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