Poisson Algebras, Symplectic Brackets, and Hamiltonian Actions

This page develops the symplectic Poisson bracket, the equivariant moment-map bridge, and the induced Lie-algebra action on smooth observables. It reuses the existing canonical Poisson and moment-map results and adds the distinct comoment/action interface authorized for this page.

Poisson algebra and symplectic bracket

In this page, a real Poisson algebra means a commutative associative real algebra with a real-bilinear Lie bracket that is a derivation in each argument. The theorem Smooth functions form a Poisson algebra proves that $C^{\mathrm{\infty }}(M,\mathbb{R})$ has this structure when $(M,\omega )$ is a symplectic manifold: $\omega$ is closed and pointwise nondegenerate (Symplectic form and symplectic manifold), multiplication is pointwise, and the bracket is the symplectic bracket below. The theorem is restricted to symplectic manifolds; it does not assert a result for a degenerate Poisson tensor.

The library fixes the Hamiltonian vector field by ${\iota }_{X_H}\omega =dH$ (Hamiltonian vector field and Hamiltonian function). Pointwise nondegeneracy of $\omega$ gives a unique smooth $X_H$ for every smooth $H$ (Hamiltonian vector fields exist uniquely for smooth functions), so the bracket is defined for every pair of smooth functions. With this convention,

$\{F,G\}=\omega (X_F,X_G)=dF(X_G)=X_G(F)=-X_F(G),$

so evolution by $H$ differentiates an observable as $X_H(F)=\{F,H\}$. The definition is Poisson bracket on a symplectic manifold. The proof that this is a Poisson algebra uses the existing biderivation and Jacobi results, The Poisson bracket is bilinear, skew, and a derivation in each entry and The Poisson bracket satisfies the Jacobi identity.

Hamiltonian actions and moment maps

For a smooth left action (Smooth left actions of Lie groups) of a finite-dimensional real Lie group $G$ on $(M,\omega )$, write $\mathfrak{g}=T_{e}G$ for its Lie algebra and ${\mathfrak{g}}^{\ast }=\mathcal{L}(\mathfrak{g},\mathbb{R})$ for its linear dual (Linear functionals and the algebraic dual $V^{\ast }=\mathcal{L}(V,F)$). Use the library fundamental field ${\xi }_M(p)={\frac{d}{dt}\mid }_0{\exp }_G(-t\xi )\cdot p$. An infinitesimal moment map $\mu :M\to {\mathfrak{g}}^{\ast }$ satisfies $d{\mu }^{\xi }=-{\iota }_{{\xi }_M}\omega$ for every $\xi \in \mathfrak{g}$, where ${\mu }^{\xi }(p)=⟨\mu (p),\xi ⟩$; an equivariant moment map also satisfies $\mu (g\cdot p)=g\cdot \mu (p)$ for the coadjoint action $g\cdot \alpha =\alpha \circ {\mathrm{Ad}}_{g^{-1}}$ (The coadjoint representation, action and orbits). These conventions are fixed by Symplectic and Hamiltonian Lie-group actions, Moment map, component Hamiltonians and infinitesimal moment maps, and Fundamental vector fields for a left action. The component equation gives $X_{{\mu }^{\xi }}=-{\xi }_M$ (Moment map components generate the negative infinitesimal action).

Under the explicitly declared assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the Axiom of Countable Choice), For connected groups, equivariance is equivalent to the moment-map Poisson bracket identity proves that equivariance implies $\{{\mu }^{\xi },{\mu }^{\eta }\}={\mu }^{[\xi ,\eta ]}$ for any Lie group, and that the converse holds when $G$ is connected in the topological sense (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets). For a disconnected group the identity detects equivariance under $G^0$, the connected component containing the identity, only; equivariance under representatives of the other components must be checked separately. Connectedness of $M$ is unnecessary when the identity is assumed on all of $M$; it is relevant only to checking the identity at one point via The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle.

