"Generalized Cartan matrix"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | * 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 |
| − | |||
| − | == | + | ==Cartan matrix of a simple Lie algebra== |
| − | * | + | * 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> | ||
| + | ** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math> | ||
| + | |||
| + | |||
| + | ==generalized Cartan matrix== | ||
| + | * A generalized Cartan matrix is a square matrix <math>A = (a_{ij})</math> with integer entries such that | ||
| + | ** For diagonal entries, <math>a_{ii} = 2</math>. | ||
| + | ** For non-diagonal entries, <math>a_{ij} \leq 0 </math>. | ||
| + | ** <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math> | ||
| + | * an <math>n\times n</math> matrix <math>A=(a_{ij})</math> is called a generalised Cartan matrix if it satisfies the conditions | ||
| + | # <math>a_{ii}=2</math> for <math>i=1,\cdots,n</math> | ||
| + | # <math>a_{ij}\in \mathbb{Z}</math> and <math>a_{ij}\leq 0</math> if <math>i\neq j</math> | ||
| + | # <math>a_{ij}=0</math> impies <math>a_{ji}=0</math> | ||
| + | |||
| + | ==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 | ||
| + | ** <math>\text{det }A\neq 0</math> | ||
| + | ** there exists <math>u>0</math> with <math>Au>0</math> | ||
| + | ** <math>Au\geq 0</math> implies <math>u>0</math> or <math>u=0</math> | ||
| + | * A GCM A has affine type if | ||
| + | ** <math>\text{corank }A=1</math> | ||
| + | ** there exists <math>u>0</math> such that <math>Au=0</math> | ||
| + | ** <math>Au\geq 0</math> implies <math>Au=0</math> | ||
| + | * A GCM A has indefinite type if | ||
| + | ** there exists <math>u>0</math> with <math>Au<0</math> | ||
| + | ** <math>Au\geq 0</math> and <math>u\geq 0</math> implies <math>u>0</math> or <math>u=0</math> | ||
| + | |||
| + | |||
| + | ====main result==== | ||
| + | * Let <math>A</math> be an indecomposable GCM. Then exactly one of the following three possibilities holds: | ||
| + | ** <math>A</math> has finite type | ||
| + | ** <math>A</math> has affine type | ||
| + | ** <math>A</math> has indefinite type | ||
| + | * Moreover the type of <math>A^t</math> is the same as the type of <math>A</math>. | ||
| + | ;cor | ||
| + | Let <math>A</math> be an indecomposable GCM. Then | ||
| + | # 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 <math>u>0</math> with <math>Au=0</math> | ||
| + | # 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 | ||
| + | * 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]] | * [[Killing form and invariant symmetric bilinear form]] | ||
| + | * [[Symmetrizable generalized Cartan matrix]] | ||
| + | |||
| + | ==computational resource== | ||
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxWGFXeWtoYTVyTlk/edit | ||
| + | [[분류:Lie theory]] | ||
| + | [[분류:migrate]] | ||
| − | == | + | ==메타데이터== |
| − | * https:// | + | ===위키데이터=== |
| + | * ID : [https://www.wikidata.org/wiki/Q2004951 Q2004951] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'cartan'}, {'LEMMA': 'matrix'}] | ||
2021년 2월 17일 (수) 02:33 기준 최신판
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
- \(a_{ii}=2\) 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
- \(\text{corank }A=1\)
- 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
- A GCM A has finite type if and only if there exists \(u>0\) with \(Au>0\)
- A GCM A has affine type if and only if there exists \(u>0\) with \(Au=0\)
- 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.
- Rank 2 generalized Cartan matrix
- Skew-symmetrizable matrix
- Killing form and invariant symmetric bilinear form
- Symmetrizable generalized Cartan matrix
computational resource
메타데이터
위키데이터
- ID : Q2004951
Spacy 패턴 목록
- [{'LOWER': 'cartan'}, {'LEMMA': 'matrix'}]