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In section 3.1 here I read that derivations are, morally, "infinitesimal automorphisms". For instance, given $\phi:(-\varepsilon, \varepsilon)\to\mathrm{Aut}(A)$ with $\phi(0)=1_A$, then the derivative at zero of $\phi$ is a derivation (by playing around with the quotient defining a derivative).

Looking to understand more I stumbled upon section 4.4 of Bertram's Differential Geometry over General Base Fields and Rings. There, an endomorphism of the tangent bundle is an infinitesimal automorphism if it's a bundle automorphism which acts on each fiber by translation. Just after the definition, the group of such things said to be isomorphic to the additive group of vector fields on the base.

  1. What's the geometric intuition for derivations? Does "infinitesimal automorphism" in this context mean something which is an automorphism on "infinitesimal neighborhoods"?
  2. What's the geometric intuition for infinitesimal automorphisms of the tangent bundle?

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I was also curious about this question. Physicists formulate your question a littlebit handwavingly, saying that (i) the smooth tangent vector fields are infinitesimal version of smooth diffeomorphisms, and (ii) as such the Lie algebra of smooth vector fields (here denoted by $X$) "act similarly as $\mathrm{igl}(T_{p})$ at a point $p$". Point (i) is clear I guess, can be found in any diffgeo text. I tried to make some mathematically less handwaving sense of (ii). This is what I managed to formulate so far: let us fix a point $p$ of the manifold, then one can define the following subsets of $X$.

$X_{p}^{-1} := X$,

$X_{p}^{k} := \{u \in X | \text{up to } k\text{-th derivative of u vanishes at }p\}$

($k=0,1,2,...$)

where the derivative is understood to be any covariant derivation. It is seen that the definition of each $X_{p}^{k}$ is independent of the choice of that covariant derivation. They form a decreasing sequence of sub linear spaces in $X$. Moreover, one has $[X_{p}^{k},X_{p}^{l}] \subset X_{p}^{max(-1,k+l)}$ $(k,l=-1,0,1,2,...)$ and therefore each of these subspaces are also sub Lie algebras, and are called the k-th order isotropy sub-Lie algebras within $X$ at $p$. Unfortunately, they are not normal sub Lie algebras, i.e. not Lie algebra ideals within $X$, therefore one cannot define for instance the quotient Lie algebra $X_{p}^{-1}/X_{p}^{0}$. However, all of the quotients $\hat{X}_{p}^{k} := X_{p}^{k}/X_{p}^{k+1}$ $(k=-1,0,1,2,...)$ are meaningful as quotient linear subspaces. One has the identity $[\hat{X}_{p}^{k},\hat{X}_{p}^{l}] \subset \hat{X}_{p}^{max(-1,k+l)}$

Therefore, one can define (I think, it is called the covering Lie algebra of $X$): $\hat{X}_{p} := \oplus_{k=-1,0,1,2,...} \;\hat{X}_{p}^{k}$

and by construction, it shall be isomorphic (not naturally) to $X$ as Lie algebra.

That means that one can inject elements of $\hat{X}_{p}$ to $X$ homomorphically. Since $\hat{X}_{p}^{-1} \equiv T_{p}$ and $\hat{X}_{p}^{0} \equiv \mathrm{Lin}(T_{p}) \equiv \mathrm{gl}(T_{p})$

one has that the direct sum of the first 2 sectors of the covering Lie algebra $\hat{X}_{p}^{-1} \oplus \hat{X}_{p}^{0}$ is isomorphic to $\mathrm{igl}(T_{p})$.

So, in this sense, one can say that elements of $\mathrm{igl}(T_{p})$ can be represented on $X$ with injective Lie algebra homomorphism, and they will end up in $\hat{X}_{p}^{-1}\oplus \hat{X}_{p}^{0}$ sector of the covering Lie algebra $\hat{X}_{p}$.

Best regards, Andras