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 SIGMA, 2016, Volume 12, 106, 30 pages (Mi sigma1188)

Andrew James Brucea, Janusz Grabowskia, Mikołaj Rotkiewiczb

a Institute of Mathematics, Polish Academy of Sciences, Poland
b Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland

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.

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

 Funding Agency Grant Number National Science Centre (Narodowe Centrum Nauki) DEC-2012/06/A/ST1/00256 Research funded by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256.

DOI: https://doi.org/10.3842/SIGMA.2016.106

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ArXiv: 1512.02345
MSC: 55R10; 58A32; 58A50
Received: December 14, 2015; in final form October 25, 2016; Published online November 2, 2016
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Citation: Andrew James Bruce, Janusz Grabowski, Mikołaj Rotkiewicz, “Polarisation of Graded Bundles”, SIGMA, 12 (2016), 106, 30 pp.

Citation in format AMSBIB
\Bibitem{BruGraRot16} \by Andrew~James~Bruce, Janusz~Grabowski, Miko\l aj~Rotkiewicz \paper Polarisation of Graded Bundles \jour SIGMA \yr 2016 \vol 12 \papernumber 106 \totalpages 30 \mathnet{http://mi.mathnet.ru/sigma1188} \crossref{https://doi.org/10.3842/SIGMA.2016.106} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000388502400001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84996488626} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, “Remarks on Contact and Jacobi Geometry”, SIGMA, 13 (2017), 059, 22 pp.
2. A. J. Bruce, K. Grabowska, J. Grabowski, “Introduction to graded bundles”, Note Mat., 37:1, S (2017), 59–74
3. A. J. Bruce, K. Grabowska, J. Grabowski, “On the concept of a filtered bundle”, Int. J. Geom. Methods Mod. Phys., 15:1 (2018), 1850013
4. M. Rotkiewicz, M. Jóźwikowski, “Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms”, SIGMA, 14 (2018), 135, 46 pp.
5. Vishnyakova E., “Graded Manifolds of Type and N-Fold Vector Bundles”, Lett. Math. Phys., 109:2 (2019), 243–293
6. Bruce A.J., “Connections Adapted to Non-Negatively Graded Structures”, Int. J. Geom. Methods Mod. Phys., 16:2 (2019), 1950021
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