"Generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 3번째 줄: | 3번째 줄: | ||
* 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 | ||
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| + | ==generalized Cartan matrix== | ||
| + | * an $n\times n$ matrix $A=(a_{ij})$ is called a generalised Cartan matrix if it satisfies the conditions | ||
| + | # $a_{ii}=2$ for $i=1,\cdots,n$ | ||
| + | # $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$ | ||
| + | # $a_{ij}=0$ impies $a_{ji}=0$ | ||
==example== | ==example== | ||
2015년 3월 10일 (화) 19:48 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
generalized Cartan matrix
- an $n\times n$ matrix $A=(a_{ij})$ is called a generalised Cartan matrix if it satisfies the conditions
- $a_{ii}=2$ for $i=1,\cdots,n$
- $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
- $a_{ij}=0$ impies $a_{ji}=0$
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