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Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that if $\alpha: G \to \text{Diff}(M)$ is a left action (a group homomorphism) then $T_e \alpha: \frak{g} \to \frak{X}$$(M)$ is an antihomorphism of Lie algebras.

It seems that this is not a typo, and he proves that this is indeed the case. Is there something I'm missing here?

If $G$ is a Lie group (a microlinear group), then the Lie algebra consists of all $X: D \to G$ such that $X(0)=e$. Here $D = \{ d \in \mathbb{R} : d^2 =0 \}$ is the "walking tangent vector." The Lie bracket of $X, Y \in \frak{g}$ is then defined to be the unique element $[X, Y] \in \frak{g}$ such that $$[X, Y](d_1 \cdot d_2) = Y(-d_2) \cdot X(-d_1) \cdot Y(d_2) \cdot X(d_1)$$ for any $d_1, d_2 \in D$. If $\text{Diff}(M)$ is microlinear, then the tangent space at $\text{id}_M$ is $\frak{X}$$(M)$, so this defines the Lie bracket on vector fields as well. This appears to be equivalent to the usual definition of the Lie bracket in terms of flows.

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    $\begingroup$ It means one or more (really, an odd number) of the author's conventions are weird; for example, maybe their definition of the Lie bracket on vector fields is the opposite of the usual one. Can you reproduce the proof? You can see another author saying the obvious thing, for example, here: math.toronto.edu/mein/teaching/action.pdf (Theorem 1.11). $\endgroup$ – Qiaochu Yuan Dec 26 '15 at 23:11
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    $\begingroup$ If the book defines the Lie bracket on $\mathfrak{g}$ and on vector fields in this same uniform way then I can't imagine how it possibly gets an antihomomorphism out of a morphism of groups. Can you quote some more text from the book? I still think there's just a weird convention somewhere. $\endgroup$ – Qiaochu Yuan Dec 27 '15 at 3:44
  • $\begingroup$ Yeah I figure one of his conventions differ from the ones I'm used to, but I can't seem to figure out what it is exactly. I'll check his convention for the action of vector fields on smooth functions when I get home. $\endgroup$ – ಠ_ಠ Dec 27 '15 at 5:09

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