Generalized Cartan matrix

introduction

Cartan matrix encodes relative lenghths and angles among vectors in the root system

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, \(a_{ij} \leq 0 \).
- \(a_{ij} = 0\) 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
- \(\text{corank }A=1\)
- 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 \(A\).

Let \(A\) be an indecomposable GCM. Then