Silting t-Structures, ℙ-Objects, and Weyl Groupoids

Abstract

The main topics of this thesis are triangulated categories with t-structures and weight structures, Koszul duality, and certain generalizations of spherical objects known as &Popf;-objects. The secondary topic are Weyl groupoids, which are a certain aspect of the structure theory of Lie superalgebras. The thesis consists of four rather independent parts. <br/> In the first part we introduce derived projective covers and relate them to the notion of enough derived projectives introduced by Genovese–Lowen–Van den Bergh. Our main result uses derived projective covers to provide an if-and-only-if criterion for a t-structure with finite-length heart to be a silting t-structure in the sense of Psaroudakis–Vitória. We also provide equivalent axioms for the ST pairs introduced by Adachi–Mizuno–Yang, and formulate the bijection between simple-minded collections and silting collections due to Koenig–Yang in terms of derived projective covers. <br/> In the second part we show that the non-positive respectively positive dg algebras obtained from silting and simple-minded collections corresponding to orthogonal weight structures and t-structures are dg Koszul dual to each other. This can be seen as a first step towards a tentative Koszul duality of weight structures and t-structures. <br/> In the third part we consider the constructible derived category of complex projective space &Popf;^<em>n</em>, equipped with the middle-perverse t-structure. We show that the simple perverse sheaf corresponding to the open stratum is a &Popf;^<em>n</em>-object in the sense of Huybrechts–Thomas, and that its associated &Popf;-twist is the inverse Serre functor. Moreover, we classify the &Popf;-like objects in the category of perverse sheaves on &Popf;ⁿ. This part is joint work with Alessio Cipriani. <br/> In the fourth part we study Weyl groupoids of contragredient Lie superalgebras. We provide a convenient graphical formulation of the definitions of Cartan graphs and Weyl groupoids introduced by Heckenberger in the context of Nichols algebras, and apply this to Lie superalgebras following Heckenberger–Yamane. We explicitly describe the Weyl groupoids of sl(<em>m</em>|<em>n</em>), osp(2<em>m</em>+1|2<em>n</em>) and osp(2<em>m</em>|2<em>n</em>) in terms of partitions. Furthermore, we compare this notion of Weyl groupoid to other similar constructions, and in particular to the root groupoid recently introduced by Gorelik–Hinich–Serganova. This part is joint work with Jonas Nehme.