Polarisation of Graded Bundles

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.

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Cited by in corpus (7)

• Remarks on Contact and Jacobi Geometry
• Introduction to graded bundles
• On the concept of a filtered bundle
• Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
• Connections Adapted to Non-Negatively Graded Structures
• Graded covering of a supermanifold I. The case of a Lie supergroup
• The functor between two categories of $\mathbb{Z}-$graded manifolds