On certain strongly quasihereditary algebras

Description

Introduction......Page 13 Overview of the chapter......Page 19 Definitions and basic properties......Page 20 Properties of Artin algebras......Page 22 Approximations......Page 23 Projectivisation......Page 24 Definition and first properties......Page 25 Hereditary and cohereditary preradicals......Page 27 Filtrations arising from preradicals......Page 31 Quasihereditary algebras......Page 32 Standard and costandard modules......Page 33 Definition of a quasihereditary algebra......Page 36 The tilting modules and the Ringel dual......Page 39 Overview of the chapter......Page 41 The ADR algebra of [A]......Page 42 The standard modules......Page 44 The ADR algebra is quasihereditary......Page 47 Definition and first properties......Page 49 The structure of an ultra strongly quasihereditary algebra......Page 55 The Ringel dual of an ultra strongly quasihereditary algebra......Page 59 The ADR algebra of a certain Brauer tree algebra......Page 61 Overview of the chapter......Page 65 [Delta]-semisimple modules and [Delta]-semisimple filtrations......Page 66 [Delta]-semisimple modules......Page 67 The preradical [delta] and [Delta]-semisimple filtrations......Page 71 [Delta]-semisimple filtrations of modules over the ADR algebra......Page 74 Theorem A......Page 77 An application of Theorem A......Page 80 Motivation......Page 87 Preliminary results......Page 89 Proof of Theorem B......Page 95 Overview of the chapter......Page 101 The ADR algebra is not a 1-faithful quasihereditary cover......Page 102 The representation type of [F(Delta)] and [F(Nabla)]......Page 104 Ringel self-duality for ADR algebras......Page 106 Representation dimension, finitistic dimension and the ADR algebra......Page 107 A class of left strongly quasihereditary algebras......Page 109 Properties of weak LUSQ algebras......Page 111 Overview of the chapter......Page 123 Construction......Page 124 Applications......Page 129 The ADR algebra and generalisations......Page 130 Iyama's construction......Page 131 Cluster-tilted algebras associated with reduced words in Coxeter groups......Page 132 Ultra strongly quasihereditary endomorphism algebras......Page 138 Results......Page 139 The relationship between [Ringel(R(A))] and [R(A op)]: an example......Page 149 The algebra [Ringel(R(A))]......Page 150 The algebra [R(A op)op]......Page 151 Comparison......Page 152 Natural ways of constructing hereditary preradicals......Page 153 Notation......Page 157 Bibliography......Page 161 Index......Page 167