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 \mathcal{O} over complex semisimple Lie algebras to the category {\mathcal{O}}^{\prime } over complex generalized reductive Lie algebras. Then, we preliminarily research the highest weight modules and the projective modules in this new category {\mathcal{O}}^{\prime }, and generalize some conclusions for the classical case. Also, we investigate the associated varieties with respect to the irreducible modules in {\mathcal{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.

Keywords

Generalized reductive Lie algebras / BGG category / irreducible modules / projective modules / associated varieties / center

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