BIREP – Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Seminar

Friday, 30 June 2017

• 14:15, Lecture Hall H10
  Otto Kerner (Duesseldorf): Thick subcategories of the stable category of modules over the exterior algebra
  Abstract: Let R be a finite dimensional exterior algebra over an algebraically closed field. R is a graded algebra in the obvious way. We consider the graded category of finite dimensional R-modules (with homomorphisms of degree zero). The corresponding stable category is a triangulated category is equivalent as a triangulated category to the derived category of coherent sheaves over the corresponding projective space). We describe the thick subcategories of this category, generated by R-modules of complexity one.
  This is joint work with Dan Zacharia.

For a regular email announcement please contact birep.

---

Future Talks

Friday, 14 July 2017

• Lecture Hall H10
  Rasool Hafezi (Isfahan): On finitely presented functors over the stable categories
  Abstract: In this talk, I will explain my recent work available on arXiv with same title as my talk here.
  In this paper, I studied the category of finitely presented functor over the stable category of some certain subcategories of an abelian category. In particular, this investigation provides a positive answer to a conjecture of M. Auslander, that is, a direct summand of a covariant Ext-functor is again of that form. I will continue this study for the subcategory of Gorenstein projective modules, and as a result this gives some criteria when the relative Auslander translation respect to this subcategory is the first syzygy functor.
  
• Lecture Hall H10
  Sondre Kvamme (Bonn): tba

Friday, 13 October 2017

• Alexander Samokhin (Düsseldorf): tba

---

Seminar Archive

Friday, 23 June 2017

• Bielefeld - Bochum Seminar
  It takes place in Bochum.

Friday, 09 June 2017

• 14:15, Lecture Hall H10
  Julia Sauter (Bielefeld): Desingularizations of orbit closures and quiver Grassmannians from tilting modules
  Abstract: We study orbit closures in representation spaces of finite-dimensional algebras and quiver Grassmannians. In both cases we construct desingularizations assuming the module is gen-finite, i.e. it has only finitely many isomorphism classes of quotients. Our construction uses tilting modules on endomorphism rings of generators, recollements and homotopy categories. This is joint work in progress with Matthew Pressland generalizing previous work from Sauter--Crawley-Boevey.
• 15:30, Lecture Hall H10
  Tobias Rossmann (Auckland): The average size of the kernel of a matrix and orbits of linear groups
  Abstract: Motivated by questions on the representation growth of unipotent algebraic groups, we study generating functions enumerating orbits of p-adic linear groups. Using Lie theory, these functions turn out to be closely related to average sizes of kernels in modules of matrices.

Friday, 02 June 2017

• 13:15, Lecture Hall H10
  Apolonia Gottwald (Bielefeld): Lattices of subobject closed subcategories
  Abstract: For an Abelian length category A, we consider the lattice S(A) of full additive subobject closed subcategories. This lattice is distributive if and only if certain conditions on the Ext-quiver of A are fulfilled. If the Ext-quiver is symmetric, then S(A) is distributive if and only if an even stronger property holds: every subobject of an indecomposable object in A is itself indecomposable. In particular, we get this equivalence, if A is the category of finitely generated modules over an Artin algebra over an algebraically closed field.
• 14:30, Lecture Hall H10
  Alex Martsinkovsky (Boston): (Co-)torsion via stable functors
  Abstract: This talk will concentrate on two new applications of stable functors (these are functors defined on injectively or projectively stable module categories). The first one is a definition of the torsion submodule of a module, which provides a simultaneous generalization of the classical torsion and, for finitely presented modules, of the 1-torsion (= the kernel of the bidualization map). The second one is a definition of the cotorsion quotient module of a module, which doesn’t seem to have a classical prototype. This is done in utmost generality: for arbitrary modules over arbitrary rings. Some of the obtained results are new even in the classical setting of abelian groups.
  The new definitions are remarkably simple and can be given without appealing to stable functors. However, one of the goals of this talk is to convince the audience that the language of functors, being simple, convenient, and natural, brings about additional insights. In that language, this talk is about the injective stabilization of the tensor product and the projective stabilization of the contravariant Hom functor.
  Time permitting, we shall see that the Auslander-Gruson-Jensen functor sends the cotorsion functor to the torsion functor. If the injective envelope of the ring is finitely presented, then the right adjoint of the AGJ functor sends the torsion functor back to the cotorsion functor. In particular, over an artin algebra, this correspondence establishes a duality between the requisite functors on the categories of all modules.
  This will be an expository talk, no prior familiarity with functor categories is assumed. This is joint work with Jeremy Russell.
• 16:00, Lecture Hall H10
  Dan Zacharia (Syracuse): Using linear modules to study exceptional sheaves on the projective n-space
  Abstract: I will talk on joint work with Otto Kerner. Let V be an n+1 dimensional vector space over the field of complex numbers, let R be the exterior algebra on V, and let S be the polynomial algebra in n+1 indeterminates. Finally, consider the category of coherent sheaves on the corresponding projective space. A coherent sheaf E is called exceptional, if it has no extensions with itself, and, in addition, E has an endomorphism ring isomorphic to the ground field. My talk will be about a possible way to reduce certain problems about coherent sheaves (in particular, exceptional ones) to working with some particularly nice modules (called linear modules) over the exterior algebra.

