Součásti dokumentu 02LIAG
Zdrojový kód
%\wikiskriptum{02LIAG}
\section{Cvičení}
\Prl{
$\mfrk{so}(3,\C)\sim\mfrk{sl}(2,\C): [L_3,L_\pm]=\pm L_\pm,\ [L_+,L_-] = 2L_3$,
\begin{align*}
&\rho(L_3) = \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, &&
\rho(L_+) = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, &&
\rho(L_-) = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \\
&\rho(L_3)\ket{\uparrow} = \frac{1}{2}\ket{\uparrow}, && \rho(L_3)\ket{\downarrow} = -\frac{1}{2}\ket{\downarrow}, && \text{váhy: } \lambda = \pm\frac{1}{2},
\end{align*}
$\rho:\mfrk{sl}(2,\C) \to \gl\left(D^{1/2}\right),\ D^{1/2} = \mrm{span}\left\{ \ket{\uparrow},\ket{\downarrow} \right\}$ .
Tenzorový součin $\rho$ se sebou samou:
\begin{align*}
(\rho\otimes\rho)(L_3) = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\otimes\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\otimes\frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
\end{align*}
\begin{align*}
&(\rho\otimes\rho)(L_3)\ket{\uparrow\uparrow} = \ket{\uparrow\uparrow} &&
(\rho\otimes\rho)(L_3)\ket{\uparrow\downarrow} = \frac{1}{2}\ket{\uparrow\downarrow} -\frac{1}{2}\ket{\uparrow\downarrow} = 0 \\
&(\rho\otimes\rho)(L_3)\ket{\downarrow\downarrow} = -\ket{\downarrow\downarrow} &&
(\rho\otimes\rho)(L_3)\ket{\downarrow\uparrow} = 0 \\
\\
&(\rho\otimes\rho)(L_-)\ket{\uparrow\uparrow} = \ket{\downarrow\uparrow} + \ket{\uparrow\downarrow} && (\rho\otimes\rho)(L_-)\big(\ket{\downarrow\uparrow} - \ket{\uparrow\downarrow}\big) = \ket{\downarrow\downarrow} - \ket{\downarrow\downarrow} = 0\\
&(\rho\otimes\rho)(L_-)\ket{\downarrow\downarrow} = 0 && \\
\\
& (\rho\otimes\rho)(L_+) \dots
\end{align*}
Váhy: $\pm 2\lambda,0;\ n_{\pm 2\lambda} = 1,\ n_0 = 2$.
}
\Prl{
$A_l = \mfrk{sl}(l+1,\C) = \left\{ A \in \C^{l+1,l+1} \middle| \Tr A = 0 \right\}$
\begin{itemize}
\item Kořeny: $\g_0 = \mrm{diag} \subset \mfrk{sl}(l+1),\ \dim \g_0 = l, [\g_0,\g_0] = 0 \rimpl \g_0$ Abelovská$\rimpl \g_0$ nilpotentní, tj. opravdu je to Cartanova podalgebra. Mějme
\begin{align*}
\ E_{ij} = \bordermatrix{
~ & & j \cr
& & \vdots \cr
i & \dots & 1 \cr},\qquad i \neq j
\end{align*}
$\Rightarrow\quad \mfrk{sl}(l+1) = \g_0 + \mrm{span}\{ E_{ij} \}$ a pro $D \in \g_0,\ D = \mrm{diag}(d_1,\dots,d_{l+1})$ máme $[D,E_{ij}] - (d_i - d_j)E_{ij}$. Nechť $\phi_j \in \mfrk{sl}^*(l+1),\ \phi_j(D) = d_j \rimpl (\phi_i - \phi_j)(D)E_{ij} = [D,E_{ij}]$, tj:
\begin{align*}
\Delta = \left\{ (\phi_i - \phi_j) \middle|\ i \neq j,\ i,j \in \widehat{l+1} \right\}
\end{align*}
Zvolíme $H_0 = \mrm{diag}(h_1,\dots,h_{l+1}),\ h_i > h_{i+1},\ (\phi_i - \phi_j)(H_0) \neq 0$, máme tedy uspoŕádání koŕenů:
\begin{align*}
\phi_1 > \phi_2 > \dots > \phi_{l+1} >0.
