Geodesic
Symmetries of the Space of Linear Symplectic Connections
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen–Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.
Keywords: symplectic connection; compatible Lie brackets; Hamiltonian action; symmetric tensors. 
@article{SIGMA_2017_13_a1,
     author = {Daniel J. F. Fox},
     title = {Symmetries of the {Space} of {Linear} {Symplectic} {Connections}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a1/}
}
TY  - JOUR
AU  - Daniel J. F. Fox
TI  - Symmetries of the Space of Linear Symplectic Connections
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a1/
LA  - en
ID  - SIGMA_2017_13_a1
ER  - 
%0 Journal Article
%A Daniel J. F. Fox
%T Symmetries of the Space of Linear Symplectic Connections
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a1/
%G en
%F SIGMA_2017_13_a1
Daniel J. F. Fox. Symmetries of the Space of Linear Symplectic Connections. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a1/

[1] Atiyah M. F., Bott R., “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, 308 (1983), 523–615 | DOI | MR | Zbl

[2] Bieliavsky P., Cahen M., Gutt S., Rawnsley J., Schwachhöfer L., “Symplectic connections”, Int. J. Geom. Methods Mod. Phys., 3 (2006), 375–420, arXiv: math.SG/0511194 | DOI | MR | Zbl

[3] Bolsinov A. V., “Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution”, Math. USSR Izv., 38 (1992), 69–90 | DOI | MR

[4] Bolsinov A. V., Borisov A. V., “Compatible Poisson brackets on Lie algebras”, Math. Notes, 72 (2002), 10–30 | DOI | MR | Zbl

[5] Bourgeois F., Cahen M., Can one define a preferred symplectic connection?, Rep. Math. Phys., 43 (1999), 35–42 | DOI | MR | Zbl

[6] Bourgeois F., Cahen M., “A variational principle for symplectic connections”, J. Geom. Phys., 30 (1999), 233–265 | DOI | MR | Zbl

[7] Cahen M., Gutt S., “Moment map for the space of symplectic connections”, Liber Amicorum Delanghe, eds. F. Brackx, H. De Schepper, Gent Academia Press, 2005, 27–36

[8] Fedosov B., Deformation quantization and index theory, Mathematical Topics, 9, Akademie Verlag, Berlin, 1996 | MR | Zbl

[9] Fedosov B., “The Atiyah–Bott–Patodi method in deformation quantization”, Comm. Math. Phys., 209 (2000), 691–728 | DOI | MR | Zbl

[10] Fox D. J. F., Critical symplectic connections on surfaces, arXiv: 1410.1468

[11] Golubchik I. Z., Sokolov V. V., “Compatible Lie brackets and integrable equations of the principal chiral field model type”, Funct. Anal. Appl., 36 (2002), 172–181 | DOI | MR | Zbl

[12] Guillemin V., “The integrability problem for $G$-structures”, Trans. Amer. Math. Soc., 116 (1965), 544–560 | DOI | MR | Zbl

[13] Gutt S., “Remarks on symplectic connections”, Lett. Math. Phys., 78 (2006), 307–328 | DOI | MR | Zbl

[14] Kobayashi S., Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 70, Springer-Verlag, Berlin–Heidelberg, 1995 | DOI | MR

[15] Kostant B., “The Weyl algebra and the structure of all Lie superalgebras of Riemannian type”, Transform. Groups, 6 (2001), 215–226, arXiv: math.RT/0106171 | DOI | MR | Zbl

[16] McDuff D., Salamon D., Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995 | MR

[17] Penrose R., Rindler W., Spinors and space-time, v. 1, Cambridge Monographs on Mathematical Physics, Two-spinor calculus and relativistic fields, Cambridge University Press, Cambridge, 1984 | DOI | MR | Zbl

[18] Sternberg S., Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964 | MR | Zbl

[19] Tamarkin D. E., “Topological invariants of connections on symplectic manifolds”, Funct. Anal. Appl., 29 (1995), 258–267 | DOI | MR | Zbl

[20] Wald R. M., General relativity, University of Chicago Press, Chicago, IL, 1984 | DOI | MR | Zbl