Double Vector Bundles Research Articles

Q-Manifolds and Mackenzie Theory

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie's original definition; we show how it fits into a bigger picture of equivalent structures on `neighbor' double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.

Double structures and jets

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.

Graded bundles and homogeneity structures

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T n Q playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0 , 1 , … , n . We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid ( R ≥ 0 , ⋅ ) of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double ( r - tuple, in general) graded bundle–a broad generalization of the concept of a double ( r -tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.

Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids

The word ‘double’ was used by Ehresmann to mean ‘an object X in the category of all X’. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid.

Double affine bundles

A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles have been recognized as a proper geometrical tool in a frame independent description of many important systems. Different approaches to the (special) double affine bundles are compared and carefully studied together with the problems of double vector bundle models and hulls, duality, and relations to associated phase spaces, contact structures, and other canonical constructions.

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB -algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB -algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB -algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.

Higher vector bundles and multi-graded symplectic manifolds

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F , or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain N n -grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known “universal Legendre transformation” T ∗ E ≃ T ∗ E ∗ , identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg’s picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin–Vilkovisky formalisms. This is also a natural framework for studying n -fold Lie algebroids and related structures.