\end{align*}
\begin{align*}
\Delta^+ &= \left\{ \phi_i - \phi_j \middle| i < j \leq l+1 \right\} \\
\Delta^p &= \big\{ \underbrace{\phi_i - \phi_ {i+1}}_{=: \alpha_i} \big| i \in \widehat{l} \big\}
\end{align*}
Ověříme, že pomocí $\Delta^p$ můžeme nakombinovat celé $\Delta$:
\begin{align*}
\phi_i - \phi_j = (\phi_i - \phi_{i+1}) + (\phi_{i+1} - \phi_{i+2}) + \dots + (\phi_{j-1} - \phi_j).
\end{align*}
\item Cartanova matice, Dynkinův diagram:
\begin{align*}
a_{\beta\alpha} = - (p+q) \overset{\alpha,\beta \in \Delta^p}{=} -q, && \{ \beta +k\alpha \}_{k=p}^q \in \Delta^+
\end{align*}
\begin{align*}
\left.\begin{array}{l}
\alpha_i := \phi_i - \phi_{i+1} \\
\alpha_j := \phi_j - \phi_{j+1}
\end{array} \right\} \rimpl \alpha_i + k \alpha_j = \phi_i - \phi_{i+1} + k( \phi_j - \phi_{j+1} ) \overset{!}{=} \phi_a - \phi_b,\ a < b
\end{align*}
\begin{align*}
\begin{array}{lll}
(i < j-1) \lor (i > j-1) &\rimpl k = 0 &\rimpl a_{ij} = 0 \\
(i = j-1) \lor (j = i-1) &\rimpl k = 0 \lor k = 1 &\rimpl a_{ij} = -1
\end{array}
\end{align*}
\begin{align*}
a =\begin{pmatrix}
2 & -1 & \\
-1 & \ddots & \ddots \\
& \ddots & 2 & -1 \\
& & -1 & 2
\end{pmatrix}, && \text{\LARGE $\underset{\text{\normalsize $1$}}{\cdot} \! - \! \underset{\text{\normalsize $2$}}{\cdot} \! - \cdots - \!\!\! \underset{\text{\normalsize $l-1$}}{\cdot} \!\!\! - \underset{\text{\normalsize $l$}}{\cdot} $}
\end{align*}
\item Adjungovaná reprezentace: váhy (kořeny): $\alpha_i = \phi_i - \phi_{i+1},\ \alpha_i(T_j) = a_{ij}$, kde
\begin{align*}
\phi_i \begin{pmatrix}
d_1 \\
& \ddots \\
& & d_{l+1}
\end{pmatrix} = d_i, && \phi_1 > \phi_2 > \dots > \phi_{l+1} >0.
\end{align*}
Z tvaru vah $\alpha_i = \phi_i - \phi_j$ a uspořádání $\phi_i$ plyne, že nejvyšší váha je $\phi_1 - \phi_{l+1} = \alpha_1 + \dots + \alpha_l$.
