"Theta functions in affine Kac-Moody algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* let $k\in \mathbb{Z}_{\geq 1}$ be the level | * let $k\in \mathbb{Z}_{\geq 1}$ be the level | ||
| + | * for $\gamma\in M$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by | ||
| + | $$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$ | ||
* definition | * definition | ||
$$ | $$ | ||
2014년 12월 5일 (금) 19:46 판
introduction
- let $k\in \mathbb{Z}_{\geq 1}$ be the level
- for $\gamma\in M$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by
$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$
- definition
$$ \begin{align} \Theta_{k,\lambda} &=e^{-|\lambda|^2\delta/2k}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu-\frac{1}{2}k\langle \mu,\mu \rangle \delta)}\\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} \end{align} $$
$A_1$ example
- level k=1, $\lambda=0$
- let $z=e^{-\alpha_1}$
$$ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots $$