The existence and classification of couplings between Lie algebra bundles and tangent bundles (Q5963973)

In the book [The general theory of Lie groupoids and Lie algebroids. Cambridge: Cambridge University Press (2005; Zbl 1078.58011)], \textit{K. C. H. Mackenzie} studied the adjoint bundles of the Lie algebra bundles associated to a transitive Lie algebroid by considering the couplings with the tangent bundle of the base manifold. He investigated also the inverse problem i.e., for a Lie algebra bundle \(L\), one looks for a transitive Lie algebroid with \(L\) as the adjoint bundle. One formulates the definition of coupling and one constructs an obstruction class, called the Mackenzie obstruction, which depends on the coupling and gives a criterion for the existence of the initial transitive Lie algebroid. Later, one of the authors of this paper (A. S. Mishchenko) has shown that the construction can be managed as a homotopy functor from the category of smooth manifolds to the category of transitive Lie algebroids. In the present paper the authors show that the coupling between Lie algebra bundle and tangent bundle plays an important role in the theory of the Mackenzie obstruction and the classification of transitive Lie algebroids. First, they study the existence of the above coupling. They define a new topology on the group \(\mathrm{Aut }\mathfrak g\) of all automorphisms of the Lie algebra \(\mathfrak g\), denoted by \(\mathrm{Aut }\mathfrak g^\delta\) and show that the tangent bundle \(TM\) can be coupled with the Lie algebra bundle \(L\) if and only if the Lie algebra bundle \(L\) admits a locally trivial structure with structure group endowed with this new topology. They present a definition of isomorphism of couplings. They show that there exists a one-to-one correspondence between the set \(\mathrm{Coup}_{\mathfrak g}(M)\) of all isomorphism classes of couplings \(|(L,\Xi)|\) and the family \(|M,B_{\mathrm{Aut }\mathfrak g^\delta} |\) of homotopy classes of continuous maps from \(M\) to \(B_{\mathrm{Aut }\mathfrak g^\delta}\), where \(B_{\mathrm{Aut }\mathfrak g^\delta}\) is the classifying space for \(\mathfrak g\)-bundles with structure group \({\mathrm{Aut }\mathfrak g^\delta}\).