ADVANCES IN REPRESENTATION THEORY OF ALGEBRAS 7 SEPTEMBER 24 28, Program

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1 ADVANCES IN REPRESENTATION THEORY OF ALGEBRAS 7 SEPTEMBER 24 28, 2018 Program Monday, September 24, :50-9:00 Board the bus. The bus leaves the hotel at 9:00 AM. 9:30h - 10:00 Registration 10:00h - 10:50 Helmut Lenzing (Universität Paderborn) Working with José Antonio 10:50h - 11:20 Coffee and registration 11:20h - 12:10 Andrea Solotar (Universidad de Buenos Aires) The Gerstenhaber structure of Hochschild cohomology 12:10h - 12:20h Coffee 12:20-13:10 Ralf Schiffler (University of Connecticut) Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers 13:10-15:00 LUNCH 15:00-15:50 Gordana Todorov (Northeastern University) Cyclic Posets and Triangulation Clusters 15:50-16:00 Coffee 16:00-16:50 Eduardo do Nascimento Marcos (Universidade de São Paulo) Hochschild-Mitchell (co)homology of skew categories and of Galois coverings 17:20 Bus leaves Math Institute to hotel. 1

2 2 ARTA7 Tuesday, September 25, :50-9:00 Board the bus. The bus leaves the hotel at 9:00 AM. 10:00h - 10:50 Birge Huisgen-Zimmermann (University of California, Santa Barbara) Truncated path algebras, a geometric and homological stepping stone. Part I 10:50h - 11:20 Coffee 11:20h - 12:10 Ken Goodearl (University of California, Santa Barbara) Truncated path algebras, a geometric and homological stepping stone. Part II 12:10h - 12:20h Coffee 12:20-13:10 Bernhard Keller (Université Paris Diderot - Paris 7) Koszul duality and Hochschild cohomology 13:10-15:00 LUNCH 15:00-15:50 Lidia Angeleri-Hügel (Università degli Studi di Verona) Comparing localizations of rings 15:50-16:00 Coffee 16:00-16:50 Thomas Brüstle (Bishop s University and Université de Sherbrooke) Matrix reduction and exact structures 17:20 Bus leaves Math Institute to hotel.

3 ARTA7 3 Wednesday, September 26, :50-9:00 Board the bus. The bus leaves the hotel at 9:00 AM. 10:00-10:50 Raymundo Bautista (CCM-UNAM) Representations of Bocses and A -modules 10:50-11:00 Coffee 11:00-11:50 Pierre-Guy Plamondon (Université de Paris Sud XI) Derived categories of gentle algebras and surface dissections 11:50-12:00 PACK LUNCH 12:00 Bus leaves Math Institute to Chapultepec Castle 12:30 Van leaves Math Institute to hotel 17:30 Bus returns from Chapultepec Castle to hotel

4 4 ARTA7 Thursday, September 27, :50-9:00 Board the bus. The bus leaves the hotel at 9:00 AM. 10:00h - 10:50 Sonia Trepode (Universidad Nacional de Mar del Plata) The representation dimension of selfinjective algebras of tilted type 10:50h - 11:20 Coffee 11:20h - 12:10 Changchang Xi (Capital Normal University) Constructing derived equivalences by Milnor patching 12:10h - 12:20h Coffee 12:20-13:10 Henning Krause (Universität Bielefeld) The Morita theory for derived categories revisited 13:10-15:00 LUNCH 15:00-15:50 Grzegorz Bobiński (Nicolaus Copernicus University) Birkhoff varieties 15:50-16:00 Coffee 16:00-16:50 Kunio Yamagata (Tokyo University of Agriculture and Technology) Selfinjective orbit algebras induced from repetitive algebras 17:20 Bus leaves Math Institute to hotel.

5 ARTA7 5 Friday, September 28, :50-9:00 Board the bus. The bus leaves the hotel at 9:00 AM. 10:00-10:50 Piotr Malicki (Nicolaus Copernicus University) Cycle-finite module categories 10:50-11:20 Coffee 11:20-12:10 Gustavo Jasso (Universität Bonn) Higher-dimensional Auslander algebras of type A and the higher-dimensional Waldhausen S-constructions 12:10-13:10 Snacks 13:10-14:00 Stanislaw Kasjan (Nicolaus Copernicus University) Tame strongly simply connected algebras 14:30 Bus leaves Math Institute to restaurant Van leaves Math Institute to hotel 20:00 Bus returns from restaurant to hotel

