Representations of symmetrizable Kac-Moody algebras

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 $\mathcal{O}$

1. $V=\oplus_{\lambda\in \mathfrak{h}^{*}}V_{\lambda}$
2. $\dim V_{\lambda}$ is finite for each $\lambda\in \mathfrak{h}^{*}$
3. 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