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Let $V$ be a "Lie vector bundle", which I define as follows:

A Lie vector bundle is a vector bundle $V$ with a Lie bracket $[\cdot,\cdot]$ on sections of $V$, turning it into a Lie algebra and that satisfies $[fv, w]=f[v,w]$ for every local scalar-valued function $f$ and local sections $v,w$. My question is: are there any non-trivial examples?

Of course one can ask the same question in different categories over different fields.

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    $\begingroup$ Of course. Take $V$ any vector bundle and consider $\mathfrak{gl}(V)$. In general bundles with fiber a Lie algebra $\mathfrak{g}$ are classified by principal $\text{Aut}(\mathfrak{g})$-bundles. $\endgroup$ Commented Mar 16 at 21:08
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    $\begingroup$ It's the group of automorphisms of $\mathfrak{g}$ as a Lie algebra. $\endgroup$ Commented Mar 16 at 21:20
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    $\begingroup$ Nothing I've said is specific to Lie algebras, this is how fiber bundles work in general. A bundle with fiber $F$ together with whatever extra structure you want is classified by a principal $\text{Aut}(F)$-bundle. See en.wikipedia.org/wiki/Associated_bundle for details. $\endgroup$ Commented Mar 16 at 21:26
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    $\begingroup$ There is a notion called Lie algebroids. $\endgroup$ Commented Mar 16 at 22:57
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    $\begingroup$ The behavior of the Lie bracket wrt scalar multiplication implies that fiberwise it is linear bundle morphism $[-,-] \colon V \otimes V \to V$, which satisfies the Lie bracket identities. There are many different Lie brackets on any vector space, some of them may even belong to non-equivalent classes (under joint $GL(V)$ action). So here you are simply choosing one Lie bracket on each fiber of $V$, up to whatever continuity/regularity conditions you want to impose on it. $\endgroup$ Commented Mar 17 at 10:57

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