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A GENERALISATION OF HIGHER-RANK GRAPHS

Part of: Special categories Selfadjoint operator algebras

Published online by Cambridge University Press:  26 July 2021

MARK V. LAWSON Open the ORCID record for MARK V. LAWSON [Opens in a new window]  and
ALINA VDOVINA Open the ORCID record for ALINA VDOVINA [Opens in a new window]
MARK V. LAWSON*
Affiliation:
Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
ALINA VDOVINA
Affiliation:
School of Mathematics and Statistics, Herschel Building, Newcastle University, Newcastle-upon-Tyne NE1 7RU, UK e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We introduce ‘generalised higher-rank k-graphs’ as a class of categories equipped with a notion of size. They extend not only higher-rank k-graphs, but also the Levi categories introduced by the first author as a categorical setting for graphs of groups. We prove that examples of generalised higher-rank k-graphs can be constructed using Zappa–Szép products of groupoids and higher-rank graphs.

Keywords

higher-rank graphsZappa–Szép productsC*-algebras

MSC classification

Primary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories)
Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 2 , April 2022 , pp. 257 - 266
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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References

Afsar, Z., Brownlowe, N., Ramagge, J. and Whittaker, M. F., ‘C *-algebras of self-similar actions of groupoids on higher-rank graphs and their equilibrium states’, Preprint, 2019.Google Scholar
Brin, M. G., ‘On the Zappa–Szép product’, Comm. Algebra 33 (2005), 393–424.CrossRefGoogle Scholar
Kumjian, A. and Pask, D., ‘Higher rank graph ${C}^{\ast }$ -algebras’, New York J. Math. 6 (2000), 1–20.Google Scholar
Lallement, G., Semigroups and Combinatorial Applications (Wiley, New York, 1979).Google Scholar
Lawson, M. V., ‘A correspondence between a class of monoids and self-similar group actions I’, Semigroup Forum 76 (2008), 89–517.10.1007/s00233-008-9052-xCrossRefGoogle Scholar
Lawson, M. V., ‘Semigroups related to subshifts of graphs’, in: Proceedings of the International Conference on Semigroups, Acts and Categories with Applications to Graphs (eds. Laan, V., Bulman-Fleming, S. and Kaschek, R.) (Estonian Mathematical Society, Tartu, 2008), 92–107.Google Scholar
Lawson, M. V., Sims, A. and Vdovina, A., ‘Higher dimensional generalizations of the Thompson groups via higher rank graphs’, Preprint, 2020.CrossRefGoogle Scholar
Lawson, M. V. and Vdovina, A., ‘Higher dimensional generalizations of the Thompson groups’, Adv. Math. 369 (2020), 107–191.10.1016/j.aim.2020.107191CrossRefGoogle Scholar
Lawson, M. V. and Wallis, A. R., ‘A correspondence between a class of monoids and self-similar group actions II’, Internat. J. Algebra Comput. 25 (2015), 633–668.10.1142/S0218196715500137CrossRefGoogle Scholar
Lawson, M. V. and Wallis, A. R., ‘Levi categories and graphs of groups’, Theory Appl. Categ. 32 (2017), 780–802.Google Scholar
Robertson, G. and Steeger, T., ‘Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras’, J. reine angew. Math. 513 (1999), 115–144.CrossRefGoogle Scholar
Spielberg, J., ‘Groupoids and ${C}^{\ast }$ -algebras for categories of paths’, Trans. Amer. Math. Soc. 366 (2014), 5771–5819.CrossRefGoogle Scholar
Spielberg, J., ‘Groupoids and ${C}^{\ast }$ -algebras for left cancellative small categories’, Indiana Univ. Math J. 69 (2020), 1579–1626.10.1512/iumj.2020.69.7969CrossRefGoogle Scholar