6 6 ARTA7 Abstracts Lidia Angeleri-Hügel. Comparing localizations of rings. The ring epimorphisms starting in a ring R encode important ring theoretic and representation theoretic information on R. We will review and compare some constructions of ring epimorphisms via localization, and we will discuss the interplay with subcategories of the category of R-modules. Our focus will lie on three particular cases: commutative noetherian rings, hereditary rings, and finite dimensional algebras. The talk is based on joint work with Frederik Marks, Jan Šťovíček, Ryo Takahashi, and Jorge Vitória. Raymundo Bautista. Representations of Bocses and A -modules. In this talk we will associate to any A algebra a graded bocs, then we will see a category of twisted complexes on the representations of this graded bocs which is equivalent to the category of A modules. This is joint work with Leonardo Salmerón. Grzegorz Bobiński. Birkhoff varieties. Birkhoff varieties consist of homomorphisms between modules over a truncated polynomial algebra. They are connected with the classical Birkhoff problem dealing with subgroups of finite abelian groups. Its version over a field attracted a lot of interest, including a series of papers by Ringel and Schmidmeier and connections to weighted projective lines due to Lenzing. The main result of the paper states that the Birkhoff varieties are irreducible. This is a report on a joint project with Jan Schröer. Thomas Brüstle. Matrix reduction and exact structures. Matrix reduction techniques have been used by the Kiev school to prove fundamental results in representation theory, such as the Brauer-Thrall conjectures or the tame and wild dichotomy. To formalize the matrix reduction techniques, Roiter introduced the notion of a bocs, which models matrix reductions by iterated change of categories - their objects and morphisms. We propose instead to model matrix reduction by keeping the same additive category, but changing the exact structure. A path of reductions is thus modeled by a path in the lattice of exact structures. This is a report on joint work with Souheila Hassoun, Denis Langford and Sunny Roy. Ken Goodearl. Truncated path algebras, a geometric and homological stepping stone. Part II. In this second of a pair of talks, we address the goal of generically understanding the representation theory of a truncated path algebra Λ. This goal incorporates (a) determination of the irreducible components of the parametrizing varieties Rep d (Λ) in representation theoretic terms, and (b) determination of generic modules for each component. Part (a) now has a complete solution in terms of a new module invariant. This solution comes hand in hand with good progress relative to (b). (This is joint work with Birge Huisgen-Zimmermann.)

7 ARTA7 7 Birge Huisgen-Zimmermann. Truncated path algebras, a geometric and homological stepping stone. Part I. Part I of this pair of lectures starts by addressing the key position held by truncated path algebras within the class of arbitrary basic finite dimensional algebras. We follow with a brief summary of the contents of the two lectures; both represent multiple projects. Then we describe the homological assets of truncated path algebras. The main focus is the theory of iterated strong tilting (in the sense of Auslander and Reiten) admitted by these algebras. Collaborators on the geometric work are E. Babson, F. M. Bleher, T. Chinburg, K. R. Goodearl, I. Shipman, R. Thomas; collaborators on the homological work are A. Dugas, J. Learned, and M. Saorin. Gustavo Jasso. Higher-dimensional Auslander algebras of type A and the higherdimensional Waldhausen S-constructions. This is a report on part of a joint project with Tobias Dyckerhoff (Hamburg). In this talk I will describe a relationship between Iyama s higher-dimensional Auslander algebras of type A and a higher-dimensional version of the Waldhausen S-construction, a construction arising in the context of algebraic K-theory. Using rudiments of the language of higher category theory, I will explain how to use this relationship to extend the higher-dimensional reflection functors of Iyama and Oppermann to representations of the higher-dimensional Auslander algebras of type A in an arbitrary stable -category (=enhanced triangulated category). If time permits I will describe certain ladders of recollements that arise in this context. These results can be seen as contributions to the abstract representation theory of Groth and Šťovíček. Stanislaw Kasjan. Tame strongly simply connected algebras. Research on relations between the representation type and the definitness of some quadratic forms associated with an algebra have long history. It has been one of the important threads of the scientific cooperation between México and Toruń. In this talk we recall the theorem due to Thomas Brüstle, Jose Antonio de la Peña and Andrzej Skowroński that a strongly simply connected algebra is tame if and only if its Tits quadratic form is weakly nonnegative. We present the historical context of the result, the main ideas of the proof and some of its consequences. Bernhard Keller. Koszul duality and Hochschild cohomology. We show that an augmented dg algebra and its Koszul dual dg coalgebra have Hochschild cohomology complexes linked by an isomorphism in the homotopy category of B-infinity algebras. Henning Krause. The Morita theory for derived categories revisited. A fundamental theorem of Rickard states for a pair of coherent rings that the categories of perfect complexes are triangle equivalent if and only if their bounded derived categories of finitely presented modules are triangle equivalent. The proofs of both directions are somewhat delicate and not treated properly in any text book. The aim of this talk is to propose new proofs, involving the notion of Cauchy completion for one direction, and the notion of a distinguished t-structure for the other direction.

