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Now let [itex]a[/itex] and [itex]g[/itex] be elements of a Lie group [itex]G[/itex], the left translation [itex]L_{a}: G \rightarrow G[/itex] of [itex]g[/itex] by [itex]a[/itex] are defined by :

[itex]L_{a}g=ag[/itex]

which induces a map [itex]L_{a*}: T_{g}G \rightarrow T_{ag}G[/itex]

Let [itex]X[/itex] be vector field on a Lie group [itex]G[/itex]. [itex]X[/itex] is said to be a left invariant vector field if [itex]L_{a*}X|_{g}=X|_{ag}[/itex]. A vector [itex]V \in T_{e}G[/itex] defines a unique left-invariant vector field [itex]X_{V}[/itex] throughout [itex]G[/itex] by:

[itex]X_{V}|_{g}= L_{g*}V[/itex], [itex]g \in G[/itex]

Now the author gives an example of the left invariant vector field of [itex]GL(n,\mathbb{R})[/itex]:

Let [itex]g={x^{ij}(g)}[/itex] and [itex]a={x^{ij}(a)}[/itex] be elements of [itex]GL(n,\mathbb{R})[/itex] where [itex]e= I_{n}=\delta^{ij}[/itex] is the unit element. The left translation is:

[itex]L_{a}g=ag=\Sigma x^{ik}(a)x^{kj}(g)[/itex]

Now take a vector [itex]V=\Sigma V^{ij}\frac{\partial}{\partial x^{ij}}|_{e} \in T_{e}G[/itex] where the [itex]V^{ij}[/itex] are the entries of [itex]V[/itex]. The left invariant vector field generated by [itex]V[/itex] is:

[itex]X_{V|_{g}}=L_{g*}V=\Sigma V^{ij}\frac{\partial}{\partial x^{ij}}|_{e}x^{kl}(g)x^{lm}(e) \frac{\partial}{x^{km}}|_{g}= \Sigma V^{ij}x^{kl}(g) \delta^{l}_{i} \delta^{m}_{j} \frac{\partial}{\partial x^{km}}|_{g}= \Sigma x^{ki}(g)V^{ij} \frac{\partial}{\partial x^{kj}}|_{g}= \Sigma (gV)^{kj} \frac{\partial}{\partial x^{kj}}|_g[/itex]

Where [itex]gV[/itex] is the usual matrix multiplication.

This is a bit over my head. What does it mean that one has a tangent vector at the unit element of a Lie group? Maybe solving this exercise may help with this question:

Let

[itex]c(s)=\begin{pmatrix} cos s & -sin s & 0 \\ sin s & cos s & 0 \\ 0 & 0 & 1 \end{pmatrix}[/itex]

be a curve in [itex]SO(3)[/itex]. Find the tangent vector to this curve at [itex]I_{3}[/itex].

And why does this induce a left invariant vector field? And btw, what is a left invariant vector field?? What does it mean geometrically? And what does it mean if a vector [itex]V^{ij}[/itex] has two indices?? Can one explain the example to me?