Momentum mapping

The momentum mapping is essentially due to S. Lie, [a5], pp. 300–343. The modern notion is due to B. Kostant [a3], J.M. Souriau [a9] and A.A. Kirillov [a2].

The setting for the momentum mapping is a smooth symplectic manifold $( M , \omega )$ or even a Poisson manifold $( M , P )$ (cf. also Poisson algebra; Symplectic structure) with the Poisson brackets on functions $\{ f , g \} = P ( d f , d g )$ (where $P = \omega ^ { - 1 } : T ^ { * } M \rightarrow T M$ is the Poisson tensor). To each function $f$ there is the associated Hamiltonian vector field $H _ { f } = P ( d f ) \in \mathfrak{X} ( M , P )$, where $\mathfrak { X } ( M , P )$ is the Lie algebra of all locally Hamiltonian vector fields $Y \in \mathfrak { X } ( M )$ satisfying $\mathcal{L} _ { Y } P = 0$ for the Lie derivative.

The Hamiltonian vector field mapping can be subsumed into the following exact sequence of Lie algebra homomorphisms:

\begin{equation*} 0 \rightarrow H ^ { 0 } ( M ) \rightarrow C ^ { \infty } ( M ) \stackrel { H } { \rightarrow } \mathfrak{X} ( M , \omega ) \stackrel { \gamma } { \rightarrow } H ^ { 1 } ( M ) \rightarrow 0, \end{equation*}

where $\gamma ( Y ) = [ i_{ Y } \omega ]$, the de Rham cohomology class of the contraction of $Y$ into $\omega$, and where the brackets not yet mentioned are all $0$.

A Lie group $G$ can act from the right on $M$ by $\alpha : M \times G \rightarrow M$ in a way which respects $\omega$, so that one obtains a homomorphism $\alpha ^ { \prime } : \mathfrak { g } \rightarrow \mathfrak { X } ( M , \omega )$, where $\frak g$ is the Lie algebra of $G$. (For a left action one gets an anti-homomorphism of Lie algebras.) One can lift $\alpha ^ { \prime }$ to a linear mapping $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ if $\gamma \circ \alpha ^ { \prime } = 0$; if not, one replaces $\frak g$ by its Lie subalgebra $\operatorname { ker } ( \gamma \circ \alpha ^ { \prime } ) \subset \mathfrak { g }$. The question is whether one can change $j$ into a homomorphism of Lie algebras. The mapping $\mathfrak { g } \ni X , Y \mapsto \{ j X , j Y \} - j ( [ X , Y ] )$ then induces a Chevalley $2$-cocycle in $H ^ { 2 } ( \mathfrak { g } , H ^ { 0 } ( M ) )$. If it vanishes one can change $j$ as desired. If not, the cocycle describes a central extension of $\frak g$ on which one may change $j$ to a homomorphism of Lie algebras.

In any case, even for a Poisson manifold, for a homomorphism of Lie algebras $j : \mathfrak { g } \rightarrow C ^ { \infty } ( M )$ (or more generally, if $j$ is just a linear mapping), by flipping coordinates one gets a momentum mapping $J$ of the $\frak g$-action $\alpha ^ { \prime }$ from $M$ into the dual $\mathfrak{g} ^ { * }$ of the Lie algebra $\frak g$,

\begin{equation*} J : M \rightarrow \mathfrak { g } ^ { * }, \end{equation*}

\begin{equation*} \langle J ( x ) , X \rangle = j ( X ) ( x ) , H _ { j ( X ) }= \alpha ^ { \prime } ( X ), \end{equation*}

\begin{equation*} x \in M , X \in \mathfrak { g }, \end{equation*}

where $\langle \cdot,\cdot\rangle$ is the duality pairing.

For a particle in Euclidean $3$-space and the rotation group acting on $T ^ { * } \mathbf{R} ^ { 3 }$, this is just the angular momentum, hence its name. The momentum mapping is infinitesimally equivariant for the $\frak g$-actions if $j$ is a homomorphism of Lie algebras. It is a Poisson morphism for the canonical Poisson structure on $\mathfrak{g} ^ { * }$, whose symplectic leaves are the co-adjoint orbits. The momentum mapping can be used to reduce the number of coordinates of the original mechanical problem, hence it plays an important role in the theory of reductions of Hamiltonian systems.

[a6], [a4] and [a7] are convenient references; [a7] has a large and updated bibliography. The momentum mapping has a strong tendency to have a convex image, and is important for representation theory, see [a2] and [a8]. There is also a recent (1998) proposal for a group-valued momentum mapping, see [a1].

References

[a1]	A. Alekseev, A. Malkin, E. Meinrenken, "Lie group valued moment maps" J. Diff. Geom. , 48 (1998) pp. 445–495
[a2]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976)
[a3]	B. Kostant, "Orbits, symplectic structures, and representation theory" , Proc. United States–Japan Sem. Diff. Geom. , Nippon Hyoronsha (1966) pp. 71
[a4]	P. Libermann, C.M. Marle, "Symplectic geometry and analytic mechanics" , Reidel (1987)
[a5]	S. Lie, "Theorie der Transformationsgruppen, Zweiter Abschnitt" , Teubner (1890)
[a6]	G. Marmo, E. Saletan, A. Simoni, B. Vitale, "Dynamical systems. A differential geometric approach to symmetry and reduction" , Wiley/Interscience (1985)
[a7]	J. Marsden, T. Ratiu, "Introduction to mechanics and symmetry" , Springer (1999) (Edition: Second)
[a8]	K.-H. Neeb, "Holomorphy and convexity in Lie theory" , de Gruyter (1999)
[a9]	J.M. Souriau, "Quantification géométrique" Commun. Math. Phys. , 1 (1966) pp. 374–398