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

SIGMA 12 (2016), 106, 30 pages      arXiv:1512.02345      https://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.

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