On Quasi-hereditary Structures

Flores Galicia M. On Quasi-hereditary Structures. Bielefeld: Universität Bielefeld; 2020.#### Abstract A quasi-hereditary algebra is an Artin algebra *A* together with the choice of a partial order on the set of isoclasses of simple *A*-modules which satisfies certain conditions. We refer to this partial order as a quasi-hereditary structure on *A*. In this thesis we discuss two approaches to investigate all the possible choices that yield quasi-hereditary structures for a given Artin algebra. The first strategy is to study total orders inducing quasi-hereditary structures via the homological poset, which is a partial order on the set of simple modules reflecting homological properties. The second approach refines the notion of a quasi-hereditary structure considering an appropriate equivalence relation. In particular we exhibit combinatorial characterisations of the homological poset of Auslander algebras arising from truncated polynomial rings and for blocks of Schur algebras of finite representation type. For the case of path algebras of Dynkin type *An* we find a complete characterisation of all the equivalence classes of quasi-hereditary structures by means of binary trees and certain quiver decompositions