K nalezení $T_j$ využijeme $\alpha_i(T_j) = a_{ij} = t_{j,i} - t_{j,i+1} \neq 0 \text{ pro } i = j-1,j,j+1:$
\begin{align*}
\left.\begin{array}{rl}
\alpha_{j-1}(T_j) &= t_{j,j-1} - t_{j,j} = -1 \\
\alpha_j(T_j) &= t_{j,j} - t_{j,j+1} = 2 \\
\alpha_{j+1}(T_j) &= t_{j,j+1} - t_{j,j+2} = -1
\end{array} \right\} \rimpl T_j = \begin{array}{cl}
\left(\begin{array}{cccccc}
\ddots \\
& 0 \\
& & 1 & \dots & \dots & \dots \\
& & & -1 \\
& & & & 0 \\
& & & & & \ddots \\
\end{array}\right) &
\begin{array}{c}
\\ \\ j \\ \\ \\ \\
\end{array}
\end{array}
\end{align*}
\item Fundamentální váhy, $\lambda_i(T_j) = \delta_{ij}$:
\begin{align*}
&\lambda_1 \left(\begin{smallmatrix}
1 \\
& -1 \\
& & 0 \\
& & & \ddots \\
& & & & 0
\end{smallmatrix}\right) = 1, &&
\lambda_1 \left(\begin{smallmatrix}
\ddots \\
& 0 \\
& & 1 \\
& & & -1 \\
& & & & 0 \\
& & & & & \ddots \\
\end{smallmatrix}\right) = 0 && \rimpl \lambda_1 = \phi_1
\end{align*}
\begin{align*}
&\lambda_2 \left(\begin{smallmatrix}
1 \\
& -1 \\
& & 0 \\
& & & \ddots \\
& & & & 0
\end{smallmatrix}\right) = 0, &&
\lambda_2 \left(\begin{smallmatrix}
0 \\
& 1 \\
& & -1 \\
& & & 0 \\
& & & & \ddots \\
\end{smallmatrix}\right) = 1, &&
&\lambda_2 \left(\begin{smallmatrix}
\ddots \\
& 0 \\
& & 1 \\
& & & -1 \\
& & & & 0 \\
& & & & & \ddots \\
\end{smallmatrix}\right) = 0
\end{align*}
$\Rightarrow\quad \lambda_2 = \phi_2 + \phi_1 \rimpl \dots \rimpl \lambda_i = \phi_1 + \dots + \phi_i$. Je vidět že pak platí $\lambda_i(T_j) = \delta_{ij}$.
\item Definující reprezentace: Mějme definující reprezentaci v standardní bázi $(e_j),\ D \in \g_0,\ \ De_j = \left(\begin{smallmatrix} d_1 \\ & \ddots \\ && d_{l+1} \end{smallmatrix} \right) e_j = d_je_j$. Její váhy $\{ \phi_1,\dots,\phi_{l+1} \},\ \phi_{l+1} = -(\phi_1 + \dots + \phi_l)$, lze zapsat jako $\{ \phi_1, \phi_1 - \alpha_1, \phi_1 - \alpha_1 - \alpha_2, \dots,\phi_1 - \alpha_1 - \dots - \alpha_l \}$. Nejvyšší váha je $\phi_1 = \lambda_1$, násobnosti $1$, $\dim\rho_1 = l+1$. $\rho_1 \land \rho_1$:
\begin{align*}
(\rho_1 \land \rho_1)(e_i \land e_j) &= (D \otimes \mathbb{1} + \mathbb{1} \otimes D)(e_i \otimes e_j - e_j \otimes e_i) = \\
&= d_ie_i \otimes e_j - d_je_j \otimes e_i + e_i \otimes d_je_j - e_j \otimes d_ie_i = (d_i+d_j)(e_i \land e_j),
\end{align*}
váhy: $\{ \phi_i + \phi_j | i \neq j \},\ \dim \rho\land\rho = \binom{l+1}{2}$, nejvyšší je $\phi_1 + \phi_2$.
Pro $\rho^{\land j}$ jsou váhy $\left\{ \phi_{i_1} + \dots + \phi_{i_j} \middle| i_1 < \dots < i_j \right\},\ \dim\rho^{\land j} = \binom{l+1}{j}$, nejvyšší váha $\lambda_j = \phi_1 + \dots + \phi_j$.
Pro $\rho^{\land l}$ jsou váhy $\left\{ \sum_{i\neq 1}\phi_i,\dots,\sum_{i\neq l+1}\phi_i \right\} = \{ -\phi_1,\dots,-\phi_{l+1} \} \overset{l\neq 1}{\neq} \{ \phi_1,\dots,\phi_{l+1} \}$. Takže nejvyšší váha je $-\lambda_{l+1}$. Když $l=1$, pak $\rho^{\land l=1} \simeq \rho$, tj. $\rho^{\land l=1}$ je izomorfní definující reprezentaci.
\end{itemize}
}
\Pzn{
Nechť $\rho$ reprezentace $\g$ na $V$, definujeme $\rho^T: \rho^T(X) = (-\rho(X))^T \rimpl \rho^{\land l} = \rho^T$.
