"Degrees and exponents"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
| − | * eigenvalues of Cartan matrices | + | * eigenvalues of Cartan matrices |
| − | * eigenvalues of incidence matrices of Dynkin diagram | + | * eigenvalues of incidence matrices of Dynkin diagram |
* http://pythagoras0.springnote.com/pages/1938682/attachments/3170605 | * http://pythagoras0.springnote.com/pages/1938682/attachments/3170605 | ||
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==Cartan matrix== | ==Cartan matrix== | ||
| − | * h : [[Coxeter number and dual Coxeter number|Coxeter number]] | + | * h : [[Coxeter number and dual Coxeter number|Coxeter number]] |
| − | * eigenvalue | + | * eigenvalue <math>4\sin^2(m_{i}\pi/2h)</math> |
| − | * <math>m_{i}</math> is called the exponents | + | * <math>m_{i}</math> is called the exponents |
| − | * <math>d_{i}=m_{i}+1</math> is called a degree | + | * <math>d_{i}=m_{i}+1</math> is called a degree |
==adjacency matrix== | ==adjacency matrix== | ||
| − | * h : Coxeter number | + | * h : Coxeter number |
| − | * eigenvalue <math>2\cos(\pi l_n/h)</math | + | * eigenvalue <math>2\cos(\pi l_n/h)</math> |
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==degree and exponent of simple Lie algebra== | ==degree and exponent of simple Lie algebra== | ||
| − | * appears in invariant theory | + | * appears in invariant theory |
| − | * can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram | + | * can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram |
| − | * for incidence matrix, the eigenvalues are given by | + | * for incidence matrix, the eigenvalues are given by:<math>2\cos(\pi l_n/h)</math> where h is the Coxeter number and <math>l_i</math>'s are the exponents |
| − | * | + | ===example=== |
| − | + | * A4 Cartan matrix has the Coxeter number 5 | |
| − | + | :<math>\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)</math> * incidence matrix:<math>\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)</math> | |
| − | + | * eigenvalues of the incidence matrix:<math>\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}</math> | |
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==homological algebraic characterization== | ==homological algebraic characterization== | ||
| − | + | * For a semisimple. Lie algebra L | |
| − | For a | + | * $H^{\bullet}(L)$ is a free super-commutative algebra with homogeneous generator in degrees $2m_1+1,\cdots,2m_l+1$ |
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==encyclopedia== | ==encyclopedia== | ||
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* http://en.wikipedia.org/wiki/Coxeter_number | * http://en.wikipedia.org/wiki/Coxeter_number | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:Lie theory]] | [[분류:Lie theory]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 7월 5일 (금) 04:51 판
introduction
- eigenvalues of Cartan matrices
- eigenvalues of incidence matrices of Dynkin diagram
- http://pythagoras0.springnote.com/pages/1938682/attachments/3170605
Cartan matrix
- h : Coxeter number
- eigenvalue \(4\sin^2(m_{i}\pi/2h)\)
- \(m_{i}\) is called the exponents
- \(d_{i}=m_{i}+1\) is called a degree
adjacency matrix
- h : Coxeter number
- eigenvalue \(2\cos(\pi l_n/h)\)
degree and exponent of simple Lie algebra
- appears in invariant theory
- can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram
- for incidence matrix, the eigenvalues are given by\[2\cos(\pi l_n/h)\] where h is the Coxeter number and \(l_i\)'s are the exponents
example
- A4 Cartan matrix has the Coxeter number 5
\[\left( \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} \right)\] * incidence matrix\[\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right)\]
- eigenvalues of the incidence matrix\[\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}\]
homological algebraic characterization
- For a semisimple. Lie algebra L
- $H^{\bullet}(L)$ is a free super-commutative algebra with homogeneous generator in degrees $2m_1+1,\cdots,2m_l+1$
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