The BGG Category for Generalized Reductive Lie Algebras

Abstract

A Lie algebra $\mathfrak{g}$ is considered generalized reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. This paper extends the BGG category ${\cal O}$ over complex semisimple Lie algebras to the category ${{\cal O}^\prime }$ over complex generalized reductive Lie algebras. Then, we preliminarily research the highest weight modules and the projective modules in this new category ${{\cal O}^\prime }$, and generalize some conclusions for the classical case. Also, we investigate the associated varieties with respect to the irreducible modules in ${{\cal O}^\prime }$ and obtain a result that extends Joseph’s result on the associated varieties for reductive Lie algebras. Finally, we study the center of the universal enveloping algebra U($\mathfrak{g}$) and independently provide a new proof of a theorem by Ou–Shu–Yao for the center in the case of enhanced reductive Lie algebras.