02LIAG:Kapitola17: Porovnání verzí
Z WikiSkripta FJFI ČVUT v Praze
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(Není zobrazena jedna mezilehlá verze od stejného uživatele.) | |||
Řádka 28: | Řádka 28: | ||
Váhy: $\pm 2\lambda,0;\ n_{\pm 2\lambda} = 1,\ n_0 = 2$. | Váhy: $\pm 2\lambda,0;\ n_{\pm 2\lambda} = 1,\ n_0 = 2$. | ||
} | } | ||
+ | |||
\Prl{ | \Prl{ | ||
− | $A_l = \mfrk{sl}(l+1,\C)$, | + | $A_l = \mfrk{sl}(l+1,\C) = \left\{ A \in \C^{l+1,l+1} \middle| \Tr A = 0 \right\}$ |
+ | \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*} | \begin{align*} | ||
a =\begin{pmatrix} | a =\begin{pmatrix} | ||
2 & -1 & \\ | 2 & -1 & \\ | ||
-1 & \ddots & \ddots \\ | -1 & \ddots & \ddots \\ | ||
− | & \ddots & | + | & \ddots & 2 & -1 \\ |
& & -1 & 2 | & & -1 & 2 | ||
− | \end{pmatrix}, && | + | \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} | \phi_i \begin{pmatrix} | ||
− | + | d_1 \\ | |
& \ddots \\ | & \ddots \\ | ||
− | & & | + | & & d_{l+1} |
− | \end{pmatrix} = \ | + | \end{pmatrix} = d_i, && \phi_1 > \phi_2 > \dots > \phi_{l+1} >0. |
\end{align*} | \end{align*} | ||
− | $\alpha_i(T_j) = a_{ij} = t_{j,i} - t_{j,i+1} \neq 0 \text{ pro } i = j-1,j,j+1:$ | + | 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*} | \begin{align*} | ||
\left.\begin{array}{rl} | \left.\begin{array}{rl} | ||
\alpha_{j-1}(T_j) &= t_{j,j-1} - t_{j,j} = -1 \\ | \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(T_j) &= t_{j,j} - t_{j,j+1} = 2 \\ | ||
− | \alpha_{j+1}(T_j) &= t_{j,j+1} - t_{j,j+2} = -1 | + | \alpha_{j+1}(T_j) &= t_{j,j+1} - t_{j,j+2} = -1 |
− | \end{array} \right\} \rimpl T_j = \begin{array}{ | + | \end{array} \right\} \rimpl T_j = \begin{array}{cl} |
\left(\begin{array}{cccccc} | \left(\begin{array}{cccccc} | ||
\ddots \\ | \ddots \\ | ||
Řádka 64: | Řádka 109: | ||
\end{array} | \end{array} | ||
\end{align*} | \end{align*} | ||
− | Fundamentální váhy, $\lambda_i(T_j) = \delta_{ij}$: | + | \item Fundamentální váhy, $\lambda_i(T_j) = \delta_{ij}$: |
\begin{align*} | \begin{align*} | ||
− | &\lambda_1 \begin{ | + | &\lambda_1 \left(\begin{smallmatrix} |
1 \\ | 1 \\ | ||
& -1 \\ | & -1 \\ | ||
Řádka 72: | Řádka 117: | ||
& & & \ddots \\ | & & & \ddots \\ | ||
& & & & 0 | & & & & 0 | ||
− | \end{ | + | \end{smallmatrix}\right) = 1, && |
− | \lambda_1 \begin{ | + | \lambda_1 \left(\begin{smallmatrix} |
\ddots \\ | \ddots \\ | ||
& 0 \\ | & 0 \\ | ||
Řádka 80: | Řádka 125: | ||
& & & & 0 \\ | & & & & 0 \\ | ||
& & & & & \ddots \\ | & & & & & \ddots \\ | ||
− | \end{ | + | \end{smallmatrix}\right) = 0 && \rimpl \lambda_1 = \phi_1 |
− | &\lambda_2\begin{ | + | \end{align*} |
+ | \begin{align*} | ||
+ | &\lambda_2 \left(\begin{smallmatrix} | ||
1 \\ | 1 \\ | ||
& -1 \\ | & -1 \\ | ||
Řádka 87: | Řádka 134: | ||
& & & \ddots \\ | & & & \ddots \\ | ||
& & & & 0 | & & & & 0 | ||
− | \end{ | + | \end{smallmatrix}\right) = 0, && |
− | \lambda_2\begin{ | + | \lambda_2 \left(\begin{smallmatrix} |
0 \\ | 0 \\ | ||
& 1 \\ | & 1 \\ | ||
Řádka 94: | Řádka 141: | ||
& & & 0 \\ | & & & 0 \\ | ||
& & & & \ddots \\ | & & & & \ddots \\ | ||
− | \end{ | + | \end{smallmatrix}\right) = 1, && |
− | &\lambda_2 \begin{ | + | &\lambda_2 \left(\begin{smallmatrix} |
\ddots \\ | \ddots \\ | ||
& 0 \\ | & 0 \\ | ||
Řádka 102: | Řádka 149: | ||
& & & & 0 \\ | & & & & 0 \\ | ||
& & & & & \ddots \\ | & & & & & \ddots \\ | ||
− | \end{ | + | \end{smallmatrix}\right) = 0 |
\end{align*} | \end{align*} | ||
