"Affine sl(2)"의 두 판 사이의 차이

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imported>Pythagoras0
 
(사용자 2명의 중간 판 6개는 보이지 않습니다)
2번째 줄: 2번째 줄:
 
* affine sl(2) <math>A^{(1)}_1</math>
 
* affine sl(2) <math>A^{(1)}_1</math>
 
* {{수학노트|url=Sl(2)의_유한차원_표현론}}
 
* {{수학노트|url=Sl(2)의_유한차원_표현론}}
 
+
  
 
==construction from semisimple Lie algebra==
 
==construction from semisimple Lie algebra==
  
* this is borrowed from [[affine Kac-Moody algebra]] entry
+
* 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>
 
* 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>
+
* 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>
+
*  Cartan matrix<math>\mathbf{A} = \begin{pmatrix} 2 \end{pmatrix}</math>
*  Find the highest root  <math>\alpha</math><br>
+
*  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.<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.
*  Construct a new Cartan matrix<br><math>A' = \begin{pmatrix} 2 & -2  \\ -2 & 2  \end{pmatrix}</math><br>
+
*  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<br>
+
*  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><br>
+
*  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
+
*  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><br>
+
:<math>\begin{pmatrix} 2 & -2 & 1\\ -2 & 2 &0 \\ 1 &0 & 0  \end{pmatrix}</math>
 
 
 
 
  
 
==basic quantities==
 
==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<br>
+
*  coxeter number 2
*  dual Coxeter number 2<br>
+
*  dual Coxeter number 2
*  Weyl vector<br>
+
*  Weyl vector
  
 
+
  
 
+
  
 
==root systems==
 
==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
 
*  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>
 
:<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==
 
==fixing a Cartan subalgebra and its dual==
58번째 줄: 56번째 줄:
 
*  basis of the dual of H : <math>\omega_0,\alpha_0,\alpha_1</math>
 
*  basis of the dual of H : <math>\omega_0,\alpha_0,\alpha_1</math>
 
*  pairing
 
*  pairing
$$
+
:<math>
 
\begin{array}{c|ccc}
 
\begin{array}{c|ccc}
 
  {} & \alpha _0 & \alpha _1 & \omega _0 \\
 
  {} & \alpha _0 & \alpha _1 & \omega _0 \\
66번째 줄: 64번째 줄:
 
  d & 1 & 0 & 0 \\
 
  d & 1 & 0 & 0 \\
 
\end{array}
 
\end{array}
$$
+
</math>
*  dual basis for H : <math>\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1</math><br>
+
*  dual basis for H : <math>\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1</math>
$$
+
:<math>
 
\begin{array}{c|ccc}
 
\begin{array}{c|ccc}
 
  {} & \omega_0 & \omega_1 & \delta \\
 
  {} & \omega_0 & \omega_1 & \delta \\
76번째 줄: 74번째 줄:
 
  d & 0 & 0 & a_0=1 \\
 
  d & 0 & 0 & a_0=1 \\
 
\end{array}
 
\end{array}
$$
+
</math>
 
*  Weyl vector : <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math>
 
*  Weyl vector : <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math>
 
+
  
 
==killing form==
 
==killing form==
*  invariant symmetric non-deg bilinear forms, $\langle h_i,h_j\rangle =A_{ij}$, $\langle h_0,d\rangle =1$, $\langle h_1,d\rangle =0$, $\langle d,d\rangle =0$,  
+
*  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 $C=h_0+h_1$), $\langle C,h_0\rangle =0$, $\langle C,h_1\rangle =0$, $\langle C,d\rangle =1$,  
+
*  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>,  
  
 
   
 
   
 
+
  
 
==explicit construction==
 
==explicit construction==
* start with a semisimple Lie algebra $\mathfrak{g}$ with invariant form $\langle \cdot,\cdot\rangle $,
+
* 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,
 
