Verma modules
imported>Pythagoras0님의 2013년 4월 13일 (토) 06:09 판 (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...)
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
\[H v_j=(\lambda -2j)v_j\] \[F v_j=f_jv_{j+1}\] \[E v_j=e_jv_{j-1}\]
- as long as $f_j e_{j+1}-f_{j-1} e_{j}=\lambda -2j$ 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