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Book: ComplexificationOfCompactRealForms

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

Understand complexification of compact real forms.

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

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


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


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Puslapis paskutinį kartą pakeistas 2020 balandžio 07 d., 20:16