}
\Prl{
$C_l = \mfrk{sp}(2l,\C) = \left\{ A \in \C^{2l,2l} \middle| JA + A^TJ = 0 \right\}$, kde $J = \left( \begin{smallmatrix} 0 & -\mathbb{1} \\ \mathbb{1} & 0 \end{smallmatrix} \right)$
\begin{itemize}
\item Cartanova podalgebra: Označme $A = \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$:
\begin{align*}
JA + A^TJ =
\begin{pmatrix}
-c & -d \\
a & b
\end{pmatrix} +
\begin{pmatrix}
c^T & -a^T \\
d^T & -b^T
\end{pmatrix} = 0 && \Rightarrow && d = -a^T,\ b = b^T,\ c = c^T
\end{align*}
\begin{align*}
\g_0 = \left\{ \left(\begin{smallmatrix} \Lambda & 0 \\ 0 & -\Lambda \end{smallmatrix} \right) \middle| \Lambda = \mrm{diag} (\lambda_1,\dots,\lambda_l) \in \C^{l,l} \right\}
\end{align*}
\begin{align*}
[\Lambda,E_{ij}] = (\lambda_i - \lambda_j)E_{ij} && \left[ \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} , \begin{pmatrix} E_{ij} & 0 \\ 0 & -E_{ij} \end{pmatrix} \right] = (\lambda_i - \lambda_j) \underbrace{ \begin{pmatrix} E_{ij} & 0 \\ 0 & -E_{ij} \end{pmatrix} }_{=: I_{ij},\ i \neq j}
\end{align*}
\begin{align*}
\left[ \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} , \begin{pmatrix} 0 & E_{ij}+E_{ji} \\ 0 & 0 \end{pmatrix} \right] = \begin{pmatrix} 0 & \Lambda(E_{ij}+E_{ji}) \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & (E_{ij}+E_{ji})\Lambda \\ 0 & 0 \end{pmatrix} = (\lambda_i + \lambda_j) \underbrace{\begin{pmatrix} 0 & E_{ij}+E_{ji} \\ 0 & 0 \end{pmatrix} }_{=: F_{ij},\ i \leq j}
\end{align*}
\begin{align*}
G_{ij} := F_{ij}^T && \Rightarrow && \left[ \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} , G_{ij} \right] = - \left[ \begin{pmatrix} \Lambda & 0 \\ 0 & \Lambda \end{pmatrix} , F_{ij} \right]^T = - (\lambda_i + \lambda_j)G_{ij}
\end{align*}
$\Rightarrow\quad \g_0$ je skutečně Cartanova podalgebra. $\phi_i \left( \begin{smallmatrix} \Lambda & 0 \\ 0 & -\lambda \end{smallmatrix} \right) := \lambda_i,\ i \in \widehat{l}$ tvoří bázi $\g_0^*$.
\item Kořeny:
\begin{align*}
\Delta = \left\{ \phi_i - \phi_j \middle| i \neq j \right\} \cup \left\{ \phi_i + \phi_j \middle| i \leq j \right\} \cup \left\{ -(\phi_i + \phi_j) \middle| i \leq j \right\}
\end{align*}
$H_0:\ \phi_i(H_0) > \phi_{i+1}(H_0) > 0, \forall i$.
\begin{align*}
\Delta^+ &= \left\{ \phi_i - \phi_j \middle| i < j \right\} \cup \left\{ \phi_i + \phi_j \middle| i \leq j \right\} \\
\Delta^- &= \left\{ \phi_i - \phi_j \middle| i > j \right\} \cup \left\{ -(\phi_i + \phi_j) \middle| i \leq j \right\} \\
\Delta^p &= \big\{ \underbrace{\phi_i - \phi_{i+1}}_{=: \alpha_i} \big| i \in \widehat{l-1} \big\} \cup \big\{ \underbrace{2\phi_l}_{=: \alpha_l} \big\}
\end{align*}
\begin{align*}
\phi_i - \phi_j &= (\phi_i - \phi_{i+1}) + \dots + (\phi_{j-1} - \phi_j) = \sum_{k=1}^{j-1} \alpha_k \\
