Momentum in Diffeology

General Idea

An intrinsic way of looking at the moment map

Given a presymplectic manifold $(M,\omega)$ where $\omega$ is also exact so $\omega = d\lambda$, and a smooth action $\rho$ of a Lie group $G$, s.t. $\rho^\ast_g \omega = \omega$ and $\rho^\ast_g\lambda = \lambda$ for $g\in G$. For every point $x\in M$, the 1-form $\lambda$ can be pulled back by the orbit map ${}^x\hat{\rho}: G\to M:g \mapsto \rho_g (x)$ and yields a left-invariant 1-form of $G$, which is in the dual $\mathfrak{g}^\ast$ of the Lie algebra $\mathfrak{g}$.

The map $\mu: M\to \mathfrak{g}^\ast: x\mapsto {}^x\hat{\rho}^*(\lambda)$ is exactly the moment map of the action of $G$ on $(M,\omega)$.

The construction doesn’t actually involve the Lie algebra. This construction can be generalized to diffeology setting by simply changing the nouns. Now call $\mathfrak{g}^\ast$, the space of left-invariant 1-forms on a diffeological group $G$, as the space of momenta of $G$. The group $G$ still acts on $\mathfrak{g}^\ast$ by the coadjoint action.