* make a vector space from it,
* Construct a Loop algbera $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]$
+
* Construct a Loop algbera <math>\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]</math>
* Let $\alpha(m)=\alpha\otimes t^m$,
+
* Let <math>\alpha(m)=\alpha\otimes t^m</math>,
* 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 $$[E(m),F(n)]=H\otimes t^{m+n}+m\delta_{m,-n}c$$
+
* 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>
$$[H(m),E(n)]=2E\otimes t^{m+n}$$
+
:<math>[H(m),E(n)]=2E\otimes t^{m+n}</math>
$$[H(m),F(n)]=-2F\otimes t^{m+n}$$
+
:<math>[H(m),F(n)]=-2F\otimes t^{m+n}</math>
$$[E(m),E(n)]=[F(m),F(n)]=0$$
+
:<math>[E(m),E(n)]=[F(m),F(n)]=0</math>
$$\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0$$
+
:<math>\langle c,\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c\rangle =0</math>
*  Add a derivation $d$, $d=t\frac{d}{dt}$ to get $\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c \oplus\mathbb{C}d$
+
*  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>
$$d(\alpha(n))=n\alpha(n)$$
+
:<math>d(\alpha(n))=n\alpha(n)</math>
$$d(c)=0$$
+
:<math>d(c)=0</math>
$$\langle c,d\rangle =0$$
+
:<math>\langle c,d\rangle =0</math>
*  Define a Lie bracket $[d,x]=d(x)$
+
*  Define a Lie bracket <math>[d,x]=d(x)</math>
  
  
141번째 줄: 139번째 줄:
 
   
 
   
  
 
+
 
==characters of irreducible representations==
 
==characters of irreducible representations==
 
* [[Weyl-Kac character formula]]
 
* [[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}}}
 
\operatorname{ch} L(\lambda)=\frac{\sum_{w\in W} (-1)^{\ell(w)}e^{w\cdot \lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}
$$
+
</math>
* Let $M=M^{*}=\mathbb{Z}\alpha_1$
+
* Let <math>M=M^{*}=\mathbb{Z}\alpha_1</math>
* the affine Weyl group $W=t(M^{*})W^{0}$ where $t(M^{*})$ is the set $t_{\alpha} : H^{*} \to H^{*}$ given by
+
* the affine Weyl group <math>W=t(M^{*})W^{0}</math> where <math>t(M^{*})</math> is the set <math>t_{\alpha} : H^{*} \to H^{*}</math> given by
$$
+
:<math>
 
t_{\alpha}(\lambda)=\lambda+\lambda(c)\alpha-\left (\langle \lambda, \alpha \rangle +\frac{1}{2}\langle \alpha,\alpha \rangle \lambda(c) \right)\delta
 
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
 
* note that this is linear
* $\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1$
+
* <math>\rho=\omega_0+\omega_1=2\omega_0+\frac{1}{2}\alpha_1</math>
* $s_{\alpha_1}(\omega_0+\omega_1)=3\omega_0-\omega_1$
+
* <math>s_{\alpha_1}(\omega_0+\omega_1)=3\omega_0-\omega_1</math>
 
* in general
 
* in general
$$
+
:<math>
 
s_{\alpha_0}(m\omega_0+n\omega_1)=-m \delta - m \omega_0 + (2 m + n) \omega_1\\
 
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
 
s_{\alpha_1}(m\omega_0+n\omega_1)=(m+2n)\omega_0-n\omega_1
$$
+
</math>
* $t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta$
+
* <math>t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta</math>
* $t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta$
+
* <math>t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta</math>
* $w\in W$ can be written as $(n\alpha_1,\pm 1)$
+
* <math>w\in W</math> can be written as <math>(n\alpha_1,\pm 1)</math>
  
  
 
===denominator formula===
 
===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 <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 $w=(n\alpha_1,-1)$, $e^{w\cdot 0}=e^{w\rho-\rho}=e^{-(2n-1)\alpha_1-n(2n-1)\delta}$
+
* 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
 