Friday, 26 May 2017

• 13:15, Lecture Hall H10
  Estanislao Herscovich (Grenoble): On some mixture conditions of monoidal structures appearing in Quantum Field Theory
  Abstract: R. Borcherds has introduced a different point of view to formalise perturbative Quantum Field Theory (pQFT). In particular, he uses several objects which behave somehow like bialgebras and comodules over them, and which are essential in his definition of Feynman measure. The former objects don’t seem however to be bialgebras in the classical sense, for their product and coproduct are with respect to two different tensor products, and similarly for comodules. Moreover, following physical motivations, these objects are given as some symmetric constructions of geometric nature.
  The aim of this talk is on the one hand to show that the “bialgebras” and “comodules” introduced by Borcherds cannot “naturally” exist, and on the other side to provide a background where a modified version of the so-called “bialgebras” and “comodules” do exist. This involves a category provided with two monoidal structures satisfying some compatibility conditions. As expected, the modified version of the mentioned “bialgebras” and “comodules” are not so far from the original one, considered by Borcherds. Moreover, we remark that these new candidates allowed us to prove the main results stated by Borcherds in his article (see my manuscript "Renormalization in Quantum Field Theory").
• 14:30, Lecture Hall H10
  Dave Benson (Aberdeen): Surface bundles over surfaces, and cohomology of finite groups
  Abstract: This is a report on joint work with Caterina Campagnolo, Andrew Ranicki and Carmen Rovi. The signature of a surface bundle over a surface is always divisible by four. We describe how to compute the signature modulo eight using the cohomology and representation theory of finite groups.
• 16:00, Lecture Hall H10
  Jon Carlson (Athens, Georgia): An obstreperous class of modules
  Abstract: We discuss several questions concerning the nature of modules over a modular group algebra having full support variety but which have dimension divisible by the prime characteristic of the coefficient field. We will mention some results and work with Paul Balmer and with Dave Benson.

Friday, 05 May 2017

• 13:15, Lecture Hall H10
  Ming Lu (Chengdu): Modified Ringel-Hall algebras and Drinfeld double
  Abstract: Inspired by the works of Bridgeland and Gorsky, we define an algebra from the Ringel-Hall algebra of the category formed by Z/2-graded complexes over a hereditary abelian category which may not have enough projective objects. Such algebra is called modified Ringel-Hall algebra. We prove it to be of some nice properties and structures. The first one is that it has a nice basis, which yields that it is a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the derived category of Z/2-graded complexes. The second one is that in twisted case it is isomorphic to the Drinfeld double Ringel-Hall algebra of the hereditary abelian category. Finally, if the category has a tilting object T, then its modified Ringel-Hall algebra is isomorphic to the Z/2-graded semi-derived Hall algebra and also the Bridgeland's Ringel-Hall algebra of the endomorphism algebra of T.
  This is a joint work with Liangang Peng, and it is available at arXiv:1608.03106.
• 14:30, Lecture Hall H10
  Jorge Vitória (London): Silting and cosilting classes in derived categories
  Abstract: A class of modules over a ring is a tilting class if and only if it is the Ext-orthogonal class to a set of compact modules of bounded projective dimension. Cotilting classes, on the other hand, are precisely the resolving and definable subcategories of the module category whose Ext-orthogonal class has bounded injective dimension.
  Silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules. In this talk we will discuss a generalisation of the characterisations above to this derived setting, with a particular focus on the silting case. This is joint work with Frederik Marks.
• 16:00, Lecture Hall H10
  Olaf Schnürer (Bonn): Geometric applications of conservative descent for semi-orthogonal decompositions
  Abstract: Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry we introduce a technique called "conservative descent” in order to establish such decompositions locally. The decompositions we have in mind are those for projective bundles, blow-ups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semi-orthogonal decompositions for Brauer-Severi varieties.
  This is joint work in progress with Daniel Bergh.

