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
- $C$ Cartan matrix
- Let $\langle \cdot,\cdot \rangle$ be the invariant inner product
on $\g$, 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 $\h$ of $\g$
- Let $r^\vee$ be the maximal number of edges connecting two vertices of
the Dynkin diagram of $\g$. Thus, $r^\vee=1$ for simply-laced $\g$, $r^\vee=2$ for $B_\el, C_\el, F_4, G_2$, 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 $\h^*$
- $D=\operatorname{diag}(d_1,\cdots, d_\ell)$ 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
$$ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) $$