Theta functions in affine Kac-Moody algebras - 편집 역사

2020년 11월 16일 (월) 12:34에 Pythagoras0님의 편집

2020-11-16T12:34:39Z

2020년 11월 14일 (토) 03:06에 imported>Pythagoras0님의 편집

2020-11-14T03:06:37Z

imported>Pythagoras0: /* notation */

2019-03-14T01:55:07Z

notation

imported>Pythagoras0: /* analytic function */

2015-05-25T12:01:30Z

analytic function

imported>Pythagoras0: /* analytic function */

2015-05-25T08:44:00Z

analytic function

imported>Pythagoras0: /* analytic function */

2015-05-25T08:43:28Z

analytic function

imported>Pythagoras0: /* introduction */

2015-05-25T08:42:06Z

introduction

imported>Pythagoras0: Pythagoras0 사용자가 Theta functions in Kac-Moody algebras 문서를 Theta functions in affine Kac-Moody algebras 문서로 옮겼습니다.

2015-04-27T01:41:38Z

Pythagoras0 사용자가 Theta functions in Kac-Moody algebras 문서를 Theta functions in affine Kac-Moody algebras 문서로 옮겼습니다.

imported>Pythagoras0: /* $A_1$ example */

2014-12-16T00:40:19Z

$A_1$ example

imported>Pythagoras0: /* computational resource */

2014-12-16T00:37:07Z

computational resource

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
 ===notation===
-* Let $M=Q^{\vee}$. This is also the $\mathbb{Z}$-span of $W\theta$ where $\theta$ is the highest root
+* Let <math>M=Q^{\vee}</math>. This is also the <math>\mathbb{Z}</math>-span of <math>W\theta</math> where <math>\theta</math> is the highest root
-* for $\gamma\in M$, define $t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}$ by
+* for <math>\gamma\in M</math>, define <math>t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}</math> by
-$$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$
+:<math>t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta </math>
 ;definition
-Let $k\in \mathbb{Z}_{\geq 1}$ be the level of $\lambda$. Note $\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta$. and $|\lambda|^2=|\bar{\lambda}|^2+2k\xi$. The theta function is defined by
+Let <math>k\in \mathbb{Z}_{\geq 1}</math> be the level of <math>\lambda</math>. Note <math>\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta</math>. and <math>|\lambda|^2=|\bar{\lambda}|^2+2k\xi</math>. The theta function is defined by
-$$
+:<math>
 \begin{align}
 \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\
@@ 12번째 줄: / 12번째 줄: @@
 &=e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{1}{2k}(k\gamma+\lambda,k\gamma+\lambda)}
 \end{align}
-$$
+</math>
 We also have
-$$
+:<math>
 \begin{align}
 \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\
@@ 20번째 줄: / 20번째 줄: @@
 &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )}
 \end{align}
-$$
+</math>
 ===analytic function===
 * FMS 605p, (14.318)
-* let $(\zeta;\tau;t)=\zeta+ \tau \Lambda_0+ t\delta\in \mathfrak{h}^{*}$
+* let <math>(\zeta;\tau;t)=\zeta+ \tau \Lambda_0+ t\delta\in \mathfrak{h}^{*}</math>
-* once we evaluate them at $\xi=-2\pi i (\zeta;\tau;t)$, we get
+* once we evaluate them at <math>\xi=-2\pi i (\zeta;\tau;t)</math>, we get
-$$
+:<math>
 \Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2}
-$$
+</math>
 ;thm
-$$
+:<math>
 \Theta_{\lambda}^{(k)}(\frac{\zeta}{\tau},-\frac{1}{\tau},t+\frac{|\zeta|^2}{2\tau})=(-\frac{i \tau}{k})^{r/2}\frac{1}{\operatorname{vol}(Q^{\vee})}\sum_{\mu\in P/k Q^{\vee}}e^{-2\pi i (\mu,\nu)/k} \Theta_{\mu}^{(k)}(\zeta,\tau,t)
-$$
+</math>
-==$A_1$ example==
+==<math>A_1</math> example==
-* let $z=e^{-\alpha_1}$
+* let <math>z=e^{-\alpha_1}</math>
 ===level k=1===
-* $\lambda=0$
+* <math>\lambda=0</math>
-$$
+:<math>
 \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots
-$$
+</math>
 ==related items==

← 이전 판		2020년 11월 14일 (토) 03:06 판
54번째 줄:		54번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:Lie theory]]		[[분류:Lie theory]]
		+	[[분류:migrate]]

@@ 5번째 줄: / 5번째 줄: @@
 $$t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta $$
 ;definition
-Let $k\in \mathbb{Z}_{\geq 1}$ be the level of $\lambda$. Note $\lambda=\bar{\lambda}+k\Lambda_0$. and $|\lambda|^2=|\bar{\lambda}|^2$. The theta function is defined by
+Let $k\in \mathbb{Z}_{\geq 1}$ be the level of $\lambda$. Note $\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta$. and $|\lambda|^2=|\bar{\lambda}|^2+2k\xi$. The theta function is defined by
 $$
 \begin{align}
-\Theta_{k,\lambda} &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\
+\Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\
-&=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{k(\gamma,\gamma)}{2}+(\gamma,\lambda)}\\
+&=e^{-\frac{|\lambda|^2\delta}{2k}}e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{k(\gamma,\gamma)}{2}+(\gamma,\lambda)}\\
 &=e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{1}{2k}(k\gamma+\lambda,k\gamma+\lambda)}
 \end{align}
@@ 16번째 줄: / 16번째 줄: @@
 $$
 \begin{align}
-\Theta_{k,\lambda} &=e^{-\frac{|\bar{\lambda}|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\
+\Theta_{k,\lambda} &=e^{-\frac{|\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( \mu,\mu ) \delta}\\
 &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )}
 \end{align}
 $$
 ===analytic function===
 * FMS 605p, (14.318)

@@ 23번째 줄: / 23번째 줄: @@
 ===analytic function===
 * FMS 605p, (14.318)
 * once we evaluate them at $\xi=-2\pi i (\zeta;\tau;t)$, we get
 $$

@@ 25번째 줄: / 25번째 줄: @@
 * once we evaluate them at $\xi=-2\pi i (\zeta;\tau;t)$, we get
 $$
-\Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}=e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2}
+\Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2}
 $$
 ;thm

← 이전 판	2015년 4월 27일 (월) 01:41 판
(차이 없음)

← 이전 판		2014년 12월 16일 (화) 00:37 판
38번째 줄:		38번째 줄:
	==computational resource==		==computational resource==
	* https://docs.google.com/file/d/0B8XXo8Tve1cxbW9iUTgtaThCM2s/edit		* https://docs.google.com/file/d/0B8XXo8Tve1cxbW9iUTgtaThCM2s/edit
		+
		+
		+	[[분류:math and physics]]
		+	[[분류:Lie theory]]