Knowlpedia

Definition

Comoment map

A Lie algebra homomorphism assigning to each infinitesimal symmetry its Hamiltonian function.

Definition

Let GG act on (M,ω)(M,\omega) by a , with g\mathfrak g, and use ιXfω=df\iota_{X_f}\omega=df. A comoment map is a

c:g⟶C∞(M),ξ⟼c(ξ),c:\mathfrak g\longrightarrow C^\infty(M),\qquad \xi\longmapsto c(\xi),

such that dc(ξ)=ιξMωdc(\xi)=\iota_{\xi_M}\omega and

c([ξ,η])={c(ξ),c(η)}c([\xi,\eta])=\{c(\xi),c(\eta)\}

for all ξ,η∈g\xi,\eta\in\mathfrak g. Thus cc is a into the , and each c(ξ)c(\xi) generates the associated with ξ\xi. The differential axiom identifies the generating ; the bracket axiom requires the chosen Hamiltonians to respect the Lie algebra exactly, rather than only up to constants.

Relationship to moment maps

A μ:M→g∗\mu:M\to\mathfrak g^* determines the linear assignment

cμ(ξ)=μξ=⟨μ,ξ⟩.c_\mu(\xi)=\mu^\xi=\langle\mu,\xi\rangle.

The differential condition for cμc_\mu is exactly the componentwise moment-map identity. Under the conventions in the core, the bracket condition is infinitesimal equivariance of μ\mu; for connected GG, it is equivalent to global coadjoint equivariance Ortega and Ratiu, §4.2.

Conversely, a comoment map defines μ\mu uniquely by ⟨μ(x),ξ⟩=c(ξ)(x)\langle\mu(x),\xi\rangle=c(\xi)(x), because cc is linear in ξ\xi.

Obstruction to bracket preservation

If a linear choice of Hamiltonians satisfies only the differential condition, then

σ(ξ,η)={c(ξ),c(η)}−c([ξ,η])\sigma(\xi,\eta) =\{c(\xi),c(\eta)\}-c([\xi,\eta])

is locally constant on MM. On a connected manifold it is a real-valued Lie-algebra 22-cocycle. Changing cc by constants changes σ\sigma by a coboundary, so the resulting cohomology class measures the obstruction to converting the weak assignment into a comoment map.

Conventions and scope
References
  1. Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. DOI record. Relevant: Chapter 3, infinitesimal Hamiltonians and moment maps.
  2. Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. DOI record. Relevant: §4.2, infinitesimal equivariance and the cocycle obstruction.