"Affine sl(2)"의 두 판 사이의 차이

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29번째 줄: 29번째 줄:
 
*  c_i=a_i^{\vee}=1<br>
 
*  c_i=a_i^{\vee}=1<br>
 
*  a_{ij}<br>
 
*  a_{ij}<br>
 +
*  coxeter number<br>
 +
**  2<br>
 
*  dual Coxeter number<br>
 
*  dual Coxeter number<br>
 +
**  2<br>
 
*  Weyl vector<br>
 
*  Weyl vector<br>
  
43번째 줄: 46번째 줄:
 
*  imaginary roots   <br>
 
*  imaginary roots   <br>
 
** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math>
 
** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math>
 +
** <math>\delta=\alpha_0+\alpha_1</math><br>
 
*  simple roots<br>
 
*  simple roots<br>
 
**   <br><math>\alpha_0,\alpha_1</math><br>
 
**   <br><math>\alpha_0,\alpha_1</math><br>
*  positive roots      <br>
+
*  positive roots<br>
* <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math><br>
+
** <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math><br>
 <br>
+
 
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2010년 3월 22일 (월) 14:57 판

Gannon 190p, 193p, 196p,371p

 

 

construction
  • this is borrowed from affine Kac-Moody algebra entry
  • Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
  • say \(\mathfrak{g}=A_1\),  \(\Phi=\{\alpha\}\)
  • Cartan matrix
    \(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\)
  • Find the highest root 
    • \(\alpha\)
  • Add another simple root \(\alpha_0\) to the root system \(\Phi\)
    • \(\alpha_0=-\alpha\)
  • Construct a new Cartan matrix
    \(A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}\)
  • Note that this matrix has rank 1 since \((1,1)\) belongs to the null space
  • construct a Lie algebra from the new Cartan matrix \(A'\)
  • Add a outer derivation\(d=-l_0\) to compensate the degeneracy of the Cartan matrix
    \(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)

 

 

basic quantities
  • a_i=1
  • c_i=a_i^{\vee}=1
  • a_{ij}
  • coxeter number
    • 2
  • dual Coxeter number
    • 2
  • Weyl vector

 

 

root systems
  • \(\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}\)
  • real roots
    • \(\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\)
  • imaginary roots   
    • \(\{n\delta|n\in\mathbb{Z},n\neq 0\}\)
    • \(\delta=\alpha_0+\alpha_1\)
  • simple roots
    •  
      \(\alpha_0,\alpha_1\)
  • positive roots
    • \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)

 

 

 

fixing a Cartan subalgebra and its dual
  • basis of the Cartan subalgebra H
    \(h_0=C-h_1\)
    \(h_1\)
    \(d=-l_0\)
  • dual basis for H
    \(\omega_0,\omega_1,\delta\)

 

 

killing form
  • invariant symmetric non-deg bilinear forms
    \(<h_i,h_j>=A_{ij}\)
    \(<h_0,d>=1\)
    \(<h_1,d>=0\)
    \(<d,d>=0\)

 

 

level k highest weight representation
  • integrable highest weight
    \(\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1\), \(\lambda_{i}\in\mathbb{N}\)
  • level
    \(k=\lambda_{0}+\lambda_{1}\)
  • therefore \(\lambda_{0}\in\{0,1,\cdots,k\}\)