Součásti dokumentu 02LIAG
Zdrojový kód
%\wikiskriptum{02LIAG}
\section{Reálné formy komplexních poloprostých algeber}
Mějme poloprostou reálnou algebru $\g,\ \g_\C = \g \oplus_\R i\g = \C \otimes_\R \g$. Z $\g_\C$ lze zpetně najít $\g$:
\begin{align*}
\phi: \g_\C \to \g_\C : \phi(u+iv) = u - iv,\ \forall u,v \in \g \rimpl \g = \{ X \in \g_\C | \phi(X) = X \}.
\end{align*}
\Pzn{
Vlastnosti $\phi$:
\begin{enumerate}
\item $\phi \circ \phi = \mrm{id} \rimpl$je involutivní,
\item $\phi(\lambda X) = \overline{\lambda}\phi(X),\ \phi(X+Y) = \phi(X) + \phi(Y) \rimpl$je antilineární,
\item $\phi\big( [u_1+iv_1,u_2 + iv_2] \big) = \phi \big( \left( [u_1,u_2] - [v_1,v_2] \right) + i \left( [v_1,u_2] + [u_1,v_2] \right) \big) = \left( [u_1,u_2] - [v_1,v_2] \right) - i \left( [v_1,u_2] + [u_1,v_2] \right) = [u_1 - iv_1,u_2 - iv_2] = \big[ \phi(u_1 + iv_1),\phi(u_2 + iv_2) \big] \rimpl$automorfismus.
\end{enumerate}
Tj. reálná forma $\g$ komplexní algebry $\g_\C$ nám určuje involutivní antilineární automorfismus $\phi$.
}
Naopak, mějme $\g_\C$ a její involutivní antilineární automorfismus $\phi$. Pak $\g = \{ X \in \g_\C | \phi(X) = X \}$ nám zadává reálnou podalgebru $\g_\phi$ v $\g_\C : (\g_\phi)_{\C} = \g_\C$.
\begin{proof}
Máme $\phi : \g_\C \to (\g_\C)_\R,\ \dim_\R(\g_\C)_\R = 2\dim_\C(\g_\C)_\R$. Uvažujme $\zuz{\phi}{(\g_\C)_\R}:(\g_\C)_\R \to (\g_\C)_\R,\ \phi^2 = \mathbb{1} \rimpl \sigma\left(\zuz{\phi}{(\g_\C)_\R}\right) = \pm 1$, takže
\begin{align*}
(\g_\C)_\R = \underbrace{\ker\left( \zuz{\phi}{(\g_\C)_\R} - \mathbb{1} \right)}_{\g} \dotplus \ker\left( \zuz{\phi}{(\g_\C)_\R} + \mathbb{1} \right)
\end{align*}
$\Rightarrow\quad X \in \g \rimpl \phi(X) = X \rimpl \phi(iX) = -iX \rimpl iX \in \ker\left( \zuz{\phi}{(\g_\C)_\R} + \mathbb{1} \right) \rimpl$obě jádra mají stejnou dimenzi a násobení $i$ zobrazuje jedno na druhé$\rimpl \dim_\R \g = \dim_\C \g_\C,\ \g_\C = \g \dotplus i\g$ a platí: $\forall X,Y \in \g,\ \phi\big( [X,Y] \big) = [\phi(X),\phi(Y)] = [X,Y] \rimpl \forall X,Y \in \g,\ [X,Y] \in \g$.
\end{proof}
\Prl{
$\mfrk{sl}(l+1,\C),\ \phi:\mfrk{sl}(l+1,\C) \to \mfrk{sl}(l+1,\C)$:
\begin{itemize}
\item $\phi(A) = \overline{A} \ \dots\ \g = \mfrk{sl}(l+1,\R)$
\item $\phi(A) = -A^+ \ \dots\ \g = \left\{ X \in \mfrk{sl}(l+1,\C) \middle| -X^+ = X \right\} = \mfrk{su}(l+1)$
\item $\phi(A) = -JA^+J$, kde $J = \mrm{diag}(\underbrace{1,\dots1}_{p},\underbrace{-1,\dots,-1}_{q})$:
\begin{align*}
\phi(\phi(A)) &= J\left( J \left( A^+ \right)^+ J \right) J = A \\
\phi\big( [A,B] \big) &= - J[A,B]^+J = J \left[ A^+,B^+ \right] J = \left[ -JA^+J,-JB^+J \right] = [\phi(A),\phi(B)]
\end{align*}
$\dots\ \g = \left\{ X \in \mfrk{sl}(l+1,\C) \middle| -JX^+J = X \right\} = \mfrk{su}(p,q),\ p+q = l+1$.
\end{itemize}
}
Uvažujme komplexní poloprostou Lieovu algebru $\g$ vyjádřenou ve Weyl-Chevalleyho bázi, tj.:
\begin{gather*}
\g = \mrm{span}\{H_\alpha\}_{\alpha \in \Delta^p} \dotplus \dot{\bigplus_{\alpha \in \Delta}}\mrm{span}\{E_\alpha\}, \\
[H,E_\alpha] = \alpha(H)E_\alpha,\ H \in \mrm{span}_\R \{ H_\alpha \}_{\alpha \in \Delta^p} = \h \rimpl \forall \alpha \in \Delta,\ \alpha(H) \in \R \\
[E_\alpha,E_{-\alpha}] = \underbrace{K(E_\alpha,E_{-\alpha})}_{\in\R}H_\alpha, \\
[E_\alpha,E_\beta] = N_{\alpha\beta}E_{\alpha+\beta},\ \N_{\alpha\beta} \in \Z,\ N_{(-\alpha)(-\beta)} = -N_{\alpha\beta}.
