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:


1. the map $\rho:E\rightarrow TM$ induce a morphism of Lie algebras $\Gamma(M,E)\rightarrow \Gamma(M,TM)$,
2. 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:



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> How is the theory of Lie algebroids useful in (better) understanding of generalized complex structures?
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Any pointers are welcome. I have done google search for ''Lie algebroids and generalised complex structure" but could not find anything specific.



Use of theory of Lie algebroids in (better) understanding of generalised complex structures
The compatibility conditions that you mention in the definition of a generalized complex structure are equivalent to the statement that the $+i$-eigenbundle $L$ of $J$ is a complex Dirac structure:


1. compatibility of $J$ with the inner product is equivalent to $L$ being 'isotropic', and
2. compatibility of $J$ with the Courant bracket is equivalent to the sections of $L$ being closed under this bracket.


A Dirac structure automatically inherits the structure of a Lie algebroid over $M$. So $L$ in this case is naturally a Lie algebroid. Many of the properties and constructions for $J$ can be stated in terms of the Lie algebroid $L$. For example:


1. The Lie algebroid cohomology of $L$ is a fundamental invariant of $J$. When $J$ comes from a symplectic form, it is the usual de Rham cohomology, whereas when $J$ comes from a complex structure, it is a sum of Dolbeault cohomology groups.
2. A generalized holomorphic bundle for $J$ can be defined to be a representation of $L$. When $J$ comes from a symplectic form, this is a bundle with flat connection, and when $J$ comes from a complex structure, this is a co-higgs bundle.


Note that the Lie algebroid structure on $L$ is somewhat less information than $J$. On the other hand, if you view $L$ as a Dirac structure in $(TM \oplus T^{\*}M)\otimes \mathbb{C}$, then it is equivalent data to $J$. So it's often better to work with it in this way, and many constructions are very naturally stated using the formalism of Dirac geometry.


