Polarisation of Graded Bundles

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2016
• Hauptverfasser: Bruce, A.J., Grabowski, J., Rotkiewicz, M.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2016
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/148538
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Polarisation of Graded Bundles / A.J. Bruce, J. Grabowski, M. Rotkiewicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 38 назв. — англ.

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 Sk. 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.