Picard group of isotropic realizations of twisted Poisson manifolds

Previous Article
Infinitesimally natural principal bundles
JGM Home
This Issue
Next Article
Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

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

Picard group of isotropic realizations of twisted Poisson manifolds

Chi-Kwong Fok ^1,

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

Received March 2015 Revised December 2015 Published June 2016

Abstract
Related Papers

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.

Keywords: Picard group., Dazord-Delzant homomorphism, twisted Poisson manifold, Almost symplectically complete isotropic realization, tropical affine structure.

Mathematics Subject Classification: Primary: 53D15, 53D17; Secondary: 55N3.

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), 131-199.
[2]	A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.
[3]	P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.
[4]	K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251-256. doi: 10.1016/S1631-073X(02)00024-9.
[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]	H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39-66, 310.
[7]	A. Catteneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.
[8]	M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.
[9]	M. Crainic and R. Fernandes, Integrablity of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.
[10]	P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.
[11]	H. Duan, Schubert calculus and cohomology of Lie groups. Part II. Compact Lie groups,, preprint, ().
[12]	J. J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.
[13]	F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. doi: 10.1063/1.2783937.
[14]	K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377-412. doi: 10.1215/S0012-7094-97-08917-1.
[15]	R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, 52, Springer-Verlag, 1977.
[16]	N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures, J. Geometric Mechanics, 5 (2013), 233-256. doi: 10.3934/jgm.2013.5.233.
[17]	P. Severa and A. Weinstein, Poisson geometry with a 3-form background, in Noncommutative Geometry and String Theory (Yokohama, 2001), ed. Y. Maeda and S. Watamura, Progr. Theoret. Phys. Suppl. 144, Kyoto Univ., Kyoto, 2001, 145-154. doi: 10.1143/PTPS.144.145.
[18]	R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry, Bull. Braz. Math. Soc., 42 (2011), 783-803. doi: 10.1007/s00574-011-0035-2.
[19]	I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, Birhaüser Verlag, 1994. doi: 10.1007/978-3-0348-8495-2.
[20]	P. Xu, Morita equivalent symplectic groupoids, in Symplectic geometry, groupoids, and integrable systems, Séminaire sud-rhodanien de géométrie à Berkeley, Math. Sci. Res. Inst. Publ., 20, Springer-Verlag, New York, 1991, 291-311. doi: 10.1007/978-1-4613-9719-9_20.

show all references

References:

[1]	A. Alekseev, H. Bursztyn and E. Meinrenken, Pure spinors on Lie groups, in Asterique, 327 (2009), 131-199.
[2]	A. Alekseev, A. Malkin and E. Meinrenken, Lie group valued moment maps, J. Differential Geom., 48 (1998), 445-495.
[3]	P. Balseiro and L. García-Naranjo, Gauge transformations, twisted Poisson brackets and Hamiltonianization of nonholonomic systems, Arch. Rat. Mech. Anal., 205 (2012), 267-310. doi: 10.1007/s00205-012-0512-9.
[4]	K. Behrend, P. Xu and B. Zhang, Equivariant gerbes over compact simple Lie groups, C. R. Acad. Sci. Paris, 336 (2003), 251-256. doi: 10.1016/S1631-073X(02)00024-9.
[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]	H. Bursztyn and A. Weinstein, Picard groups in Poisson geometry, Moscow Math. J., 4 (2004), 39-66, 310.
[7]	A. Catteneo and P. Xu, Integration of twisted Poisson structures, J. Geom. Phys., 49 (2004), 187-196. doi: 10.1016/S0393-0440(03)00086-X.
[8]	M. Crainic and R. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. doi: 10.4007/annals.2003.157.575.
[9]	M. Crainic and R. Fernandes, Integrablity of Poisson brackets, J. Differential Geom., 66 (2004), 71-137.
[10]	P. Dazord and T. Delzant, Le problème général des variables actions-angles, J. Differential Geom., 26 (1987), 223-251.
[11]	H. Duan, Schubert calculus and cohomology of Lie groups. Part II. Compact Lie groups,, preprint, ().
[12]	J. J. Duistermaat, On global action-angle variables, Comm. Pure Appl. Math., 33 (1980), 687-706. doi: 10.1002/cpa.3160330602.
[13]	F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems, J. Math. Phys., 48 (2007), 092902, 13pp. doi: 10.1063/1.2783937.
[14]	K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377-412. doi: 10.1215/S0012-7094-97-08917-1.
[15]	R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, 52, Springer-Verlag, 1977.
[16]	N. Sansonetto and D. Sepe, Twisted isotropic realizations of twisted Poisson structures, J. Geometric Mechanics, 5 (2013), 233-256. doi: 10.3934/jgm.2013.5.233.
[17]	P. Severa and A. Weinstein, Poisson geometry with a 3-form background, in Noncommutative Geometry and String Theory (Yokohama, 2001), ed. Y. Maeda and S. Watamura, Progr. Theoret. Phys. Suppl. 144, Kyoto Univ., Kyoto, 2001, 145-154. doi: 10.1143/PTPS.144.145.
[18]	R. Sjamaar, Hans Duistermaat's contributions to Poisson geometry, Bull. Braz. Math. Soc., 42 (2011), 783-803. doi: 10.1007/s00574-011-0035-2.
[19]	I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, Birhaüser Verlag, 1994. doi: 10.1007/978-3-0348-8495-2.
[20]	P. Xu, Morita equivalent symplectic groupoids, in Symplectic geometry, groupoids, and integrable systems, Séminaire sud-rhodanien de géométrie à Berkeley, Math. Sci. Res. Inst. Publ., 20, Springer-Verlag, New York, 1991, 291-311. doi: 10.1007/978-1-4613-9719-9_20.

[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 and 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]	Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008
[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]	Javier de la Cruz, Ricardo Villanueva-Polanco. Public key cryptography based on twisted dihedral group algebras. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022031
[8]	Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155
[9]	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
[10]	Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37
[11]	Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56.
[12]	Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga. Determining the global manifold structure of a continuous-time heterodimensional cycle. Journal of Computational Dynamics, 2022 doi: 10.3934/jcd.2022008
[13]	Marianito R. Rodrigo, Rogemar S. Mamon. Bond pricing formulas for Markov-modulated affine term structure models. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2685-2702. doi: 10.3934/jimo.2020089
[14]	Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[15]	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
[16]	Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939
[17]	Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267
[18]	Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
[19]	Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[20]	Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282

2020 Impact Factor: 0.857

Tools

Metrics

Other articles
by authors

on AIMS
Chi-Kwong Fok
on Google Scholar
Chi-Kwong Fok

[Back to Top]