×

The Hopf algebroids of functions on étale groupoids and their principal Morita equivalence. (English) Zbl 0986.16017

J. Pure Appl. Algebra 160, No. 2-3, 249-262 (2001); erratum ibid. 176, No. 2-3, 275-276 (2002).
The author begins with a study of the Connes convolution algebra of functions with compact support on an étale groupoid. This is seen to admit a Hopf algebroid structure which is, in general, not commutative. It has several interesting properties, which the author uses to define the notion of an ‘étale Hopf algebroid’. This concept then becomes the focus of the article, which looks at bimodules over such objects, with a view to understanding Morita equivalence in this setting.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
22A22 Topological groupoids (including differentiable and Lie groupoids)
46L52 Noncommutative function spaces
16D90 Module categories in associative algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brylinski, J.-L.; Nistor, V., Cyclic cohomology of étale groupoids, K-Theory, 8, 341-365 (1994) · Zbl 0812.19003
[2] A. Bruguières, G. Maltsiniotis, Quasi-groupoı̈des quantiques, C. R. Acad. Sci. Paris 319 (1994) 933-926.; A. Bruguières, G. Maltsiniotis, Quasi-groupoı̈des quantiques, C. R. Acad. Sci. Paris 319 (1994) 933-926. · Zbl 0836.16020
[3] A. Connes, The von Neumann algebra of a foliation, in: G. Dell’Antonio, S. Doplicher, G. Jona-Lasinio (Eds.), Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Vol. 80, Springer, New York, 1978, pp. 145-151.; A. Connes, The von Neumann algebra of a foliation, in: G. Dell’Antonio, S. Doplicher, G. Jona-Lasinio (Eds.), Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., Vol. 80, Springer, New York, 1978, pp. 145-151. · Zbl 0433.46056
[4] A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Proceedings of the Symposium on Pure Mathematics 38, Part I, American Mathematical Society, Providence, RI, 1982, pp. 521-628.; A. Connes, A survey of foliations and operator algebras, Operator algebras and applications, Proceedings of the Symposium on Pure Mathematics 38, Part I, American Mathematical Society, Providence, RI, 1982, pp. 521-628. · Zbl 0531.57023
[5] Haefliger, A., Groupoı̈des d’holonomie et classifiants, Astérisque, 116, 70-97 (1984) · Zbl 0562.57012
[6] Hilsum, M.; Skandalis, G., Stabilité des algèbres de feuilletages, Ann. Inst. Fourier Grenoble, 33, 201-208 (1983) · Zbl 0505.46043
[7] Hilsum, M.; Skandalis, G., Morphismes K-orientes d’espaces de feuilles et functorialite en theorie de Kasparov, Ann. Sci. École. Norm. Sup., 20, 325-390 (1987) · Zbl 0656.57015
[8] Lu, J.-H., Hopf algebroids and quantum groupoids, Int. J. Math., 7, 47-70 (1996) · Zbl 0884.17010
[9] Maltsiniotis, G., Groupoı̈des quantiques, C. R. Acad. Sci. Paris, 314, 249-252 (1992) · Zbl 0767.17015
[10] G. Maltsiniotis, Groupoı̈des quantiques de base non commutative, Comm. Algebra, to appear.; G. Maltsiniotis, Groupoı̈des quantiques de base non commutative, Comm. Algebra, to appear. · Zbl 0956.18002
[11] Moerdijk, I., Classifying toposes and foliations, Ann. Inst. Fourier, Grenoble, 41, 189-209 (1991) · Zbl 0727.57029
[12] Mrčun, J., Functoriality of the bimodule associated to a Hilsum-Skandalis map, K-Theory, 18, 235-253 (1999) · Zbl 0938.22002
[13] Muhly, P. S.; Renault, J.; Williams, D., Equivalence and isomorphism for groupoid \(C^*\)-algebras, J. Operator Theory, 17, 3-22 (1987) · Zbl 0645.46040
[14] Ravenel, D., Complex Cobordism and Stable Homotopy Groups of Spheres, Pure and Applied Math. Series, vol. 121 (1986), Academic Press: Academic Press San Diego · Zbl 0608.55001
[15] Renault, J., A Groupoid Approach to \(C^*\)-algebras, Lecture Notes in Math. vol. 793 (1980), Springer: Springer New York · Zbl 0433.46049
[16] Xu, P., Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris, 326, 289-294 (1998) · Zbl 0911.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.