"Theta functions in affine Kac-Moody algebras"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 4번째 줄: | 4번째 줄: | ||
$$ | $$ | ||
\begin{align} | \begin{align} | ||
| − | \ | + | \Theta_{k,\lambda} &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k(\mu-\frac{1}{2}\langle \mu,\mu \rangle \delta)}\\ |
&=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} | &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} | ||
\end{align} | \end{align} | ||
| 14번째 줄: | 14번째 줄: | ||
* let $z=e^{-\alpha_1}$ | * 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 |
$$ | $$ | ||
2014년 12월 5일 (금) 19:08 판
introduction
- let $k\in \mathbb{Z}_{\geq 1}$ be the level
- definition
$$ \begin{align} \Theta_{k,\lambda} &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k(\mu-\frac{1}{2}\langle \mu,\mu \rangle \delta)}\\ &=\sum_{\mu\in Q^{\vee}+\frac{\lambda}{k}}e^{k\mu}q^{\frac{k}{2}\langle \mu,\mu \rangle} \end{align} $$
$A_1$ example
- level k=1, $\lambda=0$
- let $z=e^{-\alpha_1}$
$$ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots $$