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Title: On Auslander-Reiten theory for algebras and derived categories
Author: Scherotzke, Sarah
ISNI:       0000 0004 2683 2825
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2009
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This thesis consists of three parts. In the first part we look at Hopf algebras. We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. These Hopf algebras were classified by Radford and Krop for fields of characteristic zero. We obtain three types of Hopf algebras presented by generators and relations. The third type is new and has not previously appeared in literature. The second part of this thesis deals with Auslander-Reiten theory of finitedimensional algebras over fields. We consider G-transitive algebras and develop necessary conditions for them to have Auslander-Reiten components with Euclidean tree class. Thereby a result in [F3, 4.6] is corrected and generalized. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras. Finally we deduce a condition for a smash product of a local basic algebra Λ with a commutative semi-simple group algebra to have components with Euclidean tree class, in terms of the components of the Auslander-Reiten quiver of Λ. In the last part we introduce and analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. Also a necessary condition for the existence of components of Euclidean tree class is deduced. We determine components that contain shift periodic complexes.
Supervisor: Erdmann, Karin Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; Algebra ; Reprsentation theory ; Auslander-Reiten theory ; Hopf algebras ; Derived Categories