On the relation of Lie algebroids to constrained systems and their BV/BFV formulation

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge. Adding a Hamiltonian to the system corresponds to a metric g on M. Evolution invariance of the constraint surface introduces a connection nabla on E and one reobtains the compatibility of g with (E,rho,nabla) found previously in the literature. The covariantization of the Hamiltonian to a function on T*E[1] serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning (E, rho, [ , ], nabla) into a Cartan-Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.