"Generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 2번째 줄: | 2번째 줄: | ||
* Cartan matrix encodes relative lenghths and angles among vectors in the root system. | * 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 | * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | ||
| + | |||
| + | |||
| + | ==example== | ||
| + | * $G_2$ Cartan matrix | ||
| + | $$ | ||
| + | A=\left( | ||
| + | \begin{array}{cc} | ||
| + | 2 & -1 \\ | ||
| + | -3 & 2 \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | $$ | ||
| + | * take $D$ as follows : | ||
| + | $$ | ||
| + | D=\left( | ||
| + | \begin{array}{cc} | ||
| + | 3 & 0 \\ | ||
| + | 0 & 1 \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | $$ | ||
| + | * $DA=AD^{t}$ | ||
2013년 10월 8일 (화) 10:31 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
- $G_2$ Cartan matrix
$$ A=\left( \begin{array}{cc} 2 & -1 \\ -3 & 2 \\ \end{array} \right) $$
- take $D$ as follows :
$$ D=\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \\ \end{array} \right) $$
- $DA=AD^{t}$
Killing form