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

\[ \begin{array}{c|ccc} {} & \alpha _0 & \alpha _1 & \omega _0 \\ \hline h_0 & 2 & -2 & 1 \\ h_1 & -2 & 2 &0 \\ d & 1 & 0 & 0 \\ \end{array} \]

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

\[ \begin{array}{c|ccc} {} & \omega_0 & \omega_1 & \delta \\ \hline h_0 & 1 & 0 & 0 \\ h_1 & 0 & 1 &0 \\ d & 0 & 0 & a_0=1 \\ \end{array} \]

• 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 \([d,x]=d(x)\)

level k highest weight representation

• integrable highest weight

\[\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1,\quad \lambda_{i}\in\mathbb{N}\]

• 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

\[\frac{\langle \rho,\rho \rangle}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}\]

• very strange formula
• conformal anomaly

\[m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}\]

 

characters of irreducible representations

• Weyl-Kac character formula

\[ \operatorname{ch} L(\lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}} \]

• 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

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

• note that this is linear
• \(\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1\)
• \(s_{\alpha_1}(\omega_0+\omega_1)=3\omega_0-\omega_1\)
• in general

\[ s_{\alpha_0}(m\omega_0+n\omega_1)=-m \delta - m \omega_0 + (2 m + n) \omega_1\\ s_{\alpha_1}(m\omega_0+n\omega_1)=(m+2n)\omega_0-n\omega_1 \]

• \(t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta\)
• \(t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta\)
• \(w\in W\) can be written as \((n\alpha_1,\pm 1)\)

denominator formula

• if \(w=(n\alpha_1,1)\), \(e^{w\cdot 0}=e^{w\rho-\rho}=e^{2n\alpha_1-n(2n+1)\delta}\)
• if \(w=(n\alpha_1,-1)\), \(e^{w\cdot 0}=e^{w\rho-\rho}=e^{-(2n-1)\alpha_1-n(2n-1)\delta}\)
• let us write down the Weyl-Kac denominator formula explicitly

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

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

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

 

basic representation

• Let \(\lambda=\omega_0\)
• let us use the Weyl-Kac formula

\[ \operatorname{ch} L(\omega_0)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \]

• if \(w=(n\alpha_1,1)\), \(e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-3 \delta n^2+3 \alpha _1 n-\delta n+\omega _0}\)
• if \(w=(n\alpha_1,-1)\), \(e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-\alpha _1-3 \delta n^2+3 \alpha _1 n+\delta n+\omega _0}\)
• we get

\[ \operatorname{ch} L(\omega_0)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot 0}} \]

• this can be rewritten as

\[ \operatorname{ch} L(\omega_0)=\frac{\sum_{\mu\in Q}e^{\omega_0+\mu-\frac{1}{2}\langle \mu,\mu \rangle \delta}}{\prod_{k>0}(1-q^k)}=\frac{e^{\omega_0}\sum _{n=-\infty }^{\infty } z^{-n} q^{n^2}}{(q;q)_{\infty }} \] where \(z=e^{-\alpha_1}, q = e^{−\delta}\).

highest weight representations

• level \(k\)
• highest weight \(\omega=(k-l)\omega_0+l\omega_1\)
• character

\[ \chi(L(\omega))=\frac{\theta_{k+2,l+1}-\theta_{k+2,-l-1}}{\theta_{2,1}-\theta_{2,-1}} \] where \[ \theta_{k,l}=\sum_{r\in \mathbb{Z}+\frac{l}{2k}}e^{kr}q^{kr^2} \]

related items

• Modular invariant partition functions of affine sl(2)
• sl(2) - orthogonal polynomials and Lie theory
• vertex algebras
• Quantum affine sl(2)

 

computational resource

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

 

books

• Gannon 190p, 193p, 196p,371p

 

articles

• Zeitlin, Anton M. “On the Unitary Representations of the Affine \(ax+b\)-Group, \(\widehat{sl}(2,\mathbb{R})\) and Their Relatives.” arXiv:1509.06072 [hep-Th, Physics:math-Ph], September 20, 2015. http://arxiv.org/abs/1509.06072.
• Bakalov, Bojko, and Daniel Fleisher. “Bosonizations of \(\widehat{\mathfrak{sl}}_2\) and Integrable Hierarchies.” arXiv:1407.5335 [math], July 20, 2014. http://arxiv.org/abs/1407.5335.
• Dong, Jilan, and Naihuan Jing. 2014. “Realizations of Affine Lie Algebra A_^(1) at Negative Levels.” arXiv:1405.0339 [hep-Th], May. doi:10.1007/978-3-642-55361-5_36. http://arxiv.org/abs/1405.0339.
• 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.