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 (GL) bundles, including double vector bundles as a particular case. On 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 VB-algebroids as defined by Gracia-Saz & Mehta and the 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.

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

“…However, these two notions of a higher algebroid are different, as can be easily seen from coordinate calculations already in order 2. The construction of [5] can be adapted to develop higher-order mechanics on N-graded manifolds [4], equivalent to our results [25] in comparable cases. So perhaps the difference here is of a philosophical nature.…”

“…The idea here is to mimic the canonical inclusion T k M → TT k−1 M , which makes the higher tangent bundle T k M a sub-object of the tangent Lie algebroid of T k−1 M . In this way the construction of [5] admits higher tangent bundles and also prolongations of algebroids as examples of higher (Lie) algebroids, similarly to our approach. However, these two notions of a higher algebroid are different, as can be easily seen from coordinate calculations already in order 2.…”

“…The most recent attempt is due to Bruce, Grabowska and Grabowski. In [5] they constructed a linearisation functor, which, to a N-graded manifold σ k : E k → M of order k, canonically associates a manifold l E k equipped with a graded-linear structure of a bi-order (k − 1, 1) over E 1 and E k−1 , respectively. Now the higher (Lie) algebroid structure on σ k is defined as a (Lie) algebroid structure on the vector bundle l E k → E k−1 compatible with the second grading.…”

The category of Z2n-supermanifolds

“…Lie groupoid or Lie algebroid structures, compatible with multi-gradations and induced multiparities in the spirit of [BGG14,BGG16], can also provide a fruitful frame for this new supergeometry.…”

Connections adapted to non-negatively graded structures

“…We show, by using the results of Křižka [32] on the existence of linear A-connections, that weighted Aconnections exist for any Lie algebroid and any graded bundle over the same base manifold. The methodology is to use the fact that all graded bundles are non-canonically isomorphic to split graded bundles, i.e., a Whitney sum of graded vector bundles (see [9]). One can then consider splittings of graded bundles as gauge transformations and use this to define a weighted A-connection given a linear A-connection on the associated Whitney sum of graded vector bundles.…”