The BGG Category for Generalised Reductive Lie Algebras

TL;DR

This paper extends the classical BGG category to generalised reductive Lie algebras, exploring highest weight modules and projective modules, and reveals that no projective modules exist in this new setting.

Contribution

It introduces the category ' for generalised reductive Lie algebras and analyzes its structure, highlighting key differences from the classical case.

Findings

No projective modules exist in ' category

Extended classical results to the generalised reductive case

Preliminary analysis of highest weight modules

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 $\mathcal {O'}$ 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 $\mathcal {O'}$, 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 $\mathcal {O'}$.