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So I have discovered that indeed that indeed $\zeta = T\lambda: \frak{g} \to \frak{X}$$(M)$.

There is a simple reason why this mapping is an antihomomorphism, even though the Lie functor should take Lie group homomorphism to Lie algebra homomorphisms.

The Lie bracket of most Lie algebras is defined via left-invariant vector fields on the Lie group, but the bracket on $\frak{X}$$(M)$ is usually defined in such a way that it corresponds to right-invariant vector fieldsright-invariant vector fields on $\text{Diff}(M)$. Thus, if we chose to define Lie brackets for all of our Lie algebras consistently using the right-invariant convention (as Olver does, for example), then the fundamental vector field mapping will be a homomorphism.

So I have discovered that indeed that indeed $\zeta = T\lambda: \frak{g} \to \frak{X}$$(M)$.

There is a simple reason why this mapping is an antihomomorphism, even though the Lie functor should take Lie group homomorphism to Lie algebra homomorphisms.

The Lie bracket of most Lie algebras is defined via left-invariant vector fields on the Lie group, but the bracket on $\frak{X}$$(M)$ is usually defined in such a way that it corresponds to right-invariant vector fields on $\text{Diff}(M)$. Thus, if we chose to define Lie brackets for all of our Lie algebras consistently using the right-invariant convention (as Olver does, for example), then the fundamental vector field mapping will be a homomorphism.

So I have discovered that indeed that indeed $\zeta = T\lambda: \frak{g} \to \frak{X}$$(M)$.

There is a simple reason why this mapping is an antihomomorphism, even though the Lie functor should take Lie group homomorphism to Lie algebra homomorphisms.

The Lie bracket of most Lie algebras is defined via left-invariant vector fields on the Lie group, but the bracket on $\frak{X}$$(M)$ is usually defined in such a way that it corresponds to right-invariant vector fields on $\text{Diff}(M)$. Thus, if we chose to define Lie brackets for all of our Lie algebras consistently using the right-invariant convention (as Olver does, for example), then the fundamental vector field mapping will be a homomorphism.

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So I have discovered that indeed that indeed $\zeta = T\lambda: \frak{g} \to \frak{X}$$(M)$.

There is a simple reason why this mapping is an antihomomorphism, even though the Lie functor should take Lie group homomorphism to Lie algebra homomorphisms.

The Lie bracket of most Lie algebras is defined via left-invariant vector fields on the Lie group, but the bracket on $\frak{X}$$(M)$ is usually defined in such a way that it corresponds to right-invariant vector fields on $\text{Diff}(M)$. Thus, if we chose to define Lie brackets for all of our Lie algebras consistently using the right-invariant convention (as Olver does, for example), then the fundamental vector field mapping will be a homomorphism.