"Representations of symmetrizable Kac-Moody algebras"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| − | * Let | + | * Let <math>L(A)</math> be a symmetrizable Kac-Moody algebra |
* the category <math>\mathcal{O}</math> | * the category <math>\mathcal{O}</math> | ||
* Integrable modules | * Integrable modules | ||
| − | ==the category | + | ==the category <math>\mathcal{O}</math>== |
| − | * | + | * <math>V</math> is an object in <math>\mathcal{O}</math> |
| − | # | + | # <math>V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}</math> |
| − | # | + | # <math>\dim V_{\lambda}</math> is finite for each <math>\lambda\in \mathfrak{h}^{*}</math> |
| − | # there exists a finite set | + | # 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== | ==integrable module== | ||
| − | * An | + | * An <math>L(A)</math>-module <math>V</math> is called integrable if |
| − | + | :<math> | |
V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda} | V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda} | ||
| − | + | </math> | |
| − | and if | + | 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 | ;Thm | ||
| − | Let | + | 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]] | * [[Weyl-Kac character formula]] | ||
2020년 11월 16일 (월) 10:07 기준 최신판
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 \(\mathcal{O}\)
- \(V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}\)
- \(\dim V_{\lambda}\) 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.