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 $A=DB$.

memo

• from https://www.sharelatex.com/project/55caaef83e9789d92821b3e8
• Let $\mathfrak g$ be a simple Lie algebra of rank $\ell$
• $C$ Cartan matrix
• Let $\langle \cdot,\cdot \rangle$ be the invariant inner product on $\mathfrak g$, normalized as in \cite{Kac}, so that the square of length of the maximal root equals $2$ with respect to the induced inner product on the dual space to the Cartan subalgebra $\mathfrak h$ of $\mathfrak g$
• Let $r^\vee$ be the maximal number of edges connecting two vertices of the Dynkin diagram of $\mathfrak g$. Thus, $r^\vee=1$ for simply-laced $\mathfrak g$, $r^\vee=2$ for $B_\ell, C_\ell, F_4, G_2$, and $r^\vee=3$ for $D_4$.
• From now on we will use the inner product

$$ (\cdot,\cdot) = r^\vee \langle \cdot,\cdot \rangle $$ on $\mathfrak h^*$

• $D=\operatorname{diag}(d_1,\cdots, d_\ell)$ such that $B:=D C$ is symmetric
• Let $B = (B_{ij})_{1\leq i,j\leq \ell}$ be the symmetric matrix

$$ B = D C, $$ i.e., $$ B_{ij} = (\alpha_i,\alpha_j) = r^\vee \langle \alpha_i,\alpha_j \rangle. $$

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