LeftInvariance

A left-invariant vector field on a Lie group ( G ) is a vector field that remains invariant under the action of left multiplication by any element of the group.

Here's a more detailed explanation:

1. Lie Group and Left Multiplication: Let ( G ) be a Lie group. For each element ( g \in G ), left multiplication by ( g ) is a map ( L_g : G \to G ) defined by ( L_g(h) = gh ) for any ( h \in G ).

2. Vector Field: A vector field on ( G ) is a smooth assignment of a tangent vector to each point in ( G ). Formally, a vector field ( X ) assigns to each point ( g \in G ) a tangent vector ( X(g) \in T_gG ), where ( T_gG ) is the tangent space of ( G ) at ( g ).

3. Left-Invariant Vector Field: A vector field ( X ) on ( G ) is called left-invariant if for every ( g, h \in G ), the following condition holds: [ (L_g)* X(h) = X(gh) ] Here, ( (L_g)* ) is the differential (or pushforward) of the map ( L_g ). This condition means that the vector field ( X ) is invariant under the action of left multiplication by any element of ( G ).

In other words, if you have a vector field ( X ) that is left-invariant, then for any ( g \in G ), the vector at ( h ) is related to the vector at ( gh ) by the left multiplication map. This invariance property makes left-invariant vector fields crucial in studying the structure of Lie groups and their corresponding Lie algebras.