Whitehead's second lemma removes the infinitesimal equivariance obstruction for semisimple actions

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, connected $G$ and $M$, an infinitesimal moment map $\mu$ for the action, and a finite-dimensional real semisimple $\mathfrak{g}$.

${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice; it is used through the moment-map, defect, and equivariance interfaces cited in [F1], [F2], and [F6].

The components satisfy $d{\mu }^{\xi }=-{\iota }_{{\xi }_M}\omega$ for all $\xi$. Moment map, component Hamiltonians and infinitesimal moment maps.

The nonequivariance defect $c(\xi ,\eta )=\{{\mu }^{\xi },{\mu }^{\eta }\}-{\mu }^{[\xi ,\eta ]}$ is a constant on $M$ and is a Chevalley--Eilenberg two-cocycle with trivial coefficients. The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle.

For a finite-dimensional semisimple $\mathfrak{g}$ over a characteristic-zero field, $H^2(\mathfrak{g},M)=0$ for every finite-dimensional module $M$, in particular $H^2(\mathfrak{g},\mathbb{R})=0$ for the trivial module. Second Whitehead lemma, Lie algebra cohomology, Simple, semisimple, and reductive Lie algebras.

With the zero-based convention, the Chevalley--Eilenberg differential of a one-cochain $b$ is $(db)(\xi ,\eta )=-b([\xi ,\eta ])$; hence $c=db$ means $c(\xi ,\eta )=-b([\xi ,\eta ])$. Chevalley–Eilenberg differential.

Constants are Poisson-central: adding a constant to a function changes no Hamiltonian vector field and the Poisson bracket of a constant with any function vanishes. Poisson bracket on a symplectic manifold.

The defect vanishes identically on the connected manifold $M$ if and only if $\mu$ is coadjoint equivariant, and for connected $G$ the bracket identity is equivalent to equivariance. The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle, For connected groups, equivariance is equivalent to the moment-map Poisson bracket identity.