Representations of symmetrizable Kac-Moody algebras - 편집 역사

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imported>Pythagoras0: 새 문서: ==introduction== * Let $L(A)$ be a symmetrizable Kac-Moody algebra * the category $\mathcal{O}$ * Integrable modules ==the category $\mathcal{O}$== * $V$ is an object in ...

2014-03-14T08:37:58Z

새 문서: ==introduction== * Let $L(A)$ be a symmetrizable Kac-Moody algebra * the category <math>\mathcal{O}</math> * Integrable modules ==the category $\mathcal{O}$== * $V$ is an object in ...

새 문서

==introduction==
* Let $L(A)$ be a symmetrizable Kac-Moody algebra
* the category <math>\mathcal{O}</math>
* Integrable modules

==the category $\mathcal{O}$==
* $V$ is an object in $\mathcal{O}$
# $V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}$
# $\dim V_{\lambda}$ is finite for each $\lambda\in \mathfrak{h}^{*}$
# there exists a finite set $\lambda_1,\cdots, \lambda_s\in \mathfrak{h}^{*}$ such that each $\lambda$ with $V_{\lambda}\neq 0$ satisfies $\lambda \prec \lambda_i$ for some $i\in \{1,\cdots, s\}$

==integrable module==
* An $L(A)$-module $V$ is called integrable if
$$
V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}
$$
and if $e_i : V\to V$ and $f_i : V\to V$ are locally nilpotent for all $i$
;Thm
Let $L(A)$ be a symmetrizable Kac-Moody algebra and $L(\lambda)$ be an irreducible $L(A)$-module in the category $\mathcal{O}$. Then $L(\lambda)$ is integrable if and only if $\lambda$ is dominant and integral.
* [[Weyl-Kac character formula]]

[[분류:Lie theory]]

← 이전 판		2020년 11월 14일 (토) 04:56 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~
−	* Let $L(A)$ be a symmetrizable Kac-Moody algebra
−	* the category <math>\mathcal{O}</math>
−	* Integrable modules

−
−	~~==the category $\mathcal{O}$==~~
−	* $V$ is an object in $\mathcal{O}$
−	~~# $V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}$~~
−	~~# $\dim V_{\lambda}$ is finite for each $\lambda\in \mathfrak{h}^{*}$~~
−	~~# there exists a finite set $\lambda_1,\cdots, \lambda_s\in \mathfrak{h}^{*}$ such that each $\lambda$ with $V_{\lambda}\neq 0$ satisfies $\lambda \prec \lambda_i$ for some $i\in \{1,\cdots, s\}$~~
−
−
−	~~==integrable module==~~
−	* An $L(A)$-module $V$ is called integrable if
−	$$
−	~~V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}~~
−	$$
−	~~and if $e_i : V\to V$ and $f_i : V\to V$ are locally nilpotent for all $i$~~
−	~~;Thm~~
−	Let $L(A)$ be a symmetrizable Kac-Moody algebra and $L(\lambda)$ be an irreducible $L(A)$-module in the category $\mathcal{O}$. Then $L(\lambda)$ is integrable if and only if $\lambda$ is dominant and integral.
−	* [[Weyl-Kac character formula]]
−
−
−	~~==related items==~~
−	* [[BGG category and BGG resolution]]
−
−
−	~~[[분류:Lie theory]]~~

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Let $L(A)$ be a symmetrizable Kac-Moody algebra
+* Let <math>L(A)</math> be a symmetrizable Kac-Moody algebra
 * the category <math>\mathcal{O}</math>
 * Integrable modules
-==the category $\mathcal{O}$==
+==the category <math>\mathcal{O}</math>==
-* $V$ is an object in $\mathcal{O}$
+* <math>V</math> is an object in <math>\mathcal{O}</math>
-# $V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}$
+# <math>V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}</math>
-# $\dim V_{\lambda}$ is finite for each $\lambda\in \mathfrak{h}^{*}$
+# <math>\dim V_{\lambda}</math> is finite for each <math>\lambda\in \mathfrak{h}^{*}</math>
-# there exists a finite set $\lambda_1,\cdots, \lambda_s\in \mathfrak{h}^{*}$ such that each $\lambda$ with $V_{\lambda}\neq 0$ satisfies $\lambda \prec \lambda_i$ for some $i\in \{1,\cdots, s\}$
+# there exists a finite set <math>\lambda_1,\cdots, \lambda_s\in \mathfrak{h}^{*}</math> such that each <math>\lambda</math> with <math>V_{\lambda}\neq 0</math> satisfies <math>\lambda \prec \lambda_i</math> for some <math>i\in \{1,\cdots, s\}</math>
 ==integrable module==
-* An $L(A)$-module $V$ is called integrable if
+* An <math>L(A)</math>-module <math>V</math> is called integrable if
-$$
+:<math>
 V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}
-$$
+</math>
-and if $e_i : V\to V$ and $f_i : V\to V$ are locally nilpotent for all $i$
+and if <math>e_i : V\to V</math> and <math>f_i : V\to V</math> are locally nilpotent for all <math>i</math>
 ;Thm
-Let $L(A)$ be a symmetrizable Kac-Moody algebra and $L(\lambda)$ be an irreducible $L(A)$-module in the category $\mathcal{O}$. Then $L(\lambda)$ is integrable if and only if $\lambda$ is dominant and integral.
+Let <math>L(A)</math> be a symmetrizable Kac-Moody algebra and <math>L(\lambda)</math> be an irreducible <math>L(A)</math>-module in the category <math>\mathcal{O}</math>. Then <math>L(\lambda)</math> is integrable if and only if <math>\lambda</math> is dominant and integral.
 * [[Weyl-Kac character formula]]

← 이전 판	2020년 11월 16일 (월) 17:57 판
(차이 없음)

Representations of symmetrizable Kac-Moody algebras - 편집 역사

2020년 11월 16일 (월) 18:07에 Pythagoras0님의 편집

2020년 11월 16일 (월) 17:57에 Pythagoras0님의 편집

2020년 11월 14일 (토) 04:56에 imported>Pythagoras0님의 편집

2020년 11월 14일 (토) 04:56에 imported>Pythagoras0님의 편집

2014년 3월 17일 (월) 03:03에 imported>Pythagoras0님의 편집

imported>Pythagoras0: 새 문서: ==introduction== * Let $L(A)$ be a symmetrizable Kac-Moody algebra * the category \mathcal{O} * Integrable modules ==the category $\mathcal{O}$== * $V$ is an object in ...

imported>Pythagoras0: 새 문서: ==introduction== * Let $L(A)$ be a symmetrizable Kac-Moody algebra * the category $\mathcal{O}$ * Integrable modules ==the category $\mathcal{O}$== * $V$ is an object in ...