TITLE: What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
QUESTION [14 upvotes]: Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence of holomorphic connections on $E$. In particular if we take $E=TX$ to be the tangent bundle of $X$ itself, we can prove that the Atiyah class $\alpha(TX)\in \text{Ext}^1(TX\otimes TX, TX)$ and this gives a Lie algebra structure
$$
TX[-1]\otimes TX[-1]\rightarrow TX[-1]
$$
 in the derived category $D(X)$.
For the details see M. Kapranov's paper "Rozansky–Witten invariants via Atiyah classes" (arXiv) and N. Markarian's paper "The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem" (arXiv).
And it is well-known that we have the Lie bracket as commutator on the tangent vector fields
$$
\Gamma(X,TX)\otimes \Gamma(X,TX)\rightarrow\Gamma(X,TX).
$$
Of course these two kinds of Lie bracket are very different. $\textbf{My question}$ is: is there any relation between them? More precisely, could we regard the Lie bracket of Atiyah class as a kind of "higher lift" of the naive Lie bracket as commutators?

REPLY [9 votes]: Starting from $X$ one can construct a sequence $(X_0,X_1,\dots)$ of groupoid objects in spaces, all having $X$ as space of objects (I am writing "space" in order not to specify the geometric context I am working with, but I actually mean "derived scheme"):

$X_0$ is the pair groupoid $X\times X$ of $X$. The Lie algebroid of this groupoid is $TX$. 
$X_1=X\times^h_{X_0}X$ is the derived self-intersection of the diagonal in $X_0$ (i.e. the derived loop space of $X$). The Lie algebroid of this groupoid is $TX[-1]$ with the Atiyah Lie bracket. It happens to be a Lie algebra (because the derived loop space is actually a group). 
In general $X_{n+1}=X\times^h_{X_n}X$ and its Lie algebroid $\mathfrak g_{n+1}$ is $TX[-n-1]$, and is an abelian Lie algebra as soon as $n\geq1$. 

In other words, we have that $\mathfrak g_n=\Omega^n_0(TX)=TX[-n]$. Hence, in particular  $TX[-1]$ is a group in Lie algebroids, and thus must be a Lie algebra. You get that the Atiyah-Kapranov Lie bracket on $TX[-1]$ is completely dertmined by the Lie algebroid structure on $TX$. 
One also gets that the Lie algebroid $\mathfrak g_2$ of $X_2$ is going to be $\Omega_0(TX[-1])$, and thus is a group in Lie algebras... hence it must be an abelian Lie algebra. 
The hierarchy Lie algebroid -> Lie algebra -> abelian Lie algebra -> ... one gets is analogous to the following one we get in algebraic topology: set -> group -> abelian group -> ...