\phi_i + \phi_j &= 2\phi_l + (\phi_i - \phi_l) + (\phi_j - \phi_l) = 2\phi_l + \sum_{k=i}^{l-1} \alpha_k + \sum_{k=j}^{l-1} \alpha_k
\end{align*}
$a_{\beta\alpha} \overset{\alpha,\beta \in \Delta^p}{=} -q$:
\begin{align*}
\begin{array}{lllll}
\{ \alpha_i + k\alpha_j \}_{i,j < l} &= (\phi_i - \phi_{i+1}) + k(\phi_j - \phi_{j+1}) & \rimpl & |i-j| > 1 &\rimpl k=0 \\
&&& |i-j| = 1 &\rimpl k = 0 \lor k = 1 \\
\{ \alpha_i + k\alpha_l \}_{i < l} &= (\phi_i - \phi_{i+1}) + 2k\phi_l & \rimpl & i < l-1 &\rimpl k=0 \\
&&& i= l-1 &\rimpl k = 0 \lor k = 1 \\
\{ \alpha_l + k\alpha_i \}_{i,j < l} &= 2\phi_l + k(\phi_i - \phi_{i+1}) & \rimpl & i < l-1 &\rimpl k=0 \\
&&& i= l-1 &\rimpl k = 0 \lor k = 1 \lor k = 2\\
\end{array}
\end{align*}
$\Rightarrow\quad a_{l-1,l} = -1 = \frac{\braket{\alpha_{l-1},\alpha_l}}{\braket{\alpha_l,\alpha_l}},\ a_{l,l-1} = -2 = \frac{\braket{\alpha_l,\alpha_{l-1}} }{\braket{\alpha_{l-1},\alpha_{l-1}} } \rimpl \norm{\alpha_l} = \sqrt{2} \norm{\alpha_{l-1}}$.
\begin{align*}
(a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & 2 & -1 \\
& & -1 & 2 & -1 \\
& & & -2 & 2
\end{pmatrix}, && \text{\LARGE $\underset{\text{\normalsize $1$}}{\cdot} \! - \! \underset{\text{\normalsize $2$}}{\cdot} \! - \cdots - \!\!\! \underset{\text{\normalsize $l-2$}}{\cdot} \!\!\! - \!\!\! \underset{\text{\normalsize $l-1$}}{\cdot} \!\!\! \Rightarrow \! \underset{\text{\normalsize $l$}}{\cdot} $}
\end{align*}
\item Definující reprezentace:
$D \in \g_0,\ \phi_i(D) = d_i$:
\begin{align*}
D = \begin{pmatrix}
d_1 \\
& \ddots \\
&& d_l \\
&&& -d_1 \\
&&&& \ddots \\
&&&&& -d_l
\end{pmatrix}
\end{align*}
Definující reprezentace má váhy $\{ \phi_1,\dots,\phi_l,\phi_{-1},\dots,\phi_{-l} \},\ \dim = 2l$, nejvyšší váha je $\phi_1$.
\item Adjungovaná reprezentace:
$\ \alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1,\ \alpha_l = 2\phi_l,\ \alpha_i(T_j) = a_{ij}$
\begin{align*}
T_j &= \begin{array}{cc}
\left(\begin{array}{ccccccccccc}
\ddots \\
& 0 \\
&& 1 & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\
&&& -1 \\
&&&& 0 \\
&&&&& \ddots \\
&&&&&& 0 \\
&&&&&&& 1 & \dots & \dots & \dots \\
&&&&&&&& -1 \\
&&&&&&&&& 0 \\
&&&&&&&&&& \ddots \\
\end{array}\right) &
\begin{array}{c}
\\
\\
j \\
\\
\\
\\
\\
\\
l+j \\
\\
\\
\\
\end{array}
\end{array} && j \leq l-1
\end{align*}
\begin{align*}
\left .\begin{array}{rl}
\alpha_i(T_l) &= 0,\ i < l-1 \\
\alpha_{l-1}(T_l) &= -1 \\
\alpha_l(T_l) &= 2
\end{array} \right\} \rimpl T_l = \begin{array}{cc}
\left(\begin{array}{ccccccc}
\ddots \\
& 0 \\
&& 1 & \dots & \dots & \dots & \dots \\
&&& 0 \\
&&&& \ddots \\
&&&&& 0 \\
&&&&&& 1 \\
\end{array}\right) &
\begin{array}{c}
\\
\\
l \\
\\
\\
\\
\\
\end{array}
\end{array}\\
\end{align*}
$\lambda_i(T_j) = \delta_{ij} \rimpl \lambda_i = \phi_1 + \dots + \phi_i,\ i \in \hat{l}$.
