Residually linear abstract groupoids

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We introduce the notion of residually linear groupoids. We characterize this class in analogy with the group-theoretic setting. Various properties are proved and a relationship with residual finiteness is investigated. From a categorical point of view, our approach extends some well-known results in the theory of discrete groups, due mainly to Mal'cev and Menal. Finally, as an application, we show that the character groupoid of the Hopf algebroid of representative functions of a transitive groupoid is always residually linear.Bibliography: 24 titles.

Sobre autores

Khalid Draoui

Sidi Mohamed Ben Abdellah University

Autor responsável pela correspondência
Email: khalid.draoui@usmba.ac.ma
ORCID ID: 0000-0001-9879-4096

PhD, no status

Hanan Choulli

Sidi Mohamed Ben Abdellah University

Email: hanan.choulli@usmba.ac.ma

Doctor of Science, Professor

Hakima Mouanis

Sidi Mohamed Ben Abdellah University

Email: hakima.mouanis@usmba.ac.ma

Bibliografia

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  10. P. Menal, “Residual linearity for certain nilpotent groups”, Proc. Amer. Math. Soc., 68:1 (1978), 27–31
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  13. H. Choulli, K. Draoui, H. Mouanis, “Residually linear groups”, Proc. Jangjeon Math. Soc., 27:2 (2024), 271–288
  14. M. Amini, “Tannak–Krein duality for compact groupoids II, duality”, Oper. Matrices, 4:4 (2010), 573–592
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  17. H.-J. Baues, M. Jibladze, “Classification of Abelian track categories”, $K$-theory, 25:3 (2002), 299–311
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  19. F. Komura, “Quotients of Etale groupoids and the abelianizations of groupoid $C^*$-algebras”, J. Aust. Math. Soc., 111:1 (2021), 56–75
  20. L. El Kaoutit, L. Spinosa, “On Burnside theory for groupoids”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 66(114):1 (2023), 41–87
  21. L. El Kaoutit, Representative functions on discrete groupoids and duality with Hopf algebroids, 2013
  22. L. El Kaoutit, “On geometrically transitive Hopf algebroids”, J. Pure Appl. Algebra, 222:11 (2018), 3483–3520
  23. L. El Kaoutit, J. Gomez-Torrecillas, “On the finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard–Vessiot theory”, Bull. Belg. Math. Soc. Simon Stevin, 28:1 (2021), 53–121
  24. A. J. Berrick, “Groups with no nontrivial linear representations”, Bull. Aust. Math. Soc., 50:1 (1994), 1–11

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