Friday, 28 April 2017

• 14:15, Lecture Hall H10
  Chun-Ju Lai (Bonn): Affine Hecke algebras and quantum symmetric pairs
  Abstract: In an influential work of Beilinson, Lusztig and MacPherson, they provide a construction for (idempotented) quantum groups of type A together with its canonical basis. Although a geometric method via partial flags and dimension counting is applied, it can also be approached using Hecke algebras and combinatorics. In this talk I will focus on the Hecke algebraic approach and present our work on a generalization to affine type C, which produces favorable bases for q Schur algebras and certain coideal subalgebras of quantum groups of affine type A. We further show that these algebras are examples of quantum symmetric pairs, which are quantization of symmetric pairs consisting of a Lie algebra and its fixed-point subalgebra associated to an involution.
  This is a joint work (arXiv:1609.06199) with Z. Fan, Y. Li, L. Luo, and W. Wang.
• 15:30, Lecture Hall H10
  Charles Vial (Bielefeld): Numerical obstructions to the existence of exceptional collections on surfaces
  Abstract: I will give a complete classification of smooth projective complex surfaces that admit a numerically exceptional collection of maximal length. I will also give arithmetic constraints for the existence of such collections on surfaces defined over non-algebraically closed field.

Friday, 21 April 2017

• 13:15, Lecture Hall H10
  Pieter Belmans (Antwerpen): Hochschild cohomology of noncommutative planes and quadrics
  Abstract: The derived category of P^2 has a full exceptional collection, described by the famous Beilinson quiver and its relations. One can compute the Hochschild cohomology of P^2 as the Hochschild cohomology of this finite-dimensional algebra, for which many tools are available. Changing the relations in the quiver corresponds to considering derived categories of noncommutative planes.
  I will explain how these (and the noncommutative analogues of the quadric surface) are described using Artin-Schelter regular (Z-)algebras, and how one can use their classification to compute the Hochschild cohomology of all finite-dimensional algebras obtained in this way, exhibiting an interesting dimension drop.
  If time permits I will explain how it is expected that the fully faithful functor between the derived category of P^2 and its Hilbert scheme of two points relates their Hochschild cohomologies, and how this is expected to behave under deformation.
• 14:30, Lecture Hall H10
  Shiquan Ruan (Beijing): Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras
  Abstract: Let coh-X be the category of coherent sheaves over a weighted projective line X and let D^b(coh-X) be its bounded derived category. In this talk we will focus on the study of the right and left mutation functors arising in D^b(coh-X) attached to certain line bundles. We first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver Q associated with X. By further dealing with the Ringel–Hall algebra of X, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra g_Q of Q, as well as for Lusztig’s symmetries of the quantum enveloping algebra of g_Q.
• 16:00, Lecture Hall H10
  Thorsten Weist (Wuppertal): Normal forms for quiver representations induced by tree modules
  Abstract: With a fixed tree module of a quiver and a fixed tree-shaped basis of its group of self-extensions, one can associate an affine space of representations with the same dimension vector. If all representations in this cell are indecomposable and pairwise non-isomorphic, this induces a normal form for these representations. Even if this is general not true, it is possible to state conditions which are sufficient to obtain cells of pairwise non-isomorphic indecomposable representations. For instance this is the case if the tree module is a torus fixed point under a certain torus action. But also recursive constructions of indecomposable representations turn out to be very useful. Now it also might happen that the same representation lies in different cells. Nevertheless, for certain roots this machinery can be used to give a full description of the set of indecomposable representations. In any case, this gives a concept which can be used to deduce normal forms for indecomposable representations of quivers based on the notion of tree modules.
  We also discuss connections to a rather vague conjecture of Kac which says that the set of isomorphism classes of indecomposable representations of quivers admits a cell decomposition into locally closed subvarieties which are affine cells.