8 8 ARTA7 Helmut Lenzing. Working with José Antonio. I will discuss aspects of the longstanding cooperation with José Antonio concerning writing mathematical papers and organizing conferences and workshops. Instead of trying to give a complete report of our collaboration or the written output, I will instead focus on characteristic aspects and will, even there, be quite selective. The aim is to draw a picture of an outstanding scientist and science-organizer. Piotr Malicki. Cycle-finite module categories. Let A be a basic indecomposable artin algebra over a commutative artin ring K and mod A the category of finitely generated right A-modules. Recall that a cycle in a module category mod A is a sequence: f 1 f n M 0 M1 M n 1 Mn = M 0 of nonzero nonisomorphisms between indecomposable modules in mod A, and the cycle is called finite if the homomorphisms f 1,..., f n do not belong to the infinite Jacobson radical of mod A. Then mod A (respectively, A) is called cycle-finite if all cycles in mod A are finite. The aim of the talk is to describe the structure of an arbitrary cycle-finite module category mod A as well as the support algebras of its indecomposable modules. Moreover, some homological properties of indecomposable modules over cycle-finite algebras will be presented. Eduardo do Nascimento Marcos. Hochschild-Mitchell (co)homology of skew categories and of Galois coverings. Let C be a category over a commutative ring k, its Hochschild-Mitchell homology and cohomology are denoted respectively HH (C) and HH (C). Let G be a group acting on C, and C[G] be the skew category. We provide decompositions of the (co)homology of C[G] along the conjugacy classes of G. For Hochschild homology of a k-algebra, this corresponds to the decomposition obtained by M. Lorenz. If the coinvariants and invariants functors are exact, we obtain isomorphisms (HH (C)) G (C[G]) and (HH (C)) G HH{1} (C[G]), where {1} is the trivial conjugacy class of G. We first obtain these isomorphisms in case the action of G is free on the objects of C. Then we introduce an auxiliary category M G (C) with an action of G which is free on its objects, related to the infinite matrix algebra considered by J. Cornick. This category enables us to show that the isomorphisms hold in general, and in particular for the Hochschild (co)homology of a k-algebra with an action of G by automorphisms. We infer that (HH (C)) G is a canonical direct summand of HH (C[G]). This provides a frame for monomorphisms obtained previously, and which have been described in low degrees. This is joint work with Claude Cibils. HH {1} Pierre-Guy Plamondon. Derived categories of gentle algebras and surface dissections. The class of gentle algebras has been extensively studied from the point of view of its representations theory. Recently, this class of algebras has appeared in connection with various subjects, such as cluster theory (in the guise of certain Jacobian algebras arising from triangulated or dissected surfaces) and Fukaya categories (as in the recent work of Haiden- Katzarkov-Kontsevich).

9 ARTA7 9 In this talk, we will see how the derived category of a gentle algebra may be understood using bordered surfaces with marked points. We will see how one can construct a dissected surface from a gentle algebra. With this model, indecomposable objects become graded curves on the surfaces and morphisms become intersection points. Other information can be studied on the surface, such as the derived invariant of Avella-Alaminos and Geiss, the Koszul dual, the perfect objects and their Auslander-Reiten translations, and some silting and tilting objects as well as their mutation. Ralf Schiffler. Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers. We give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of an algebraic variety X(Q), called frieze variety, associated to the quiver Q. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of preprojective cluster variables in the cluster algebra associated to the quiver. We show that an acyclic quiver Q is representation finite, tame, or wild, respectively, if and only if dim X(Q)=0,1 or 2, respectively. Andrea Solotar. The Gerstenhaber structure of Hochschild cohomology. Homological methods provide important information about the structure of associative algebras, revealing sometimes hidden connections amongst them. The Gerstenhaber bracket in Hochschild co-homology of unital associative algebras over a field is an invariant preserved by derived equivalences. There has been a significant amount of effort expended by many authors in order to study this structure, specially in recent times. In particular, the Gerstenhaber bracket provides the first Hochschild cohomology space of a Lie algebra structure. The computation of Hochschild cohomology requires a resolution of the algebra considered as a bimodule over itself. Of course, there is always a canonical resolution available, the bar resolution, very useful from a theoretical point of view, but not very satisfactory in practice: the complexity of this resolution rarely allows explicit calculations to be carried out. The use of alternative resolutions is not well adapted to the computation of the Gerstenhaber bracket. However, some results bywitherspoon?negron,volkov and Suárez AÁlvarez provide useful tools to solve this problem. I will illustrate how, using these methods, it is possible to describe the first Hochschild cohomology spaces of some families of algebras -both finite and infinite dimensional - as Lie algebras, and relate this to the structure of the algebras. Gordana Todorov. Cyclic Posets and Triangulation Clusters. Triangulated categories coming from cyclic posets were originally introduced in [IT15] as a generalization of the constructions of various triangulated categories with cluster structures. We will give an overview, and analyze triangulation clusters which are those corresponding to topological triangulations of the 2- disk. Locally finite non-triangulation clusters give topological triangulations of the cactus space associated to the cactus cyclic poset. This is joint work with Kiyoshi Igusa. [IT15] Continuous Cluster Categories, Algebras and Representation Theory, Vol.18, (2015), pp (arxiv: ).

