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

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

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

$$ s_{\alpha_1}(m\omega_0+n\omega_1)=(m+2n)\omega_0-n\omega_1 $$

$t_{n\alpha_1}\omega_0=\omega_0+n\alpha_1-n^2\delta$
$t_{n\alpha_1}\alpha_1=\alpha_1-2n\delta$
$w\in W$ 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