Modular Classes of Lie Groupoid Representations up to Homotopy

We describe a construction of the modular class associated to a representation up to homotopy of a Lie groupoid. In the case of the adjoint representation up to homotopy, this class is the obstruction to the existence of a volume form, in the sense of Weinstein's "The volume of a differentiable stack"

Q-groupoids and their cohomology

We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q- and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid "matched pairs".Comment: v2 is the published version. Significant revision over previous version, particularly in section 5.

VB-groupoids and representation theory of Lie groupoids

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids. Under this correspondence, the tangent bundle of a Lie groupoid G corresponds to the "adjoint representation" of G. The value of this point of view is that the tangent bundle is canonical, whereas the adjoint representation is not. We define a cochain complex that is canonically associated to any VB-groupoid. The cohomology of this complex is isomorphic to the groupoid cohomology with values in the corresponding representations up to homotopy. When applied to the tangent bundle of a Lie groupoid, this construction produces a canonical complex that computes the cohomology with values in the adjoint representation. Finally, we give a classification of regular 2-term representations up to homotopy. By considering the adjoint representation, we find a new cohomological invariant associated to regular Lie groupoids.Comment: v5: Introduction is completely rewritten, many other improvements in the exposition. v6: Implements numerous corrections and changes suggested by referees, most notably a significant simplification of the calculations in Appendix A.2. Final version, to appear in J. Symp. Geo

On Homotopy Poisson Actions and Reduction of Symplectic Q-Manifolds

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field

Q-Groupoids and Their Cohomology

We approach Mackenzie\u27s L{script}A{script}-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of L{script}A{script}-groupoids to the category of Q-groupoids. We associate to every Qgroupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q-and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebroid matched pairs

The Atiyah class of a dg-vector bundle

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary bracket, and the Atiyah cocycle corresponding to a torsion-free affine connection as binary bracket.Comment: 7 pages. To appear in Compte Rendus Mat

Courant cohomology, Cartan calculus, connections, curvature, characteristic classes

We give an explicit description, in terms of bracket, anchor, and pairing, of the standard cochain complex associated to a Courant algebroid. In this formulation, the differential satisfies a formula that is formally identical to the Cartan formula for the de Rham differential. This perspective allows us to develop the theory of Courant algebroid connections in a way that mirrors the classical theory of connections. Using a special class of connections, we construct secondary characteristic classes associated to any Courant algebroid.Comment: v2: new and improved version implementing comments from refere

L ∞-Algebra Actions

We define the notion of action of an L -algebra g on a graded manifold M, and show that such an action corresponds to a homological vector field on g[1]×M of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particular, we consider actions of g on a second L -algebra, leading to a notion of semidirect product of L -algebras more general than those we found in the literature