\end{itemize}
}
\Prl{
$D_l = \mfrk{so}(2l,\C) = \left\{ A \in \C^{2l,2l} \middle| A^TJ + JA = 0 \right\}$, kde $J = \left( \begin{smallmatrix} 0& \mathbb{1} \\ \mathbb{1} & 0 \end{smallmatrix} \right),\ l >1$
Označme $A = \left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)$
\begin{align*}
A^TJ + JA =
\begin{pmatrix}
c^T & a^T \\
d^T & b^T
\end{pmatrix} +
\begin{pmatrix}
c & d \\
a & b
\end{pmatrix} = 0 && \Rightarrow && d = -a^T,\ b = -b^T,\ c = -c^T
\end{align*}
\begin{itemize}
\item Cartanova podalgebra:
Ukážeme že $\g_0 = \left\{ H = \mrm{diag}(\lambda_1\sigma_2,\dots,\lambda_l\sigma_2) \right\},\ \sigma_2 = \left( \begin{smallmatrix} 0& -i \\ i & 0 \end{smallmatrix} \right)$. Nechť $X \in \C^{2,2},\ X = \begin{pmatrix}
x_{11} & x_{12} \\
x_{21} & x_{22}
\end{pmatrix}$
\begin{align*}
\lambda_i\sigma_2 X - \lambda_j X \sigma_2 = i\lambda_i \begin{pmatrix}
-x_{21} & -x_{22} \\
x_{11} & x_{12}
\end{pmatrix} - i\lambda_j\begin{pmatrix}
x_{12} & -x_{11} \\
x_{22} & -x_{21}
\end{pmatrix} = c(\lambda_i,\lambda_j)\begin{pmatrix}
x_{11} & x_{12} \\
x_{21} & x_{22}
\end{pmatrix}
\end{align*}
Zapíšeme ve tvaru:
\begin{align*}
i\begin{pmatrix}
ic & -\lambda_j & -\lambda_i & 0 \\
\lambda_j & ic & 0 & -\lambda_i \\
\lambda_i & 0 & ic & -\lambda_j \\
0 & \lambda_i & \lambda_j & ic
\end{pmatrix}
\begin{pmatrix}
x_{11} \\ x_{12} \\ x_{21} \\ x_{22}
\end{pmatrix} = 0
\end{align*}
Z požadavku řešitelnosti soustavy ($\det = 0$) dostaneme $c_{1,2,3,4} = \pm(\lambda_i \pm \lambda_j)$. Pro $c_1 = \lambda_i + \lambda _j$ najdeme $X_1 = \left( \begin{smallmatrix} 1 & i \\ i & -1 \end{smallmatrix} \right) = \sigma_3 + i\sigma_1$.
\begin{align*}
\widetilde{F} := X_1, && F_{ij} :=
\bordermatrix{
~ & & i & j & \cr
& & \vdots & \vdots & \cr
i & \dots & & \widetilde{F} & \cr
j & \dots & -\widetilde{F}^T & & \cr
& & & & \cr
} , \quad i<j ,&& [H,F_{ij}] = (\lambda_i+\lambda_j)F_{ij} \overset{\exists i,j}{\neq} 0
\end{align*}
\begin{align*}
\left[ H,F_{ij}^+ \right] = \left[ H^+,F_{ij}^+ \right] = -\left[ H,F_{ij} \right]^+ = - (\lambda_i +\lambda_j)F_{ij}^+
\end{align*}
Pro $c_2 = \lambda_i -\lambda_j$ dostaneme:
\begin{align*}
\widetilde{G} := \mathbb{1} + \sigma_2, && G_{ij} := \bordermatrix{
~ & & i & j & \cr
& & \vdots & \vdots & \cr
i & \dots & & \widetilde{G} & \cr
j & \dots & -\widetilde{G}^T & & \cr
& & & & \cr
} , \quad i<j
\end{align*}
\begin{align*}
[H,G_{ij}] = (\lambda_i+\lambda_j) G_{ij} , &&[H,G_{ij}] = (\lambda_i+\lambda_j) G_{ij}
\end{align*}
\item Kořeny: $\phi_j \in g_0^*,\ \phi_j(H) = \lambda_j$:
