Understanding Lie algebra of matrix Lie group

differential-geometrylie-algebraslie-groupstangent-spaces

In my lecture, we gave a very sloppy (physics people …) proof of the fact that the Lie algebra $\mathfrak{g}$ of a matrix Lie group $G$ is a subspace of $\text{Mat}_n(\mathbb{F})$.
I am not satisfied and I would love to understand this fact in the language of differential geometry that I am learning right now.

What I know:

The Lie algebra $\mathfrak{g}$ of a matrix Lie group $G$ is the set of all left-invariant vector fields on $G$. This set is isomorphic to the tangent space at the identity of the Lie group $G$. The elements $v$ of the tangent space at the identity are according to my definition of the tangent space real-valued functions
$$v:F(G)\to \mathbb{R}$$
where $F(M)$ is the set of all smooth-real valued functions on the manifold $G$. The tangent vectors are $\mathbb{R}$ linear and fulfil a Leibniz rule and the tangent space at $p$, namely $T_pG$ is simply the set of all these tangent vectors.

What I want to know

Given the above definitions, how can I formally understand that the members of the tangent space at the identity of a matrix Lie group are matrices?

Best Answer

A matrix Lie group is a closed subgroup of $\text{GL}(n,\mathbb{K})$ where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$

Now $\text{GL}(n,\mathbb{K})$ is an open subset of the vector space $\mathbb{K}^{n\times n},$ and you may use:

  1. If $V$ is a vector space of dimension $n,$ any norm in $V$ determines a topology independent of the choice of the norm, and $V$ has a standard smooth structure making it into an $n$-dimensional manifold.
  2. If $V$ is a vector space with its standard manifold structure, the tangent space at a point $p\in V$ is naturally isomorphic to $V,$ that is, there is a canonical isomorphism $\varphi_V:V \to T_pV:$ This isomorphism is such that if $L:V\to W$ is a linear map between two vector spaces then $dL_p\circ \varphi_V=\varphi_W\circ L:V\to T_{Lp}W$ where $dL_p:T_pV\to T_{Lp}W$ is the differential of $L$ at $p.$ This allow us to identify the tangent space $T_pV$ with $V$ itself.
  3. If $\iota:N\to M$ is the inclusion of a submanifold $N\subset M$ and $p\in N$ then $d\iota_p:T_pN\to T_pM$ gives a inclusion of $T_pN$ into $T_pM.$ If $N$ is an open submanifold of $M$ then $d\iota_p$ is an isomorphism, so we can identify $T_pN$ with $T_pM.$

Now let $G\subset \text{GL}(n,\mathbb{K})$ be a matrix Lie group and $I\in G$ be the identity. By 2, the tangent space $T_I(\mathbb{K}^{n\times n})$ is identified with $\mathbb{K}^{n\times n}$ itself. By 3, the tangent space $T_I(\text{GL}(n,\mathbb{K}))$ can be identified with $T_I(\mathbb{K}^{n\times n}),$ hence with $\mathbb{K}^{n\times n}.$ Hence by the first statement in 3, the tangent space $T_IG$ is a subspace of $\mathbb{K}^{n\times n}.$

Related Question