Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / ComplexificationOfCompactRealForms

See: {$A_2$}, Lie algebra to group

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.

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

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

Keith Conrad. Complexification.
Representation theory of SU(2): Real and complexified Lie algebras
Wikipedia: Complexification of a real Lie group
Wikipedia: Classical group
Lie group - Lie algebra correspondence Includes the properties of compact Lie groups.
The Lie Algebras su(N), Walter Pfeifer
Complexification of Real Analytic Groups, G.Hochschild