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Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids. (English) Zbl 1181.22008

An étale Lie groupoid \(G\) is completely determined by its convolution algebra \(\mathcal C_c^{\infty }(G)\) equipped with the natural Hopf algebroid structure.
The paper has two principal results: 1) any principal \(H\)-bundle \(P\) over \(G\) is uniquely determined by the associated \(\mathcal C_c^{\infty }(G)-\mathcal C_c^{\infty }(H)\)-bimodule \(\mathcal C_c^{\infty }(P)\) equipped with the natural coalgebra structure, and 2) the functor \(\mathcal C_c^{\infty }\) gives an equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
58H05 Pseudogroups and differentiable groupoids
16T05 Hopf algebras and their applications
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References:

[1] Blohmann, C.; Weinstein, A., Group-like objects in Poisson geometry and algebra (2007), Preprint, arXiv: math.SG/0701499
[2] Blohmann, C.; Tang, X.; Weinstein, A., Hopfish structure and modules over irrational rotation algebras (2006), Preprint, arXiv: math.QA/0604405
[3] Böhm, G.; Szlachányi, K., Hopf algebroids with bijective antipodes: axioms, integrals, and, duals, J. Algebra, 274, 708-750 (2004) · Zbl 1080.16035
[4] Cannas da Silva, A.; Weinstein, A., Geometric Models for Noncommutative Algebras, (Berkeley Mathematics Lecture Notes, vol. 10 (1999), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1135.58300
[5] Connes, A., (Noncommutative Geometry (1994), Academic Press: Academic Press San Diego) · Zbl 0681.55004
[6] Crainic, M.; Moerdijk, I., A homology theory for étale groupoids, J. Reine Angew. Math., 521, 25-46 (2000) · Zbl 0954.22002
[7] Crainic, M.; Moerdijk, I., Foliation groupoids and their cyclic homology, Adv. Math., 157, 177-197 (2001) · Zbl 0989.22010
[8] Haefliger, A., Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes, Comment. Math. Helv., 32, 248-329 (1958) · Zbl 0085.17303
[9] Haefliger, A., Groupoids and foliations. Groupoids in analysis, geometry, and physics, Contemp. Math., 282, 83-100 (2001) · Zbl 0994.57025
[10] Hilsum, M.; Skandalis, G., Morphismes K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. Ecole Norm. Sup., 20, 325-390 (1987) · Zbl 0656.57015
[11] Kapranov, M., Free Lie algebroids and the space of paths, Selecta Math. (N.S.), 13, 277-319 (2007) · Zbl 1149.14003
[12] Lu, J.-H., Hopf algebroids and quantum groupoids, International J. Math., 7, 47-70 (1996) · Zbl 0884.17010
[13] Mackenzie, K. C.H., General Theory of Lie Groupoids and Lie Algebroids, (London Math. Soc. Lecture Note Ser., vol. 213 (2005), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1078.58011
[14] Maltsiniotis, G., Groupoïdes quantiques de base non commutative, Comm. Algebra, 28, 3441-3501 (2000) · Zbl 0956.18002
[15] Moerdijk, I., Classifying toposes and foliations, Ann. Inst. Fourier (Grenoble), 41, 189-209 (1991) · Zbl 0727.57029
[16] Moerdijk, I., Orbifolds as groupoids: an introduction, Contemp. Math., 310, 205-222 (2002) · Zbl 1041.58009
[17] Moerdijk, I.; Mrčun, J., Introduction to Foliations and Lie Groupoids, (Cambridge Studies in Advanced Mathematics, vol. 91 (2003), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1029.58012
[18] Moerdijk, I.; Mrčun, J., Lie groupoids, sheaves and cohomology, (Poisson Geometry, Deformation Quantisation and Group Representations, London Math. Soc. Lecture Note Ser., vol. 323 (2005), Cambridge University Press: Cambridge University Press Cambridge), 145-272 · Zbl 1082.58018
[19] Mrčun, J., Functoriality of the bimodule associated to a Hilsum-Skandalis map, K-Theory, 18, 235-253 (1999) · Zbl 0938.22002
[20] Mrčun, J., The Hopf algebroids of functions on étale groupoids and their principal Morita equivalence, J. Pure Appl. Algebra, 160, 249-262 (2001) · Zbl 0986.16017
[21] Mrčun, J., On duality between étale groupoids and Hopf algebroids, J. Pure Appl. Algebra, 210, 267-282 (2007) · Zbl 1115.22003
[22] Mrčun, J., Sheaf coalgebras and duality, Topology Appl., 154, 2795-2812 (2007) · Zbl 1182.16027
[23] Renault, J., A Groupoid Approach to \(C^*\) -algebras, (Lecture Notes in Math., vol. 793 (1980), Springer: Springer New York) · Zbl 0433.46049
[24] Takeuchi, M., Groups of algebras over A ⊗ \(Ā\), J. Math. Soc. Japan, 29, 459-492 (1977) · Zbl 0349.16012
[25] Xu, P., Quantum groupoids and deformation quantization, C. R. Acad. Sci. Paris, 326, 289294 (1998) · Zbl 0911.17012
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