* 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}
 
\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
 
* the LHS of \ref{WK} can be written as
$$
+
:<math>
 
\begin{align}
 
\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_{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}\\
179번째 줄: 177번째 줄:
 
& =\sum_{m}(-1)^m z^{m}q^{m(m-1)/2}
 
& =\sum_{m}(-1)^m z^{m}q^{m(m-1)/2}
 
\end{align}
 
\end{align}
$$
+
</math>
where $z=e^{-\alpha_1}$ and $q=e^{-\delta}$
+
where <math>z=e^{-\alpha_1}</math> and <math>q=e^{-\delta}</math>
 
* the RHS of \ref{WK} can be written as
 
* the RHS of \ref{WK} can be written as
$$
+
:<math>
 
\begin{align}
 
\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_{\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)
 
& = \prod _{n=1}^{\infty } \left(1-zq^{n-1}\right)\left(1-z^{-1}q^n\right)\left(1-q^n\right)
 
\end{align}
 
\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>
 
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)}}
 
* we obtain {{수학노트|url=자코비_삼중곱(Jacobi_triple_product)}}
 
+
  
 
===basic representation===
 
===basic representation===
* Let $\lambda=\omega_0$
+
* Let <math>\lambda=\omega_0</math>
 
* let us use the Weyl-Kac formula
 
* 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}}
 
\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 $w=(n\alpha_1,1)$, $e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-3 \delta  n^2+3 \alpha _1 n-\delta  n+\omega _0}$
+
* if <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 $w=(n\alpha_1,-1)$, $e^{w\cdot \lambda}=e^{w(\lambda+\rho)-\rho}=e^{-\alpha _1-3 \delta  n^2+3 \alpha _1 n+\delta  n+\omega _0}$
+
* 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
 
* 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}}
 
\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
 
* 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 }}
 
\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 $z=e^{-\alpha_1}, q = e^{−\delta}$.
+
where <math>z=e^{-\alpha_1}, q = e^{−\delta}</math>.
  
 
===highest weight representations===
 
===highest weight representations===
* level $k$
+
* level <math>k</math>
* highest weight $\omega=(k-l)\omega_0+l\omega_1$
+
* highest weight <math>\omega=(k-l)\omega_0+l\omega_1</math>
 
* character
 
* character
$$
+
:<math>
 
\chi(L(\omega))=\frac{\theta_{k+2,l+1}-\theta_{k+2,-l-1}}{\theta_{2,1}-\theta_{2,-1}}
 
\chi(L(\omega))=\frac{\theta_{k+2,l+1}-\theta_{k+2,-l-1}}{\theta_{2,1}-\theta_{2,-1}}
$$
+
</math>
 
where
 
where
$$
+
:<math>
 
\theta_{k,l}=\sum_{r\in \mathbb{Z}+\frac{l}{2k}}e^{kr}q^{kr^2}
 
\theta_{k,l}=\sum_{r\in \mathbb{Z}+\frac{l}{2k}}e^{kr}q^{kr^2}
$$
+
</math>
  
 
==related items==
 
==related items==
226번째 줄: 224번째 줄:
 
* [[sl(2) - orthogonal polynomials and Lie theory]]
 
* [[sl(2) - orthogonal polynomials and Lie theory]]
 
* [[vertex algebras]]
 
* [[vertex algebras]]
 
+
* [[Quantum affine sl(2)]]
 
+
  
 
==computational resource==
 
==computational resource==
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit
  
 
+
  
 
==books==
 
==books==
  
* Gannon 190p, 193p, 196p,371p
+
* Gannon 190p, 193p, 196p,371p
  
 
+
  
 
==articles==
 
==articles==
* 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.
+
* 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.
 
* 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].
  
 
[[분류:Lie theory]]
 
[[분류:Lie theory]]
 +
[[분류:migrate]]

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.