"Generalized Cartan matrix"의 두 판 사이의 차이

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** For non-diagonal entries, <math>a_{ij} \in {0,-1,-2,-3}</math>
 
** For non-diagonal entries, <math>a_{ij} \in {0,-1,-2,-3}</math>
 
** If <math>a_{ij} = -2\text{ or }-3</math> then <math>a_{ji} = 0</math>
 
** If <math>a_{ij} = -2\text{ or }-3</math> then <math>a_{ji} = 0</math>
−
** $a_{ij} = 0$ if and only if $a_{ji} = 0$
+
** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
  
  
 
==generalized Cartan matrix==
 
==generalized Cartan matrix==
−
* A generalized Cartan matrix is a square matrix $A = (a_{ij})$ with integer entries such that
+
* A generalized Cartan matrix is a square matrix <math>A = (a_{ij})</math> with integer entries such that
−
** For diagonal entries, $a_{ii} = 2$.
+
** For diagonal entries, <math>a_{ii} = 2</math>.
−
** For non-diagonal entries, $a_{ij} \leq 0 $.
+
** For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
−
** $a_{ij} = 0$ if and only if $a_{ji} = 0$
+
** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
−
* an $n\times n$ matrix $A=(a_{ij})$ is called a generalised Cartan matrix if it satisfies the conditions
+
* an <math>n\times n</math> matrix <math>A=(a_{ij})</math> is called a generalised Cartan matrix if it satisfies the conditions
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# $a_{ii}=2$ for $i=1,\cdots,n$
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# <math>a_{ii}=2</math> for <math>i=1,\cdots,n</math>
−
# $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
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# <math>a_{ij}\in \mathbb{Z}</math> and <math>a_{ij}\leq 0</math> if <math>i\neq j</math>
−
# $a_{ij}=0$ impies $a_{ji}=0$
+
# <math>a_{ij}=0</math> impies <math>a_{ji}=0</math>
  
 
==classification of generalized Cartan matrix==
 
==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 is called indecomposable if it is not equivalent to a diagonal sum of two smaller GCMs.
 
* A GCM A has finite type if
 
* A GCM A has finite type if
−
** $\text{det }A\neq 0$
+
** <math>\text{det }A\neq 0</math>
−
** there exists $u>0$ with $Au>0$
+
** there exists <math>u>0</math> with <math>Au>0</math>
−
** $Au\geq 0$ implies $u>0$ or $u=0$
+
** <math>Au\geq 0</math> implies <math>u>0</math> or <math>u=0</math>
 
* A GCM A has affine type if
 
* A GCM A has affine type if
−
** $\text{corank }A=1$
+
** <math>\text{corank }A=1</math>
−
** there exists $u>0$ such that $Au=0$
+
** there exists <math>u>0</math> such that <math>Au=0</math>
−
** $Au\geq 0$ implies $Au=0$
+
** <math>Au\geq 0</math> implies <math>Au=0</math>
 
* A GCM A has indefinite type if
 
* A GCM A has indefinite type if
−
** there exists $u>0$ with $Au<0$
+
** there exists <math>u>0</math> with <math>Au<0</math>
−
** $Au\geq 0$ and $u\geq 0$ implies $u>0$ or $u=0$
+
** <math>Au\geq 0</math> and <math>u\geq 0</math> implies <math>u>0</math> or <math>u=0</math>
  
  
 
====main result====
 
====main result====
−
* Let $A$ be an indecomposable GCM. Then exactly one of the following three possibilities holds:
+
* Let <math>A</math> be an indecomposable GCM. Then exactly one of the following three possibilities holds:
−
** $A$ has finite type
+
** <math>A</math> has finite type
−
** $A$ has affine type
+
** <math>A</math> has affine type
−
** $A$ has indefinite type
+
** <math>A</math> has indefinite type
−
* Moreover the type of $A^t$ is the same as the type of $A$.
+
* Moreover the type of <math>A^t</math> is the same as the type of <math>A</math>.
  
 
;cor
 
;cor
−
Let $A$ be an indecomposable GCM. Then
+
Let <math>A</math> be an indecomposable GCM. Then
−
# A GCM A has finite type if and only if there exists $u>0$ with $Au>0$
+
# A GCM A has finite type if and only if there exists <math>u>0</math> with <math>Au>0</math>
−
# A GCM A has affine type if and only if there exists $u>0$ with $Au=0$
+
# A GCM A has affine type if and only if there exists <math>u>0</math> with <math>Au=0</math>
−
# A GCM A has indefinite type if and only if there exists $u>0$ with $Au<0$
+
# A GCM A has indefinite type if and only if there exists <math>u>0</math> with <math>Au<0</math>
  
 
* R.Carter's 'Lie algebras of finite and affine type' 337~344p
 
* R.Carter's 'Lie algebras of finite and affine type' 337~344p

2020년 11월 16일 (월) 05:31 판

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

computational resource

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