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Quadratic sl2 comoment action on the symplectic plane

Example

Let $M = R^{2}$ with coordinates $(q, p)$ and $ω = d q \land d p$ . Let $g = {s l}_{2} (R) = {(\begin{array}{cc} a & b \\ c & - a \end{array}) : a, b, c \in R}$ with bracket $[A, B] = A B - B A$ . Use the basis $H = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ , $E = (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array})$ , and $F = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array})$ . Define the real-linear map $J : g \to C^{\infty} (M, R)$ by

$J (H) = - q p, J (E) = \frac{q^{2}}{2}, J (F) = - \frac{p^{2}}{2} .$

Write $\partial_{q}$ and $\partial_{p}$ for the coordinate derivative operators. Then $J$ is a comoment map under Symplectic comoment maps and Hamiltonian observable actions, and its observable operators are

$ρ_{J} (H) = q \partial_{q} - p \partial_{p}, ρ_{J} (E) = q \partial_{p}, ρ_{J} (F) = p \partial_{q} .$

These operators give the real three-dimensional matrix Lie algebra its observable representation. By Comoment maps act on observables by Poisson derivations, they preserve both pointwise multiplication and the Poisson bracket on all smooth functions; the Verification also checks the commutators on an arbitrary smooth test function and gives explicit product and bracket-preservation calculations.

Facts & Assumptions

Given: The standard symplectic plane $(R^{2}, d q \land d p)$ , the real traceless $2 \times 2$ matrices with the commutator bracket, the displayed basis, and the map $J$ in the Example. Assume ${A C}_{ω}$ only when applying the coordinate-formula supplier.

[A1]

${A C}_{ω}$ is the Axiom of Countable Choice; it is the sole choice assumption used by the coordinate-formula supplier. No arbitrary-index choice is used here. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

A symplectic form is closed and restricts to a nondegenerate alternating form on every tangent space. Symplectic form and symplectic manifold

[F2]

A finite-dimensional real symplectic vector space has an alternating bilinear form whose map to the dual is an isomorphism; equivalently, only the zero vector pairs to zero with every vector. Symplectic vector space

[F3]

For a real vector space $V$ , ${End}_{R} (V)$ has the commutator Lie bracket, and a Lie-algebra representation is a bracket-preserving linear map into it. Representations of Lie algebras

[F4]

A real Lie algebra has a bilinear alternating bracket satisfying Jacobi. Lie algebras over a field

[F5]

In canonical $(q, p)$ coordinates and under ${A C}_{ω}$ , ${A, B} = A_{q} B_{p} - A_{p} B_{q}$ . Coordinate formula for the Poisson bracket

[F6]

The comoment identity and observable action are $J ([ξ, η]) = {J (ξ), J (η)}$ and $ρ_{J} (ξ) A = {J (ξ), A}$ . Symplectic comoment maps and Hamiltonian observable actions

[F7]

A comoment map defines a Lie-algebra representation on functions, and each operator is a Poisson derivation. Comoment maps act on observables by Poisson derivations

Verification

1.1givenF1F2algebra

The form $ω = d q \land d p$ is smooth and alternating, and $d ω = 0$ because $d (d q) = d (d p) = 0$ . For $v = a \partial_{q} + b \partial_{p}$ , $ι_{v} ω = a d p - b d q$ ; if this is zero, then $a = b = 0$ . Thus $ω$ is nondegenerate at every point and is symplectic by [F1]–[F2].

1.2givenF3F4algebra

Every real traceless matrix is uniquely $a H + b E + c F$ , so $(H, E, F)$ is a basis of the stated vector space. Matrix multiplication gives $H E = E$ , $E H = - E$ , $H F = - F$ , $F H = F$ , $E F = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ , and $F E = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array})$ . Consequently $[H, E] = 2 E$ , $[H, F] = - 2 F$ , and $[E, F] = H$ ; the span is closed under commutators. The ambient endomorphism algebra has the commutator Lie bracket by [F3], and the closed subspace therefore inherits its bilinearity, alternation, and Jacobi identity by [F4]. This verifies the stated ${s l}_{2} (R)$ convention without identifying it with the library's complex ${s l}_{2} (C)$ .

1.3givenA1F4F5F6algebra

Differentiating the three quadratic functions gives $(\partial_{q} J (H), \partial_{p} J (H)) = (- p, - q)$ , $(\partial_{q} J (E), \partial_{p} J (E)) = (q, 0)$ , and $(\partial_{q} J (F), \partial_{p} J (F)) = (0, - p)$ . By [F5], ${J (H), J (E)} = q^{2} = 2 J (E) = J ([H, E])$ , ${J (H), J (F)} = p^{2} = - 2 J (F) = J ([H, F])$ , and ${J (E), J (F)} = - q p = J (H) = J ([E, F])$ . The matrix bracket is alternating by [F4], and swapping the inputs in the coordinate formula [F5] negates the Poisson bracket; both are bilinear. Since $(H, E, F)$ is a basis, these three pairs, their reversals, and the zero diagonal pairs establish the comoment identity for every $ξ, η \in g$ .

1.4givenA1F5F6F7algebra

Applying [F5] to arbitrary $u \in C^{\infty} (R^{2}, R)$ gives $ρ_{J} (H) u = q u_{q} - p u_{p}$ , $ρ_{J} (E) u = q u_{p}$ , and $ρ_{J} (F) u = p u_{q}$ . The result in [F7] supplies the two Poisson-derivation identities for every pair of smooth functions.

2.1givenstep 1.2algebra

Write $u_{q p} = \partial_{q} \partial_{p} u$ and $u_{p q} = \partial_{p} \partial_{q} u$ ; for smooth $u$ these are equal. Direct composition gives $[ρ_{J} (H), ρ_{J} (E)] u = (q u_{p} + q^{2} u_{q p} - p q u_{p p}) - (q^{2} u_{p q} - q u_{p} - p q u_{p p}) = 2 q u_{p} = 2 ρ_{J} (E) u$ , $[ρ_{J} (H), ρ_{J} (F)] u = (p q u_{q q} - p u_{q} - p^{2} u_{q p}) - (p u_{q} + p q u_{q q} - p^{2} u_{p q}) = - 2 p u_{q} = - 2 ρ_{J} (F) u$ , and $[ρ_{J} (E), ρ_{J} (F)] u = (q u_{q} + p q u_{q p}) - (p u_{p} + p q u_{p q}) = q u_{q} - p u_{p} = ρ_{J} (H) u$ . These are exactly the three matrix brackets from step 1.2; bilinearity extends them from basis elements to all of $g$ .

2.2givenA1F5step 1.4algebra

For $u = p$ and $v = q + p$ , $ρ_{J} (E) (u v) = q \partial_{p} (p (q + p)) = q (q + 2 p) = q^{2} + 2 p q$ , while $ρ_{J} (E) (u) v + u ρ_{J} (E) (v) = q (q + p) + p q = q^{2} + 2 p q$ . Thus the product rule is directly verified on this nontrivial pair.

3.1givenA1F5F7step 1.4algebra∎

For $u = q^{2} / 2$ and $v = p^{2} / 2$ , the coordinate bracket gives ${u, v} = q p$ , so $ρ_{J} (H) {u, v} = 0$ . Also $ρ_{J} (H) u = q^{2}$ and $ρ_{J} (H) v = - p^{2}$ , whence ${ρ_{J} (H) u, v} + {u, ρ_{J} (H) v} = {q^{2}, p^{2} / 2} + {q^{2} / 2, - p^{2}} = 2 q p - 2 q p = 0$ . This checks bracket preservation directly on a nonconstant pair, while [F7] establishes it for all smooth functions.

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