"Affine sl(2)"의 두 판 사이의 차이
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(사용자 2명의 중간 판 45개는 보이지 않습니다) | |||
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− | + | ==introduction== | |
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* affine sl(2) <math>A^{(1)}_1</math> | * affine sl(2) <math>A^{(1)}_1</math> | ||
+ | * {{수학노트|url=Sl(2)의_유한차원_표현론}} | ||
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− | + | ==construction from semisimple Lie algebra== | |
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− | + | * this is borrowed from [[affine Kac-Moody algebra]] | |
+ | * Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math>\langle \cdot,\cdot \rangle</math> | ||
+ | * say <math>\mathfrak{g}=A_1</math>, <math>\Phi=\{\alpha,-\alpha\}</math> | ||
+ | * Cartan matrix<math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math> | ||
+ | * Find the highest root <math>\alpha</math> | ||
+ | * Add another simple root <math>\alpha_0</math> to the root system <math>\Phi</math> which is <math>\alpha_0=-\alpha</math>, but we regard this as an independent one now. | ||
+ | * Construct a new Cartan matrix<math>A' = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix}</math> | ||
+ | * Note that this matrix has rank 1 since <math>(1,1)</math> belongs to the null space | ||
+ | * construct a Lie algebra from the new Cartan matrix <math>A'</math> | ||
+ | * Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix | ||
+ | :<math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0 \end{pmatrix}</math> | ||
− | + | ==basic quantities== | |
− | * a_i=1< | + | * <math>a_i=1</math> |
− | * c_i=a_i^{\vee}=1< | + | * <math>c_i=a_i^{\vee}=1</math> |
− | * a_{ij}< | + | * <math>a_{ij}</math> |
− | * coxeter number 2 | + | * coxeter number 2 |
− | * dual Coxeter number 2 | + | * dual Coxeter number 2 |
− | * Weyl vector | + | * Weyl vector |
− | + | ||
− | + | ||
− | + | ==root systems== | |
* <math>\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | * <math>\Phi=\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}\cup \{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | ||
− | * real roots | + | * real roots |
− | ** <math>\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}</math | + | ** <math>\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}</math> |
− | * imaginary | + | * imaginary roots |
** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | ** <math>\{n\delta|n\in\mathbb{Z},n\neq 0\}</math> | ||
− | ** <math>\delta=\alpha_0+\alpha_1</math | + | ** <math>\delta=\alpha_0+\alpha_1</math> |
− | * simple roots | + | * simple roots |
− | ** <math>\alpha_0,\alpha_1</math | + | ** <math>\alpha_0,\alpha_1</math> |
− | * positive roots | + | * positive roots |
− | + | :<math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math> | |
− | + | ||
− | + | ||
− | + | ==fixing a Cartan subalgebra and its dual== | |
* H is a 3-dimensional space | * H is a 3-dimensional space | ||
+ | * basis of the Cartan subalgebra H (this defines C and l_0 also) | ||
+ | :<math>h_0=C-h_1 \\ | ||
+ | h_1\\d=-l_0</math> | ||
+ | * basis of the dual of H : <math>\omega_0,\alpha_0,\alpha_1</math> | ||
+ | * pairing | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | * dual basis for H : <math>\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1</math> | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | * Weyl vector : <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math> | ||
+ | |||
− | + | ==killing form== | |
− | + | * invariant symmetric non-deg bilinear forms, <math>\langle h_i,h_j\rangle =A_{ij}</math>, <math>\langle h_0,d\rangle =1</math>, <math>\langle h_1,d\rangle =0</math>, <math>\langle d,d\rangle =0</math>, | |
− | + | * with centers (note that <math>C=h_0+h_1</math>), <math>\langle C,h_0\rangle =0</math>, <math>\langle C,h_1\rangle =0</math>, <math>\langle C,d\rangle =1</math>, | |
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− | * invariant symmetric non-deg bilinear forms | ||
− | * with centers (note that | ||
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− | * [ | + | ==explicit construction== |
