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{
$A_l = \mfrk{sl}(l+1,\C)$, kořeny: $\alpha_i = \phi_i - \phi_{i+1},\ \alpha_i(T_j) = a_{ij}$,
\begin{align*}
a =\begin{pmatrix}
2 & -1 & \\
-1 & \ddots & \ddots \\
& \ddots & \ddots & -1 \\
& & -1 & 2
\end{pmatrix}, &&
\phi_i \begin{pmatrix}
\lambda_1 \\
& \ddots \\
& & \lambda_{l+1}
\end{pmatrix} = \lambda_i.
\end{align*}
$\alpha_i(T_j) = a_{ij} = t_{j,i} - t_{j,i+1} \neq 0 \text{ pro } i = j-1,j,j+1:$
\end{align*}
\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}{cc}
\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*}
Fundamentální váhy, $\lambda_i(T_j) = \delta_{ij}$:
\begin{align*}
&\lambda_1 \begin{pmatrix}
1 \\
& -1 \\
& & 0 \\
& & & \ddots \\
& & & & 0
\end{pmatrix} = 1, &&
\lambda_1 \begin{pmatrix}
\ddots \\
& 0 \\
& & 1 \\
& & & -1 \\
& & & & 0 \\
& & & & & \ddots \\
\end{pmatrix} = 0 && \rimpl \lambda_1 = \phi_1 \\
&\lambda_2\begin{pmatrix}
1 \\
& -1 \\
& & 0 \\
& & & \ddots \\
& & & & 0
\end{pmatrix} = 0, &&
\lambda_2\begin{pmatrix}
0 \\
& 1 \\
& & -1 \\
& & & 0 \\
& & & & \ddots \\
\end{pmatrix} = 1, \\
&\lambda_2 \begin{pmatrix}
\ddots \\
& 0 \\
& & 1 \\
& & & -1 \\
& & & & 0 \\
& & & & & \ddots \\
\end{pmatrix} = 0 &&\rimpl \lambda_2 = \phi_2 + \phi_1
\end{align*}
$\Rightarrow\quad \lambda_i = \phi_1 + \dots + \phi_i$. Je vidět že pak platí $\lambda_i(T_j) = \delta_{ij}$.
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 $\left\{ \sum_{i\neq 1}\phi_i,\dots,\sum_{i\neq l+1}\phi_i \right\} = \{ -\phi_1,\dots,-\phi_{l+1} \} \overset{l\neq 1}{\neq} \{ \phi_1,\dots,\phi_{l+1} \}$. Když $l=1$, pak $\rho^{\land l=1} \simeq \rho$, tj. $\rho^{\land l=1}$ je izomorfní definující reprezentaci.
}
\Pzn{
Nechť $\rho$ reprezentace $\g$ na $V$, definujeme $\rho^T: \rho^T(X) = (-\rho(X))^T \rimpl \rho^{\land l} = \rho^T$.
}
\Prl{
$C_l = \mfrk{sp}(2l,\C),\ D \in \g_0,\ \phi_i(D) = d_i,\ \alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1,\ \alpha_l = 2\phi_l,\ \alpha_i(T_j) = a_{ij}$:
\begin{align*}
D = \begin{pmatrix}
d_1 \\
& \ddots \\
&& d_l \\
&&& -d_1 \\
&&&& \ddots \\
&&&&& -d_l
\end{pmatrix} && (a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & \ddots & \ddots \\
& & -1 & 2 & -1 \\
& & & -2 & 2
\end{pmatrix}
\end{align*}
\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},\text{ pro } 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*}
$\lambda_i(T_j) = \delta_{ij} \rimpl \lambda_i = \phi_1 + \dots + \phi_i,\ i \in \hat{l}$. Definující reprezentace má váhy $\{ \phi_1,\dots,\phi_l,\phi_{-1},\dots,\phi_{-l} \},\ \dim = 2l$, nejvyšší váha je $\phi_1$.
}
\Prl{
$D_l = \mfrk{so}(2l,\C)$.
\begin{align*}
H = \begin{pmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2
\end{pmatrix} = H(d_1,\dots,d_l) &&
(a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & 2 & -1 & -1 \\
& & -1 & 2 & 0 \\
& & -1 & 0 & 2
\end{pmatrix}
\end{align*}
\begin{align*}
&\phi_i(H) = d_i && \alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1 \\
&T_i = H(0,\dots,0,\underset{i}{1},\underset{i+1}{-1},0,\dots,0),\ i \leq l-1 && \alpha_l = \phi_{l-1}+\phi_l
\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 $\{ \phi_1,\dots,\phi_l,-\phi_1,\dots, -\phi_l \}$.
}
\Prl{
$B_l = \mfrk{so}(2l + 1)$.
\begin{align*}
H = \begin{pmatrix}
d_1\sigma_2 \\
& \ddots \\
&& d_l\sigma_2 \\
&&& 0
\end{pmatrix} &&
(a_{ij}) = \begin{pmatrix}
2 & -1 \\
-1 & \ddots & \ddots \\
& \ddots & \ddots & -1 \\
& & \ddots & 2 & -2 \\
& & & -1 & 2
\end{pmatrix}
\end{align*}
\begin{align*}
&\phi_i(H) = d_i && \alpha_i = \phi_i - \phi_{i+1},\ i \leq l-1 \\
&T_i = H(0,\dots,0,\underset{i}{1},\underset{i+1}{-1},0,\dots,0) && \alpha_l = \phi_l
\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*}
}