Seminar
Friday, 28 April 2017

14:15, Lecture Hall H10
ChunJu 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 fixedpoint 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 nonalgebraically closed field.
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Future Talks
Friday, 05 May 2017

13:15, Lecture Hall H10
Ming Lu (Chengdu): Modified RingelHall algebras and Drinfeld double
Abstract: Inspired by the works of Bridgeland and Gorsky, we define an algebra from the RingelHall algebra of the category formed by Z/2graded complexes over a hereditary abelian category which may not have enough projective objects. Such algebra is called modified RingelHall 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/2graded complexes. The second one is that in twisted case it is isomorphic to the Drinfeld double RingelHall algebra of the hereditary abelian category. Finally, if the category has a tilting object T, then its modified RingelHall algebra is isomorphic to the Z/2graded semiderived Hall algebra and also the Bridgeland's RingelHall 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 Extorthogonal 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 Extorthogonal 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 semiorthogonal decompositions
Abstract: Motivated by the local flavor of several wellknown semiorthogonal 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, blowups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semiorthogonal decompositions for BrauerSeveri varieties.
This is joint work in progress with Daniel Bergh.
Friday, 23 June 2017

Bielefeld  Bochum Seminar
It takes place in Bochum.
Seminar Archive
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 finitedimensional 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 ArtinSchelter regular (Z)algebras, and how one can use their classification to compute the Hochschild cohomology of all finitedimensional 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 cohX be the category of coherent sheaves over a weighted projective line X and let D^b(cohX) 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(cohX) 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 treeshaped basis of its group of selfextensions, one can associate an affine space of representations with the same dimension vector. If all representations in this cell are indecomposable and pairwise nonisomorphic, 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 nonisomorphic 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 U2113
Robert Marsh (Leeds): Rigid and Schurian modules over tame clustertilted algebras
Abstract: We classify the indecomposable rigid and Schurian modules over a clustertilted algebra of tame representation type. Such a clustertilted 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 Bmodule. 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 Bmodules.
The talk is based on joint work with Idun Reiten.
Thursday, 09 March 2017

16:15, Room U2135
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 3fold 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 V2213
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 socalled 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 U2135
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 finitedimensional 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 clustertilted algebras studied by GeißLeclerc–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 U2135
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 AuslanderReiten quiver, we study the distance W(C) between the two nonintersecting 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 quasirank 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 U2135
Kevin De Laet (Antwerpen): The connection between Sklyanin algebras and the finite Heisenberg groups
Abstract: The 3dimensional 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 C01226
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 Kalgebras, 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 U2135
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 selfinjective 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 quasihereditary structure on the Auslander algebra if and only if our original algebra is uniserial.

14:30, Room U2135
Florian Gellert (Bielefeld): Maximum antichains in subrepresentation posets
Abstract: For indecomposable representations of Dynkin quivers, the structure of indecomposable morphisms is given by the AuslanderReiten 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 U2135
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 nonselfinjective 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 2n2, 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 representationfinite 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 V2210/216
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 V2210/216
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 finitedimensional hereditary algebra over a finite field. Moreover, we prove that this number is positive whenever the Grassmannian is nonempty, 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 nonnegative coefficients.

NWDR Workshop Winter 2016
14:00, Room V2210/216
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 ChurchFarb. 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 V2210/216
PierreGuy 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 V2210/216
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 multiplicityfree 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 U2135
Matthew Pressland (Bonn): Dominant dimension and canonical tilts
Abstract: Any finite dimensional algebra with dominant dimension d admits a 'canonical' ktilting 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 representationfinite algebra A, CrawleyBoevey and Sauter (generalising Cerulli Irelli, Feigin and Reineke) used the tilt B_1 to construct desingularisations of certain varieties of Amodules. More generally, for d at least 2, any algebra of dominant dimension d is the endomorphism algebra of a generatingcogenerating 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 dAuslander (or, more generally, dAuslander–Gorenstein) algebras. This is joint work with Julia Sauter.
Friday, 18 November 2016

14:15, Room U2135
Sondre Kvamme (Bonn): A generalization of finitedimensional IwanagaGorenstein algebras
Abstract: We will introduce certain wellbehaved 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 finitedimensional algebra. We will also define what it means for the comonad to be Gorenstein, and state analogues of some classical results for IwanagaGorenstein algebras. We will illustrate the constructions and results on specific examples.

15:30, Room U2135
Rosanna Laking (Bonn): KrullGabriel dimension, Ziegler spectra of module categories and applications to compactly generated triangulated categories.
Abstract: We will begin by defining the notion of KrullGabriel (KG) dimension for the module category of a ring R (with many objects) and outlining how this relates to a topology on the indecomposable pureinjective Rmodules. Using examples, we aim to explain how the KGdimension "measures" transfinite factorisations of morphisms in Rmod.
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 deriveddiscrete algebra A and calculate its KGdimension. 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 U2135
Shengfei Geng (Chengdu): Tilting modules and support tautilting modules over preprojective algebras associated with symmetrizable generalized Cartan matrices
Abstract: For each skewsymmetrizable generalized Cartan matrix, GeissLeclecSchröer 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 nonDynkin type. Based on this, we proved that there is a bijection between the sets of support tautilting 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 U2135
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 1Gorenstein 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 clustertilted algebras and representations of quivers over local rings.
Friday, 28 October 2016

13:15, Room U2135
Gustavo Jasso (Bonn): Mesh categories of type Ainfinity and tubes in higher AuslanderReiten theory
Abstract: This is a report on joint work with Julian Külshammer. We construct higher analogues of mesh categories of type Ainfinity and of the tubes from the viewpoint of Iyama's higher AuslanderReiten 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 U2135
Julian Külshammer (Stuttgart): Spherical objects in higher AuslanderReiten 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 mCalabiYau triangulated category with almost split triangles. Moreover, its AuslanderReiten quiver has m1 connected components of type ZAinfinity. Building upon work of Amiot, Guo, Keller, and OppermannThomas, for each positive integer d we construct an mdCalabiYau (d+2)angulated category with almost split (d+2)angles. Moreover, its higher AuslanderReiten quiver has m1 connected components of higher mesh type Ainfty. For m=2, our construction is analogous to the cluster a category of type Ainfinity introduced by HolmJørgensen.

16:00, Room U2135
SvenAke 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 socalled quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical tstructure 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 U2135
Michael K. Brown (Bonn): Topological Ktheory of dg categories of graded matrix factorizations
Abstract: Topological Ktheory of complexlinear dg categories is a notion recently introduced by A. Blanc. The main goal of the talk is to discuss a calculation of the topological Ktheory of the dg category of graded matrix factorizations associated to a complex quasihomogeneous 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.