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I am wondering whether there is a Lie algebraic version of principal bundle for Lie group over a given manifold $M$. The first thing I try to think of is group cocycle picture of principal bundle.

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    $\begingroup$ Try Atyiah algebroid. $\endgroup$
    – Vít Tuček
    Commented Jan 31, 2012 at 22:13

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A very comprehensive source is:

MR2157566 Mackenzie, Kirill C. H. General theory of Lie groupoids and Lie algebroids. London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005. xxxviii+501 pp.

Starting from the total space, there is also the notion of the action of a Lie algebra on a manifold, which one can extend to a Lie group action by enlarging the manifold (and loosing Hausdorff in general), see here.

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Building upon Peter's answer, An Atiyah algebroid, or transitive Lie algebroid is one answer to your question.

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