🧠📚 Mitch's Mind Tangle

Search

SearchSearch

Lie algebra action

Mar 19, 20252 min read

Definition

Given a Lie algebra , and a smooth manifold , a Lie algebra action of on is a Lie algebra morphism

where denotes the space of all vector fields on .

From Lie groups

Every Lie group action of on induces a Lie algebra action defined by

these are the fundamental vector fields on a manifold.

If an infinitesimal action is induced by an action of on then the vector fields are complete. So, not every Lie algebra action comes from a Lie group action, but we do have a connection between some:

Theorem (Lie-Palais): Given a manifold and a 1-connected Lie group with Lie algebra , given the equation above for induced Lie algebra actions, we have a 1-1 correspondence

Missing \begin{Bmatrix} or extra \end{Bmatrix}\alpha: \mathfrak{g} \to \mathfrak{X}(M)\end{Bmatrix} \leftrightarrow \begin{Bmatrix} \text{Lie group actions} \\ \mathcal{A}: G \to \text{Diff}(M)\end{Bmatrix}$$

Graph View

Backlinks

  • No backlinks found