Generalized Cartan matrix

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

1. $a_{ii}=2$ for $i=1,\cdots,n$
2. $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
3. $a_{ij}=0$ 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

$$ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) $$

related items

• Rank 2 generalized Cartan matrix
• Skew-symmetrizable matrix

Killing form

• Killing form and invariant symmetric bilinear form

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxdlBHdXA5THp3SFE/edit