Mackenzie obstruction for the existence of a transitive Lie algebroid

Abstract

Let g be a finite-dimensional Lie algebra and L be a Lie algebra bundle (LAB). A given coupling Ξ between the LAB L and the tangent bundle TM of a manifold M generates the so-called Mackenzie obstruction Obs(Ξ) ∈ H 3 (M; ZL) to the existence of a transitive Lie algebroid (K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, 2005, p. 279). We present two cases of evaluating the Mackenzie obstruction. In the case of a commutative algebra g, the group Aut(g) δ is isomorphic to the group of all matrices GL(g) with the discrete topology. We show that the Mackenzie obstruction for coupling Obs(Ξ) vanishes. The other case describes the Mackenzie obstruction for simply connected manifolds. We prove that, for simply connected manifolds, the Mackenzie obstruction is also trivial, i.e. Obs(Ξ) = 0 ∈ H 3(M; ZL; ∇ Z ).