Affine sl(2)

introduction

• affine sl(2) \(A^{(1)}_1\)
• 틀:수학노트

 

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 \(\langle \cdot,\cdot \rangle\)
• 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 the dual of H \[\omega_0,\alpha_0,\alpha_1\]
  
• pairing

$$ \begin{array}{cccc} {} & \alpha _0 & \alpha _1 & \omega _0 \\ h_0 & 2 & -2 & 1 \\ h_1 & -2 & 2 &0 \\ d & 1 & 0 & 0 \\ \end{array} $$

• dual basis for H \[\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1\]
  

$$ \begin{array}{cccc} {} & \omega_0 & \omega_1 & \delta \\ h_0 & 1 & 0 & 0 \\ h_1 & 0 & 1 &0 \\ d & 0 & 0 & a_0=1 \\ \end{array} $$

• Weyl vector \[\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1\]

 

killing form

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

 

explicit construction

• start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $,
• make a vector space from it,
• Construct a Loop algbera $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$
• Let $\alpha(m)=\alpha\otimes t^m$,
• Add a central element to get a central extension $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c$, and give a bracket $$[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c$$

$$[H(m),E(n)]=2E\otimes t^{m+n}$$ $$[H(m),F(n)]=-2F\otimes t^{m+n}$$ $$[E(m),E(n)]=[F(m),F(n)]=0$$ $$\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0$$

• Add a derivation $d$, $d=t\frac{d}{dt}$ to get $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$

$$d(\alpha(n))=n\alpha(n)$$ $$d(c)=0$$ $$\langle c,d\rangle =0$$

• Define a Lie bracket $[d,x]=d(x)$

 

denominator formula

• Let $M=M^{*}=\mathbb{Z}\alpha_1$
• the affine Weyl group $W=t(M^{*})W^{0}$ where $t(M^{*})$ is the set $t_{\alpha} : H^{*} \to H^{*}$ given by

$$ t_{\alpha}(\lambda)=\lambda+\lambda(c)\alpha-\left (\langle \lambda, \alpha \rangle +\frac{1}{2}\langle \alpha,\alpha \rangle \lambda(c) \right)\delta $$

• so $w\in W$ can be written as $(n\alpha_1,\pm \alpha_1)$
• if $w=(n\alpha_1,\alpha_1)$, $w(e^{\rho})-e^{\rho}=2n\alpha_1-n(2n+1)\delta$
• if $w=(n\alpha_1,-\alpha_1)$, $w(e^{\rho})-e^{\rho}=-(2n+1)\alpha_1-n(2n+1)\delta$
• let us write down the Weyl-Kac character formula explicitly

$$ {\sum_{w\in W} (-1)^{\ell(w)}(w(e^{\rho})-e^{\rho}) = \prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}\label{WK} $$

• the LHS of \ref{WK} can be written as

$$ \begin{align} \sum_{w\in W} (-1)^{\ell(w)}(w(e^{\rho})-e^{\rho})&=\sum_{n}e^{2n\alpha_1-n(2n+1)\delta}-\sum_{n}e^{-(2n+1)\alpha_1-n(2n+1)\delta}\\ & =\sum_{n}z^{-2n}q^{n(2n+1)}-\sum_{n}z^{2n+1}q^{n(2n+1)}\\ & =\sum_{m}(-1)^m z^{m}q^{m(m-1)/2} \end{align} $$ where $z=e^{-\alpha_1}$ and $q=e^{-\delta}$

• the RHS of \ref{WK} can be written as

$$ \begin{align} \prod_{\alpha\in \Phi^{+}}(1-e^{-\alpha})&=(1-e^{-\alpha_1})\prod_{n=1}^{\infty}(1-e^{-\alpha_1-n\delta})(1-e^{\alpha_1-n\delta})(1-e^{-n\delta})\\ & = \prod _{n=1}^{\infty } \left(1-zq^{n-1}\right)\left(1-z^{-1}q^n\right)\left(1-q^n\right) \end{align} $$ from \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)

• we obtain 틀:수학노트

 

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

 

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit

 

books

• Gannon 190p, 193p, 196p,371p

 

articles

• Lepowsky, James, and Robert Lee Wilson. 1978. “Construction of the affine Lie algebraA 1 (1)”. Communications in Mathematical Physics 62 (1): 43-53. doi:10.1007/BF01940329.