The BGG Category for Generalised Reductive Lie Algebras

Abstract

A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category ${\mathcal{O}}^{\prime }$ over complex generalised reductive Lie algebras. Then we make a preliminary research on the highest weight modules and the projective modules in this new category ${\mathcal{O}}^{\prime }$, and generalize some conclusions in the classical case. As a critical difference from the complex semisimple Lie algebra case, we prove that there is no projective module in ${\mathcal{O}}^{\prime }$.

arXiv comment14 pages, a primary version