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Constructions in higher-dimensional Auslander-Reiten theoryPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Description

##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics , 2019. , p. 42
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 114
##### Keywords [en]

Representation theory, higher-dimensional Auslander-Reiten theory, Postnikov diagram, 2-representation finite algebra, self-injective algebra, quiver with potential, skew group algebra
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-377405ISBN: 978-91-506-2754-1 (print)OAI: oai:DiVA.org:uu-377405DiVA, id: diva2:1301812
##### Public defence

2019-06-03, Room 4001, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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##### Supervisors

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#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt446",{id:"formSmash:j_idt446",widgetVar:"widget_formSmash_j_idt446",multiple:true}); Available from: 2019-05-13 Created: 2019-04-03 Last updated: 2019-05-13
##### List of papers

This thesis consists of an introduction and five research articles about representation theory of algebras.

Papers I and II focus on the tensor product of algebras from the point of view of higher-dimensional Auslander-Reiten theory. In Paper I we consider the tensor product Λ of two algebras which are *n- *respectively *m-*representation finite. In the case when Λ itself is *(n+m)*-representation finite, we construct its *(n+m)*-almost split sequences explicitly in function of the *n-* and *m-*almost split sequences of the factors. In Paper II we use the constructions of Paper I to prove the following result: the tensor product of an *n-* and an *m-*complete acyclic algebras (in the sense of Iyama) is *(n+m)*-complete and acyclic.

Papers III and IV deal with the combinatorics of Postnikov diagrams, or equivalently of the Grassmannian cluster category. This is motivated by 2-dimensional Auslander-Reiten theory: we are interested in constructing self-injective Jacobian algebras as they are the 3-preprojective algebras of 2-representation finite algebras. In Paper III we investigate when the stable Jacobian algebra associated to a *(k,n)-*Postnikov diagram is self-injective. We prove that this happens if and only if the Postnikov diagram is invariant under rotation by 2π*k **⁄ n. *In Paper IV (joint with Thörnblad and Zimmermann) we determine a necessary and sufficient condition on *(k,n)* for such a symmetric Postnikov diagram to exist, namely *k ≡ *-1, 0* *or 1 modulo n ⁄ *GCD(k,n).* As a corollary, we prove that there exist self-injective planar quivers with potential with Nakayama automorphism of any prescribed order, answering a question by Herschend and Iyama.

Paper V (joint with Giovannini) is about skew group algebras. Let *G *be a finite group acting on a quiver with potential *(Q, W), *such that certain assumptions hold. We construct a quiver with potential *(Q*_{G}*, W*_{G}*) *such that the skew group algebra of the Jacobian algebra of *(Q, W) *is Morita equivalent to the Jacobian algebra of *(Q*_{G}*, W*_{G}*). *Moreover, we show that this construction is a duality if *G* is abelian. We also apply our results to quivers with potential associated to Postnikov diagrams.

1. Tensor products of higher almost split sequences$(function(){PrimeFaces.cw("OverlayPanel","overlay1056720",{id:"formSmash:j_idt495:0:j_idt499",widgetVar:"overlay1056720",target:"formSmash:j_idt495:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Tensor product of n-complete algebras$(function(){PrimeFaces.cw("OverlayPanel","overlay1230007",{id:"formSmash:j_idt495:1:j_idt499",widgetVar:"overlay1230007",target:"formSmash:j_idt495:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Self-injective algebras from Postnikov diagrams$(function(){PrimeFaces.cw("OverlayPanel","overlay1230009",{id:"formSmash:j_idt495:2:j_idt499",widgetVar:"overlay1230009",target:"formSmash:j_idt495:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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5. Skew group algebras of Jacobian algebras$(function(){PrimeFaces.cw("OverlayPanel","overlay1230010",{id:"formSmash:j_idt495:4:j_idt499",widgetVar:"overlay1230010",target:"formSmash:j_idt495:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

isbn
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