# Why do we need left invariant vector fields?

Here Wikipedia Says

$$1-$$Vector fields on any smooth manifold $$M$$ can be thought of as derivations $$X$$ of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket $$[X, Y] = XY − YX,$$ because the Lie bracket of any two derivations is a derivation.

$$2-$$We apply this construction to the case when the manifold M is the underlying space of a Lie group $$G$$, with $$G$$ acting on $$G = M$$ by left translations $$L_g(h) = gh$$. This shows that the space of left invariant vector fields on a Lie group is a Lie algebra under the Lie bracket of vector fields.

As we can see that for any smooth manifold $$M$$, the vector fields for a Lie Algebra under the Lie bracket.

When we are considering a smooth manifold $$M$$ which is a group as well(that is it is a lie group) why are we suddenly interested in the left invariant vector fields instead of all the vector fields?

Is the following true? The collection of all left invariant vector fields is a sub Lie Algebra of the collection of vector fields?

• The answer to this really involve as least a whole introductory book on Lie group and Lie algebra. The short and boring answer is that the Lie algebra of a Lie group is finite dimensional, which is easier to work with. Commented Jun 7, 2021 at 11:46
• @ArcticChar: Your comment does not mention left-invariance, the topic of this question. Commented Jun 7, 2021 at 11:53
• In my comment, the Lie algebra of a Lie group refers to the set of all left-invariant vector fields. (is that what you are asking?) @WillR Commented Jun 7, 2021 at 11:55
• @WillR, It is the way the lie algebra of lie group is defined. So it was understandable. Commented Jun 7, 2021 at 12:32
• One thing that makes left-invariant vector fields better than the space of all vector fields is that they are naturally identified with the tangent space at the identity. This identification is mutually beneficial: on one hand, it turns the tangent space at the identity into a Lie algebra, on the other hand, it makes this construction functorial (a Lie group homomorphism induces a Lie algebra homomorphism between the tangent spaces). Neither of these properties is that clear from the other perspective and this interplay is the starting point of the relevance of the Lie algebra to Lie groups. Commented Jun 7, 2021 at 22:16

For instance: Suppose that $$G_1, G_2$$ are simply-connected Lie groups and $$\phi: {\mathfrak g}_1\to {\mathfrak g}_2$$ is an isomorphism of their Lie algebras. Then there exists an isomorphism of Lie groups $$f: G_1\to G_2$$ such that $$df_e=\phi$$.
Another useful property of the Lie algebra of left-invariant vector fields is that a basis for the latter always forms a global frame for the tangent bundle of the Lie group. While the space of all vector fields is quite large, it still has a "basis" such that any $$X\in\mathfrak{X}(G)$$ is a finite linear combination $$X=f_1\cdot X_1+\dots+f_n\cdot X_n$$ where $$X_1,\dots,X_n$$ is a basis of the Lie algebra and $$f_1,\dots,f_n\in\mathcal{C}^\infty(G)$$. So it shows how the tangent bundle of a Lie group is trivializable.