Quasi-hereditary covers and derived equivalences of higher zigzag algebras

Bocca, Gabriele (2018) Quasi-hereditary covers and derived equivalences of higher zigzag algebras. Doctoral thesis, University of East Anglia.

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In this thesis we look at higher zigzag algebras Zds of type A as a generalization of Brauer tree algebras whose tree is a line. We recall the presentation of these algebras as path algebras with relations and their relation with higher preprojective algebras of d- representation finite and Koszul algebras. The algebras Zds are not Koszul, since simple modules do not have linear projective resolutions. To overcome this lack of regularity we give an explicit construction of a quasi-hereditary cover for Zds as a quotient algebra of Zds+1 and we study different Koszul properties of these quasi-hereditary algebras. We prove that they are Koszul in the classical sense, standard Koszul and, endowed with an appropriate grading, Koszul with respect to the standard module ∆. This more general Koszul property leads to a well-defined notion of duality, generalizing the classical Koszul duality. We will show that the ∆-Koszul dual of our quasi-hereditary cover is again a Koszul algebra in the classical sense. Using the fact that Koszul algebras are quadratic we will be able to give a presentation of the ∆-Koszul dual algebras as path algebras with relations.
The last chapter of this thesis will be about derived Morita theory and silting objects for higher zigzag algebras. Since in the case of Brauer tree algebras the Okuyama{ Rickard method to obtain two-term tilting complexes leads to the complete classification of the derived equivalence class, we focus our attention on two-term tilting complexes in the derived category Db(Zd s ). For Zds we give a more explicit description of irreducible Okuyama-Rickard mutations of Zds. To conclude we describe the derived equivalence class of Z23 by showing all the algebras derived equivalent to it.

Item Type: Thesis (Doctoral)
Additional Information: Unfortunately some of the equations in the abstract have lost their formatting when copying from the thesis pdf
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Jackie Webb
Date Deposited: 05 Oct 2018 13:16
Last Modified: 05 Oct 2018 13:16
URI: https://ueaeprints.uea.ac.uk/id/eprint/68418


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