"Generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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* Cartan matrix encodes relative lenghths and angles among vectors in the root system. | * 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 | * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | ||
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| + | * In semi-simple Lie theory, a cartan matrix is a square matrix | ||
| + | ** For diagonal entries, <math>a_{ii} = 2</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> | ||
| + | ** $a_{ij} = 0$ 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 $a_{ji} = 0$ | ||
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# $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$ | # $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$ | ||
# $a_{ij}=0$ impies $a_{ji}=0$ | # $a_{ij}=0$ impies $a_{ji}=0$ | ||
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| + | ==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 | ||
| + | ** $\text{rank }A=1$ | ||
| + | ** 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 $u=0$ | ||
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| + | |||
| + | ====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 $A$. | ||
| + | * 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. | ||
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==example== | ==example== | ||
2015년 4월 1일 (수) 19:11 판
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 $a_{ji} = 0$
generalized Cartan matrix
- an $n\times n$ matrix $A=(a_{ij})$ is called a generalised Cartan matrix if it satisfies the conditions
- $a_{ii}=2$ for $i=1,\cdots,n$
- $a_{ij}\in \mathbb{Z}$ and $a_{ij}\leq 0$ if $i\neq j$
- $a_{ij}=0$ 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
- $\text{rank }A=1$
- 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 $u=0$
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 $A$.
- 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.
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
$$ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) $$
Killing form