\end{gather*}
Označíme
\begin{align*}
\g_\text{split} := \mrm{span}_\R \{ H_\alpha \}_{\alpha \in \Delta^p} \dotplus \dot{\bigplus}_{\alpha \in \Delta}\mrm{span}_\R \{ E_\alpha \}.
\end{align*}
$\g_\text{split}$ je reálná forma $\g$, tj. $\phi(H_\alpha) = H_\alpha,\ \phi(E_\alpha) = E_\alpha,\ \forall H_\alpha,E_\alpha \in \g_\text{split},$. Zjistíme signaturu Killingovy formy $K$ pro $\g_\text{split}$ (Killingova forma je dobrá pro rozlišení reálnych algeber).
\begin{align*}
&K(H,E_\alpha) = 0 \text{ protože }\h\perp\mrm{span}\{ E_\alpha \} \\
&\!\!\zuz{K}{\h} \text{ je pozitivně definitní} \\
&K(E_\alpha,E_\beta) = 0,\ \alpha+\beta \neq 0 \\
&E_\alpha,E_{-\alpha}: \begin{pmatrix}
K(E_\alpha,E_\alpha) & K(E_\alpha,E_{-\alpha}) \\
K(E_{-\alpha},E_\alpha) & K(E_{-\alpha},E_{-\alpha})
\end{pmatrix} = \begin{pmatrix}
0 & \lambda \\
\lambda & 0
\end{pmatrix}\text{, kde }\lambda \neq 0 \ \dots\ \mrm{sgn} = (1,1,0)
\end{align*}
$\Rightarrow\quad \mrm{sgn}\zuz{K}{\g_\text{split}} = \left( l+\frac{n-l}{2},\frac{n-l}{2},0 \right)$. Dále prozkoumáme
\begin{align*}
\g_\text{komp} := \underbrace{\mrm{span}_\R \{ iH_\alpha \}_{\alpha \in \Delta^p}}_{i\h} \dotplus \dot{\bigplus_{\alpha \in \Delta^+}}\mrm{span}\left\{ \frac{E_\alpha - E_{-\alpha}}{\sqrt{2}},\frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}} \right\}.
\end{align*}
$\forall H_\alpha,E_\alpha \in \g_\text{komp},\ \phi(H_\alpha) = -H_\alpha,\ \phi(E_\alpha) = -E_{-\alpha}$. Zřejmě platí taky $\phi^2 = \mathbb{1}$. Dále máme:
\begin{align*}
&\phi \big( [H_\alpha,H_\beta] \big) = [\phi(H_\alpha),\phi(H_\beta)] \text{ protože } [H_\alpha,H_\beta] = 0 \\
&\phi \big( [H_\alpha,E_\beta] \big) = \underbrace{\beta(H_\alpha)}_{\in\R}\phi(E_\beta) = -\beta(H_\alpha)E_{-\beta} = [H_\alpha,E_{-\beta}] = [\phi(H_\alpha),\phi(E_\beta)] \\
&\phi\big( [E_\alpha,E_\beta] \big) = N_{\alpha\beta}\phi(E_{\alpha+\beta}) = -N_{\alpha\beta}E_{-\alpha-\beta} = N_{(-\alpha)(-\beta)}E_{-\alpha-\beta} = [-E_{-\alpha},-E_{-\beta}] = \\
&\qquad=[\phi(E_\alpha),\phi(E_\beta)] \\
&\phi\big( [E_\alpha,E_{-\alpha}] \big) = K(E_\alpha,E_{-\alpha})\phi(H_\alpha) = -K(E_\alpha,E_{-\alpha})H_\alpha = -[E_\alpha,E_{-\alpha}] = [-E_{-\alpha},-E_\alpha] \\
&\qquad= [\phi(E_\alpha),\phi(E_{-\alpha})] \\
&\phi\left( \frac{E_\alpha - E_{-\alpha}}{\sqrt{2}} \right) = \frac{-E_{-\alpha} + E_\alpha}{\sqrt{2}} \\
&\phi\left( \frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}} \right) = \frac{-i(-E_{-\alpha} - E_{\alpha})}{\sqrt{2}} = \frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}} \\
&\phi(iH_\alpha) = -i(-H_\alpha) = iH_\alpha \\
&\!\!\zuz{K}{i\h}\text{ je negativně definitní} \\
&\!\!\!\!\! \left. \begin{array}{l}\displaystyle
\displaystyle K\left( \frac{E_\alpha - E_{-\alpha}}{\sqrt{2}},\frac{E_\alpha - E_{-\alpha}}{\sqrt{2}} \right) = -K(E_\alpha,E_{-\alpha}) \\
\displaystyle K\left( \frac{E_\alpha - E_{-\alpha}}{\sqrt{2}},\frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}} \right) = 0 \\
\displaystyle K\left( \frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}},\frac{i(E_\alpha + E_{-\alpha})}{\sqrt{2}} \right) = -K(E_\alpha,E_{-\alpha})
\end{array}\right\}\mrm{sgn} = (0,2,0) \text{ pro volbu } K(E_\alpha,E_{-\alpha}) > 0
\end{align*}
$\Rightarrow\quad \mrm{sgn}\zuz{K}{\g_\text{kompl}} = (0,n,0)$.
\Vet{(Weyl)
Buď $\g$ reálná poloprostá Lieova algebra, $G$ jí odpovídající souvislá a jednoduše souvislá Lieova grupa. Pak $G$ je kompaktní$\quad\Leftrightarrow\quad$Killingova forma $\g$ je negativně definitní. Bez důkazu.
}
