Description of coupling in the category of transitive Lie algebroids

TL;DR

This paper refines the understanding of couplings in transitive Lie algebroids by establishing a one-to-one correspondence between couplings and local trivial structures, aiding in classifying such algebroids.

Contribution

It proves a bijective relationship between all couplings of a Lie algebra bundle with a fixed Lie algebra and classes of local trivial structures with a specific topology.

Findings

Establishes a one-to-one correspondence between couplings and local trivial structures.

Provides a geometric construction for the classifying space of transitive Lie algebroids.

Clarifies the categorical description of characteristic classes for these algebroids.

Abstract

In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on the group $\Aut(\rg)$ of all automorphisms of the Lie algebra $\rg$, say $\Aut(\rg)^{\delta}$, 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. But the question how many couplings exist under these conditions still remains open. Here we make the result more accurate and prove that there is a one-to-one correspondence between the family $Coup(L)$ of all coupling of the Lie algebra bundle $L$ with fixed finite dimensional Lie algebra $\rg$ as the fiber and the structural group $\Aut(\rg)$ of all automorphisms of Lie algebra $\rg$ and the tangent bundle $TM$ and the family $LAB^{\delta}(L)$ of equivalent classes of local trivial structures with structural group $\Aut(\rg)$ endowed with new topology $\Aut(\rg)^{\delta}$. This result gives a way for geometric construction of the classifying space for transitive Lie algebroids with fixed structural finite dimensiaonal Lie algebra $\rg$. Hence we can clarify a categorical description of the characteristic classes for transitive Lie algebroids and a comparison with that by J. Kubarski.