The Existence of Coupling in the Category of Transitive Lie Algebroid

The coupling of the tangent bundle TM with the Lie algebra bundle L ([5], Definition 7.2.2) plays the crucial role in the classification of the transitive Lie algebroids for Lie algebra bundle L with fixed finite dimensional Lie algebra g as a fiber of L. Here we give a necessary and sufficient condition for the existence of such a coupling. Namely we define a new topology on the group Aut (g) of all automorphisms of Lie algebra g and show that tangent bundle TM can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology.