A Q-algebroid is a Lie superalgebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the BRST model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of...

We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H , and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra as...

We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid G in terms of the cohomology of S * (G), the cosphere bundle of A(G), where A(G) is the Lie algebroid of G. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space A * (G) 0 ∼ = S * (G) × (0, ∞), the dual of A(G) with the zero section removed. ...

For a manifold $M$ with an integral closed 3-form $\omega$, we construct $PU(H)$-bundle and Lie groupoid over its total space, together curving in the sense of gerbes. If form is non-degenerate, furthermore give natural 2-algebra quasi-isomorphism from observables $(M,\omega)$ to weak symmetries above geometric structure, generalising prequantisation map Kostant Souriau.

Given a Lie groupoid G over a manifold M , we show that multiplicative 2forms on G relatively closed with respect to a closed 3-form φ on M correspond to maps from the Lie algebroid of G into T ∗M satisfying an algebraic condition and a differential condition with respect to the φ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to φ-twist...

We give an explicit description of a 1-1 correspondence between Morita equivalence classes of, on the one hand, principal 2-group [G → Aut(G)]bundles over Lie groupoids (i.e. [G → Aut(G)]-bundles over differentiable stacks) and, on the other hand, centralG-extensions of Lie groupoids (i.e. Ggerbes over differentiable stacks). We also introduce universal characteristic classes for 2-group bundle...

The classical Serre-Swan’s theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its convolution algebra tha...

Abstract The van Est map is a from Lie groupoid cohomology (with respect to sheaf taking values in representation) algebroid cohomology. We generalize the allow for more general sheaves, namely sheaves of sections (smooth or holomorphic) $G$-module, where $G$-modules are structures, which differentiate representations. Many geometric structures involving groupoids and stacks classified by not r...