Let the Lie group $G$ act on the smooth manifold $X$ with the map $(g,x)\to gx$. In any point $x\in X$, the differential of this map induces a linear map: $$ \mu:T_e G \to T_xX\;, $$ and globally, if $\mathfrak{g}$ is the Lie algebra of $G$, a map $\mathfrak{g}\to \mathfrak{X}(X)$. (If I'm not mistaken, this should be even a Lie algebra homomorphism.) If $X$ is symplectic and $G$ acts by symplectomorphisms, this map is known as the moment map, because it gives the momenta of physics.

What is its name in general? And where can I find information about it?


EDIT (see Mike Miller's comment below):

Note that a linear map $\mathfrak{g}\to T_xX$ is (basically) the same as an element of $\mathfrak{g}^*\otimes T_xX$, which is (basically) the same as a map $T^*_xX\to \mathfrak{g}^*$.

In the case of symplectic manifolds, the map $m:T^*_xX\to \mathfrak{g}^*$ is composed with the symplectic gradient $gr:f\mapsto(df)^\sharp$, to give: $$ gr^*m: X\to T^*X\to \mathfrak{g}^*\;, $$ and this is called the moment map (Souriau, "Structure of Dynamical Systems"). But this works only for symplectic manifolds, because in general there is no canonical way to map functions $\mathfrak{g}\to \mathfrak{X}(X)$ into maps $X\to \mathfrak{g}^*$. The first one is just the differential of the group action, and so it is naturally defined in general.

  • $\begingroup$ 1) You mean $\mathfrak g \to \mathfrak X(M)$, the latter being the space of vector fields, yes? Otherwise I don't know what you mean. 2) As far as I know, the moment map is a map $M \to \mathfrak g^*$, and only exists for Hamiltonian actions. $\endgroup$ – user98602 Dec 9 '15 at 17:22
  • $\begingroup$ @MikeMiller Oh yes, corrected. $\endgroup$ – geodude Dec 9 '15 at 17:31
  • $\begingroup$ Can you address 2), maybe with a reference to somewhere this is called the moment map? $\endgroup$ – user98602 Dec 9 '15 at 17:33
  • $\begingroup$ @MikeMiller yes, edited. $\endgroup$ – geodude Dec 9 '15 at 17:48
  • $\begingroup$ Great, thanks. I don't see people talking much about the Lie algebra homomorphism you mention. The only place I've seen it is in Ch 1 of Kobayashi-Nomizu. I don't think it'll say anything satisfying there but it might be worth a look. $\endgroup$ – user98602 Dec 9 '15 at 17:50

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