"Generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 5번째 줄: | 5번째 줄: | ||
==example== | ==example== | ||
| − | * $G_2$ | + | * Cartan matrix of $G_2$ |
$$ | $$ | ||
A=\left( | A=\left( | ||
| 23번째 줄: | 23번째 줄: | ||
\right) | \right) | ||
$$ | $$ | ||
| − | * $DA= | + | * Then $DA=A^{t}D$ is a symmetric matrix |
| + | $$ | ||
| + | \left( | ||
| + | \begin{array}{cc} | ||
| + | 6 & -3 \\ | ||
| + | -3 & 2 \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | $$ | ||
2013년 10월 8일 (화) 10:35 판
introduction
- Cartan matrix encodes relative lenghths and angles among vectors in the root system.
- symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms
example
- Cartan matrix of $G_2$
$$ A=\left( \begin{array}{cc} 2 & -1 \\ -3 & 2 \\ \end{array} \right) $$
- take $D$ as follows :
$$ D=\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \\ \end{array} \right) $$
- Then $DA=A^{t}D$ is a symmetric matrix
$$ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) $$
Killing form