LieAlgebra

The construction of a Lie algebra from a Lie group via left-invariant vector fields goes like this:

1. Lie Group and Lie Algebra: A Lie group is a group that is also a differentiable manifold, meaning that the group operations (multiplication and inversion) are smooth. The Lie algebra associated with a Lie group is a vector space equipped with a bracket operation that encodes the structure of the Lie group near the identity element.

2. Left-Invariant Vector Fields: A vector field on a Lie group is called left-invariant if it is invariant under the action of left multiplication by any element of the group. Mathematically, if ( g \in G ) is an element of the Lie group and ( X ) is a vector field, then ( X ) is left-invariant if ( L_g^* X = X ), where ( L_g ) denotes left multiplication by ( g ).

3. Bracket Operation: The Lie bracket of two left-invariant vector fields is itself a left-invariant vector field. This bracket operation on the space of left-invariant vector fields gives the Lie algebra structure.

4. Differential Geometric Perspective: From a differential geometric viewpoint, the Lie bracket on the Lie algebra can be derived from the commutator of the corresponding left-invariant vector fields. This is done by differentiating the group multiplication and utilizing the properties of the left-invariant vector fields.

The process you're describing involves taking the differential geometric structure of the Lie group and translating it into the algebraic structure of the Lie algebra through the use of left-invariant vector fields. Before plugging in specific vectors, you see the abstract form of this structure. Once you plug in vectors, the abstract Lie algebra elements become concrete vectors in the tangent space at the identity, equipped with the induced Lie bracket operation.