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Constructions in higher-dimensional Auslander-Reiten theory
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Description
Abstract [en]

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 2πk ⁄ 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 (QG, WG) such that the skew group algebra of the Jacobian algebra of (Q, W) is Morita equivalent to the Jacobian algebra of (QG, WG). 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.

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
Supervisors
Available from: 2019-05-13 Created: 2019-04-03 Last updated: 2019-05-13
List of papers
1. Tensor products of higher almost split sequences
Open this publication in new window or tab >>Tensor products of higher almost split sequences
2017 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 221, no 3, p. 645-665Article in journal (Refereed) Published
Abstract [en]

We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama give in [6] a criterion for when the tensor product of an n-representation finite algebra and an m-representation finite algebra is (n + m)-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suitable chain map and using a natural notion of tensor product for chain maps.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-309779 (URN)10.1016/j.jpaa.2016.07.010 (DOI)000387191700006 ()
Available from: 2016-12-15 Created: 2016-12-07 Last updated: 2019-04-03
2. Tensor product of n-complete algebras
Open this publication in new window or tab >>Tensor product of n-complete algebras
(English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376Article in journal (Other academic) In press
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-355624 (URN)
Available from: 2018-07-02 Created: 2018-07-02 Last updated: 2019-04-03
3. Self-injective algebras from Postnikov diagrams
Open this publication in new window or tab >>Self-injective algebras from Postnikov diagrams
(English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079Article in journal (Other academic) Submitted
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-355625 (URN)
Available from: 2018-07-02 Created: 2018-07-02 Last updated: 2019-04-03
4. Existence of symmetric maximal noncrossing collections of k-element sets
Open this publication in new window or tab >>Existence of symmetric maximal noncrossing collections of k-element sets
(English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Other academic) Submitted
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-377300 (URN)
Available from: 2019-02-18 Created: 2019-02-18 Last updated: 2019-04-03
5. Skew group algebras of Jacobian algebras
Open this publication in new window or tab >>Skew group algebras of Jacobian algebras
2019 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 526, p. 112-165Article in journal (Refereed) Published
Abstract [en]

For a quiver with potential (Q, W) with an action of a finite cyclic group G, we study the skew group algebra Lambda G of the Jacobian algebra Lambda = P(Q, W). By a result of Reiten and Riedtmann, the quiver Q(G) of a basic algebra eta(Lambda G)eta Morita equivalent to Lambda G is known. Under some assumptions on the action of G, we explicitly construct a potential W-G on Q(G) such that eta(Lambda G)eta similar or equal to P(Q(G),W-G). The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of G. If Lambda is self-injective, then Lambda G is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on (Q, W) behave with respect to our construction.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-355626 (URN)10.1016/j.jalgebra.2019.02.005 (DOI)000463309100008 ()
Funder
Swedish Research Council
Available from: 2018-07-02 Created: 2018-07-02 Last updated: 2019-04-25Bibliographically approved

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