| Title:
|
Representations of étale Lie groupoids and modules over Hopf algebroids (English) |
| Author:
|
Kališnik, Jure |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
653-672 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The classical Serre-Swan's theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them. (English) |
| Keyword:
|
étale Lie groupoids |
| Keyword:
|
Hopf algebroids |
| Keyword:
|
representations |
| Keyword:
|
modules |
| Keyword:
|
equivalence |
| Keyword:
|
Morita category |
| MSC:
|
16D40 |
| MSC:
|
16D90 |
| MSC:
|
19L47 |
| MSC:
|
22A22 |
| MSC:
|
58H05 |
| idZBL:
|
Zbl 1249.22003 |
| idMR:
|
MR2853081 |
| DOI:
|
10.1007/s10587-011-0037-7 |
| . |
| Date available:
|
2011-09-22T14:34:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141628 |
| . |
| Reference:
|
[1] Adem, A., Leida, J., Ruan, Y.: Orbifolds and Stringy Topology.Cambridge Tracts in Mathematics 171, Cambridge University Press, Cambridge (2007). Zbl 1157.57001, MR 2359514 |
| Reference:
|
[2] Abad, C. Arias, Crainic, M.: Representations up to homotopy and Bott's spectral sequence for Lie groupoids.Preprint arXiv: 0911.2859 (2009). MR 3107517 |
| Reference:
|
[3] Atiyah, M. F., Anderson, D. R.: K-Theory.With reprints of M. F. Atiyah: Power operations in K-theory. New York-Amsterdam, W. A. Benjamin (1967). Zbl 0159.53401, MR 0224083 |
| Reference:
|
[4] Bos, R.: Continuous representations of groupoids.Preprint arXiv: 0612639 (2006). MR 2844451 |
| Reference:
|
[5] Blohmann, Ch., Tang, X., Weinstein, A.: Hopfish structure and modules over irrational rotation algebras.Contemporary Mathematics 462 (2008), 23-40. Zbl 0216.33803, MR 2444366, 10.1090/conm/462/09059 |
| Reference:
|
[6] Connes, A.: A survey of foliations and operator algebras.Proc. Symp. Pure Math. 38 (1982), 521-628. Zbl 0531.57023, MR 0679730 |
| Reference:
|
[7] Connes, A.: Noncommutative Geometry.Academic Press, San Diego (1994). Zbl 0818.46076, MR 1303779 |
| Reference:
|
[8] Crainic, M., Moerdijk, I.: A homology theory for étale groupoids.J. Reine Angew. Math. 521 (2000), 25-46. Zbl 0954.22002, MR 1752294 |
| Reference:
|
[9] Crainic, M., Moerdijk, I.: Foliation groupoids and their cyclic homology.Adv. Math. 157 (2001), 177-197. Zbl 0989.22010, MR 1813430, 10.1006/aima.2000.1944 |
| Reference:
|
[10] Crainic, M., Moerdijk, I.: Čech-De Rham theory for leaf spaces of foliations.Math. Ann. 328 (2004), 59-85. Zbl 1043.57012, MR 2030370, 10.1007/s00208-003-0473-2 |
| Reference:
|
[11] Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature, and Cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles.Pure and Applied Mathematics 47, Volume 1, Academic Press, New York (1972). Zbl 0322.58001, MR 0336650 |
| Reference:
|
[12] Haefliger, A.: Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes.Comment. Math. Helv. 32 (1958), 248-329. Zbl 0085.17303, MR 0100269, 10.1007/BF02564582 |
| Reference:
|
[13] Haefliger, A.: Groupoïdes d'holonomie et classifiants.Astérisque 116 (1984), 70-97. Zbl 0562.57012, MR 0755163 |
| Reference:
|
[14] Hilsum, M., Skandalis, G.: Morphismes $K$-orientés d'espaces de feuilles et fonctorialité en théorie de Kasparov.Ann. Sci. Éc. Norm. Supér. 20 (1987), 325-390. Zbl 0656.57015, MR 0925720, 10.24033/asens.1537 |
| Reference:
|
[15] Kališnik, J.: Representations of orbifold groupoids.Topology Appl. 155 (2008), 1175-1188. MR 2421827, 10.1016/j.topol.2008.02.004 |
| Reference:
|
[16] Kališnik, J., Mrčun, J.: Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids.Indag. Math., New Ser. 19 (2008), 73-96. MR 2466395, 10.1016/S0019-3577(08)80016-X |
| Reference:
|
[17] Kamber, F., Tondeur, P.: Foliated Bundles and Characteristic Classes.Lecture Notes in Mathematics 493, Berlin-Heidelberg-New York, Springer-Verlag (1975). Zbl 0308.57011, MR 0402773 |
| Reference:
|
