Sorting out minus signs in differential geometry

in the Lie algebra of vector fields on a smooth manifold §

Bug

The usual definition of commutator $[X,Y]:=XY-YX$ as derivations of ${\mathcal{C}}^{\infty }(M)$ and infinitesimal generators of Lie group actions

$$V\mapsto \text{dvf}t{|}_{t=0}\exp (tV)(-)$$

produces an anti-homomorphism $\text{goth}g\to \mathrm{Vec}(M)$. This is expected because $\mathrm{LDiff}(M)$ acts on ${\mathcal{C}}^{\infty }(M)$ on the right.

• settle with the $[X,Y]=XY-YX$ brackets
  • [?] does it agree with brackets in $\text{mor}End\text{cat}Vec$?
  • settle with the anti-homomorphism
  • use $V\mapsto \text{dvf}t{|}_{t=0}\exp (-tV)(-)$ as definition of infinitesimal generators
    • [!] this seems to produce $-X$ as the infinitesimal generator of flow of $X$
• use the brackets $-[X,Y]$, possibly defined using flows/adjoint action

in the Hamiltonian vector field of a smooth function on a symplectic manifold §

Hamiltonian vector field of $H\in {\mathcal{C}}^{\infty }(M)$ is

$$-{\omega }^{-1}\text{dd}H$$

where $\omega$ is the symplectic form.

Why the minus sign?