Projective-injective modules, Serre functors and symmetric algebras

TL;DR

This paper characterizes Serre functors in categories related to semi-simple Lie algebras, highlighting the role of projective-injective modules and proving conjectures about their structure in parabolic category O.

Contribution

It introduces a new description of Serre functors using projective-injective modules and proves three conjectures of Khovanov in the context of parabolic category O.

Findings

Serre functors are described for generalized category O.

Projective-injective modules control Serre functors in certain quasi-hereditary algebras.

Three conjectures of Khovanov about projective-injective modules are proved.

Abstract

We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.