Brst Quantization of Quasi-symplectic Manifolds and Beyond

A class of factorizable Poisson brackets is studied which includes almost all reasonable Poisson manifolds. In the simplest case these brackets can be associated with symplectic Lie algebroids (or, in another terminology, with triangular Lie bialgebroids associated to a nondegenerate r-matrix). The BRST theory is applied to describe the geometry underlying these brackets and to develop a covariant deformation quantization scheme in this particular case. The proposed quantization procedure can be viewed as an extension of the Fedosov deformation quantization to a wide class of irregular Poisson structures. In more general case, the factorizable Poisson brackets are shown to be connected with the notion of n-algebroid which can be viewed as particular example of reducible gauge algebra well studied in the BRST theory. A simple description is suggested for the geometry underlying the factorizable Poisson brackets, which is based on the construction of odd Poisson algebra bundle equipped with an Abelian connection. It is shown that the zero-curvature condition for this connection generates all the structure relations for the n-algebroid as well as a generalization of the Yang-Baxter equation for the symplectic structure.