− | $\Rightarrow\quad \lambda_i = \phi_1 + \dots + \phi_i$. Je vidět že pak platí $\lambda_i(T_j) = \delta_{ij}$. | + | $\Rightarrow\quad \lambda_2 = \phi_2 + \phi_1 \rimpl \dots \rimpl \lambda_i = \phi_1 + \dots + \phi_i$. 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*} | \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) = \\ | (\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) = \\ | ||
Řádka 115: | Řádka 161: | ||
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 $\rho^{\land j}$ 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 $\left\{ \sum_{i\neq 1}\phi_i,\dots,\sum_{i\neq l+1}\phi_i \right\} = \{ -\phi_1,\dots,-\phi_{l+1} \} \overset{l\neq 1}{\neq} \{ \phi_1,\dots,\phi_{l+1} \}$. Když $l=1$, pak $\rho^{\land l=1} \simeq \rho$, tj. $\rho^{\land l=1}$ je izomorfní definující reprezentaci. | + | 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{ | \Pzn{ | ||
Nechť $\rho$ reprezentace $\g$ na $V$, definujeme $\rho^T: \rho^T(X) = (-\rho(X))^T \rimpl \rho^{\land l} = \rho^T$. | Nechť $\rho$ reprezentace $\g$ na $V$, definujeme $\rho^T: \rho^T(X) = (-\rho(X))^T \rimpl \rho^{\land l} = \rho^T$. | ||
} | } | ||
+ | |||
\Prl{ | \Prl{ | ||
− | $C_l = \mfrk{sp}(2l,\C) | + | $C_l = \mfrk{sp}(2l,\C) = \left\{ A \in \C^{2l,2l} \middle| JA + A^TJ = 0 \right\}$, 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*} | \begin{align*} | ||
D = \begin{pmatrix} | D = \begin{pmatrix} | ||
Řádka 130: | Řádka 241: | ||
&&&& \ddots \\ | &&&& \ddots \\ | ||
&&&&& -d_l | &&&&& -d_l | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
\end{pmatrix} | \end{pmatrix} | ||
\end{align*} | \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*} | \begin{align*} | ||
T_j &= \begin{array}{cc} | T_j &= \begin{array}{cc} | ||
Řádka 167: | Řádka 275: | ||
\\ | \\ | ||
\end{array} | \end{array} | ||
− | \end{array} | + | \end{array} && j \leq l-1 |
\end{align*} | \end{align*} | ||
\begin{align*} | \begin{align*} | ||
Řádka 195: | Řádka 303: | ||
\end{array}\\ | \end{array}\\ | ||
\end{align*} | \end{align*} | ||
− | $\lambda_i(T_j) = \delta_{ij} \rimpl \lambda_i = \phi_1 + \dots + \phi_i,\ i \in \hat{l}$. | + | $\lambda_i(T_j) = \delta_{ij} \rimpl \lambda_i = \phi_1 + \dots + \phi_i,\ i \in \hat{l}$. |
+ | \end{itemize} | ||
} | } | ||
\Prl{ | \Prl{ | ||
− | $D_l = \mfrk{so}(2l,\C)$. | + | $D_l = \mfrk{so}(2l,\C) = \left\{ A \in \C^{2l,2l} \middle| A^TJ + JA = 0 \right\}$, 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*} | \begin{align*} | ||
− | |||
− | |||
− | |||
− | |||
− | |||
(a_{ij}) = \begin{pmatrix} | (a_{ij}) = \begin{pmatrix} | ||
2 & -1 \\ | 2 & -1 \\ | ||
Řádka 211: | Řádka 401: | ||
& & -1 & 2 & 0 \\ | & & -1 & 2 & 0 \\ | ||
& & -1 & 0 & 2 | & & -1 & 0 & 2 | ||
− | \end{pmatrix} | + | \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*} | \end{align*} | ||
− | \begin{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*} | \end{align*} | ||
$T_l$: | $T_l$: | ||
Řádka 233: | Řádka 436: | ||
\end{align*} | \end{align*} | ||
Definující reprezentace má váhy $\{ \phi_1,\dots,\phi_l,-\phi_1,\dots, -\phi_l \}$. | Definující reprezentace má váhy $\{ \phi_1,\dots,\phi_l,-\phi_1,\dots, -\phi_l \}$. | ||
− | } | + | \end{itemize} |
+ | } | ||
\Prl{ | \Prl{ | ||
− | $B_l = \mfrk{so}(2l + 1)$ | + | $B_l = \mfrk{so}(2l + 1,\C)$ |
\begin{align*} | \begin{align*} | ||
− | H = \begin{ | + | \g_0 = \left\{H = \left( \begin{smallmatrix} |
d_1\sigma_2 \\ | d_1\sigma_2 \\ | ||
& \ddots \\ | & \ddots \\ | ||
&& d_l\sigma_2 \\ | && d_l\sigma_2 \\ | ||
&&& 0 | &&& 0 | ||
