On Lie algebra actions (Q2180360)

This paper defines a Lie algebra action on a smooth manifold and explores some of its consequences. Although specific examples are already well known (for example, the \(\operatorname{so}(4)\) symmetry arising from the Kepler problem of classical mechanics), the authors look at the question more broadly and consider the relationships between actions of a Lie algebra on a manifold and the corresponding actions of the associated Lie group. The general setting is as follows. Take \(G\) to be a connected Lie group with Lie algebra \(\mathfrak{g}\) and take \(M\) to be a smooth Hausdorff manifold with \((\mathcal{X} (M), - [ \; , \;])\) the Lie algebra of smooth vector fields on \(M\). A domain \(D\) is an open subset of \(G \times M\) such that for each \(p \in M\) the set \(D_p = \{g \in G: (g,p) \in D \}\) is a connected open neighborhood of the identity in \(G\). A local action of \(G\) on \(M\) is a smooth mapping \(\Phi\) from \(D\) to \(M\) that satisfies the expected group-type properties. The infinitesimal generator of a local \(G\)-action \(\Phi\) is a mapping \(\varphi\) from \(\mathfrak{g}\) to \(\mathcal{X}(M)\) sending \(\xi\) to \(\varphi(\xi) = X_\xi\) where \(X_\xi (p) = T_e\Phi_p \xi\) for every \(p \in M\) and every \(\xi \in \mathfrak{g}\). A Lie algebra action of \(\mathfrak{g}\) on \(M\) is a homomorphism of the Lie algebra (\(\mathfrak{g}, [\; , \;] )\) into the Lie algebra \((\mathcal{X}(M), - [\; , \;])\). The infinitesimal generator \(\varphi\) of the local \(G\)-action \(\Phi\) on \(M\) with domain \(D\) is then a \(\mathfrak{g}\)-action on \(M\). If \(\varphi\) is a \(\mathfrak{g}\)-action on \(M\), then for each \(p \in M\) the orbit \(O_p\) of \(\varphi\) through \(p\) is the orbit of the family \(X_\mathfrak{g} = \{\varphi(\xi) \in \mathcal{X}(M) : \xi \in \mathfrak{g}\}\) of vector fields on M through \(p\). A \(\mathfrak{g}\)-action \(\varphi\) is called proper if the local \(G\)-action \(\Phi\) on \(M\) with domain \(D\) generated by \(\varphi\) is proper. One of the authors' main results is that when a proper \(\mathfrak{g}\) action on \(M\) generates a local \(G\)-action with domain \(D\), then at each point \(p\) in \(M\), the \(G\)-orbit is equal to the \(\mathcal{X}_\mathfrak{g}\) orbit \(O_p\). Another main result is that if \(\varphi\) is a proper \(\mathfrak{g}\)-action on \(M\), then the \(\mathfrak{g}\)-orbit space \(M/X_\mathfrak{g}\) arising from the projection mapping of \(M\) to \(M/X_\mathfrak{g}\) that takes \(p\) in \(M\) to \(O_p\) is a locally Euclidean differential space. So every point in \(M/X_\mathfrak{g}\) has an open neighborhood diffeomorphic to a subset of Euclidean space. The authors also look at a specific application that occurs in the reduction of symmetries for integrable Hamiltonian systems.