Nonlinear Equations - UFRJ

[SEC. 6.2: THE MOMENTUM MAP 75
6.2 The momentum map
Let M = C n /(2π √ −1 Z n ). Let A ⊂ Z n ≥0 ⊂ (Rn ) ∗ be finite, and let
F A = {f : x ↦→ f(x) = ∑ a∈A f ae ax }.
If we set z i = e xi , then elements of F A are actually polynomials
in z. (The roots that have a real negative coordinate z i are irrelevant
for this section).
We assume an inner product on F A of the form.
{
〈e ax , e bx ca if a = b
〉 =
0 otherwise
where the variances c a are arbitrary.
In this context,
K(x, y) = ∑ a∈A
c −1
a e a(x+ȳ) .
Notice the property that for any purely imaginary vector g, K(x+
g, y + g) = K(x, y). In particular, K i·(x, x) is always real. This
is a particular case of toric action which arises in a more general
context. Properly speaking, the n-torus (R n /2πR n , +) acts on M by
x ↦→ θ
x + iθ).
The momentum map m : M → (R n ) ∗ for this action is defined by
m x = 1 d log K(x, x) (6.1)
2
The terminology momentum arises because it corresponds to the
angular momentum of the Hamiltonian system
˙q i =
∂
∂p i
H(x)
ṗ i = − ∂
∂q i
H(x)
where x i = p i + √ −1q i and H(x) = m x · ξ. The definition for an
arbitrary action is more elaborate, see [75].
Proposition 6.7. 1. The image {m x : x ∈ M} of m is the the
interior Å of the convex hull A of A.