Generalized geometry finds many applications in the mathematical description of some aspects string theory. In a nutshell, it explores various structures on generalized tangent bundle associated to given manifold. particular, several integrability conditions can be formulated terms canonical Dorfman bracket, an example Courant algebroid. On other hand, smooth manifolds involve functions $\mathb...

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...

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...

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is als...

Supergroupoids, double structures, and equivariant cohomology by Rajan Amit Mehta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Weinstein, Chair Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Ma...

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid H = (HL,HR)) is cleft if and only if it is HR-Galois and has a normal basis property relative to the base ring L of HL. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the ...

The purpose of this note is to give a “coordinate free” construction and prove the uniqueness of the vertex algebroid which gives rise to the chiral de Rham complex of [GMS]. In order to do that, we adapt the strategy of [BD] to the setting of vertex agebroids. To this end we show that the stack EVAX of exact vertex algebroids on X is a torsor under the stack in Picard groupoids ECAX of exact C...