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

In semi-simple Lie theory, a cartan matrix is a square matrix
- For diagonal entries, $a_{ii} = 2$.
- For non-diagonal entries, $a_{ij} \in {0,-1,-2,-3}$
- If $a_{ij} = -2\text{ or }-3$ then $a_{ji} = 0$
- $a_{ij} = 0$ if and only if $a_{ji} = 0$
A generalized Cartan matrix is a square matrix $A = (a_{ij})$ with integer entries such that
- For diagonal entries, $a_{ii} = 2$.
- For non-diagonal entries, $a_{ij} \leq -1 $.
- $a_{ij} = 0$ if and only if $a_{ji} = 0$

an $n\times n$ matrix $A=(a_{ij})$ is called a generalised Cartan matrix if it satisfies the conditions

A GCM is called indecomposable if it is not equivalent to a diagonal sum of two smaller GCMs.
A GCM A has finite type if
- $\text{det }A\neq 0$
- there exists $u>0$ with $Au>0$
- $Au\geq 0$ implies $u>0$ or $u=0$
A GCM A has affine type if
- $\text{rank }A=1$
- there exists $u>0$ such that $Au=0$
- $Au\geq 0$ implies $Au=0$
A GCM A has indefinite type if
- there exists $u>0$ with $Au<0$
- $Au\geq 0$ and $u\geq 0$ implies $u>0$ or $u=0$

Let $A$ be an indecomposable GCM. Then exactly one of the following three possibilities holds:
- $A$ has finite type
- $A$ has affine type
- $A$ has indefinite type
Moreover the type of $A^t$ is the same as the type of $A$.
R.Carter's 'Lie algebras of finite and affine type' 337~344p
Now we turn to the classification of GCM of affine and finite type.

$$ A=\left( \begin{array}{cc} 2 & -1 \\ -3 & 2 \\ \end{array} \right) $$

$$ D=\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \\ \end{array} \right) $$

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