Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Consider {$\mathrm{SU}(n)$}:
Consider the simplest case: {$\mathrm{SU}(2)$}
Exercises
{$ \text{Lie}(G) = \{X\in M(n;\mathbb{C})e^{tX}\in G \text{ for all } t \in \mathbb{R}\}$} The complexification of a compact real group can be realized concretely as a closed subgroup of the complex general linear group. It consists of linear transformations with polar decomposition {$g = u \cdot e^{iX}$}, where {$u=e^iH$} is a unitary matrix in the compact group, {$iH$} is a skewadjoint matrix, and {$X$} is a skewadjoint matrix (with purely imaginary eigenvalues) in the compact group's Lie algebra. The matrices {$H$} and {$iX$} are selfadjoint matrices with real eigenvalues. The term {$e^{iX}$} is a positivedefinite Hermitian matrix, which means that its eigenvalues are all positive real numbers. {$\begin{matrix} \text{complex Lie group} & \text{compact Lie group} & \text{Lie algebra}\\ u \cdot e^{iX} & \text{unitary: } u & \text{skewadjoint: } X \\ \mathrm{SL}(n,\mathbb{C}) & \mathrm{SU}(n) & \mathfrak{su}(n) \\ \mathrm{SO}(n,\mathbb{C}) & \mathrm{SO}(n) & \mathfrak{so}(n)\\ \mathrm{Sp}(n,\mathbb{C}) & \mathrm{Sp}(n) & \mathfrak{sp}(n) \end{matrix}$} Properties of compact Lie groups If {$G$} is a connected Lie group with finite center. Then: {$G$} is a compact Lie group {$\iff$} there exists an invariant inner product on {$\frak{g}$} {$\iff$} there exists an embedding into some {$O(n,\mathbb{R})$}. Readings
