September  2021, 13(3): 355-384. doi: 10.3934/jgm.2021024

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 Published  September 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, 2021, 13 (3) : 355-384. 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. 

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

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. 

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

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