\begin{align*}
\Delta &= \left\{ \phi_i + \phi_j \middle| i < j \right\} \cup \left\{ \phi_i - \phi_j \middle| i \neq j \right\} \cup \left\{ -(\phi_i + \phi_j) \middle| i < j \right\}
\end{align*}
$H_0 = \mrm{diag}(\lambda_1,\dots,\lambda_l),\ \lambda_1 > \lambda_2 > \dots > \lambda_l > 0$:
\begin{align*}
\Delta^+ &= \left\{ \phi_i + \phi_j \middle| i < j \right\} \cup \left\{ \phi_i - \phi_j \middle| i < j \right\} \\
\Delta^p &= \big\{ \underbrace{\phi_i - \phi_{i+}}_{=: \alpha_i} \big| i \in \widehat{l-i} \big\} \cup \big\{ \underbrace{\phi_{l-1} + \phi_l}_{=: \alpha_l} \big\}
\end{align*}
\begin{align*}
\begin{array}{rll}
\alpha_i + k\alpha_{i+1} &= (\phi_i - \phi_{i+1}) + k(\phi+1 - \phi_{i+2}) ,\ i \in \widehat{l-1} &\rimpl k = 0 \lor k = 1 \\
\alpha_{l-2} + k\alpha_l &= (\phi_{l-2} - \phi_{l-1}) + k(\phi_{l-1} + \phi_l) &\rimpl k = 0 \lor k = 1 \\
\alpha_{l-1} + k\alpha_l &= (\phi_{l-1} - \phi_l) + k(\phi_{l-1} + \phi_l) &\rimpl k=0
\end{array}
\end{align*}
\begin{align*}
(a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & 2 & -1 & -1 \\
& & -1 & 2 & 0 \\
& & -1 & 0 & 2
\end{pmatrix}, && \text{\LARGE $\underset{\text{\normalsize $1$}}{\cdot} \! - \! \underset{\text{\normalsize $2$}}{\cdot} \! - \cdots - \!\!\! \underset{\text{\normalsize $l-3$}}{\cdot} \!\!\! - \!\!\! \underset{\text{\normalsize $l-2$}}{\cdot} \!\!\! < \!\!\! \text{\small $ \begin{array}{ll}
\text{\LARGE $\cdot$} & \text{\normalsize $\!\!\! l-1$}\\
\text{\LARGE $\cdot$} & \text{\normalsize $\!\!\! l$}
\end{array}$} $}
\end{align*}
\item Váhy:
\begin{align*}
H = \begin{pmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2
\end{pmatrix} = H(d_1,\dots,d_l) , &&
\begin{array}{l}
\phi_i(H) = d_i \\
\alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1 \\
\alpha_l = \phi_{l-1}+\phi_l \\
T_i = H(0,\dots,0,\underset{i}{1},\underset{i+1}{-1},0,\dots,0),\ i \leq l-1x
\end{array}
\end{align*}
$T_l$:
\begin{align*}
\left .\begin{array}{rll}
\alpha_{l-2}(T_l) &= -1 &= d_{l-2} - d_{l-1} \\
\alpha_{l-1}(T_l) &= 0 &= d_{l-1} - d_l \\
\alpha_l(T_l) &= 2 &= \phi_{l-1}(T_l) + \phi_l(t_l) = d_{l-1} + d_l
\end{array}\right\} \rimpl T_l = H(0,\dots,0,1,1)
\end{align*}
$\lambda_i(T_j) = \delta_{ij}$:
\begin{align*}
\lambda_1 &= \phi_1 \\
\lambda_i &= \phi_1 + \dots + \phi_i,\ i \leq l-2 \\
\lambda_{l-1} &= \frac{1}{2}(\phi_1 + \dots + \phi_{l-1} - \phi_l) \\
\lambda_l &= \frac{1}{2}(\phi_1 + \dots + \phi_l)
\end{align*}
Definující reprezentace má váhy $\{ \phi_1,\dots,\phi_l,-\phi_1,\dots, -\phi_l \}$.
