The Existence of Coupling in the Category of Transitive Lie Algebroid

The coupling of the tangent bundle $TM$ with the Lie algebra bundle $L$ (K.Mackenzie,2005, 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 $\rg$ as a fiber of $L$. Here we give a necessary and sufficient condition for existence such a coupling. Namely we define a new topology on the group $\Aut(\rg)$ of all automorphisms of Lie algebra $\rg$ 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. The second version of the article is different from the first because of the improvement of the text by invaluable efforts of professor J.Stasheff, to whom we submit our sincere gratitude.