We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms $\BV_\infty$-operators on space Berezinian half-densities. prove that quantizing such a manifold is equivalent to lifting consecutive sequences Maurer-Cartan elements short exact differential graded Lie algebras, where obstruction certain class second cohomology. Consequently, quantizable if cohomology ...

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of s...

We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of β-diffeomorphisms and βtransformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac ...

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras 1 , in the theory of integrable system 2 , and in field theory 3 . But for instance, in 2 , the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In 4, 5 we have defined such a test functional space in the case of a linear Poisson...

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.