The BGG Category for Generalized Reductive Lie Algebras

A Lie algebra \(\mathfrak{g}\) $$\mathfrak{g}$$ 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 \({\cal O}\) $$\mathcal{O}$$ over complex semisimple Lie algebras to the category \({{\cal O}^\prime }\) $${\mathcal{O}}^{\prime }$$ over complex generalized reductive Lie algebras. Then, we preliminarily research the highest weight modules and the projective modules in this new category \({{\cal O}^\prime }\) $${\mathcal{O}}^{\prime }$$ , and generalize some conclusions for the classical case. Also, we investigate the associated varieties with respect to the irreducible modules in \({{\cal O}^\prime }\) $${\mathcal{O}}^{\prime }$$ 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( \(\mathfrak{g}\) $$\mathfrak{g}$$ ) and independently provide a new proof of a theorem by Ou–Shu–Yao for the center in the case of enhanced reductive Lie algebras.