− | \end{ | + | \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} | (a_{ij}) = \begin{pmatrix} | ||
2 & -1 \\ | 2 & -1 \\ | ||
-1 & \ddots & \ddots \\ | -1 & \ddots & \ddots \\ | ||
− | & \ddots & | + | & \ddots & 2 & -1 \\ |
− | & & | + | & & -1 & 2 & -2 \\ |
& & & -1 & 2 | & & & -1 & 2 | ||
− | \end{pmatrix} | + | \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*} | \end{align*} | ||
\begin{align*} | \begin{align*} | ||
− | &\phi_i(H) = d_i | + | 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*} | \end{align*} | ||
$T_l$: | $T_l$: |
Aktuální verze z 6. 8. 2016, 04:42
[ znovu generovat, | výstup z překladu ] | Kompletní WikiSkriptum včetně všech podkapitol. | |
PDF Této kapitoly | [ znovu generovat, | výstup z překladu ] | Přeložení pouze této kaptioly. |
ZIP | Kompletní zdrojový kód včetně obrázků. |
Součásti dokumentu 02LIAG
součást | akce | popis | poslední editace | soubor | |||
---|---|---|---|---|---|---|---|
Hlavní dokument | editovat | Hlavní stránka dokumentu 02LIAG | Hazalmat | 3. 8. 2016 | 21:54 | ||
Řídící stránka | editovat | Definiční stránka dokumentu a vložených obrázků | Hazalmat | 7. 7. 2016 | 07:04 | ||
Header | editovat | Hlavičkový soubor | Hazalmat | 10. 7. 2016 | 22:12 | header.tex | |
Kapitola0 | editovat | Úvod | Hazalmat | 3. 8. 2016 | 22:12 | LIAG_Kapitola0.tex | |
Kapitola1 | editovat | Definice Lieovy grupy a Lieovy algebry | Hazalmat | 5. 8. 2016 | 18:02 | LIAG_Kapitola1.tex | |
Kapitola2 | editovat | Vztah mezi Lieovou grupou a její algebrou | Hazalmat | 5. 8. 2016 | 18:27 | LIAG_Kapitola2.tex | |
Kapitola3 | editovat | Nástin teorie integrabilních distribucí | Hazalmat | 30. 7. 2016 | 15:10 | LIAG_Kapitola3.tex | |
Kapitola4 | editovat | Akce grupy na varietě | Hazalmat | 17. 7. 2016 | 20:23 | LIAG_Kapitola4.tex | |
Kapitola5 | editovat | Reprezentace Lieových grup a algeber | Hazalmat | 4. 8. 2016 | 18:21 | LIAG_Kapitola5.tex | |
Kapitola6 | editovat | Souvislost Lieových grup a algeber | Hazalmat | 4. 8. 2016 | 19:51 | LIAG_Kapitola6.tex | |
Kapitola7 | editovat | Lieovy algebry | Hazalmat | 5. 8. 2016 | 02:06 | LIAG_Kapitola7.tex | |
Kapitola8 | editovat | Cartanova kritéria | Hazalmat | 5. 8. 2016 | 18:29 | LIAG_Kapitola8.tex | |
Kapitola9 | editovat | Klasifikace pomocí kořenů | Hazalmat | 5. 8. 2016 | 18:34 | LIAG_Kapitola9.tex | |
Kapitola10 | editovat | Kořenové diagramy, Cartanova martice | Hazalmat | 31. 7. 2016 | 16:32 | LIAG_Kapitola10.tex | |
Kapitola11 | editovat | Dynkinovy diagramy | Hazalmat | 5. 8. 2016 | 18:39 | LIAG_Kapitola11.tex | |
Kapitola12 | editovat | Reálné formy komplexních poloprostých algeber | Hazalmat | 1. 8. 2016 | 00:39 | LIAG_Kapitola12.tex | |
Kapitola13 | editovat | Význam kompaktních Lieových grup | Hazalmat | 1. 8. 2016 | 00:45 | LIAG_Kapitola13.tex | |
Kapitola14 | editovat | Reprezentace poloprostých Lieových algeber | Hazalmat | 1. 8. 2016 | 13:45 | LIAG_Kapitola14.tex | |
Kapitola15 | editovat | Spinorové reprezentace | Hazalmat | 27. 7. 2016 | 21:38 | LIAG_Kapitola15.tex | |
Kapitola16 | editovat | Symetrie v QM | Hazalmat | 27. 7. 2016 | 22:21 | LIAG_Kapitola16.tex | |
Kapitola17 | editovat | Cvičení | Hazalmat | 6. 8. 2016 | 04:42 | LIAG_Kapitola17.tex |
Vložené soubory
soubor | název souboru pro LaTeX |
---|---|
Image:liag-1.pdf | liag-1.pdf |
Image:su3_1.pdf | su3_1.pdf |
Image:su3_2.pdf | su3_2.pdf |
Image:su3_3.pdf | su3_3.pdf |
Image:su3_4.pdf | su3_4.pdf |
Image:su3_5.pdf | su3_5.pdf |
Image:su3_6.pdf | su3_6.pdf |
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*} }