Category O

Category O (or category ${\displaystyle {\mathcal {O}}}$) is a mathematical object in representation theory of semisimple Lie algebras. It is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

${\displaystyle M_{\lambda }=\{v\in M;\,\,\forall \,h\in {\mathfrak {h}}\,\,h\cdot v=\lambda (h)v\}.}$

Definition of category O

The objects of category O are ${\displaystyle {\mathfrak {g}}}$-modules ${\displaystyle M}$ such that

Morphisms of this category are the ${\displaystyle {\mathfrak {g}}}$-homomorphisms of these modules.