Generalized Cartan matrix

introduction

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

Cartan matrix of a simple Lie algebra

• 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 $a_{ji} = 0$

generalized Cartan matrix

• 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

1. $a_{ii}=2$ for $i=1,\cdots,n$
2. $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
3. $a_{ij}=0$ impies $a_{ji}=0$

classification of generalized Cartan matrix

• 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 $u=0$

main result

• 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 $A$.

cor

Let $A$ be an indecomposable GCM. Then

1. A GCM A has finite type if and only if there exists $u>0$ with $Au>0$
2. A GCM A has affine type if and only if there exists $u>0$ with $Au=0$
3. A GCM A has indefinite type if and only if there exists $u>0$ with $Au<0$

• 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.

related items

• Rank 2 generalized Cartan matrix
• Skew-symmetrizable matrix
• Killing form and invariant symmetric bilinear form
• Symmetrizable generalized Cartan matrix

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxWGFXeWtoYTVyTlk/edit