On certain strongly quasihereditary algebras

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