Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 106, 30 pages      arXiv:1512.02345      http://dx.doi.org/10.3842/SIGMA.2016.106

Polarisation of Graded Bundles

Andrew James Bruce a, Janusz Grabowski a and Mikołaj Rotkiewicz b
a) Institute of Mathematics, Polish Academy of Sciences, Poland
b) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

Received December 14, 2015, in final form October 25, 2016; Published online November 02, 2016

Abstract
We construct the full linearisation functor which takes a graded bundle of degree $k$ (a particular kind of graded manifold) and produces a $k$-fold vector bundle. We fully characterise the image of the full linearisation functor and show that we obtain a subcategory of $k$-fold vector bundles consisting of symmetric $k$-fold vector bundles equipped with a family of morphisms indexed by the symmetric group ${\mathbb S}_k$. Interestingly, for the degree 2 case this additional structure gives rise to the notion of a symplectical double vector bundle, which is the skew-symmetric analogue of a metric double vector bundle. We also discuss the related case of fully linearising $N$-manifolds, and how one can use the full linearisation functor to ''superise'' a graded bundle.

Key words: graded manifolds; $N$-manifolds; $k$-fold vector bundles; polarisation; supermanifolds.

pdf (593 kb)   tex (46 kb)

References

  1. Batchelor M., The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329-338.
  2. Bertram W., Souvay A., A general construction of Weil functors, Cah. Topol. Géom. Différ. Catég. 55 (2014), 267-313, arXiv:1111.2463.
  3. Bonavolontà G., Poncin N., On the category of Lie $n$-algebroids, J. Geom. Phys. 73 (2013), 70-90, arXiv:1207.3590.
  4. Bruce A.J., Grabowska K., Grabowski J., Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090, 25 pages, arXiv:1502.06092.
  5. Bruce A.J., Grabowska K., Grabowski J., Higher order mechanics on graded bundles, J. Phys. A: Math. Theor. 48 (2015), 205203, 32 pages, arXiv:1412.2719.
  6. Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101 (2016), 71-99, arXiv:1409.0439.
  7. Bursztyn H., Cattaneo A., Mehta R., Zambon M., Reduction of Courant algebroids via supergeometry, work in progress.
  8. Chen Z., Liu Z.J., Sheng Y.H., On double vector bundles, Acta Math. Sin. 30 (2014), 1655-1673, arXiv:1103.0866.
  9. Covolo T., Grabowski J., Poncin N., The category of ${\mathbb Z}_2^n$-supermanifolds, J. Math. Phys. 57 (2016), 073503, 16 pages, arXiv:1602.03312.
  10. Covolo T., Grabowski J., Poncin N., Splitting theorem for $\mathbb{Z}_{2}^{n}$-supermanifolds, J. Geom. Phys. 110 (2016), 393-401, arXiv:1602.03671.
  11. de León M., Oubiña J.A., Rodrigues P.R., Salgado M., Almost $s$-tangent manifolds of higher order, Pacific J. Math. 154 (1992), 201-213.
  12. del Carpio-Marek F., Geometric structure on degree 2 manifolds, Ph.D. Thesis, IMPA, Rio de Janeiro, 2015.
  13. Dufour J.-P., Introduction aux tissus, in Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1990-1991 (Montpellier, 1990-1991), University Montpellier II, Montpellier, 1992, 55-76.
  14. Eliopoulos H.A., Structures $r$-tangentes sur les variétés différentiables, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A413-A416.
  15. Gawędzki K., Supersymmetries - mathematics of supergeometry, Ann. Inst. H. Poincaré Sect. A 27 (1977), 335-366.
  16. Grabowski J., Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013), 27-58, arXiv:1112.0759.
  17. Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.
  18. Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.
  19. Grabowski J., Urbański P., Tangent lifts of Poisson and related structures, J. Phys. A: Math. Gen. 28 (1995), 6743-6777, math.DG/0701076.
  20. Gracia-Saz A., Mackenzie K.C.H., Duality functors for triple vector bundles, Lett. Math. Phys. 90 (2009), 175-200, arXiv:0901.0203.
  21. Gracia-Saz A., Mackenzie K.C.H., Duality functors for $n$-fold vector bundles, arXiv:1209.0027.
  22. Jotz Lean M., $N$-manifolds of degree $2$ and metric double vector bundles, arXiv:1504.00880.
  23. Jóźwikowski M., Rotkiewicz M., A note on actions of some monoids, Differential Geom. Appl. 47 (2016), 212-245, arXiv:1602.02028.
  24. Konieczna K., Urbański P., Double vector bundles and duality, Arch. Math. (Brno) 35 (1999), 59-95, dg-ga/9710014.
  25. Kostant B., Graded manifolds, graded Lie theory, and prequantization, in Differential Geometrical Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977, 177-306.
  26. Mackenzie K.C.H., Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein, Progr. Math., Vol. 232, Birkhäuser Boston, Boston, MA, 2005, 455-481.
  27. Mackenzie K.C.H., Xu P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452.
  28. Molotkov V., Infinite-dimensional and colored supermanifolds, J. Nonlinear Math. Phys. 17 (2010), suppl. 1, 375-446.
  29. Morimoto A., Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J. 40 (1970), 99-120.
  30. Pradines J., Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1523-1526.
  31. Roytenberg D., On the structure of graded symplectic supermanifolds and Courant algebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 169-185, math.SG/0203110.
  32. Ševera P., Some title containing the words ''homotopy'' and ''symplectic'', e.g. this one, in Travaux mathématiques. Fasc. XVI, Trav. Math., Vol. 16, University Luxembourg, Luxembourg, 2005, 121-137, math.SG/0105080.
  33. Tulczyjew W.M., The Legendre transformation, Ann. Inst. H. Poincaré Sect. A 27 (1977), 101-114.
  34. Vishnyakova E.G., Graded manifolds and $n$-fold vector bundles, supergeometric point of view, work in progress.
  35. Voronov Th.Th., Graded manifolds and Drinfeld doubles for Lie bialgebroids, in Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002, 131-168, math.DG/0105237.
  36. Voronov Th.Th., $Q$-manifolds and higher analogs of Lie algebroids, in XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., Vol. 1307, Amer. Inst. Phys., Melville, NY, 2010, 191-202, arXiv:1010.2503.
  37. Voronov Th.Th., $Q$-manifolds and Mackenzie theory, Comm. Math. Phys. 315 (2012), 279-310, arXiv:1206.3622.
  38. Yano K., Ishihara S., Tangent and cotangent bundles: differential geometry, Pure and Applied Mathematics, Vol. 16, Marcel Dekker, Inc., New York, 1973.

Previous article  Next article   Contents of Volume 12 (2016)