupoid generalisation of Leavitt path algebras”, Semigroup Forum 89:3 (2014), 501–517.
  • L. O. Clark, C. Edie-Michell, A. an Huef, and A. Sims, “Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras”, preprint, 2016.
  • L. O. Clark, R. Exel, and E. Pardo, “A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras”, Forum Math. (online publication August 2017).
  • M. Cohen and S. Montgomery, “Group-graded rings, smash products, and group actions”, Trans. Amer. Math. Soc. 282:1 (1984), 237–258.
  • D. Drinen and M. Tomforde, “The $C^*$-algebras of arbitrary graphs”, Rocky Mountain J. Math. 35:1 (2005), 105–135.
  • C. Faith, Algebra: rings, modules and categories, I, Grundlehren der Math. Wissenschaften 190, Springer, 1973.
  • K. R. Goodearl, “Leavitt path algebras and direct limits”, pp. 165–187 in Rings, modules and representations, edited by N. V. Dung et al., Contemp. Math. 480, American Mathematical Society, Providence, RI, 2009.
  • E. L. Green, “Graphs with relations, coverings and group-graded algebras”, Trans. Amer. Math. Soc. 279:1 (1983), 297–310.
  • R. Hazrat, “The graded Grothendieck group and the classification of Leavitt path algebras”, Math. Ann. 355:1 (2013), 273–325.
  • R. Hazrat, “A note on the isomorphism conjectures for Leavitt path algebras”, J. Algebra 375 (2013), 33–40.
  • R. Hazrat, Graded rings and graded Grothendieck groups, London Mathematical Society Lecture Note Series 435, Cambridge University Press, 2016.
  • R. Hazrat and K. M. Rangaswamy, “On graded irreducible representations of Leavitt path algebras”, J. Algebra 450 (2016), 458–486.
  • I. Kaplansky, Rings of operators, W. A. Benjamin, New York, 1968.
  • A. Kumjian and D. Pask, “$C^*$-algebras of directed graphs and group actions”, Ergodic Theory Dynam. Systems 19:6 (1999), 1503–1519.
  • A. Kumjian and D. Pask, “Higher rank graph $C^\ast$-algebras”, New York J. Math. 6 (2000), 1–20.
  • A. Kumjian, D. Pask, I. Raeburn, and J. Renault, “Graphs, groupoids, and Cuntz–Krieger algebras”, J. Funct. Anal. 144:2 (1997), 505–541.
  • S. X. Liu and F. Van Oystaeyen, “Group graded rings, smash products and additive categories”, pp. 299–310 in Perspectives in ring theory (Antwerp, 1987), edited by F. Van Oystaeyen and L. Le Bruyn, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 233, Kluwer, Dordrecht, 1988.
  • P. Menal and J. Moncasi, “Lifting units in self-injective rings and an index theory for Rickart $C^\ast$-algebras”, Pacific J. Math. 126:2 (1987), 295–329.
  • C. Năstăsescu and F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics 1836, Springer, 2004.
  • A. L. T. Paterson, “Graph inverse semigroups, groupoids and their $C^\ast$-algebras”, J. Operator Theory 48:3, suppl. (2002), 645–662.
  • N. C. Phillips, “A classification theorem for nuclear purely infinite simple $C^*$-algebras”, Doc. Math. 5 (2000), 49–114.
  • I. Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics 103, American Mathematical Society, Providence, RI, 2005.
  • I. Raeburn, A. Sims, and T. Yeend, “The $C^*$-algebras of finitely aligned higher-rank graphs”, J. Funct. Anal. 213:1 (2004), 206–240.
  • K. M. Rangaswamy, “Leavitt path algebras with finitely presented irreducible representations”, J. Algebra 447 (2016), 624–648.
  • J. Renault, A groupoid approach to $C\sp{\ast} $-algebras, Lecture Notes in Mathematics 793, Springer, 1980.
  • A. Sims, “Gauge-invariant ideals in the $C^*$-algebras of finitely aligned higher-rank graphs”, Canad. J. Math. 58:6 (2006), 1268–1290.
  • A. Sims, “The co-universal $C^\ast$-algebra of a row-finite graph”, New York J. Math. 16 (2010), 507–524.
  • B. Steinberg, “A groupoid approach to discrete inverse semigroup algebras”, Adv. Math. 223:2 (2010), 689–727.
  • M. Tomforde, “Uniqueness theorems and ideal structure for Leavitt path algebras”, J. Algebra 318:1 (2007), 270–299.
  • S. B. G. Webster, “The path space of a directed graph”, Proc. Amer. Math. Soc. 142:1 (2014), 213–225.
  • R. Wisbauer, Foundations of module and ring theory, Algebra, Logic and Applications 3, Gordon and Breach, Philadelphia, PA, 1991.