Symmetrizable generalized Cartan matrix

introduction

• Generalized Cartan matrix
• symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms

def

A generalized Cartan matrix $A$ is symmetrisable if there exists a non-singular diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$.

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

• Killing form and invariant symmetric bilinear form

computational resource

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