Affine sl(2)

introduction

• affine sl(2) \(A^{(1)}_1\)

 

construction from semisimple Lie algebra

• 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 \(\mathfrak{g}=A_1\),  \(\Phi=\{\alpha,-\alpha\}\)
• Cartan matrix
  \(\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}\)
  
• Find the highest root  \(\alpha\)
  
• Add another simple root \(\alpha_0\) to the root system \(\Phi\) which is \(\alpha_0=-\alpha\), but we regard this as an independent one now.
  
• 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

• H is a 3-dimensional space

• basis of the Cartan subalgebra H (this defines C and l_0 also)
  \(h_0=C-h_1\)
  \(h_1\)
  \(d=-l_0\)
  
• basis of dual Cartan algebra
  \(\omega_0,\alpha_0,\alpha_1\)
  
• dual basis for H
  \(\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1\)
  
• Weyl vector
  \(\rho=\omega_0+\omega_1\)
  

• pairing
  \(\alpha_0(h_0)=2\)
  \(\alpha_0(h_1)=-2\)
  \(\alpha_0(d)=1\)
  \(\alpha_1(h_0)=-2\)
  \(\alpha_1(h_1)=2\)
  \(\alpha_1(d)=0\)
  \(\omega_0(h_0)=1\)
  \(\omega_0(h_1)=0\)
  \(\omega_0(d)=0\)
  

 

 

killing form

• invariant symmetric non-deg bilinear forms
  \(<h_i,h_j>=A_{ij}\)
  \(<h_0,d>=1\)
  \(<h_1,d>=0\)
  \(<d,d>=0\)
  
• with centers (note that C=h_0+h_1)
  \(<C,h_0>=0\)
  \(<C,h_1>=0\)
  \(<C,d>=1\)
  

 

 

explicit construction

• start with a semisimple Lie algebra \(\mathfrak{g}\) with invariant form \(<\cdot,\cdot>\)
  
• make a vector space from it
  
• Construct a Loop algbera
  \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\)
  \(\alpha(m)=\alpha\otimes t^m\)
  
•  
  Add a central element to get a central extension and give a bracket
  \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\)
  \([E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c\)
  \([H(m),E(n)]=2E\otimes t^{m+n}\)
  \([H(m),F(n)]=-2F\otimes t^{m+n}\)
  \([E(m),E(n)]=[F(m),F(n)]=0\)
  \(<c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c>=0\)
  
• Add a derivation \(d\)
  \(d=t\frac{d}{dt}\)
  \(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d\)
  \(d(\alpha(n))=n\alpha(n)\)
  \(d(c)=0\)
  \(<c,d>=0\)
  
• Define a Lie bracket
  \([d,x]=d(x)\)
  

 

 

denominator formula

• Weyl-Kac character formula
  \({\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\)
  

 

 

 

level k highest weight representation

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

 

 

central charge

• unitary representations of affine Kac-Moody algebras

• central charge (depends on the level only)
  \(c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}\)
  
• conformal weight
  \(h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}\)
  
• definition of conformal anomaly
  \(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}\)
  

• strange formula
  \(\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}\)
  
• very strange formula
  
• conformal anomaly 
  \(m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c(\lambda)}{24}\)
  

 

 

 

 

 

 

related items

• sl(2) - orthogonal polynomials and Lie theory
• vertex algebras