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 $<\cdot,\cdot>$
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

$$ \begin{array}{cccc} {} & \alpha _0 & \alpha _1 & \omega _0 \\ h_0 & 2 & -2 & 1 \\ h_1 & -2 & 2 &0 \\ d & 1 & 0 & 0 \\ \end{array} $$

dual basis for H \[\omega_0,\omega_1=\omega_0+\frac{1}{2}\alpha_1,\delta=\alpha_0+\alpha_1\]

$$ \begin{array}{cccc} {} & \omega_0 & \omega_1 & \delta \\ h_0 & 1 & 0 & 0 \\ h_1 & 0 & 1 &0 \\ d & 0 & 0 & a_0=1 \\ \end{array} $$

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 $[d,x]=d(x)$

denominator formula

Weyl-Kac character formula
${\sum_{w\in W} (-1)^{\ell(w)}w(e^{\rho}) = e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha})^{m_{\alpha}}}$

level k highest weight representation

integrable highest weight
$\lambda=\lambda_{0}\omega_0+\lambda_{1}\omega_1$, $\lambda_{i}\in\mathbb{N}$
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
$\frac{<\rho,\rho>}{2h^{\vee}}=\frac{\operatorname{dim}\mathfrak{g}}{24}$
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

related items

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxMVltb0d1OUlFY00/edit

books

Gannon 190p, 193p, 196p,371p

articles

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.