Friday, 07 April 2017

• 14:15, Room U2-113
  Robert Marsh (Leeds): Rigid and Schurian modules over tame cluster-tilted algebras
  Abstract: We classify the indecomposable rigid and Schurian modules over a cluster-tilted algebra of tame representation type. Such a cluster-tilted algebra B has an associated cluster algebra A(Q), where Q is the quiver of the algebra. Answering a question of T. Nakanishi, we show that A(Q) can have a denominator vector which is not the dimension vector of any indecomposable B-module. Using the above classification, we show that every denominator vector of A(Q) is the sum of the dimension vectors of at most three indecomposable rigid B-modules.
  The talk is based on joint work with Idun Reiten.

Thursday, 09 March 2017

• 16:15, Room U2-135
  Michael Wemyss (Glasgow): Faithful actions in algebra and geometry
  Abstract: In many algebraic and geometric contexts, the associated derived category admits an action by the fundamental group of some reasonable topological space. In other words, there is a group homomorphism from the fundamental group to the group of autoequivalences. The most famous examples are actions by braid groups, but there are many more general examples, including actions induced by 3-fold flops in algebraic geometry.
  
  I will explain one technique, based on exploiting the partial order on tilting modules, that can be used to deduce when the action is faithful, that is, when the group homomorphism is injective. This algebraic framework applies in various settings, and can be used to extract geometric corollaries, including some in the motivating example of flops. This is joint work with Yuki Hirano.

Tuesday, 28 February 2017

• 14:15, Room V2-213
  Ralf Schiffler (Storrs, Connecticut): Cluster algebras, snake graphs and continued fractions
  Abstract: This talk is on a combinatorial realization of continued fractions in terms of perfect matchings of the so-called snake graphs, which are planar graphs that have first appeared in expansion formulas for the cluster variables in cluster algebras from triangulated marked surfaces. ​I will also explain applications to cluster algebras, as well as​ to​ elementary number theory. This is​ a​ joint work with Ilke Canakci​.​

Friday, 03 February 2017

• 14:15, Room U2-135
  Teresa Conde (Stuttgart): Strongly quasihereditary algebras
  Abstract: Quasihereditary algebras are abundant in mathematics. They classically occur as blocks of the category O and as Schur algebras.
  They also occur as endomorphism algebras associated to modules endowed with special filtrations. The quasihereditary algebras produced in these cases are very often strongly quasihereditary in the sense of Ringel. The ADR algebra of a finite-dimensional algebra A is an example of such an algebra. Other examples of strongly quasihereditary algebras include: the Auslander algebras; the endomorphism algebras constructed by Iyama, used in his proof of the finiteness of the representation dimension; certain cluster-tilted algebras studied by Geiß-Leclerc–Schröer and Iyama–Reiten.
  In this talk I will start by introducing the ADR algebra and by describing its neat quasihereditary structure. I will then look at larger classes of strongly quasihereditary algebras and describe some of their properties.

Friday, 20 January 2017

• 14:15, Room U2-135
  Daniel Bissinger (Kiel): Invariants of regular components for wild Kronecker algebras
  Abstract: Motivated by the work on modular representation theory of finite group schemes, Worch introduced the categories of modules with the equal images property and the equal kernels property for the generalized Kronecker algebra.
  Given a regular component C of the Auslander-Reiten quiver, we study the distance W(C) between the two non-intersecting cones in C given by modules with the equal images and the equal kernels property.
  We show that W(C) is closely related to the quasi-rank rk(C) of C. Utilizing covering theory, we discuss how to construct for each natural number n a regular component C_n with W(C_n) = n.
• 15:30, Room U2-135
  Kevin De Laet (Antwerpen): The connection between Sklyanin algebras and the finite Heisenberg groups
  Abstract: The 3-dimensional quadratic Sklyanin algebras are noncommutative graded analogues of the polynomial ring in 3 variables and have been studied by people like Artin, Tate, Van den Bergh, ... In this talk, I am going to show how these algebras can be constructed using the representation theory of the finite Heisenberg group of order 27 such that this group acts on these algebras as gradation preserving automorphisms.
  This action will then be used to prove certain results regarding the central element of degree 3 of such algebras. This talk is based on my paper https://arxiv.org/abs/1612.06158

