Affine sl(2)

Gannon 190p, 193p, 196p,371p

 

 

construction

• Let \(\mathfrak{g}\) be a semisimple Lie algebra with root system \(\Phi\) and the invariant form \(<\cdot,\cdot>\)
• say \(\mathfrak{g}=A_2\),  \(\Phi=\{\alpha_1,\alpha_2\}\)
• Cartan matrix
  \(\mathbf{A} = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}\)
  
• Find the highest root \(\sum a_l\alpha_l\)
  
  • \(\alpha_1+\alpha_2\)
    
• Add another simple root \(\alpha_0\) to the root system \(\Phi\)
  
  • \(\alpha_0=-\alpha_1-\alpha_2\)
    
• Construct a new Cartan matrix
  \(A' = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}\)
  
• Note that this matrix has rank 2 since \((1,1,1)\) belongs to the null space