Chapter 5 The Lie Algebra of a Matrix Lie Group

Now we going to discuss the Lie Algebra \(\mathfrak{g}\) that associated with Matrix Lie group \(G\).

Definition 5.1 If \(G \subset GL(n, \mathbb{C})\) is a matrix Lie group, then the Lie algebra \(\mathfrak{g}\) of \(G\) is defined as follows: \[ \mathfrak{g} = \left\{ X \in M_n(\mathbb{C}) \;\middle|\; e^{tX} \in G \;\; \text{for all } t \in \mathbb{R} \right\}. \]

Proposition 5.1 For any matrix Lie group \(G\), the Lie algebra \(\mathfrak{g}\) of \(G\) has the following properties:

• The zero matrix \(0\) belongs to \(\mathfrak{g}\).
• For all \(X \in \mathfrak{g}\), \(tX \in \mathfrak{g}\) for all real numbers \(t\).
• For all \(X, Y \in \mathfrak{g}\), \(X + Y \in \mathfrak{g}\).
• For all \(A \in G\) and \(X \in \mathfrak{g}\) we have \(AXA^{-1} \in \mathfrak{g}\).
• For all \(X, Y \in \mathfrak{g}\), the commutator \([X, Y] := XY - YX\) belongs to \(\mathfrak{g}\).

Proof. Write the proof here

Remark. Write about real vector spaces.