Let $M$ be a smooth manifold. A Lie algebroid over $M$ is a vector bundle $E\rightarrow M$ over $M$, with a Lie bracket on $\Gamma(M,E)$, a morphism of vector bundles $\rho:E\rightarrow TM$, such that, the following conditions are satisfied:
- the map $\rho:E\rightarrow TM$ induce a morphism of Lie algebras $\Gamma(M,E)\rightarrow \Gamma(M,TM)$,
- the Lie algebra structure on $\Gamma(M,E)$ is “compatible” with the $C^\infty(M)$-algebra structure on $\Gamma(M,E)$, upto a correction; this goes by the name Leibniz condition.
A generalized complex structure on a manifold $M$ is a morphism of vector bundles $J:TM\oplus TM^*\rightarrow TM\oplus TM^*$ such that it is compatible with some bracket operation and some ''inner product'' on $\Gamma(M,TM\oplus TM^*)$.
In most of the references about generalized complex structures, they introduce the notion of Lie algebroid. I could not see detailed justification of introducing Lie algebroid over $M$ when discussing generalized complex structure on $M$. So, I am thinking of the following question:
How is the theory of Lie algebroids useful in (better) understanding of generalized complex structures?
Any pointers are welcome. I have done google search for ''Lie algebroids and generalised complex structure" but could not find anything specific.