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Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, there is no map $[\_,\_]:TX\times_XTX\to TX$ over $TX\to X$ that would induce brackets on vector fields.

Although I have heard that this non-existence is the reason why people study various kinds of algebroids, I must confess I have no idea why exactly such a bracket does not exist in general. It must have to do with integrability somehow but I don't know how. I only vaguely remember that some sort of algebroids produces a cohomology class that can be viewed as an obstruction to something I do not really remember.

Does all this make sense at all? Can anyone point to a reference where such obstructions are described in detail?

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  • $\begingroup$ May be a related post could be the following math.stackexchange.com/questions/1351116/… $\endgroup$
    – Ali Taghavi
    Commented Oct 10, 2017 at 18:44
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    $\begingroup$ Yes it is related, although there might be Lie brackets that do not relate to the "standard" bracket on vector fields $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 10, 2017 at 20:51
  • $\begingroup$ Yes. But there is no a lie bracket structure on fibers which induce the standard lie algebra structure on vector firlds because the standard lie bracket is not a tensorial 2- linear map. $\endgroup$
    – Ali Taghavi
    Commented Oct 10, 2017 at 21:16
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    $\begingroup$ @AliTaghavi My question is just this - why? Is there a calculable obstruction to this? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 11, 2017 at 5:00
  • $\begingroup$ I was thinking to a related question some times ago. I asked it at RG as follows researchgate.net/post/… $\endgroup$
    – Ali Taghavi
    Commented Oct 12, 2017 at 2:12

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