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
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

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