\end{itemize}
}
\Prl{
$B_l = \mfrk{so}(2l + 1,\C)$
\begin{align*}
\g_0 = \left\{H = \left( \begin{smallmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2 \\
&&& 0
\end{smallmatrix}\right) \right\}, && \phi_i H = \left( \begin{smallmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2 \\
&&& 0
\end{smallmatrix}\right) = \lambda_i, && X := \left(\begin{array}{ccc|c}
&&& \\
&&& v \\
&&& \\ \cline{1-4}
& v^T && 0
\end{array}\right)
\end{align*}
\begin{align*}[H,X] = \left(\begin{array}{ccc|c}
&&& \\
&&& \lambda_i\sigma_1v \\
&&& \\ \cline{1-4}
&&& 0
\end{array}\right) - \left(\begin{array}{ccc|c}
&&& \\
&&& v \\
&&& \\ \cline{1-4}
& -\lambda_i (\sigma_1v)^T && 0
\end{array}\right) = \lambda_i \left(\begin{array}{ccc|c}
&&& \\
&&& \sigma_1 v \\
&&& \\ \cline{1-4}
& (\sigma_1 v)^T && 0
\end{array}\right)
\end{align*}
Za $v$ můžeme volit vlastní vektory $\sigma_1$. Dále zvolíme $H_0:\ \lambda_1 > \dots > \lambda_l,\ \lambda_i = \phi(H_0)$.
\begin{align*}
\Delta &= \left\{ \phi_i + \phi_j \middle| i < j \right\} \cup \left\{ \phi_i - \phi_j \middle| i \neq j \right\} \cup \left\{ -(\phi_i + \phi_j) \middle| i < j \right\} \cup \left\{ \phi_i \right\} \cup \left\{ -\phi_i \right\} \\
\Delta^+ &= \left\{ \phi_i + \phi_j \middle| i < j\right\} \cup \left\{ \phi_i - \phi_j \middle| i<j \right\} \cup \left\{ \phi_i \right\} \\
\Delta^p &= \big\{ \underbrace{\phi_i - \phi_{i+1}}_{=: \alpha_i} \big| i \in \widehat{l-1} \big\} \cup \big\{ \underbrace{\phi_l}_{=: \alpha_l} \big\}
\end{align*}
\begin{align*}
\begin{array}{rll}
\alpha_{l-2} + k\alpha_l &= (\phi_{l-2} - \phi_{l-1}) + k\phi_l &\rimpl k = 0 \\
\alpha_{l-1} + k\alpha_l &= (\phi_{l-1} - \phi_l) + k\phi_l &\rimpl k=0 \lor k = 1 \lor k = 2 \\
\alpha_l + k\alpha_{l-1} &= \phi_l + k(\phi_{l-1}-\phi_l) &\rimpl k = 0 \lor k = 1
\end{array}
\end{align*}
\begin{align*}
(a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & 2 & -1 \\
& & -1 & 2 & -2 \\
& & & -1 & 2
\end{pmatrix}, && \text{\LARGE $\underset{\text{\normalsize $1$}}{\cdot} \! - \! \underset{\text{\normalsize $2$}}{\cdot} \! - \cdots - \!\!\! \underset{\text{\normalsize $l-2$}}{\cdot} \!\!\! - \!\!\! \underset{\text{\normalsize $l-1$}}{\cdot} \!\!\! \Leftarrow \! \underset{\text{\normalsize $l$}}{\cdot} $}
\end{align*}
\begin{align*}
H = \begin{pmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2 \\
&&& 0
\end{pmatrix} && \begin{array}{l}
\phi_i(H) = d_i \\
\alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1 \\
\alpha_l = \phi_l \\
T_i = H(0,\dots,0,\underset{i}{1},\underset{i+1}{-1},0,\dots,0)
\end{array}
\end{align*}
$T_l$:
\begin{align*}
\left.\begin{array}{rl}
\alpha_{l-1}(T_l) &= -2 \\
\alpha_l(t_l) &= 2
\end{array}\right\} \rimpl T_l = H(0,\dots,0,2)
\end{align*}
$\lambda_i(T_j)=\delta_{ij}$:
\begin{align*}
\lambda_i &= \phi_1 + \dots + \phi_i,\ i \leq l-1 \\
\lambda_l &= \frac{1}{2}(\phi_1 + \dots + \phi_l)
\end{align*}
}