"Affine sl(2)"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
 
(사용자 2명의 중간 판 45개는 보이지 않습니다)
1번째 줄: 1번째 줄:
<h5>introduction</h5>
+
==introduction==
 
 
 
* affine sl(2) <math>A^{(1)}_1</math>
 
* affine sl(2) <math>A^{(1)}_1</math>
 +
* {{수학노트|url=Sl(2)의_유한차원_표현론}}
 +
  
 
+
==construction from semisimple Lie algebra==
 
 
 
 
 
 
<h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">construction from semisimple Lie algebra</h5>
 
 
 
* this is borrowed from [[affine Kac-Moody algebra]] entry
 
* Let <math>\mathfrak{g}</math> be a semisimple Lie algebra with root system <math>\Phi</math> and the invariant form <math><\cdot,\cdot></math>
 
* say <math>\mathfrak{g}=A_1</math>,  <math>\Phi=\{\alpha,-\alpha\}</math>
 
*  Cartan matrix<br><math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math><br>
 
*  Find the highest root  <math>\alpha</math><br>
 
*  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.<br>
 
*  Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & -2  \\ -2 & 2  \end{pmatrix}</math><br>
 
*  Note that this matrix has rank 1 since <math>(1,1)</math> belongs to the null space<br>
 
*  construct a Lie algebra from the new Cartan matrix <math>A'</math><br>
 
*  Add a outer derivation<math>d=-l_0</math> to compensate the degeneracy of the Cartan matrix<br><math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0  \end{pmatrix}</math><br>
 
 
 
 
 
  
 
+
* 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>
  
<h5 style="margin: 0px; line-height: 2em;">basic quantities</h5>
+
==basic quantities==
  
*  a_i=1<br>
+
<math>a_i=1</math>
*  c_i=a_i^{\vee}=1<br>
+
<math>c_i=a_i^{\vee}=1</math>
*  a_{ij}<br>
+
<math>a_{ij}</math>
*  coxeter number 2<br>
+
*  coxeter number 2
*  dual Coxeter number 2<br>
+
*  dual Coxeter number 2
*  Weyl vector<br>
+
*  Weyl vector
  
 
+
  
 
+
  
<h5>root systems</h5>
+
==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<br>
+
*  real roots
** <math>\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}</math><br>
+
** <math>\{\alpha+n\delta|\alpha\in\Phi^{0},n\in\mathbb{Z}\}</math>
*  imaginary roots   <br>
+
*  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><br>
+
** <math>\delta=\alpha_0+\alpha_1</math>
*  simple roots<br>
+
*  simple roots
** <math>\alpha_0,\alpha_1</math><br>
+
** <math>\alpha_0,\alpha_1</math>
*  positive roots<br>
+
*  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><br>
+
:<math>\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup  (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}</math>
  
 
+
  
 
+
  
<h5 style="margin: 0px; line-height: 2em;">fixing a Cartan subalgebra and its dual</h5>
+
==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>
 +
  
*  basis of the Cartan subalgebra H (this defines C and l_0 also)<br><math>h_0=C-h_1</math><br><math>h_1</math><br><math>d=-l_0</math><br>
+
==killing form==
*  basis of dual Cartan algebra<br><math>\omega_0,\alpha_0,\alpha_1</math><br>
+
*  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>,
*  dual basis for H<br><math>\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1</math><br>
+
*  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>,  
*  Weyl vector<br><math>\rho=\omega_0+\omega_1</math><br>
 
 
 
*  pairing<br><math>\alpha_0(h_0)=2</math><br><math>\alpha_0(h_1)=-2</math><br><math>\alpha_0(d)=1</math><br><math>\alpha_1(h_0)=-2</math><br><math>\alpha_1(h_1)=2</math><br><math>\alpha_1(d)=0</math><br><math>\omega_0(h_0)=1</math><br><math>\omega_0(h_1)=0</math><br><math>\omega_0(d)=0</math><br>
 
 
 
 
 
 
 
 
 
 
 
<h5>killing form</h5>
 
 
 
*  invariant symmetric non-deg bilinear forms<br><math><h_i,h_j>=A_{ij}</math><br><math><h_0,d>=1</math><br><math><h_1,d>=0</math><br><math><d,d>=0</math><br>
 
*  with centers (note that C=h_0+h_1)<br><math><C,h_0>=0</math><br><math><C,h_1>=0</math><br><math><C,d>=1</math><br>
 
 
 
 
 
 
 
 
 
 
 
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">explicit construction</h5>
 
 
 
*  start with a semisimple Lie algebra <math>\mathfrak{g}</math> with invariant form <math><\cdot,\cdot></math><br>
 
