"Degrees and exponents"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
imported>Pythagoras0
4번째 줄: 4번째 줄:
 
*  eigenvalues of incidence matrices of Dynkin diagram<br>
 
*  eigenvalues of incidence matrices of Dynkin diagram<br>
 
* http://pythagoras0.springnote.com/pages/1938682/attachments/3170605
 
* http://pythagoras0.springnote.com/pages/1938682/attachments/3170605
 +
  
 
==Cartan matrix==
 
==Cartan matrix==
11번째 줄: 12번째 줄:
 
* <math>m_{i}</math> is called the exponents<br>
 
* <math>m_{i}</math> is called the exponents<br>
 
* <math>d_{i}=m_{i}+1</math> is called a degree<br>
 
* <math>d_{i}=m_{i}+1</math> is called a degree<br>
 +
  
 
==adjacency matrix==
 
==adjacency matrix==
16번째 줄: 18번째 줄:
 
*  h : Coxeter number<br>
 
*  h : Coxeter number<br>
 
*  eigenvalue <math>2\cos(\pi l_n/h)</math><br>
 
*  eigenvalue <math>2\cos(\pi l_n/h)</math><br>
 +
 +
 +
 +
==degree and exponent of simple Lie algebra==
 +
 +
*  appears in invariant theory<br>
 +
*  can also be seen as eigenvalues of Cartan matrix or incidence matrix of the Dynkin diagram<br>
 +
*  for incidence matrix, the eigenvalues are given by<br><math>2\cos(\pi l_n/h)</math><br> where h is the Coxeter number and <math>l_i</math>'s are the exponents<br>
 +
*  example : A4 Cartan matrix has the Coxeter number 5<br><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><br> incidence matrix<br><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><br> eigenvalues of the incidence matrix<br><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><br> you can evalutate the following in mathematica to get the same set<br>
 +
*#  Table[2 Cos[Pi*l/5], {l, 1, 4}]<br> <br>
 +
 +
  
 
   
 
   

2013년 1월 27일 (일) 15:23 판

introduction


Cartan matrix

  • h : Coxeter number
  • eigenvalue
    \(4\sin^2(\frac{m_{i}\pi}{2h})\)
  • \(m_{i}\) 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
    \(\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),\frac{1}{2} \left(1+\sqrt{5}\right),\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1+\sqrt{5}\right)\right\}\)
    you can evalutate the following in mathematica to get the same set
    1. Table[2 Cos[Pi*l/5], {l, 1, 4}]



  1. Table[Simplify[2 Cos[Pi*l/5]], {l, 1, 4}]
    Table[Simplify[4 Sin[Pi*l/10]^2], {l, 1, 4}]



homological algebraic characterization

For a s.s. Lie algebra L

(a)H'(L) is a free super- commutative algebra with homogeneous generator in degrees 2m_1+1,\cdots,2m_l+1





history



related items

encyclopedia




books




articles

원본 주소 "https://wiki.mathnt.net/index.php?title=Degrees_and_exponents&oldid=42006"