Affine sl(2)

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
  \(\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}\)
  

 

 

level k highest weight representation

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