Title:

Transitive Lie algebroids and Qmanifolds.

We will start by giving an outline of the fundamentals ofsupergeometry and Qmanifolds. Then, we will give local descriptionof the Atiyah sequence of a principal bundle. We will constructlocal charts for the manifolds involved and write the expressions explicitly in local form. The Atiyah sequence encodes the different notions in connection theory in a compact way. For example a section of the Atiyah algebroid will give a connection in the principal bundle and curvature will be the failure of this section to be a Lie algebroid morphism. We will describe the Lie brackets for the Atiyah algebroid $\frac{TP}{G}$ and the adjoint bundle $\frac{P\tm \li{g}}{G}$ in local coordinates. After that, we will describe the Lie algebroid of derivations $\li{D}(E)$. We will see how the curvature of some section gives the curvature on the vector bundle $E$ and when expressed locally gives the corresponding local connection forms. On the other hand we will give an explicit expression of the morphism from $\frac{TP}{G}$ to $\li{D}\lf(\frac{P\tm V}{G}\rt)$ where $V$ is some vector space on which $G$ acts. As a corollary we will get an isomorphism between $\frac{TFE}{G}$ and $\li{D}(E)$ where $FE$ is the frame bundle of $E$ and $G$ is the general linear group of the fibre $V$. We will establish an explicit equivalence between curvature and field strength in a more general sense. We will recall constructions from the paper of Kotov and Strobl~\cite{kot1} that describe the construction of characteristic classes associated with a section (connection=gauge field) of a $Q$bundle$\cl{E}(\cl{M},\cl{F},\pi)$. Finally, we state and prove the nonabelian Poincar\'{e} lemma in the case when $G=\diff(F)$, the diffeomorphism group of some supermanifold $F$, which has the space of vector fields on $F$, $\li{X}(F)$, as its super Lie algebra. The diffeomorphism group is generally infinite dimensional. It is this that makes the nonabelian Poincar\'{e} lemma more interesting to applications. Then we will show how it is applied to prove that every transitive Lie algebroid is locally trivial.