+ | * start with a semisimple Lie algebra <math>\mathfrak{g}</math> with invariant form <math>\langle \cdot,\cdot\rangle </math>, | ||
+ | * make a vector space from it, | ||
+ | * Construct a Loop algbera <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math> | ||
+ | * Let <math>\alpha(m)=\alpha\otimes t^m</math>, | ||
+ | * Add a central element to get a central extension <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math>, and give a bracket :<math>[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c</math> | ||
+ | :<math>[H(m),E(n)]=2E\otimes t^{m+n}</math> | ||
+ | :<math>[H(m),F(n)]=-2F\otimes t^{m+n}</math> | ||
+ | :<math>[E(m),E(n)]=[F(m),F(n)]=0</math> | ||
+ | :<math>\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0</math> | ||
+ | * Add a derivation <math>d</math>, <math>d=t\frac{d}{dt}</math> to get <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d</math> | ||
+ | :<math>d(\alpha(n))=n\alpha(n)</math> | ||
+ | :<math>d(c)=0</math> | ||
+ | :<math>\langle c,d\rangle =0</math> | ||
+ | * Define a Lie bracket <math>[d,x]=d(x)</math> | ||
− | |||
− | + | ==level k highest weight representation== | |
− | + | * integrable highest weight | |
− | + | :<math>\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1,\quad \lambda_{i}\in\mathbb{N}</math> | |
− | + | * level | |
− | + | :<math>k=\lambda_{0}+\lambda_{1}\in\mathbb{N}</math> | |
− | * integrable highest weight | ||
− | * level | ||
* therefore <math>\lambda_{0}\in\{0,1,\cdots,k\}</math> | * therefore <math>\lambda_{0}\in\{0,1,\cdots,k\}</math> | ||
− | + | ||
− | + | ||
− | + | ===central charge=== | |
* [[unitary representations of affine Kac-Moody algebras]] | * [[unitary representations of affine Kac-Moody algebras]] | ||
+ | * central charge (depends on the level only) | ||
+ | :<math>c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}</math> | ||
+ | * conformal weight | ||
+ | :<math>h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}</math> | ||
+ | * definition of conformal anomaly | ||
+ | :<math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}</math> | ||
+ | * strange formula | ||
+ | :<math>\frac{\langle \rho,\rho \rangle}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}</math> | ||
+ | * very strange formula | ||
+ | * conformal anomaly | ||
+ | :<math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}</math> | ||
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− | + | ||
− | + | ===vertex operator construction=== | |
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+ | ==characters of irreducible representations== | ||
+ | * [[Weyl-Kac character formula]] | ||
+ | :<math> | ||
+ | \operatorname{ch} L(\lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}} | ||
+ | </math> | ||
+ | * Let <math>M=M^{*}=\mathbb{Z}\alpha_1</math> | ||
+ | * the affine Weyl group <math>W=t(M^{*})W^{0}</math> where <math>t(M^{*})</math> is the set <math>t_{\alpha} : H^{*} \to H^{*}</math> given by | ||
+ | :<math> | ||
+ | t_{\alpha}(\lambda)=\lambda+\lambda(c)\alpha-\left (\langle \lambda, \alpha \rangle +\frac{1}{2}\langle \alpha,\alpha \rangle \lambda(c) \right)\delta | ||
+ | </math> | ||
+ | * note that this is linear | ||
+ | * <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math> | ||
+ | * <math>s_{\alpha_1}(\omega_0+\omega_1)=3\omega_0-\omega_1</math> | ||
+ | * in general | ||
+ | :<math> | ||
+ | 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 | ||
+ | </math> | ||
+ | * <math>t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta</math> | ||
+ | * <math>t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta</math> | ||
+ | * <math>w\in W</math> can be written as <math>(n\alpha_1,\pm 1)</math> | ||
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− | + | ===denominator formula=== | |
+ | * if <math>w=(n\alpha_1,1)</math>, <math>e^{w\cdot 0}=e^{w\rho-\rho}=e^{2n\alpha_1-n(2n+1)\delta}</math> | ||
+ | * if <math>w=(n\alpha_1,-1)</math>, <math>e^{w\cdot 0}=e^{w\rho-\rho}=e^{-(2n-1)\alpha_1-n(2n-1)\delta}</math> | ||
+ | * let us write down the Weyl-Kac denominator formula explicitly | ||
