Double Vector Bundles Research Articles

A geometrisation of [formula omitted]-manifolds

This paper proposes a geometrisation of N-manifolds of degree n as n-fold vector bundles equipped with a (signed) Sn-symmetry. More precisely, it proves an equivalence between the categories of [n]-manifolds and the category of (signed) symmetric n-fold vector bundles, by finding that symmetric n-fold vector bundle cocycles and [n]-manifold cocycles are identical.This extends the already known equivalences of [1]-manifolds with vector bundles, and of [2]-manifolds with involutive double vector bundles, where the involution is understood as an S2-action.

Natural affinors and torsion of connections on Weil like functors on double vector bundles

We describe completely all natural affinors on product preserving gauge bundle functors on double vector bundles. Next, we study torsion of double-linear connections.

Modules and representations up to homotopy of Lie n-algebroids

This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general nin {mathbb {N}}. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.

Description of jet like functors on double vector bundles by means of admissible systems

Description of jet like functors on double vector bundles by means of admissible systems

Quotients of double vector bundles and multigraded bundles

<p style='text-indent:20px;'>We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.</p>

Complete lifting of double-linear semi-basic tangent valued forms to Weil like functors on double vector bundles

Complete lifting of double-linear semi-basic tangent valued forms to Weil like functors on double vector bundles

Double principal bundles

We define double principal bundles (DPBs), for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples. It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections in DPBs are investigated.

The Weil Algebra of a Double Lie Algebroid

Abstract Given a double vector bundle $D\to M$, we define a bigraded bundle of algebras $W(D)\to M$ called the “Weil algebra bundle”. The space ${\mathcal{W}}(D)$ of sections of this algebra bundle ”realizes” the algebra of functions on the supermanifold $D[1,1]$. We describe in detail the relations between the Weil algebra bundles of $D$ and those of the double vector bundles $D^{\prime},\ D^{\prime\prime}$ obtained from $D$ by duality operations. We show that ${\mathcal{V}\mathcal{B}}$-algebroid structures on $D$ are equivalent to horizontal or vertical differentials on two of the Weil algebras and a Gerstenhaber bracket on the 3rd. Furthermore, Mackenzie’s definition of a double Lie algebroid is equivalent to compatibilities between two such structures on any one of the three Weil algebras. In particular, we obtain a ”classical” version of Voronov’s result characterizing double Lie algebroid structures. In the case that $D=TA$ is the tangent prolongation of a Lie algebroid, we find that ${\mathcal{W}}(D)$ is the Weil algebra of the Lie algebroid, as defined by Mehta and Abad–Crainic. We show that the deformation complex of Lie algebroids, the theory of IM forms and IM multi-vector fields, and 2-term representations up to homotopy all have natural interpretations in terms of our Weil algebras.

Lifting double linear vector fields to Weil like functors on double vector bundles

AbstractThe complete description is given of the product preserving gauge bundle functors F on the category of double vector bundles in terms of the ‐bilinear maps , where are Weil algebras and are finite dimensional (over ) ‐modules. Then the so‐called “iteration” problem is solved (i.e. by means of and is computed). That any two product preserving gauge bundle functors on commute is derived. All gauge natural operators lifting double linear vector fields to product preserving gauge bundle functors F are completely described.

From modelling of systems with constraints to generalized geometry and back to numerics

AbstractIn this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints – we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators.On a test example of a simple pendulum in a gravity field we compare the Dirac integrators with classical explicit and implicit methods, we pay special attention to conservation of constrains. Then, on a more advanced example of the Ziegler‐type system we show that the choice of numerical methods can indeed affect the conclusions of qualitative analysis of the dynamics of mechanical systems. We also tell why we think that Dirac integrators are appropriate for this kind of systems by explaining the relation with the notions of geometric degree of non‐conservativity and kinematic structural stability.

Connections adapted to non-negatively graded structures

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted[Formula: see text]-connection on a graded bundle. In a natural sense weighted [Formula: see text]-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear [Formula: see text]-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted[Formula: see text]-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.

The geometrization of N-manifolds of degree 2

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sections of the metric double vector bundle that is dual to a given involutive double vector bundle are the generators of a graded manifold of degree $2$ over the double base. We discuss how split Poisson $\mathbb N$-manifolds of degree $2$ are equivalent to \emph{self-dual representations up to homotopy} and so, following Gracia-Saz and Mehta, to linear splittings of a certain class of VB-algebroids. In other words, the equivalence of categories above induces an equivalence between so called \emph{Poisson involutive double vector bundles}, which are the dual objects to metric VB-algebroids, and Poisson $\mathbb N$-manifolds of degree $2$.

Graded manifolds of type $$\Delta $$ Δ and n-fold vector bundles

Vector bundles and double vector bundles, or twofold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of supergeometry and $$\mathbb {Z}$$ -graded manifolds of degree $$\le 2$$ . Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of $$\mathbb {Z}$$ -graded manifolds of degree $$\le 2$$ and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of $$\mathbb {Z}^r$$ -graded manifolds of type $$\Delta $$ , where $$\Delta $$ is a certain weight system and r is the rank of $$\Delta $$ . We establish an equivalence between a subcategory of the category of n-fold vector bundles and the category of graded manifolds of type $$\Delta $$ .

On the concept of a filtered bundle

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.

Duality for Graded Manifolds

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different category than the initial ones, namely graded polynomial (co)algebra bundles and free graded Weil (co)algebra bundles. Our results are then applied to obtain elegant characterizations of double vector bundles and graded bundles of degree 2. All these results have their supergeometric counterparts. For instance, we give a simple proof of a nice characterisation of $N$-manifolds of degree 2, announced in the literature.

Polarisation of Graded Bundles

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.

A note on actions of some monoids

Smooth actions of the multiplicative monoid (R,⋅) of real numbers on manifolds lead to an alternative, and for some reasons simpler, definitions of a vector bundle, a double vector bundle and related structures like a graded bundle (Grabowski and Rotkiewicz (2011) [10]). For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid (R,⋅). Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps (R,0)→(R,0), smooth actions of the monoid of real 2 by 2 matrices and smooth actions of the multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

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

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.

On double vector bundles

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theory of double vector bundles in term of the corresponding short exact sequences. Examples including the jet bundle and the Atiyah algebroid are discussed.

Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings

The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration space is TM, is based on the existence of canonical symplectic isomorphisms of double vector bundles T*TM, T*TM, and TT*M. We show that there exist an analogous picture in the dynamics of objects for which the configuration space is the vector bundle of n-vectors, if we make use of certain graded bundle structures of degree n, i.e. objects generalizing vector bundles (for which n=1). For instance, the role of TT*M is played in our approach by the vector bundle of n-vectors on the bundle of n-covectors, which is canonically a graded bundle of degree n over the bundle of n-vectors. Dynamics of strings and the Plateau problem in statics are particular cases of this framework.