Let $E\rightarrow M$ be a vector bundle.
Kirill Mackenzie in the book General theory of Lie groupoids and Lie algebroids associates a Lie algebroid to $E\rightarrow M$ in the following steps:
- talk about zero-th and first order differential operators on $E\rightarrow M$, which are some nice maps of sections $\Gamma(M,E)\rightarrow \Gamma(M,E)$.
- realise these first order differential operators as sections of a vector bundle $\text{Diff}^1(M)\rightarrow M$.
- do some pullback along some morphism of vector bundles. Call the pullback as the Lie algebroid of derivations on $E\rightarrow M$.
The book also talks about Linear vector fields and says how these are related to the Lie algebroid of derivations.
I do not fully understand the idea of linear vector fields and first/zeroth order differential operators. But, that is not what I want to ask.
Given a principal bundle $P\rightarrow M$, one can consider the morphism of tangent bundles $TP\rightarrow TM$. As the action of $G$ on $TP$ is nice, this would induce a morphism of vector bundles $(TP)/G\rightarrow TM$. The Lie bracket on $\mathfrak{X}(P)=\Gamma(P,TP)$ is nice enough to induce a Lie bracket on $\Gamma(M,(TP)/G)$. Thus, we have a vector bundle over $M$, a morphism of vector bundles to the tangent bundle of $M$, a Lie bracket on sections of this vector bundle, and some more extra nice properties. This is a Lie algebroids. This vector bundle $(TP)/G\rightarrow M$ is called the Atiyah vector bundle and the Lie algebroid is called the Atiyah Lie algebroid.
Given a vector bundle $E\rightarrow M$, one can consider the associated principal bundle $GL(E)\rightarrow M$ and consider the construction mentioned above which would result in a Lie algebroid over $M$.
Are these two procedures giving same Lie algebroid? I think the answer to this is positive, but I am not sure.
Why would one want to involve differential operators and linear vector fields to associate a Lie algebroid when there is an easier way of considering the associated Atiyah Lie algebroid?
Even though I mentioned that I am not asking about the idea behind linear vector fields/differential operators, please see if you can say a few lines which might make my understand better.