+ | :<math> | ||
+ | \sum_{w\in W} (-1)^{\ell(w)}e^{w\rho-\rho} = \prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}\label{WK} | ||
+ | </math> | ||
+ | * the LHS of \ref{WK} can be written as | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | where <math>z=e^{-\alpha_1}</math> and <math>q=e^{-\delta}</math> | ||
+ | * the RHS of \ref{WK} can be written as | ||
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | from <math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math> | ||
+ | * we obtain {{수학노트|url=자코비_삼중곱(Jacobi_triple_product)}} | ||
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− | < | + | ===basic representation=== |
+ | * Let <math>\lambda=\omega_0</math> | ||
+ | * let us use the Weyl-Kac formula | ||
+ | :<math> | ||
+ | \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}} | ||
+ | </math> | ||
+ | * if <math>w=(n\alpha_1,1)</math>, <math>e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-3 \delta n^2+3 \alpha _1 n-\delta n+\omega _0}</math> | ||
+ | * if <math>w=(n\alpha_1,-1)</math>, <math>e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-\alpha _1-3 \delta n^2+3 \alpha _1 n+\delta n+\omega _0}</math> | ||
+ | * we get | ||
+ | :<math> | ||
+ | \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}} | ||
+ | </math> | ||
+ | * this can be rewritten as | ||
+ | :<math> | ||
+ | \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 }} | ||
+ | </math> | ||
+ | where <math>z=e^{-\alpha_1}, q = e^{−\delta}</math>. | ||
+ | ===highest weight representations=== | ||
+ | * level <math>k</math> | ||
+ | * highest weight <math>\omega=(k-l)\omega_0+l\omega_1</math> | ||
+ | * character | ||
+ | :<math> | ||
+ | \chi(L(\omega))=\frac{\theta_{k+2,l+1}-\theta_{k+2,-l-1}}{\theta_{2,1}-\theta_{2,-1}} | ||
+ | </math> | ||
+ | where | ||
+ | :<math> | ||
+ | \theta_{k,l}=\sum_{r\in \mathbb{Z}+\frac{l}{2k}}e^{kr}q^{kr^2} | ||
+ | </math> | ||
+ | |||
+ | ==related items== | ||
+ | * [[Modular invariant partition functions of affine sl(2)]] | ||
* [[sl(2) - orthogonal polynomials and Lie theory]] | * [[sl(2) - orthogonal polynomials and Lie theory]] | ||
* [[vertex algebras]] | * [[vertex algebras]] | ||
+ | * [[Quantum affine sl(2)]] | ||
+ | |||
− | + | ==computational resource== | |
− | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit | |
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− | + | ==books== | |
− | + | * Gannon 190p, 193p, 196p,371p | |
− | + | ||
+ | ==articles== | ||
+ | * Zeitlin, Anton M. “On the Unitary Representations of the Affine <math>ax+b</math>-Group, <math>\widehat{sl}(2,\mathbb{R})</math> 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 <math>\widehat{\mathfrak{sl}}_2</math> 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)”. <em>Communications in Mathematical Physics</em> 62 (1): 43-53. doi:[http://dx.doi.org/10.1007/BF01940329 10.1007/BF01940329]. | * Lepowsky, James, and Robert Lee Wilson. 1978. “Construction of the affine Lie algebraA 1 (1)”. <em>Communications in Mathematical Physics</em> 62 (1): 43-53. doi:[http://dx.doi.org/10.1007/BF01940329 10.1007/BF01940329]. | ||
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− | + | [[분류:Lie theory]] | |
− | + | [[분류:migrate]] | |
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2020년 12월 28일 (월) 05:24 기준 최신판
introduction
- affine sl(2) \(A^{(1)}_1\)
- 틀:수학노트
construction from semisimple Lie algebra
- this is borrowed from affine Kac-Moody algebra
- 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
\[m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c_{\lambda}}{24}\]
vertex operator construction
characters of irreducible representations
\[ \operatorname{ch} L(\lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}} \]
- 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 \[ \theta_{k,l}=\sum_{r\in \mathbb{Z}+\frac{l}{2k}}e^{kr}q^{kr^2} \]
- Modular invariant partition functions of affine sl(2)
- sl(2) - orthogonal polynomials and Lie theory
- vertex algebras
- Quantum affine sl(2)
computational resource
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.