*  make a vector space from it<br>
 
*  Construct a Loop algbera<br><math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math><br><math>\alpha(m)=\alpha\otimes t^m</math><br>
 
*   <br> Add a central element to get a central extension and give a bracket<br><math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c</math><br><math>[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c</math><br><math>[H(m),E(n)]=2E\otimes t^{m+n}</math><br><math>[H(m),F(n)]=-2F\otimes t^{m+n}</math><br><math>[E(m),E(n)]=[F(m),F(n)]=0</math><br><math><c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c>=0</math><br>
 
*  Add a derivation <math>d</math><br><math>d=t\frac{d}{dt}</math><br><math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d</math><br><math>d(\alpha(n))=n\alpha(n)</math><br><math>d(c)=0</math><br><math><c,d>=0</math><br>
 
*  Define a Lie bracket<br><math>[d,x]=d(x)</math><br>
 
 
 
 
 
 
 
 
 
  
<h5>denominator formula</h5>
+
 +
  
* [[Weyl-Kac character formula]]<br><math>{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}</math><br>
+
==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>
<h5>level k highest weight representation</h5>
+
*  level
 
+
:<math>k=\lambda_{0}+\lambda_{1}\in\mathbb{N}</math>
*  integrable highest weight<br><math>\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1</math>, <math>\lambda_{i}\in\mathbb{N}</math><br>
 
*  level<br><math>k=\lambda_{0}+\lambda_{1}\in\mathbb{N}</math><br>
 
 
* therefore <math>\lambda_{0}\in\{0,1,\cdots,k\}</math>
 
* therefore <math>\lambda_{0}\in\{0,1,\cdots,k\}</math>
  
 
+
  
 
+
  
<h5>central charge</h5>
+
===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>
  
*  central charge (depends on the level only)<br><math>c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}</math><br>
 
*  conformal weight<br><math>h_{\lambda}=\frac{(\lambda|\lambda+2\rho)}{2(k+h^{\vee})}</math><br>
 
*  definition of conformal anomaly<br><math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}</math><br>
 
  
*  strange formula<br><math>\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}</math><br>
 
*  very strange formula<br>
 
*  conformal anomaly <br><math>m_{\Lambda}=\frac{(\Lambda+\rho)^2}{2(k+h^{\vee})}-\frac{\rho^2}{2h^{\vee}}=h_{\lambda}-\frac{c(\lambda)}{24}</math><br>
 
  
 
+
  
 
+
===vertex operator construction===
  
 
+
  
<h5 style="line-height: 2em; margin: 0px;">vertex operator construction</h5>
+
  
 
+
 +
==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>
  
 
 
  
 
+
===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)}}
 +
  
<h5>related items</h5>
+
===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
 
 
 
 
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
 
 
* http://en.wikipedia.org/wiki/
 
* http://www.scholarpedia.org/
 
* http://www.proofwiki.org/wiki/
 
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
 
 
 
 
 
 
 
 
 
 
<h5>books</h5>
 
  
* Gannon 190p, 193p, 196p,371p
+
* [[2010년 books and articles]]<br>
 
* http://gigapedia.info/1/
 
* http://gigapedia.info/1/
 
* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
 
+
==books==
  
 
+
* Gannon 190p, 193p, 196p,371p
  
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
+
  
 +
==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].
* http://www.ams.org/mathscinet
 
* http://www.zentralblatt-math.org/zmath/en/
 
* http://arxiv.org/
 
* http://www.pdf-search.org/
 
* http://pythagoras0.springnote.com/
 
* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
* http://dx.doi.org/
 
 
 
 
 
 
 
 
<h5>question and answers(Math Overflow)</h5>
 
 
* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
 
 
 
 
 
 
 
 
<h5>blogs</h5>
 
 
*  구글 블로그 검색<br>
 
**  http://blogsearch.google.com/blogsearch?q=<br>
 
** http://blogsearch.google.com/blogsearch?q=
 
* http://ncatlab.org/nlab/show/HomePage
 
 
 
 
 
 
 
 
<h5>experts on the field</h5>
 
 
* http://arxiv.org/
 
 
 
 
 
 
 
 
<h5>links</h5>
 
  
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+
[[분류:Lie theory]]
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
+
[[분류:migrate]]
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
* http://functions.wolfram.com/
 

2020년 12월 28일 (월) 05:24 기준 최신판

introduction


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

\[c_{\lambda}=\frac{k}{k+h^{\vee}}\text{dim }\mathfrak{\bar{g}}\]

  • 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 \(\Phi^{+}=\{\alpha+n\delta|\alpha\in\Phi^{0},n>0\}\cup (\Phi^{0})^{+}\cup \{n\delta|n\in\mathbb{Z},n> 0\}\)


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


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.