Friday, 13 January 2017

• 14:15, Room C01-226
  Liran Shaul (Bielefeld): A well behaved category of derived commutative rings over a noetherian ring
  Abstract: Given a commutative noetherian ring K, the goal of this talk is to present a category of derived commutative rings over K which includes the finite type K-algebras, and is closed under the operations of localization, derived tensor products, and derived adic completion. To do this we introduce the homotopy category of derived commutative rings with an adic topology, and explain how to perform these various operations in this category. In particular, we construct the derived adic completion of a derived commutative ring A with respect to a finitely generated ideal of the ring H^0(A).

Friday, 16 December 2016

• 13:15, Room U2-135
  Ögmundur Eiriksson (Bielefeld): From submodule categories to the stable Auslander algebra
  Abstract: C. Ringel and P. Zhang have studied a pair of functors from the submodule category of a truncated polynomial ring over a field to a preprojective algebra of type A. We present the analogous process starting with any self-injective algebra of finite representation type over a field k.
  To this end we study two functors from the submodule category to the module category of the stable Auslander algebra. The functors are compositions of objective functors, and both factor through the module category of the Auslander algebra. We are able to describe the kernels of these functors, both of which have finitely many indecomposables.
  One of the functors factors through the subcategory of torsionless modules over the Auslander algebra. That subcategory arises as the subcategory of objects with a filtration by standard modules for a quasi-hereditary structure on the Auslander algebra if and only if our original algebra is uniserial.
• 14:30, Room U2-135
  Florian Gellert (Bielefeld): Maximum antichains in subrepresentation posets
  Abstract: For indecomposable representations of Dynkin quivers, the structure of indecomposable morphisms is given by the Auslander-Reiten quiver. Very different posets can be formed if one considers the (not necessarily indecomposable) monomorphisms alone; the poset by inclusion admits particular properties. In this talk we study the latter poset for various orientations of type A quivers. We construct maximum antichains and obtain formulas for the widths of the respective posets. This is joint work with Philipp Lampe.
• 16:00, Room U2-135
  Rene Marczinzik (Stuttgart): On dominant dimensions of algebras
  Abstract: The famous Nakayama conjecture states that every nonselfinjective finite dimensional algebra has finite dominant dimension. A stronger conjecture is the conjecture of Yamagata: The dominant dimension of non-selfinjective algebras with a fixed number s of simple modules is bounded by a finite function depending on s. We show that Yamagata's conjecture is true for monomial algebras with n simple modules. In fact an explicit optimal bound is given by 2n-2, which answers a conjecture of Abrar who conjectured bounds in the special case of Nakayama algebras. We end the talk with several conjectures and open questions related to dominant dimension. Those conjectures are mainly motivated by computer calculations with the GAP package QPA. The conjectures are related to a new construction of higher Auslander algebras and periodic algebras from representation-finite hereditary algebras and a new interplay between homological dimensions and the combinatorics of Dyck paths and circular codes for special classes of algebras.

Friday, 02 December 2016

• NWDR Workshop Winter 2016
  10:30, Room V2-210/216
  Gabor Elek (Lancaster): Convergence and limits of finite dimensional representations of algebras
  Abstract: Motivated by the limit theory of finite graphs I will introduce the notion of metric convergence of finite dimensional representations of algebras over a countable field. It turns out that the limit points are infinite dimensional representations and together with the finite dimensional representations they form a compact metric space. I will also talk about the notion of hyperfiniteness for finite dimensional algebras and its relation with the classical notion of amenability.
  