Comoment maps and observable representations

For this page, a comoment map is a real-linear Lie-algebra map $J:\mathfrak{g}\to C^{\mathrm{\infty }}(M,\mathbb{R})$ for the symplectic Poisson bracket; its observable action is ${\rho }_J(\xi )F=\{J(\xi ),F\}$. The exact definition is Symplectic comoment maps and Hamiltonian observable actions. The proposition Comoment maps act on observables by Poisson derivations proves directly from bilinearity, Jacobi, and the biderivation rule that ${\rho }_J$ is a Lie-algebra representation and that each operator obeys Leibniz rules for both pointwise multiplication and the Poisson bracket. The first assertion applies to any real Lie algebra, with no finite-dimensional hypothesis. It neither asserts an integrated group action nor a converse construction of $J$.

When $J(\xi )={\mu }^{\xi }$ comes from an equivariant moment map, the same proposition identifies ${\rho }_J(\xi )F={\xi }_M(F)$ in the library's negative-exponential convention. This forward implication carries ${\mathrm{A}\mathrm{C}}_{\omega }$ through the actual moment-map suppliers and does not require $G$ connected. No converse for arbitrary $G$-representations by derivations is claimed.

The diagram summarizes the definition and proposition: the middle node is the image $J(\mathfrak{g})\subseteq C^{\mathrm{\infty }}(M,\mathbb{R})$, and the proposition proves that each such function acts by a Poisson derivation. The diagonal arrow equals the composite by the definition of ${\rho }_J$; the diagram summarizes the result and is not used as a proof.

\mathfrak g \arrow[r,"J"] \arrow[dr,"\rho_J"'] & J(\mathfrak g) \arrow[d,"a\mapsto\{a,-\}"] \\
& \operatorname{Der}_{\mathrm{Pois}}(C^\infty(M,\mathbb R))

Reading prerequisites and source notes

The current plan requires algebraic-representation-and-tensor-symmetry-map and moment-maps-and-symplectic-reduction as page-level reading routes. The second is published and contains the moment-map definition and proposition used above. At this handoff, the batch-32 A page contains two draft carriers, Representations of unital associative algebras and Associative algebra representations are unital modules, and its current page body explains the associative-representation/module interface. I inspected both carrier bodies, including the theorem proof. They remain draft and are not claimed independently accepted; none of this page's Poisson or moment-map claims uses them as an item-level logical supplier.

The following full-text treatments were inspected for their stated locators. Their conventions are translated here explicitly; citations provide context, while the linked library items carry the local arguments.

• Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture 18, §§18.3–18.4 (printed pp. 108–109), and Lecture 22, §22.1 (printed pp. 133–135). This full lecture-note book states the symplectic Poisson bracket and Poisson-algebra axioms and identifies the comoment map with a Lie-algebra homomorphism for connected Hamiltonian actions. Its Jacobi and Leibniz verifications are exercises; the local published suppliers prove those prerequisites, and the new observable-action proposition gives its complete local calculation.
• Eckhard Meinrenken, Symplectic Geometry, §4.2 (printed pp. 46–47) and §7.3, Definition 7.12, Lemma 7.14 and complete proof (printed pp. 83–85). The notes use the same negative-exponential generating field as the library. They prove the connected-group moment/comoment criterion and the Poisson Lie-bracket identities; the new proposition supplies the explicit arbitrary-comoment action and both derivation identities locally.
• Nima Moshayedi, Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory, §§2.7.3–2.7.15, 2.8.4, and 3.1.1 (PDF pp. 57–59, 63–64, 75–76). These notes give independent definitions and examples; the Jacobi and Leibniz verifications in §2.7 are exercises, so they are not used as substitutes for the linked local proofs. For actions, Moshayedi uses the positive-exponential generator $X^{\mathrm{\#}}$ and $d{\mu }^X={\iota }_{X^{\mathrm{\#}}}\omega$; since $X^{\mathrm{\#}}=-{\xi }_M$ here, this is the library equation $d{\mu }^{\xi }=-{\iota }_{{\xi }_M}\omega$.

Companion: poisson-algebras-and-hamiltonian-representations-examples.