"Theta functions in affine Kac-Moody algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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3번째 줄: | 3번째 줄: | ||
* for $\gamma\in M=Q^{\vee}$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by | * for $\gamma\in M=Q^{\vee}$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by | ||
$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$ | $$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$ | ||
+ | * note $\lambda=\bar{\lambda}+k\Lambda_0$ | ||
+ | * note that $|\lambda|^2=|\bar{\lambda}|^2$ | ||
* definition | * definition | ||
$$ | $$ | ||
\begin{align} | \begin{align} | ||
− | \Theta_{k,\lambda} &=e^{-|\lambda|^2\delta/2k}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ | + | \Theta_{k,\lambda} &=e^{-|\bar{\lambda}|^2\delta/2k}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ |
− | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu-\frac{1}{2}k\langle \mu,\mu \rangle \delta)}\\ | + | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k\langle \mu,\mu \rangle \delta)}\\ |
− | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} | + | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} |
\end{align} | \end{align} | ||
$$ | $$ |
2014년 12월 5일 (금) 20:13 판
introduction
- let $k\in \mathbb{Z}_{\geq 1}$ be the level
- for $\gamma\in M=Q^{\vee}$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by
$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$
- note $\lambda=\bar{\lambda}+k\Lambda_0$
- note that $|\lambda|^2=|\bar{\lambda}|^2$
- definition
$$ \begin{align} \Theta_{k,\lambda} &=e^{-|\bar{\lambda}|^2\delta/2k}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k\langle \mu,\mu \rangle \delta)}\\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} \end{align} $$
$A_1$ example
- level k=1, $\lambda=0$
- let $z=e^{-\alpha_1}$
$$ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots $$