Linear duals of graded bundles and higher analogues of (Lie) algebroids

Abstract

Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle. They are examples of double structures, graded-linear ($\mathcal{GL}$) bundles, including double vector bundles as a particular case. On $\mathcal{GL}$-bundles we define what we shall call weighted algebroids, which are to be understood as algebroids in the category of graded bundles. They can be considered as a geometrical framework for higher order Lagrangian mechanics. Canonical examples are reductions of higher tangent bundles of Lie groupoids. Weighted algebroids represent also a generalisation of $\mathcal{VB}$-algebroids as defined by Gracia-Saz & Mehta and the $\mathcal{LA}$-bundles of Mackenzie. The resulting structures are strikingly similar to Voronov's higher Lie algebroids, however our approach does not require the initial structures to be defined on supermanifolds.