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. These vector fields play a crucial role in understanding the structure of Lie groups and their corresponding Lie algebras.

Here's a 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 $G$.

4. Correspondence with Lie Algebra: Left-invariant vector fields on a Lie group G are in one-to-one correspondence with elements of the Lie algebra g of G. Each left-invariant vector field is uniquely determined by its value at the identity element e of G.

5. Generating the Lie Algebra: The set of all left-invariant vector fields on G forms a vector space that is isomorphic to the Lie algebra g. This isomorphism provides a concrete realization of the abstract Lie algebra in terms of geometric objects (vector fields) on the group manifold.

6. Flow and One-Parameter Subgroups: The flow of a left-invariant vector field generates a one-parameter subgroup of G. This connection is crucial in understanding the exponential map from the Lie algebra to the Lie group.

7. Algebraic Properties: The Lie bracket of two left-invariant vector fields is again a left-invariant vector field. This property allows us to define the Lie bracket operation on the Lie algebra g in terms of vector fields on G.

8. Coordinate-Free Description: Left-invariant vector fields provide a coordinate-free way to describe the structure of a Lie group. This is particularly useful when working with abstract or infinite-dimensional Lie groups.

9. Relation to Right-Invariant Vector Fields: While we've focused on left-invariant fields, one can similarly define right-invariant vector fields. The relationship between left- and right-invariant fields often reveals important symmetries of the Lie group.

In summary, left-invariant vector fields are fundamental objects in the study of Lie groups, providing a bridge between the local structure (Lie algebra) and the global structure (Lie group) of these mathematical entities.