Aktuální verze z 6. 8. 2016, 04:42

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*}	
	}