"Verma modules"의 두 판 사이의 차이
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imported>Pythagoras0 (새 문서: ==introduction== * <math>V=\oplus_{\lambda\in\mathbb{F}}V_{\lambda}</math>, <math>V_{\lambda}=\{v\in V|Hv=\lambda v\}</math> ==infinite in both direction== * How to construct a repr...) |
imported>Pythagoras0 |
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| 7번째 줄: | 7번째 줄: | ||
===brute force=== | ===brute force=== | ||
* impose the following conditions | * impose the following conditions | ||
| − | :<math>H v_j= | + | :<math>H v_j=c_j v_j</math> |
| − | :<math>F v_j= | + | :<math>F v_j=b_jv_{j+1}</math> |
| − | :<math>E v_j= | + | :<math>E v_j=a_jv_{j-1}</math> |
| − | * as long as $ | + | * we get the following conditions |
| − | + | $$ | |
| + | \begin{align} | ||
| + | a_j b_{j-1}-a_{j+1} b_j+c_j=0 \\ | ||
| + | a_j \left(c_{j-1}-c_j-2\right)=0\\ | ||
| + | b_j \left(-c_j+c_{j+1}+2\right)=0 | ||
| + | \end{align} | ||
| + | $$ | ||
| + | * Fix $c_j=\lambda-2j$. Then as long as $b_j a_{j+1}-b_{j-1} a_{j}=\lambda -2j$ is satisfied, we get a $U$-module structure on the space spanned by <math>\{v_j|j\in \mathbb{Z}\}</math> | ||
===symmetrical choice=== | ===symmetrical choice=== | ||
2013년 4월 13일 (토) 09:04 판
introduction
- \(V=\oplus_{\lambda\in\mathbb{F}}V_{\lambda}\), \(V_{\lambda}=\{v\in V|Hv=\lambda v\}\)
infinite in both direction
- How to construct a representation with basis \(\{v_j|j\in \mathbb{Z}\}\)
brute force
- impose the following conditions
\[H v_j=c_j v_j\] \[F v_j=b_jv_{j+1}\] \[E v_j=a_jv_{j-1}\]
- we get the following conditions
$$ \begin{align} a_j b_{j-1}-a_{j+1} b_j+c_j=0 \\ a_j \left(c_{j-1}-c_j-2\right)=0\\ b_j \left(-c_j+c_{j+1}+2\right)=0 \end{align} $$
- Fix $c_j=\lambda-2j$. Then as long as $b_j a_{j+1}-b_{j-1} a_{j}=\lambda -2j$ is satisfied, we get a $U$-module structure on the space spanned by \(\{v_j|j\in \mathbb{Z}\}\)
symmetrical choice
\[H v_j=(\lambda -2j)v_j\] \[F v_j=(j-\frac{\lambda }{2})v_{j+1}\] \[E v_j=(\frac{\lambda }{2}-j)v_{j-1}\]
semi-infinite case : Verma module
- How to construct a representation $V(\lambda)$ with basis \(\{v_j|j\geq 0\}\)
- \(\lambda\in \mathbb{F}\) 에 대하여, highest weight vector \(v_0\) 를 정의
\[Ev_0=0\]\[Hv_0=\lambda v_0\]
- impose the following conditions
\[H v_j=(\lambda -2j)v_j\]\[F v_j=(j+1)v_{j+1}\]\[E v_j=(\lambda -j+1)v_{j-1}\]
finite representation
- \(\{v_j|j\geq 0\}\) 가 생성하는 벡터공간 $V(\lambda)$ 이 유한차원인 L-모듈이 되려면, \(\lambda\in\mathbb{Z}, \lambda\geq 0\) 이 만족되어야 한다