Tangent lifts of higher order of multiplicative Dirac structures

Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms

“…Higher tangent lifts of (Lie) algebroids. It is well-known that if a VB σ : E → M carries a (Lie) algebroid structure, then its k th -tangent lift T k σ : T k E → T k M has the so-called lifted (Lie) algebroid structure (e.g., [30]). It can be elegantly described in terms of the VBC κ related with the initial algebroid structure on σ.…”

Graded Bundles in the Category of Lie Groupoids

“…As a 'corollary' we see that VB-groupoids are the objects that integrate VBalgebroids. T k A(G), see for example [24] for details. From [3] we know that that T k A(G) is a weighted Lie algebroid.…”

“…Section 2. , which we can consider as a trivial weighted Poisson-Lie groupoid. Recall that T k G ⇒ T k M is a weighted Lie groupoid of degree k. Furthermore, in [24] it was shown that the higher tangent lift of a multiplicative Poisson structure is a multiplicative Poisson structure itself. By inspection we see that Poisson structure lifted to the k-th order tangent bundle Λ := d k T π is of weight −k, with respect to the natural homogeneity structure on the higher tangent bundle.…”

“…If (A, A * ) is a Lie bi-algebroid, then (T k A, T k A * ) is a Lie bi-algebroid of degree k + 1. See for details [24], although the authors do not make mention of the graded structure.…”

“…With this in mind, let us outline the classical construction. It can be shown that there are two natural isomorphisms Πϑ : ΠA(T * G) −→ T * (ΠA(G)), Πj : ΠT(ΠA(G)) −→ ΠA(TG), which follow from application of the parity reversion functor to the standard non-super isomorphisms (see [28,30] and [24], where the higher tangent versions are also discussed). Let (G, Λ) be a Poisson-Lie groupoid, then as we have Λ # : T * G → TG, we also have the 'superised' version:…”