Lie algebras of GL(n,R) and differentials

Question

This question comes from a proof in John Lee's Introduction to Smooth Manifolds, page 194. I am questioning a line in the proof of the following proposition:

The composition of the maps

$\text{Lie}(GL(n,\mathbb{R}))\rightarrow T_{I}(GL(n,\mathbb{R}))\rightarrow\mathfrak{gl}(n,\mathbb{R})$

gives a Lie algebra isomorphism between $\text{Lie}(GL(n,\mathbb{R}))$ and the matrix algebra $\mathfrak{gl}(n,\mathbb{R})$

He uses the standard coordinates, $X^i_j$, on $GL(n,\mathbb{R})\subset \mathfrak{gl}(n,\mathbb{R})$. As I understand it, these $n^2$ coordinate functions take $A\in GL(n,\mathbb{R})$ to the $ij$-th entry of the matrix representation of $A$.

He writes any $A=(A^i_j)\in \mathfrak{gl}(n,\mathbb{R})$ determines a left-invariant vector field $A^l\in \mathfrak{g}$, given by

$A^L|_X=(dL_X)_I (A)=(dL_X)_I\left(A^i_j\frac{\partial}{\partial X^i_j}\bigg|_I\right)$.

This is fine. It is his next few lines which confuse me. He says

Since $L_X$ is the restriction to $GL(n,\mathbb{R})$ of the linear map $A\mapsto XA$ on $\mathfrak{gl}(n,\mathbb{R})$, its differential is represented in coordinates by exactly the same linear map. In other words, the left-invariant vector field $A^L$ determined by $A$ is the one whose value at $X\in GL(n,\mathbb{R})$ is

$ \begin{align}A^L|_X=X^i_j A^j_k \frac{\partial}{\partial X^i_k}\bigg|_X\end{align}$.

I understand the fact that the matrix representation of the differential of a linear map is just the matrix representation of the linear map itself. And I do believe this is essentially what is going on here. The reason I am confused, is because I feel he is using two different meanings for the same notation $X^{\alpha}_{\beta}$.

As I understand it, the $X^i_k$ in the $\partial/\partial X^i_k$ are the global coordinates defined on $GL(n,\mathbb{R})$. Whereas, I feel that the $X^i_j$ in the coefficient of each basis vector is the $ij$-th entry of the matrix representation of $X$. Wouldn't it be more appropriate to write,

$A^L|_Y = (dL_y)_I(A)=Y^i_jA^j_k\frac{\partial}{\partial X^I_k}\bigg|_Y$, where $Y^i_j$ is the $ij$-th entry of the matrix representation of $Y\in GL(n,\mathbb{R})$?

I don't know if I am being stupid or pedantic. Probably both. But if someone is able to clear this up for me it would be much appreciated!

Thanks.

Answer 1

The computation of $[A^L, B^L]$ that follows on the same page shows that the choice of notation is intentional, as Lee uses the fact that $\partial X^p_q/\partial X^i_k$ is equal to $1$ whenever $p=i$ and $q=k$, and is equal to $0$ otherwise. For better or worse, differential geometry is full of (mostly) harmless abuses of notation like this.

It’s worth thinking about Lee’s remarks on page 63, where he discusses change of coordinates:

Here we are indulging in a typical abuse of notation: in the expression $\tilde x^i(x)$, we think of $\tilde x^i$ as a coordinate function (whose domain is an open subset of $M$, identified with an open subset of ${\bf R}^n$ or ${\bf H}^n$); but we think of $x$ as representing a point (in this case, in $\varphi(U\cap V)$).

(For reference, other treatments of this computation can be found in Loring Tu’s An Introduction to Manifolds, page 184, or Michael Spivak’s A Comprehensive Introduction to Differential Geometry, volume 1, page 377.)