[18] Mackenzie, K. C. H.: The General Theory of Lie Groupoids and Lie Algebroids.LMS Lecture Note Series 213, Cambridge University Press, Cambridge (2005). Zbl 1078.58011, MR 2157566 |
| Reference:
|
[19] Milnor, J. W., Stasheff, J. D.: Characteristic Classes.Annals of Mathematics Studies 76. Princeton, N.J. Princeton University Press and University of Tokyo Press (1974). Zbl 0298.57008, MR 0440554 |
| Reference:
|
[20] MacLane, S.: Categories for the Working Mathematician.4th corrected printing, Graduate Texts in Mathematics, 5. New York etc., Springer-Verlag (1988). Zbl 0705.18001, MR 1712872 |
| Reference:
|
[21] Moerdijk, I.: The classifying topos of a continuous groupoid I.Trans. Am. Math. Soc. 310 (1988), 629-668. Zbl 0706.18007, MR 0973173, 10.1090/S0002-9947-1988-0973173-9 |
| Reference:
|
[22] Moerdijk, I.: Classifying toposes and foliations.Ann. Inst. Fourier 41 (1991), 189-209. Zbl 0727.57029, MR 1112197, 10.5802/aif.1254 |
| Reference:
|
[23] Moerdijk, I.: Orbifolds as groupoids: an introduction.Contemp. Math. 310 (2002), 205-222. Zbl 1041.58009, MR 1950948, 10.1090/conm/310/05405 |
| Reference:
|
[24] Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids.Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge (2003). MR 2012261 |
| Reference:
|
[25] Moerdijk, I., Mrčun, J.: Lie groupoids, sheaves and cohomology.Poisson Geometry, Deformation Quantisation and Group Representations, London Mathematical Society Lecture Note Series 323, Cambridge University Press, Cambridge (2005), 145-272. MR 2166453 |
| Reference:
|
[26] Mrčun, J.: Stability and invariants of Hilsum-Skandalis maps.PhD Thesis, Utrecht University (1996). |
| Reference:
|
[27] Mrčun, J.: Functoriality of the bimodule associated to a Hilsum-Skandalis map.K-Theory 18 (1999), 235-253. MR 1722796, 10.1023/A:1007773511327 |
| Reference:
|
[28] Mrčun, J.: The Hopf algebroids of functions on étale groupoids and their principal Morita equivalence.J. Pure Appl. Algebra 160 (2001), 249-262. MR 1836002, 10.1016/S0022-4049(00)00071-2 |
| Reference:
|
[29] Mrčun, J.: On duality between étale groupoids and Hopf algebroids.J. Pure Appl. Algebra 210 (2007), 267-282. MR 2311185, 10.1016/j.jpaa.2006.09.006 |
| Reference:
|
[30] Pradines, J.: Morphisms between spaces of leaves viewed as fractions.Cah. Topologie Géom. Différ. Catég. 30 (1989), 229-246. Zbl 0686.57013, MR 1029626 |
| Reference:
|
[31] Pronk, D., Scull, L.: Translation groupoids and orbifold cohomology.Can. J. Math. 62 (2010), 614-645. Zbl 1197.57026, MR 2666392, 10.4153/CJM-2010-024-1 |
| Reference:
|
[32] Renault, J.: A Groupoid Approach to $C^{\ast}$-algebras.Lecture Notes in Mathematics 793, Berlin-Heidelberg-New York, Springer-Verlag (1980). Zbl 0433.46049, MR 0584266 |
| Reference:
|
[33] Rieffel, M. A.: $C*$-algebras associated with irrational rotations.Pac. J. Math. 93 (1981), 415-429. Zbl 0499.46039, MR 0623572, 10.2140/pjm.1981.93.415 |
| Reference:
|
[34] Satake, I.: On a generalization of the notion of manifold.Proc. Natl. Acad. Sci. USA 42 (1956), 359-363. Zbl 0074.18103, MR 0079769, 10.1073/pnas.42.6.359 |
| Reference:
|
[35] Segal, G.: Equivariant $K$-theory.Publ. Math., Inst. Hates Étud. Sci. 34 (1968), 129-151. Zbl 0199.26202, MR 0234452, 10.1007/BF02684593 |
| Reference:
|
[36] Serre, J.-P.: Faisceaux algébriques cohérents.Ann. Math. 61 (1955), 197-278. Zbl 0067.16201, MR 0068874, 10.2307/1969915 |
| Reference:
|
[37] Swan, R. G.: Vector bundles and projective modules.Trans. Am. Math. Soc. 105 (1962), 264-277. Zbl 0109.41601, MR 0143225, 10.1090/S0002-9947-1962-0143225-6 |
| Reference:
|
[38] Trentinaglia, G.: On the role of effective representations of Lie groupoids.Adv. Math. 225 (2010), 826-858. Zbl 1216.58005, MR 2671181, 10.1016/j.aim.2010.03.014 |
| Reference:
|
[39] Winkelnkemper, H. E.: The graph of a foliation.Ann. Global Anal. Geom. 1 (1983), 51-75. Zbl 0526.53039, MR 0739904, 10.1007/BF02329732 |
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