Polarisation of Graded Bundles

[摘要] 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.

[关键词] graded manifolds;N-manifolds;k-fold vector bundles;polarisation;supermanifolds