Lie Algebras of Matrix Groups

Let \(G \leq \mathrm{GL}(n;\mathbb{R})\) be a Lie group. Denote the Lie algebras of \(G\) and \(\mathrm{GL}(n;\mathbb{R})\) by \(\mathfrak{g}\) and \(\mathfrak{gl}(n;\mathbb{R})\) respectively. We usually define the Lie algebra \(\mathfrak{g}\) to be the space of left \(G\)-invariant vector fields on \(G\), ie. the set of vector fields \(\mathcal{X}\) such that, for all \(g,a\in G\) we have

\begin{equation} \mathcal{X}_{g\cdot a} = (L_g)_\ast \mathcal{X}_a \end{equation}

Here the diffeomorphism \(L_g \in \mathrm{Diff}(G)\) is left multiplication by \(g\). It is well known that \(\mathfrak{g}\) is isomorphic to the tangent space \(T_{\mathrm{id}}G\) and these two spaces are usually identified. In this note we prove a further equivalence of the form

\begin{equation*} \mathfrak{g} \cong \{ A \in \mathrm{Mat}(n; \mathbb{R}) ~|~ \exp(tA) \in G \text{ for all } t \in \mathbb{R}\} \end{equation*}

which holds for matrix groups like \(G\).

Let us first consider the manifold \(\mathrm{Mat}(n;\mathbb{R})\) with coordinates \((x_{ij})\). The corresponding derivations \(\{\frac{\partial}{\partial x^{ij}}\}\) span the tangent space \(T_{\mathrm{id}}\mathrm{Mat}(n;\mathbb{R})\), which we can therefore think of as another copy of \(\mathrm{Mat}(n;\mathbb{R})\). Since \(\mathfrak{g}\) is a subspace of \(T_{\mathrm{id}}\mathrm{Mat}(n;\mathbb{R})\), this gives us a way to identify elements of \(\mathfrak{g}\) with \(n\times n\)-matrices.

Exploiting the identification above, suppose we are given some \(n\times n\) matrix \(A\) representing a vector in \(\mathfrak{g}\). Viewing \(\mathfrak{g}\) as the space of left invariant vector fields on \(G\), \(A\) corresponds to the matrix differential equation

\begin{equation*} \frac{\mathrm{d}X}{\mathrm{d}t} = A X \end{equation*}

which has as solution the matrix exponential \(t \mapsto \exp(tA)\). Since \(\exp(tA)\) is a flow on the manifold \(G\), it follows that \(\exp(tA) \in G\) for all \(t \in \mathbb{R}\). Conversely, suppose we are given a matrix \(A\) such that \(\exp(tA) \in G\) for all real \(t\). By the uniqueness theorem for systems of differential equations, \(A\) must be the coordinate representation of a vector field tangent to the manifold \(G\). In other words, \(A \in \mathfrak{g}\).