• NWDR Workshop Winter 2016
  11:45, Room V2-210/216
  Andrew Hubery (Bielefeld): Euler characteristics of quiver Grassmannians
  Abstract: We show how to define an Euler characteristic for the Grassmannian of submodules of a rigid module, for any finite-dimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is non-empty, generalising the known case of quiver Grassmannians. As an application, this shows that, for any symmetrisable acyclic cluster algebra, the Laurent expansion of a cluster variable always has non-negative coefficients.
• NWDR Workshop Winter 2016
  14:00, Room V2-210/216
  Peter Patzt (Berlin): Representation stability for the general linear groups
  Abstract: The notion of representation stability for the symmetric groups, the general linear groups and the symplectic groups was introduced by Church-Farb. We give a criterion for a sequence of algebraic representations of the general linear groups to be representation stable. With it we prove that the factors of the lower central series of the Torelli subgroups of the automorphism groups of free groups are representation stable.
• NWDR Workshop Winter 2016
  15:30, Room V2-210/216
  Pierre-Guy Plamondon (Paris): Multiplication formulas
  Abstract: In the past decade, the study of cluster algebras via representations of quivers has proved a successful way to tackle some of the problems in the theory. In this talk, I will review the theory of cluster characters and present new multiplication formulas relating them.
• NWDR Workshop Winter 2016
  16:45, Room V2-210/216
  Christine Bessenrodt (Hannover): Kronecker products of characters of the symmetric groups and their double covers
  Abstract: Decomposing Kronecker products of irreducible characters of the symmetric groups (or equivalently, of inner products of Schur functions) is a longstanding central problem in representation theory and algebraic combinatorics. The talk will focus on special Kronecker products and related problems for skew characters, in particular on the recent classification of multiplicity-free Kronecker products of irreducible characters of the symmetric groups, conjectured in 1999. Also related conjectures and results on spin characters of the double cover groups will be discussed, and the connection between them will be illustrated by some applications of spin characters towards results for symmetric groups.

Friday, 25 November 2016

• 14:15, Room U2-135
  Matthew Pressland (Bonn): Dominant dimension and canonical tilts
  Abstract: Any finite dimensional algebra with dominant dimension d admits a 'canonical' k-tilting module for each k from 0 to d, each giving a derived equivalence with some algebra B_k. These tilts have very special properties; for example, they never increase the global dimension. In the case of the Auslander algebra of a representation-finite algebra A, Crawley-Boevey and Sauter (generalising Cerulli Irelli, Feigin and Reineke) used the tilt B_1 to construct desingularisations of certain varieties of A-modules. More generally, for d at least 2, any algebra of dominant dimension d is the endomorphism algebra of a generating-cogenerating module M over some algebra A, and many of the results for Auslander algebras have analogues in this setting. In particular, we may realise each B_k as an endomorphism algebra in the homotopy category of A, an observation which we can exploit to describe rank varieties, of arbitrary finite dimensional modules over arbitrary finite dimensional algebras, as affine quotient varieties. We may also use the canonical tilting modules to give a new characterisation of d-Auslander (or, more generally, d-Auslander–Gorenstein) algebras. This is joint work with Julia Sauter.
  

Friday, 18 November 2016

• 14:15, Room U2-135
  Sondre Kvamme (Bonn): A generalization of finite-dimensional Iwanaga-Gorenstein algebras
  Abstract: We will introduce certain well-behaved comonads on abelian categories, which generalize features of the module category of a finite dimensional algebra. For example, we will define Gorenstein flat objects relative to the comonad, which generalize Gorenstein projective modules for a finite-dimensional algebra. We will also define what it means for the comonad to be Gorenstein, and state analogues of some classical results for Iwanaga-Gorenstein algebras. We will illustrate the constructions and results on specific examples.
• 15:30, Room U2-135
  Rosanna Laking (Bonn): Krull-Gabriel dimension, Ziegler spectra of module categories and applications to compactly generated triangulated categories.
  Abstract: We will begin by defining the notion of Krull-Gabriel (KG-) dimension for the module category of a ring R (with many objects) and outlining how this relates to a topology on the indecomposable pure-injective R-modules. Using examples, we aim to explain how the KG-dimension "measures" transfinite factorisations of morphisms in R-mod.
  We will then consider analogous notions for compactly generated triangulated categories. We will show that one can work in the context of an associated module category, and hence one can directly make use of the tools described in the first part of the talk. Using this insight we will describe the Ziegler spectrum of the bounded derived category of a derived-discrete algebra A and calculate its KG-dimension. These techniques lead to a classification of the indecomposable objects in the (unbounded) homotopy category of A.
  This talk is based on joint work with K. Arnesen, D. Pauksztello and M. Prest.

