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June  2012, 4(2): 111-127. doi: 10.3934/jgm.2012.4.111

Symmetries and reduction of multiplicative 2-forms

Henrique Bursztyn 1, and  Alejandro Cabrera 2,

1. 

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil
2. 

Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, CEP 21941-909, Rio de Janeiro - RJ, Brazil

Received  March 2011 Revised  July 2011 Published  August 2012

This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu [11] on Poisson actions.
Keywords: Dirac structures, Lie groupoids, reduction., multiplicative forms.
Mathematics Subject Classification: Primary: 53D1.
Citation: Henrique Bursztyn, Alejandro Cabrera. Symmetries and reduction of multiplicative 2-forms. Journal of Geometric Mechanics, 2012, 4 (2) : 111-127. doi: 10.3934/jgm.2012.4.111
References:
[1]

C. Arias Abad and M. Crainic, The Weil algebra and the Van Est isomorphism, Ann. Inst. Fourier, 61 (2011), 927-970.

[2]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Annalen, 353 (2012), 663-705.

[3]

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

[4]

H. Bursztyn, G. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Advances in Math., 211 (2007), 726-765. doi: 10.1016/j.aim.2006.09.008.

[5]

H. Bursztyn, M. Crainic, A. 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]

A. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in "Quantization of Singular Symplectic Quotients," Progr. Math., 198, Birkhäuser, Basel, (2001), 61-93.

[7]

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), i-ii, 1-62.

[8]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[9]

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

[10]

M. Crainic and R. Fernandes, Integrability of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.

[11]

R. Fernandes, J.-P. Ortega and T. Ratiu, The momentum map in Poisson geometry, Amer. J. of Math., 131 (2009), 1261-1310 doi: 10.1353/ajm.0.0068.

[12]

M. Jotz and T. Ratiu, Induced Dirac structures on isotropy type manifolds, Transform. Groups, 16 (2011), 175-191. doi: 10.1007/s00031-011-9123-z.

[13]

M. Jotz, T. Ratiu and J. Sniatycki, Singular reduction of Dirac structures, Trans. Amer. Math. Soc., 363 (2011), 2967-3013. doi: 10.1090/S0002-9947-2011-05220-7.

[14]

, J.-H. Lu, private communication.

[15]

K. Mackenzie and P. Xu, Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford Ser. (2), 49 (1998), 59-85.

[16]

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

[17]

K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci., 24 (1988), 121-140. doi: 10.2977/prims/1195175328.

[18]

I. Moerdijk and J. Mrcun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115. doi: 10.1016/j.aim.2005.05.011.

[19]

P. Ševera, Some title containing the words "homotopy'' and"symplectic'', e.g. this one, in "Travaux Mathématiques. Fasc. XVI," Trav. Math., XVI, Univ. Luxemb., Luxembourg, (2005), 121-137.

[20]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101-104.

show all references

References:
[1]

C. Arias Abad and M. Crainic, The Weil algebra and the Van Est isomorphism, Ann. Inst. Fourier, 61 (2011), 927-970.

[2]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Annalen, 353 (2012), 663-705.

[3]

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

[4]

H. Bursztyn, G. Cavalcanti and M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Advances in Math., 211 (2007), 726-765. doi: 10.1016/j.aim.2006.09.008.

[5]

H. Bursztyn, M. Crainic, A. 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]

A. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, in "Quantization of Singular Symplectic Quotients," Progr. Math., 198, Birkhäuser, Basel, (2001), 61-93.

[7]

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), i-ii, 1-62.

[8]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1.

[9]

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

[10]

M. Crainic and R. Fernandes, Integrability of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.

[11]

R. Fernandes, J.-P. Ortega and T. Ratiu, The momentum map in Poisson geometry, Amer. J. of Math., 131 (2009), 1261-1310 doi: 10.1353/ajm.0.0068.

[12]

M. Jotz and T. Ratiu, Induced Dirac structures on isotropy type manifolds, Transform. Groups, 16 (2011), 175-191. doi: 10.1007/s00031-011-9123-z.

[13]

M. Jotz, T. Ratiu and J. Sniatycki, Singular reduction of Dirac structures, Trans. Amer. Math. Soc., 363 (2011), 2967-3013. doi: 10.1090/S0002-9947-2011-05220-7.

[14]

, J.-H. Lu, private communication.

[15]

K. Mackenzie and P. Xu, Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford Ser. (2), 49 (1998), 59-85.

[16]

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

[17]

K. Mikami and A. Weinstein, Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci., 24 (1988), 121-140. doi: 10.2977/prims/1195175328.

[18]

I. Moerdijk and J. Mrcun, On the integrability of Lie subalgebroids, Adv. Math., 204 (2006), 101-115. doi: 10.1016/j.aim.2005.05.011.

[19]

P. Ševera, Some title containing the words "homotopy'' and"symplectic'', e.g. this one, in "Travaux Mathématiques. Fasc. XVI," Trav. Math., XVI, Univ. Luxemb., Luxembourg, (2005), 121-137.

[20]

A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101-104.

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