Highest weight sl_2-categorifications II: structure theory

Abstract

This paper continues the study of highest weight categorical sl_2-actions started in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the refined sense. Then we prove that any highest weight sl_2-categorification can be filtered in such a way that the successive quotients are so called basic highest weight sl_2-categorifications. For a basic highest weight categorification we determine minimal projective resolutions of standard objects. We use this, in particular, to examine the structure of tilting objects in basic categorifications and to show that the Ringel duality is given by the Rickard complex. We finish by discussing open problems.