June  2016, 8(2): 179-197. doi: 10.3934/jgm.2016003

Picard group of isotropic realizations of twisted Poisson manifolds

1. 

Department of Mathematics, Cornell University, Ithaca, NY 14853, United States

Received  March 2015 Revised  December 2015 Published  June 2016

Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of [10]. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `Néron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.
Citation: Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003
References:
[1]

A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups,, in Asterique, 327 (2009), 131.   Google Scholar

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps,, J. Differential Geom., 48 (1998), 445.   Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups,, C. R. Acad. Sci. Paris, 336 (2003), 251.  doi: 10.1016/S1631-073X(02)00024-9.  Google Scholar

[5]

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

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry,, Moscow Math. J., 4 (2004), 39.   Google Scholar

[7]

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

[8]

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

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles,, J. Differential Geom., 26 (1987), 223.   Google Scholar

[11]

H. Duan, Schubert calculus and cohomology of Lie groups. Part II. Compact Lie groups,, preprint, ().   Google Scholar

[12]

J. J. Duistermaat, On global action-angle variables,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles,, Duke Math. J., 89 (1997), 377.  doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

R. Hartshorne, Algebraic Geometry,, Graduate Text in Mathematics, 52 (1977).   Google Scholar

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures,, J. Geometric Mechanics, 5 (2013), 233.  doi: 10.3934/jgm.2013.5.233.  Google Scholar

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background,, in Noncommutative Geometry and String Theory (Yokohama, (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry,, Bull. Braz. Math. Soc., 42 (2011), 783.  doi: 10.1007/s00574-011-0035-2.  Google Scholar

[19]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[20]

P. Xu, Morita equivalent symplectic groupoids,, in Symplectic geometry, 20 (1991), 291.  doi: 10.1007/978-1-4613-9719-9_20.  Google Scholar

show all references

References:
[1]

A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups,, in Asterique, 327 (2009), 131.   Google Scholar

[2]

A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps,, J. Differential Geom., 48 (1998), 445.   Google Scholar

[3]

P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems,, Arch. Rat. Mech. Anal., 205 (2012), 267.  doi: 10.1007/s00205-012-0512-9.  Google Scholar

[4]

K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups,, C. R. Acad. Sci. Paris, 336 (2003), 251.  doi: 10.1016/S1631-073X(02)00024-9.  Google Scholar

[5]

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

[6]

H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry,, Moscow Math. J., 4 (2004), 39.   Google Scholar

[7]

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

[8]

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

[9]

M. Crainic and R. Fernandes, Integrablity of Poisson brackets,, J. Differential Geom., 66 (2004), 71.   Google Scholar

[10]

P. Dazord and T. Delzant, Le problème général des variables actions-angles,, J. Differential Geom., 26 (1987), 223.   Google Scholar

[11]

H. Duan, Schubert calculus and cohomology of Lie groups. Part II. Compact Lie groups,, preprint, ().   Google Scholar

[12]

J. J. Duistermaat, On global action-angle variables,, Comm. Pure Appl. Math., 33 (1980), 687.  doi: 10.1002/cpa.3160330602.  Google Scholar

[13]

F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2783937.  Google Scholar

[14]

K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles,, Duke Math. J., 89 (1997), 377.  doi: 10.1215/S0012-7094-97-08917-1.  Google Scholar

[15]

R. Hartshorne, Algebraic Geometry,, Graduate Text in Mathematics, 52 (1977).   Google Scholar

[16]

N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures,, J. Geometric Mechanics, 5 (2013), 233.  doi: 10.3934/jgm.2013.5.233.  Google Scholar

[17]

P. Severa and A. Weinstein, Poisson geometry with a 3-form background,, in Noncommutative Geometry and String Theory (Yokohama, (2001), 145.  doi: 10.1143/PTPS.144.145.  Google Scholar

[18]

R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry,, Bull. Braz. Math. Soc., 42 (2011), 783.  doi: 10.1007/s00574-011-0035-2.  Google Scholar

[19]

I. Vaisman, Lectures on the Geometry of Poisson Manifolds,, Progress in Mathematics, (1994).  doi: 10.1007/978-3-0348-8495-2.  Google Scholar

[20]

P. Xu, Morita equivalent symplectic groupoids,, in Symplectic geometry, 20 (1991), 291.  doi: 10.1007/978-1-4613-9719-9_20.  Google Scholar

[1]

Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233

[2]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699

[3]

Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61

[4]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385

[5]

Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155

[6]

Anna Maria Candela, J.L. Flores, M. Sánchez. A quadratic Bolza-type problem in a non-complete Riemannian manifold. Conference Publications, 2003, 2003 (Special) : 173-181. doi: 10.3934/proc.2003.2003.173

[7]

Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37

[8]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453

[9]

Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56.

[10]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

[11]

Nuh Aydin, Nicholas Connolly, Markus Grassl. Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes. Advances in Mathematics of Communications, 2017, 11 (1) : 245-258. doi: 10.3934/amc.2017016

[12]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291

[13]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109

[14]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159

[15]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282

[16]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

[17]

Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic & Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043

[18]

Mostapha Benhenda. Nonstandard smooth realization of translations on the torus. Journal of Modern Dynamics, 2013, 7 (3) : 329-367. doi: 10.3934/jmd.2013.7.329

[19]

Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084

[20]

A. Kononenko. Twisted cocycles and rigidity problems. Electronic Research Announcements, 1995, 1: 26-34.

2018 Impact Factor: 0.525

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]