Polarisation of Graded Bundles

Andrew James Bruce ; Janusz Grabowski ; Mikołaj Rotkiewicz

Symmetry, integrability and geometry: methods and applications, Tome 12 (2016)

Résumé

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.

Détail

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

Bibliographie

[1] Batchelor M., “The structure of supermanifolds”, Trans. Amer. Math. Soc., 253 (1979), 329–338 | DOI | MR | Zbl

[2] Bertram W., Souvay A., “A general construction of Weil functors”, Cah. Topol. Géom. Différ. Catég., 55 (2014), 267–313, arXiv: 1111.2463 | MR | Zbl

[3] Bonavolontà G., Poncin N., “On the category of Lie $n$-algebroids”, J. Geom. Phys., 73 (2013), 70–90, arXiv: 1207.3590 | DOI | MR | Zbl

[4] Bruce A. J., Grabowska K., Grabowski J., “Graded bundles in the category of Lie groupoids”, SIGMA, 11 (2015), 090, 25 pp., arXiv: 1502.06092 | DOI | MR | Zbl

[5] Bruce A. J., Grabowska K., Grabowski J., “Higher order mechanics on graded bundles”, J. Phys. A: Math. Theor., 48 (2015), 205203, 32 pp., arXiv: 1412.2719 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[9] Covolo T., Grabowski J., Poncin N., “The category of ${\mathbb Z}_2^n$-supermanifolds”, J. Math. Phys., 57 (2016), 073503, 16 pp., arXiv: 1602.03312 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 http://projecteuclid.org/euclid.pjm/1102635616 | DOI | MR

[12] del Carpio-Marek F., Geometric structure on degree 2 manifolds, Ph.D. Thesis, IMPA, Rio de Janeiro, 2015 | Zbl

[13] Dufour J.-P., “Introduction aux tissus”, Séminaire Gaston Darboux de Géométrie et Topologie Différentielle 1990–1991 (Montpellier, 1990–1991), University Montpellier II, Montpellier, 1992, 55–76 | MR

[14] Eliopoulos H. A., “Structures $r$-tangentes sur les variétés différentiables”, C. R. Acad. Sci. Paris Sér. A-B, 263 (1966), A413–A416 | MR

[15] Gawȩdzki K., “Supersymmetries – mathematics of supergeometry”, Ann. Inst. H. Poincaré Sect. A, 27 (1977), 335–366 | MR

[16] Grabowski J., “Graded contact manifolds and contact Courant algebroids”, J. Geom. Phys., 68 (2013), 27–58, arXiv: 1112.0759 | DOI | MR | Zbl

[17] Grabowski J., Rotkiewicz M., “Higher vector bundles and multi-graded symplectic manifolds”, J. Geom. Phys., 59 (2009), 1285–1305, arXiv: math.DG/0702772 | DOI | MR | Zbl

[18] Grabowski J., Rotkiewicz M., “Graded bundles and homogeneity structures”, J. Geom. Phys., 62 (2012), 21–36, arXiv: 1102.0180 | DOI | MR | Zbl

[19] Grabowski J., Urbański P., “Tangent lifts of Poisson and related structures”, J. Phys. A: Math. Gen., 28 (1995), 6743–6777, arXiv: math.DG/0701076 | DOI | MR | Zbl

[20] Gracia-Saz A., Mackenzie K. C. H., “Duality functors for triple vector bundles”, Lett. Math. Phys., 90 (2009), 175–200, arXiv: 0901.0203 | DOI | MR | Zbl

[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 | DOI | MR

[24] Konieczna K., Urbański P., “Double vector bundles and duality”, Arch. Math. (Brno), 35 (1999), 59–95, arXiv: dg-ga/9710014 | MR | Zbl

[25] Kostant B., “Graded manifolds, graded Lie theory, and prequantization”, Differential Geometrical Methods in Mathematical Physics, Proc. Sympos. (Univ. Bonn, Bonn, 1975), Lecture Notes in Math., 570, Springer, Berlin, 1977, 177–306 | DOI | MR

[26] Mackenzie K. C. H., “Duality and triple structures”, The Breadth of Symplectic and Poisson Geometry, Festschrift in Honor of Alan Weinstein, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005, 455–481 | DOI | MR | Zbl

[27] Mackenzie K. C. H., Xu P., “Lie bialgebroids and Poisson groupoids”, Duke Math. J., 73 (1994), 415–452 | DOI | MR | Zbl

[28] Molotkov V., “Infinite-dimensional and colored supermanifolds”, J. Nonlinear Math. Phys., 17:1 (2010), 375–446 | DOI | MR

[29] Morimoto A., “Liftings of tensor fields and connections to tangent bundles of higher order”, Nagoya Math. J., 40 (1970), 99–120 | DOI | MR | Zbl

[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 | MR | Zbl

[31] Roytenberg D., “On the structure of graded symplectic supermanifolds and Courant algebroids”, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, 169–185, arXiv: math.SG/0203110 | DOI | MR | Zbl

[32] Ševera P., “Some title containing the words “homotopy” and “symplectic”, e.g. this one”, Travaux mathématiques, Fasc. XVI, Trav. Math., 16, University Luxembourg, Luxembourg, 2005, 121–137, arXiv: math.SG/0105080 | MR

[33] Tulczyjew W. M., “The Legendre transformation”, Ann. Inst. H. Poincaré Sect. A, 27 (1977), 101–114 | MR | Zbl

[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”, Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002, 131–168, arXiv: math.DG/0105237 | DOI | MR | Zbl

[36] Voronov Th. Th., “$Q$-manifolds and higher analogs of Lie algebroids”, XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., 1307, Amer. Inst. Phys., Melville, NY, 2010, 191–202, arXiv: 1010.2503 | DOI | MR | Zbl

[37] Voronov Th. Th., “$Q$-manifolds and Mackenzie theory”, Comm. Math. Phys., 315 (2012), 279–310, arXiv: 1206.3622 | DOI | MR | Zbl

[38] Yano K., Ishihara S., Tangent and cotangent bundles: differential geometry, Pure and Applied Mathematics, 16, Marcel Dekker, Inc., New York, 1973 | MR | Zbl