Verma modules

introduction

• $M_{\lambda}=U_q(\mathfrak{g})\otimes_{U_q(\mathfrak{b})}\mathbb{C}_{\lambda}$
• \(V=\oplus_{\lambda\in\mathbb{F}}V_{\lambda}\), \(V_{\lambda}=\{v\in V|Hv=\lambda v\}\)

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

related items

• Tilting modules for quantum groups

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxRUtEQlV4M2FxazQ/edit

articles

• Bernšteĭn, I. N., I. M. Gel\cprimefand, and S. I. Gel\cprimefand. ‘Structure of Representations That Are Generated by Vectors of Highest Weight’. Akademija Nauk SSSR. Funkcional\cprime Nyi Analiz I Ego Priloženija 5, no. 1 (1971): 1–9.
• Verma, Daya-Nand. ‘Structure of Certain Induced Representations of Complex Semisimple Lie Algebras’. Bulletin of the American Mathematical Society 74 (1968): 160–66.