10 10 ARTA7 Sonia Trepode. The representation dimension of selfinjective algebras of tilted type. Our objective in this talk is to explore the relation between the representation theory of an algebra, or more precisely the shape of its Auslander-Reiten components, and its homological invariants. We are in particular interested here in the representation dimension of an algebra, introduced by Auslander, which measures in some way the complexity of the morphisms of the module category. There were several attempts to understand, or compute, this invariant. Special attention was given to algebras of representation dimension three. The reason for this interest is twofold. Firstly, it is related to the finitistic dimension conjecture: Igusa and Todorov have proved that algebras of representation dimension three have a finite finitistic dimension. Secondly, because Auslander s expectation was that the representation dimension would measure how far an algebra is from being representation-finite, there is a standing conjecture that the representation dimension of a tame algebra is at most three. Indeed, while there exist algebras of arbitrary, but finite, representation dimension, most of the best understood classes of algebras have representation dimension three. This is the case, for instance, for algebras obtained by means of tilting, such as tilted algebras, iterated tilted algebras and quasitilted algebras. In ths talk we consider algebras which are the orbit algebra of the repetitive algebra of some tilted algebra under the action of an infinite cyclic group of automorphisms. We prove that the representation dimension of a selfinjective algebra of Euclidean or wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander-Reiten component then its representation dimension is equal to three. This is joint work with Ibrahim Assem and Andrzej Skowroński. Changchang Xi. Constructing derived equivalences by Milnor patching. One of the fundamental questions on derived categories of algebras is how to construct tilting complexes and derived equivalences between two algebras. In this talk, we shall present methods to construct derived equivalences for pullback algebras by Milnor patching. In fact, we will show the following π Theorem. Suppose that A 1 π 1 2 A0 A2 are homomorphisms of Artin algebras with π 1 surjective. Let T i be a basic, radical tilting complex over A i with B i the endomorphism algebra of T i for 0 i 2. If there is an isomorphism A 0 Ai T i T 0 of complexes for i = 1, 2 and T 0 is a direct sum of shifts of projective A 0 -modules, then there exist homomorphisms η 1 η 2 B 1 B0 B2 of Artin algebras with η 1 surjective such that the pullback algebra B of η 1 and η 2 is derived equivalent to the pullback algebra A of π 1 and π 2. As applications of this result, we can repeatedly construct derived equivalences from given ones by gluing vertices, unifying arrows and identifying socle elements. This talk will present parts of a joint work with W. Hu (see arxiv: ). Kunio Yamagata. Selfinjective orbit algebras induced from repetitive algebras. This is a report on a joint work with A. Skowroński. Study of selfinjective algebras is one of the main themes in representation theory of finite dimensional algebras. As initiated by Riedtmann and Hughes-Waschbusch on representaton-finite selfinjective algebras and well

11 ARTA7 11 developed by Skowronski and his group in Torun on representation-tame selfinjective algebras, a prominent role is played by repetitive algebras and their selfinjecitve orbit algebras by admissible groups. Thus it is natural to ask when a selfinjective algebra A is obtained as an orbit algebra of the repetitive algebra of an algebra B. In the second half of 1990s we had a ring theoretical criterion theorem for a selfinjective algebra to be an orbit algebra by a particular admissible group. The theorem is applied practically to the Auslander-Reiten quiver of a given selfinjective algebra. In my talk I will explain some recent results on the problem obtained by such an application, by concentrating on the case B being tilted. Scientific Committee Ibrahim Assem (Université de Sherbrooke) Andrzej Skowroński (Nicolaus Copernicus University) Christof Geiss (IMUNAM) Daniel Labardini (IMUNAM) Octavio Mendoza (IMUNAM) Corina Sáenz (FCUNAM) Local Committee