The existence and classification of couplings between Lie algebra bundles and tangent bundles

In this paper we investigate the existence and classification of couplings between Lie algebra bundles and tangent bundles. We first give a sufficient and necessary condition on the existence of coupling between Lie algebra bundle (LAB) and the tangent bundle; i.e., we define a new topology on the group Autg of all automorphisms of the Lie algebra gg, denoted by Autgδ, and show that the tangent bundle TM can be coupled with the Lie algebra bundle L if and only if L   admits a local trivial structure with structural group endowed with such new topology. Then we show that there is a bijection between Coupg(M)Coupg(M) the set of all isomorphism classes of couplings (L,Ξ)(L,Ξ) and the set LABgδ(M) of isomorphism classes of gg-bundles with structural group Autgδ.