Lie algebroid modules and representations up to homotopy

Abstract

We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaintrob's Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.

v3: Final version, to appear in Indag. Math

References in corpus (2)

• Supergroupoids, double structures, and equivariant cohomology
• Introduction to supergeometry

Cited by in corpus (14)

• Higher Extensions of Lie Algebroids
• L-infinity algebra actions
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Hochschild cohomology of dg manifolds associated to integrable distributions
• The geometry of graded cotangent bundles
• Modular Classes of Lie Groupoid Representations up to Homotopy
• The modular class of a singular foliation
• Atiyah classes of strongly homotopy Lie pairs
• $L_{\infty }$-actions of Lie algebroids
• Lie algebroids as $L_{\infty }$ spaces
• Obstructions to representations up to homotopy and ideals
• Atiyah and Todd classes of regular Lie algebroids
• Higher Vector Bundles
• Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids