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

Abstract

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.

References in corpus (7)

• Higher vector bundles and multi-graded symplectic manifolds
• Supergroupoids, double structures, and equivariant cohomology
• Double Lie algebroids and the double of a Lie bialgebroid
• Representations up to homotopy of Lie algebroids
• Notions of double for Lie algebroids
• Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
• Duality and triple structures

Cited by in corpus (53)

• Vector bundles over Lie groupoids and algebroids
• VB-groupoids and representation theory of Lie groupoids
• On Regular Courant Algebroids
• The Atiyah class of a dg-vector bundle
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• Higher Extensions of Lie Algebroids
• Lie algebroid modules and representations up to homotopy
• Integration of Exact Courant Algebroids
• L-infinity algebra actions
• VB-algebroid morphisms and representations up to homotopy
• Van Est isomorphism for homogeneous cochains
• Morita equivalences of vector bundles
• Pre-Courant Algebroids
• N-manifolds of degree 2 and metric double vector bundles
• The Geometrisation of $\mathbb{N}$-manifolds of degree 2
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Double Lie algebroids and representations up to homotopy
• Differential forms with values in VB-groupoids
• A Local Resolution of the Problem of Time. VII. Constraint Closure
• Graded Bundles in the Category of Lie Groupoids
• Courant cohomology, Cartan calculus, connections, curvature, characteristic classes
• Lie 2-algebroids and matched pairs of 2-representations - a geometric approach
• The geometry of graded cotangent bundles
• VB-structures and generalizations
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Deformations of Linear Lie Brackets
• Multiple vector bundles: cores, splittings and decompositions
• Glanon groupoids
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Double Principal Bundles
• Dorfman connections and Courant algebroids
• Modular Classes of Lie Groupoid Representations up to Homotopy
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• Shifted Poisson structures on differentiable stacks
• Local formulas for multiplicative forms
• Poisson-Nijenhuis groupoids
• Poisson double structures
• On Hausdorff integrations of Lie algebroids
• Deformations of Lie brackets and representations up to homotopy
• Deformations of symplectic groupoids
• Higher omni-Lie algebroids
• $L_{\infty }$-actions of Lie algebroids
• Deformation Cohomology of Lie Algebroids and Morita Equivalence
• $A_{\infty }$-Algebras from Lie Pairs
• The standard cohomology of regular Courant algebroids
• Linear generalised complex structures
• Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids
• On Double Vector Bundles
• Categorification of VB-Lie algebroids and VB-Courant algebroids
• Manin triples for double Lie bialgebroids
• Obstructions to representations up to homotopy and ideals
• Atiyah and Todd classes of regular Lie algebroids