doi: 10.3934/jgm.2021024
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Local and global integrability of Lie brackets

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA

* Corresponding author: Rui L. Fernandes

Received  April 2021 Revised  August 2021 Early access September 2021

Fund Project: This work was partially supported by NSF grants DMS-1710884 and DMS-2003223

We survey recent results on the local and global integrability of a Lie algebroid, as well as the integrability of infinitesimal multiplicative geometric structures on it.

Citation: Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021024
References:
[1]

R. Almeida and P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 13-15.   Google Scholar

[2]

M. Bailey and M. Gualtieri, Integration of generalized complex structures, preprint, arXiv: 1611.03850. Google Scholar

[3]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Ann., 353 (2012), 663-705.  doi: 10.1007/s00208-011-0697-5.  Google Scholar

[4]

H. BursztynA. Cabrera and C. Ortiz, Linear and multiplicative 2-forms, Lett. Math. Phys., 90 (2009), 59-83.  doi: 10.1007/s11005-009-0349-9.  Google Scholar

[5]

H. BursztynM. CrainicA. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[6]

H. Bursztyn and T. Drummond, Lie theory of multiplicative tensors, Math. Ann., 375 (2019), 1489-1554.  doi: 10.1007/s00208-019-01881-w.  Google Scholar

[7]

A. Cabrera, I. Mǎrcuţ and M. A. Salazar, Local formulas for multiplicative forms, Transformation Groups, (2020). doi: 10.1007/s00031-020-09607-y.  Google Scholar

[8]

A. CabreraI. Mǎrcuţ and M. A. Salazar, On local integration of Lie brackets, J. Reine Angew. Math., 760 (2020), 267-293.  doi: 10.1515/crelle-2018-0011.  Google Scholar

[9]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001, 61–93. doi: 10.1007/978-3-0348-8364-1_4.  Google Scholar

[10]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[11]

A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, in Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, 87-2, Univ. Claude-Bernard, Lyon, 1987, 1–62.  Google Scholar

[12]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[13]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.  doi: 10.4310/jdg/1090415030.  Google Scholar

[14]

M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, in Lectures on Poisson Geometry, Geom. Topol. Monogr., 17, Geom. Topol. Publ., Coventry, 2011, 1–107. doi: 10.2140/gt.  Google Scholar

[15]

M. CrainicM. A. Salazar and I. Struchiner, Multiplicative forms and Spencer operators, Math. Z., 279 (2015), 939-979.  doi: 10.1007/s00209-014-1398-z.  Google Scholar

[16]

T. Drummond and L. Egea, Differential forms with values in VB-groupoids and its Morita invariance, J. Geom. Phys., 135 (2019), 42-69.  doi: 10.1016/j.geomphys.2018.08.019.  Google Scholar

[17]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[18]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.  doi: 10.1006/aima.2001.2070.  Google Scholar

[19]

R. L. Fernandes and D. Michiels, Associativity and integrability, Trans. Amer. Math. Soc., 373 (2020), 5057-5110.  doi: 10.1090/tran/8073.  Google Scholar

[20]

R. L. Fernandes and I. Struchiner, The classifying Lie algebroid of a geometric structure I: Classes of coframes, Trans. Amer. Math. Soc., 366 (2014), 2419-2462.  doi: 10.1090/S0002-9947-2014-05973-4.  Google Scholar

[21]

D. Iglesias-PonteC. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids, J. Eur. Math. Soc. (JEMS), 14 (2012), 681-731.  doi: 10.4171/JEMS/315.  Google Scholar

[22]

M. Jotz Lean and C. Ortiz, Foliated groupoids and infinitesimal ideal systems, Indag. Math. (N.S.), 25 (2014), 1019-1053.  doi: 10.1016/j.indag.2014.07.009.  Google Scholar

[23]

M. V. Karasëv and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization, Translations of Mathematical Monographs, 119, American Mathematical Society, Providence, RI, 1993.  Google Scholar

[24]

C. Laurent-GengouxM. Stiénon and P. Xu, Integration of holomorphic Lie algebroids, Math. Ann., 345 (2009), 895-923.  doi: 10.1007/s00208-009-0388-7.  Google Scholar

[25]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[26]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[27]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467.  doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[28]

A. Malcev, Sur les groupes topologiques locaux et complets, C. R. (Doklady) Acad. Sci. URSS (N.S.), 32 (1941), 606-608.   Google Scholar

[29]

P. J. Olver, Non-associative local Lie groups, J. Lie Theory, 6 (1996), 23-51.   Google Scholar

[30]

C. Ortiz, Multiplicative Dirac structures, Pacific J. Math., 266 (2013), 329-365.  doi: 10.2140/pjm.2013.266.329.  Google Scholar

[31]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245–A248.  Google Scholar

[32]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A907–A910.  Google Scholar

[33]

J. Pradines, Troisième théorème de Lie les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A-B, 267 (1968), A21–A23.  Google Scholar

[34]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, in Travaux Mathématiques. Fasc. XVI, Trav. Math., 16, Univ. Luxemb., Luxembourg, 2005,121–137.  Google Scholar

[35]

W. T. van Est, Une démonstration de É. Cartan du troisième théorème de Lie, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie (Lyon, 1986), Travaux en Cours, 27, Hermann, Paris, 1988, 83–96.  Google Scholar

[36]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101-104.  doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[37]

O. Yudilevich, The role of the Jacobi identity in solving the Maurer-Cartan structure equation, Pacific J. Math., 282 (2016), 487-510.  doi: 10.2140/pjm.2016.282.487.  Google Scholar

show all references

References:
[1]

R. Almeida and P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 13-15.   Google Scholar

[2]

