In differential geometry, a Lie group action is a group action adapted to the smooth setting: G {\displaystyle G} is a Lie group, M {\displaystyle M} is a smooth manifold, and the action map is differentiable.
Definition Let σ : G × M → M , ( g , x ) ↦ g ⋅ x {\displaystyle \sigma :G\times M\to M,(g,x)\mapsto g\cdot x} be a (left) group action of a Lie group G {\displaystyle G} on a smooth manifold M {\displaystyle M} ; it is called a Lie group action (or smooth action) if the map σ {\displaystyle \sigma } is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} . A smooth manifold endowed with a Lie group action is also called a G {\displaystyle G} -manifold.
Properties The fact that the action map σ {\displaystyle \sigma } is smooth has a couple of immediate consequences:
the stabilizers G x ⊆ G {\displaystyle G_{x}\subseteq G} of the group action are closed, thus are Lie subgroups of G {\displaystyle G}
the orbits G ⋅ x ⊆ M {\displaystyle G\cdot x\subseteq M} of the group action are immersed submanifolds. Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.
Examples For every Lie group G {\displaystyle G} , the following are Lie group actions:
the trivial action of G {\displaystyle G} on any manifold; the action of G {\displaystyle G} on itself by left multiplication, right multiplication or conjugation; the action of any Lie subgroup H ⊆ G {\displaystyle H\subseteq G} on G {\displaystyle G} by left multiplication, right multiplication or conjugation; the adjoint action of G {\displaystyle G} on its Lie algebra g {\displaystyle {\mathfrak {g}}} . Other examples of Lie group actions include:
the action of R {\displaystyle \mathbb {R} } on M {\displaystyle M} given by the flow of any complete vector field; the actions of the general linear group GL ( n , R ) {\displaystyle \operatorname {GL} (n,\mathbb {R} )} and of its Lie subgroups G ⊆ GL ( n , R ) {\displaystyle G\subseteq \operatorname {GL} (n,\mathbb {R} )} on R n {\displaystyle \mathbb {R} ^{n}} by matrix multiplication; more generally, any Lie group representation on a vector space; any Hamiltonian group action on a symplectic manifold; the transitive action underlying any homogeneous space; more generally, the group action underlying any principal bundle.
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