Friday, 04 November 2016

• 14:15, Room U2-135
  Shengfei Geng (Chengdu): Tilting modules and support tau-tilting modules over preprojective algebras associated with symmetrizable generalized Cartan matrices
  Abstract: For each skew-symmetrizable generalized Cartan matrix, Geiss-Leclec-Schröer defined a class of preprojective algebra which concide with the classical preprojective algebra when the Cartan matrix is symmetric and the symmetrizer is an identity matrix. In this paper, we proved that there is a bijection between the sets of cofinite tilting ideals with global dimension at most one of such preprojective algebra and the corresponding Weyl group when the preprojective algebra is non-Dynkin type. Based on this, we proved that there is a bijection between the sets of support tau-tilting modules of the preprojective algebra and the corresponding Weyl group when the preprojective algebra is of Dynkin type. Here the preprojective algebras of Dynkin type contain not only types of A,D,E, but also contain types of B,C,G,F. These results generalized the results over classical preprojective algebras.
• 15:30, Room U2-135
  Ming Lu (Chengdu): Singularity categories of positively graded Gorenstein algebras
  Abstract: This is a report on ongoing work with Bin Zhu. We discuss the existence of silting objects and tilting objects in the singularity categories of graded modules over positively graded Gorenstein algebras. By generalizing a result of Yamaura for positively graded selfinjective algebras, we prove that for a positively graded 1-Gorenstein algebra A such that A_0 has finite global dimension, its singularity category of graded modules has a silting object. Under some conditions, this silting object is even a tilting object. After that, we apply it to cluster-tilted algebras and representations of quivers over local rings.

Friday, 28 October 2016

• 13:15, Room U2-135
  Gustavo Jasso (Bonn): Mesh categories of type A-infinity and tubes in higher Auslander-Reiten theory
  Abstract: This is a report on joint work with Julian Külshammer. We construct higher analogues of mesh categories of type A-infinity and of the tubes from the viewpoint of Iyama's higher Auslander-Reiten theory. Our construction relies on unpublished work by Darpö and Iyama. We relate these constructions to higher Nakayama algebras, which we also introduce.
  
• 14:30, Room U2-135
  Julian Külshammer (Stuttgart): Spherical objects in higher Auslander-Reiten theory
  Abstract: This is a report on ongoing work with Gustavo Jasso. Let D be the triangulated category associated to a spherical object of dimension m
  greater than or equal to 2. By work of Jørgensen, this is an m-Calabi-Yau triangulated category with almost split triangles. Moreover, its Auslander-Reiten quiver has m-1 connected components of type ZA-infinity. Building upon work of Amiot, Guo, Keller, and Oppermann-Thomas, for each positive integer d we construct an md-Calabi-Yau (d+2)-angulated category with almost split (d+2)-angles. Moreover, its higher Auslander-Reiten quiver has m-1 connected components of higher mesh type A-infty. For m=2, our construction is analogous to the cluster a category of type A-infinity introduced by Holm-Jørgensen.
• 16:00, Room U2-135
  Sven-Ake Wegner (Wuppertal): Is functional analysis a special case of tilting theory?
  Abstract: The objects of functional analysis together with the corresponding morphisms don't form abelian categories. Classical examples, e.g., the category of Banach spaces, satisfy almost all axioms of an abelian category but the canonical morphism between the cokernel of the kernel and the kernel of the cokernel of a given map fails in general to be an isomorphism. In 2003, Bondal and van den Bergh showed that there is a correspondence between (co-)tilting torsion pairs and so-called quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical t-structure on its derived category. In this talk we discuss examples of categories appearing in functional analysis that are not quasiabelian but which carry a natural exact structure. The derived category is thus defined. We are interested in an extension of the aforementioned equivalence as this could be a step towards a successful categorification of certain analytic problems.

Friday, 21 October 2016

• 14:15, Room U2-135
  Michael K. Brown (Bonn): Topological K-theory of dg categories of graded matrix factorizations
  Abstract: Topological K-theory of complex-linear dg categories is a notion recently introduced by A. Blanc. The main goal of the talk is to discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a complex quasi-homogeneous polynomial in terms of a classical topological invariant of a hypersurface: the Milnor fiber and its monodromy. I will also discuss some applications of this calculation, and, if time permits, some future directions.

For information on earlier talks please check the complete seminar archive.

• © 2010–2017 Bielefeld University
• Last file change on 21 Jul 2015 by Andre Beineke
• Contact
• Legal notice