Equivalence of the categories of modules over Lie algebroids

Abstract

AbstractWe study geometric representation theory of Lie algebroids. A new equivalence relationfor integrable Lie algebroids is introduced and investigated. It is shown that two equivalentLie algebroids have equivalent categories of infinitesimal actions of Lie algebroids. As anapplication, it is also shown that the Hamiltonian categories for gauge equivalent Dirac struc-tures are equivalent as categories. Mathematics Subject Classification (2000). 53D17Keywords. Poisson manifolds, Lie groupoids and algebroids, Dirac structures, Geometric Moritaequivalence. 1 Introduction Poisson geometry is considered to be intermediate between differential geometry and non-commutative geometry in the sense that it provides us with powerful techniques to study manygeometric objects related to noncommutative algebras.If (Q, Π Q ) and (P, Π P ) are Poisson manifolds, then a Poisson map J : Q →P induces a Liealgebra homomorphism byC ∞ (P) −→X(Q) ⊂End(C ∞ (Q)), f −→−Π Q (·, J ∗ df ). (1.1)From (1.1), C ∞ (Q) can be regarded as a C