# Book: ComplexificationOfCompactRealForms

Understand complexification of compact real forms.

• In the compact Lie group, is every element unitary?
• In the Lie algebra of a compact Lie group, is every element a skew-adjoint operator?
• Calculate {$e^{iX}$} for various skew-Hermitian matrices {$X$}. Can I draw any conclusions? Do the Lie algebras given by those conditions for which these expressions calculate nicely?
• What is the relationship between the Lie algebra of a compact Lie group, and the complexification of that Lie algebra?
• Demonstrate that various complex Lie groups are complex manifolds.
• How do the special linear groups relate to their maximal compact subgroups?

Consider {$\mathrm{SU}(n)$}:

• How do the special linear groups {$\mathrm{SL}(n,\mathbb{C})$} relate to their maximal compact subgroups {$\mathrm{SU}(n)$}?

Consider the simplest case: {$\mathrm{SU}(2)$}

• What is the Lie algebra of {$\mathrm{SU}(2)$}?
• Relate the elements of its Lie algebra {$\mathfrak{su}(2)$} with its Dynkin diagram.
• Express {$\mathrm{SL}(2)$} as the complexification of {$\mathrm{SU}(2)$}.

Exercises

• Given a complex Lie group, calculate its Lie algebra using the formula:

{$\text{Lie}(G) = \{X\in M(n;\mathbb{C})|e^{tX}\in G \text{ for all } t \in \mathbb{R}\}$}

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 skew-adjoint matrix, and {$X$} is a skew-adjoint matrix (with purely imaginary eigenvalues) in the compact group's Lie algebra. The matrices {$H$} and {$iX$} are self-adjoint matrices with real eigenvalues. The term {$e^{iX}$} is a positive-definite Hermitian matrix, which means that its eigenvalues are all positive real numbers.

{$\begin{matrix} \text{complex Lie group} & \text{compact Lie group} & \text{Lie algebra}\\ u \cdot e^{iX} & \text{unitary: } u & \text{skew-adjoint: } X \\ \mathrm{SL}(n,\mathbb{C}) & \mathrm{SU}(n) & \mathfrak{su}(n) \\ \mathrm{SO}(n,\mathbb{C}) & \mathrm{SO}(n) & \mathfrak{so}(n)\\ \mathrm{Sp}(n,\mathbb{C}) & \mathrm{Sp}(n) & \mathfrak{sp}(n) \end{matrix}$}

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 {$O(n,\mathbb{R})$}.