M. Bailey and M. Gualtieri, Integration of generalized complex structures, preprint, arXiv: 1611.03850. Google Scholar

[3]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Ann., 353 (2012), 663-705.  doi: 10.1007/s00208-011-0697-5.  Google Scholar

[4]

H. BursztynA. Cabrera and C. Ortiz, Linear and multiplicative 2-forms, Lett. Math. Phys., 90 (2009), 59-83.  doi: 10.1007/s11005-009-0349-9.  Google Scholar

[5]

H. BursztynM. CrainicA. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J., 123 (2004), 549-607.  doi: 10.1215/S0012-7094-04-12335-8.  Google Scholar

[6]

H. Bursztyn and T. Drummond, Lie theory of multiplicative tensors, Math. Ann., 375 (2019), 1489-1554.  doi: 10.1007/s00208-019-01881-w.  Google Scholar

[7]

A. Cabrera, I. Mǎrcuţ and M. A. Salazar, Local formulas for multiplicative forms, Transformation Groups, (2020). doi: 10.1007/s00031-020-09607-y.  Google Scholar

[8]

A. CabreraI. Mǎrcuţ and M. A. Salazar, On local integration of Lie brackets, J. Reine Angew. Math., 760 (2020), 267-293.  doi: 10.1515/crelle-2018-0011.  Google Scholar

[9]

A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, Progr. Math., 198, Birkhäuser, Basel, 2001, 61–93. doi: 10.1007/978-3-0348-8364-1_4.  Google Scholar

[10]

A. S. Cattaneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196.  doi: 10.1016/S0393-0440(03)00086-X.  Google Scholar

[11]

A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, in Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, 87-2, Univ. Claude-Bernard, Lyon, 1987, 1–62.  Google Scholar

[12]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2), 157 (2003), 575-620.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[13]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.  doi: 10.4310/jdg/1090415030.  Google Scholar

[14]

M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, in Lectures on Poisson Geometry, Geom. Topol. Monogr., 17, Geom. Topol. Publ., Coventry, 2011, 1–107. doi: 10.2140/gt.  Google Scholar

[15]

M. CrainicM. A. Salazar and I. Struchiner, Multiplicative forms and Spencer operators, Math. Z., 279 (2015), 939-979.  doi: 10.1007/s00209-014-1398-z.  Google Scholar

[16]

T. Drummond and L. Egea, Differential forms with values in VB-groupoids and its Morita invariance, J. Geom. Phys., 135 (2019), 42-69.  doi: 10.1016/j.geomphys.2018.08.019.  Google Scholar

[17]

J. J. Duistermaat and J. A. C. Kolk, Lie Groups, Universitext, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-56936-4.  Google Scholar

[18]

R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. Math., 170 (2002), 119-179.  doi: 10.1006/aima.2001.2070.  Google Scholar

[19]

R. L. Fernandes and D. Michiels, Associativity and integrability, Trans. Amer. Math. Soc., 373 (2020), 5057-5110.  doi: 10.1090/tran/8073.  Google Scholar

[20]

R. L. Fernandes and I. Struchiner, The classifying Lie algebroid of a geometric structure I: Classes of coframes, Trans. Amer. Math. Soc., 366 (2014), 2419-2462.  doi: 10.1090/S0002-9947-2014-05973-4.  Google Scholar

[21]

D. Iglesias-PonteC. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids, J. Eur. Math. Soc. (JEMS), 14 (2012), 681-731.  doi: 10.4171/JEMS/315.  Google Scholar

[22]

M. Jotz Lean and C. Ortiz, Foliated groupoids and infinitesimal ideal systems, Indag. Math. (N.S.), 25 (2014), 1019-1053.  doi: 10.1016/j.indag.2014.07.009.  Google Scholar

[23]

M. V. Karasëv and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization, Translations of Mathematical Monographs, 119, American Mathematical Society, Providence, RI, 1993.  Google Scholar

[24]

C. Laurent-GengouxM. Stiénon and P. Xu, Integration of holomorphic Lie algebroids, Math. Ann., 345 (2009), 895-923.  doi: 10.1007/s00208-009-0388-7.  Google Scholar

[25]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, London Mathematical Society Lecture Note Series, 124, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[26]

K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, vol. 213 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9781107325883.  Google Scholar

[27]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467.  doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[28]

A. Malcev, Sur les groupes topologiques locaux et complets, C. R. (Doklady) Acad. Sci. URSS (N.S.), 32 (1941), 606-608.   Google Scholar

[29]

P. J. Olver, Non-associative local Lie groups, J. Lie Theory, 6 (1996), 23-51.   Google Scholar

[30]

C. Ortiz, Multiplicative Dirac structures, Pacific J. Math., 266 (2013), 329-365.  doi: 10.2140/pjm.2013.266.329.  Google Scholar

[31]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A245–A248.  Google Scholar

[32]

J. Pradines, Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A907–A910.  Google Scholar

[33]

J. Pradines, Troisième théorème de Lie les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A-B, 267 (1968), A21–A23.  Google Scholar

[34]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, in Travaux Mathématiques. Fasc. XVI, Trav. Math., 16, Univ. Luxemb., Luxembourg, 2005,121–137.  Google Scholar

[35]

W. T. van Est, Une démonstration de É. Cartan du troisième théorème de Lie, in Action Hamiltoniennes de Groupes. Troisième Théorème de Lie (Lyon, 1986), Travaux en Cours, 27, Hermann, Paris, 1988, 83–96.  Google Scholar

[36]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101-104.  doi: 10.1090/S0273-0979-1987-15473-5.  Google Scholar

[37]

O. Yudilevich, The role of the Jacobi identity in solving the Maurer-Cartan structure equation, Pacific J. Math., 282 (2016), 487-510.  doi: 10.2140/pjm.2016.282.487.  Google Scholar

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