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Grabowski, Janusz

Modular classes revisited. (English) Zbl 1343.53082

Int. J. Geom. Methods Mod. Phys. 11, No. 9, Article ID 1460042, 11 p. (2014).
Given a linear Poisson structure, the modular class can be understood as the obstruction to the existence of a homogeneous measure which is invariant by all the Hamiltonian flows.
The article under review follows work by the author in [J. Grabowski, Transform. Groups 17, No. 4, 989–1010 (2012; Zbl 1287.53072)], where the modular class was studied in the context of Q-manifolds and identified with the divergence of the associated homological vector field. The language of Q-manifolds describes several interesting structures, such as Poisson manifolds, (usual) Lie algebroids as well as general (skew) Lie algebroids, where the Jacobi identity is dropped. The current article introduces a concept of relative modular class for Dirac structures for a certain type of Courant algebroids, called projectable. This provides a geometric interpretation for modular classes of twisted Poisson manifolds.
Reviewer: Iakovos Androulidakis (Athína)

Cited in 5 Documents

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
17B70 Graded Lie (super)algebras
58A32 Natural bundles
58C50 Analysis on supermanifolds or graded manifolds

Keywords:

Lie algebroid; Berezinian density; Courant algebroid; Dirac structure

Citations:

Zbl 1287.53072
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Full Text: DOI arXiv

References:

[1] Alekseev A., J. Differ. Geom. 56 pp 133– (2000)
[2] DOI: 10.4153/CJM-2002-001-5 · Zbl 1006.53072 · doi:10.4153/CJM-2002-001-5
[3] DOI: 10.1007/s11425-010-4142-0 · Zbl 1248.53067 · doi:10.1007/s11425-010-4142-0
[4] H. Bursztyn, M. Crainic and P. Ševera, Travaux Mathématiques 16 (University of Luxembourg, Luxembourg, 2005) pp. 41–52.
[5] DOI: 10.1007/s11005-007-0164-0 · Zbl 1206.53085 · doi:10.1007/s11005-007-0164-0
[6] DOI: 10.1063/1.2958180 · doi:10.1063/1.2958180
[7] DOI: 10.5802/aif.2804 · Zbl 1432.53115 · doi:10.5802/aif.2804
[8] DOI: 10.1088/1751-8113/40/44/012 · Zbl 1204.53068 · doi:10.1088/1751-8113/40/44/012
[9] DOI: 10.2307/2001258 · Zbl 0850.70212 · doi:10.2307/2001258
[10] DOI: 10.5802/aif.2346 · Zbl 1147.53065 · doi:10.5802/aif.2346
[11] DOI: 10.1093/qjmath/50.200.417 · Zbl 0968.58014 · doi:10.1093/qjmath/50.200.417
[12] DOI: 10.1006/aima.2001.2070 · Zbl 1007.22007 · doi:10.1006/aima.2001.2070
[13] DOI: 10.1088/1751-8113/41/17/175204 · Zbl 1137.70011 · doi:10.1088/1751-8113/41/17/175204
[14] DOI: 10.1016/j.geomphys.2011.06.018 · Zbl 1223.37064 · doi:10.1016/j.geomphys.2011.06.018
[15] DOI: 10.1142/S0219887806001259 · doi:10.1142/S0219887806001259
[16] DOI: 10.1007/s00031-012-9197-2 · Zbl 1287.53072 · doi:10.1007/s00031-012-9197-2
[17] DOI: 10.1063/1.3049752 · Zbl 1200.37057 · doi:10.1063/1.3049752
[18] DOI: 10.5802/aif.2172 · Zbl 1141.17018 · doi:10.5802/aif.2172
[19] DOI: 10.1016/S0034-4877(97)85916-2 · Zbl 1005.53061 · doi:10.1016/S0034-4877(97)85916-2
[20] DOI: 10.1016/S0393-0440(99)00007-8 · Zbl 0954.17014 · doi:10.1016/S0393-0440(99)00007-8
[21] DOI: 10.1515/crll.1999.043 · Zbl 1034.53083 · doi:10.1515/crll.1999.043
[22] J. L. Koszul, Astérisque, Numéro Hors Série (1985) pp. 257–271.
[23] Y. Kosmann-Schwarzbach, Poisson Geometry, Banach Center Publications 51, eds. J. Grabowski and P. Urbański (Polish Academy of Sciences, 2000) pp. 109–129.
[24] Y. Kosmann-Schwarzbach, The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics 232 (Birkhäuser, Boston, MA, 2005) pp. 363–389.
[25] DOI: 10.5802/aif.1892 · Zbl 1054.53094 · doi:10.5802/aif.1892
[26] Y. Kosmann-Schwarzbach and C. Laurent-Gengoux, Travaux Mathématiques 16 (University of Luxembourg, Luxembourg, 2005) pp. 315–339.
[27] DOI: 10.1007/s00031-008-9032-y · Zbl 1167.53068 · doi:10.1007/s00031-008-9032-y
[28] DOI: 10.1016/j.crma.2005.09.010 · Zbl 1080.22001 · doi:10.1016/j.crma.2005.09.010
[29] J. Kubarski, Lie Algebroids and Related Topics in Differential Geometry, Banach Center Publications 54 (Polish Academy of Sciences, Institute of Mathematics, 2001) pp. 135–173. · Zbl 1014.53048
[30] DOI: 10.1016/j.geomphys.2010.01.008 · Zbl 1188.58003 · doi:10.1016/j.geomphys.2010.01.008
[31] Liu Z.-J., J. Differ. Geom. 45 pp 547– (1997) · Zbl 0885.58030 · doi:10.4310/jdg/1214459842
[32] DOI: 10.1215/S0012-7094-94-07318-3 · Zbl 0844.22005 · doi:10.1215/S0012-7094-94-07318-3
[33] DOI: 10.3934/jgm.2010.2.243 · Zbl 1282.53071 · doi:10.3934/jgm.2010.2.243
[34] DOI: 10.1023/A:1011965919259 · Zbl 1002.70013 · doi:10.1023/A:1011965919259
[35] DOI: 10.1090/conm/315/05479 · doi:10.1090/conm/315/05479
[36] DOI: 10.1023/A:1020708131005 · Zbl 1027.53104 · doi:10.1023/A:1020708131005
[37] P. Ševera, Travaux Mathématiques 16 (University of Luxembourg, Luxembourg, 2005) pp. 121–137.
[38] DOI: 10.1143/PTPS.144.145 · doi:10.1143/PTPS.144.145
[39] DOI: 10.1070/RM1997v052n02ABEH001802 · Zbl 0955.58017 · doi:10.1070/RM1997v052n02ABEH001802
[40] DOI: 10.1016/j.difgeo.2010.01.001 · Zbl 1202.53079 · doi:10.1016/j.difgeo.2010.01.001
[41] DOI: 10.1016/0034-4877(94)90038-8 · Zbl 0817.70010 · doi:10.1016/0034-4877(94)90038-8
[42] DOI: 10.1016/S0393-0440(97)80011-3 · Zbl 0902.58013 · doi:10.1016/S0393-0440(97)80011-3
[43] Weinstein A., Fields Inst. Commun. 7 pp 207– (1996)
[44] DOI: 10.1007/s002200050540 · Zbl 0941.17016 · doi:10.1007/s002200050540
[45] DOI: 10.4310/JSG.2012.v10.n4.a4 · Zbl 1260.53134 · doi:10.4310/JSG.2012.v10.n4.a4
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