Paderborn Representation Theory Seminar

The Seminar is organised by 

• Prof. Dr. Henning Krause
• PD Dr. Dirk Kussin
• Prof. Dr. Helmut Lenzing

If not announced otherwise, the Seminar meets on Thursday at 14.15 - 18.00 in room D1.320. Talks last usually between 60 and 90 minutes. A regular announcement is sent by email (contact hkrause"at"math.upb.de to get on the mailing list).

Archive:  2008 / 2007 / 2006 / 2005 / 2004 / 2003

Programme 2009:

• 28.01.2010, 14.15, Alexander Zimmermann (Université de Picardie, Amiens)
  Distinguishing deformed preprojective algebras
  Abstract: Bialkowski, Erdmann and Skowronski classified self-injective algebras with the property that the third syzygy of any simple module S is isomorphic to S again. These algebras are deformations in a certain sense of preprojective algebras of quivers of type A, D, E or L. We study the question which deformation parameters lead to algebras in different derived equivalence class.

• 21.01.2010, 16.15, Christof Geiss (Mexico)
  Cluster Algebra Structures for unipotent cells and preprojective algebras
  Abstract: This is report on a joint project with B. Leclerc and J. Schroer.
  For a simple complex Lie group $G$ and an element $w$ of the Weyl group, Berenstein, Fomin and Zelevinsky introduced for the unipotent cell $N^w=N\cap B_-w B_-$ a cluster algebra structure on its coordinate ring, with each cluster providing a test for total positivity. We provide an alternantive proof of this result via categorification which extentds the results to the symmetric Kac-Moody setting.

• 14.01.2010, 15.45, Xiao-Wu Chen (Paderborn)
  Triangulated categories of singularities for Gorenstein algebras, II
  Abstract: In this talk I will explain several applications of Orlov' Theorem  2.5: applications to AS-regular algebras, to graded Frobenius algebras and to  (commutative) projective Gorenstein vareities; in particular, complete intersections.

• 14.01.2010, 14.15, Guodong Zhou (Paderborn)
  Triangulated categories of singularities for  Gorenstein algebras, I
  
  

• 07.01.2010, 16.15, Steffen Oppermann (Köln)
  n-representation finite algebras and n-APR tilting
  Abstract: This talk is a report on joint work with Osamu Iyama. I will introduce the notion of n-representation finite algebras, which are defined to be algebras for which the module category has a finite n-cluster tilting subcategory. This setup is particularly well-suited for n-dimensional Auslander-Reiten-Theory.
  In this talk I will focus on n-APR tilting, a higher analog of APR-tilting (or BGP reflection functors). We will see that n-APR tilts act on n-cluster tilting subcategories of module categories very similarly to the way usual APR-tilts act on the module category of a hereditary algebra. That is, they essentially remove one simple projective module from the category, and in return add a new simple injective.
  Finally I will look at some derived category aspects of n-representation finiteness and n-APR tilting. Using this point of view we will be able to classify all iterated n-APR tilts of higher Auslander algebras of linearly oriented A_n.

• 17.12.2009, 14.15, David Pauksztello (Leeds)
  Generalised Moore Spectra in a Triangulated Category
  Abstract: In this talk we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which 'approximates' objects from the module category of the endomorphism algebra of C in T . This is an analogue of a construction of Jorgensen which appears in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding a module category in a triangulated category. As an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category of u > 2.

• 10.12.2009, 15.30, Henning Krause (Paderborn)
  Colocalizing subcategories and cosupport of modules and complexes (after Neeman)
  Abstract: Recent work of Neeman shows how colocalizing subcategories of the derived category of a commutative noetherian ring can be classified. Colocalizing subcategories are triangulated subcategories that are closed under taking arbitrary products.  The classification is remarkably simple and beautiful, despite all set theoretic complications one might expect. I will explain Neeman's result. Some consequences for the classification of representations of finite groups will be mentioned at the end.

• 10.12.2009, 14.15, Xiao-Wu Chen (Paderborn)
  The stable monomorphism category of a Frobenius category
  Abstract: In this talk, we will study the stable monomorphism category of a Frobenius abelian category, which is a triangulated category. Our motivating example is the stable category of Ringel-Schmidmeier. We will give two characterizations to the stable category of Ringel-Schmidmeier: one is related to bounded derived category of a finite dimensional algebra and another is related to the graded singularity category of a finite dimensional graded algebra. This work is related to  recent works by Kussin-Lenzing-Meltzer and Ladkani.

• 26.11.2009, 16.00, Jan Schröer (Bonn)
  Generating functions for Euler characteristics
  Abstract: In the categorical approach to cluster algebras there appear several "cluster characters", which map an object of a Frobenius category to a generating function for Euler characteristics of flag varieties or quiver Grassmannians. We discuss some of the properties of these functions. This is joint work with C. Geiss and B. Leclerc.

• 26.11.2009, 14.15, Claudia Köhler (Paderborn)
  Higher Auslander-Reiten theory 4
  Abstract:In my talk I will discuss two examples of 2-Calabi-Yau Frobenius categories with 2-cluster tilting objects. One class is constructed from factor algebras of preprojective algebras. The other one consists of the category of maximal Cohen-Macaulay modules over a one-dimensional hypersurface singularity.
  

• 20.11.2009, 14.15 im Raum E2.310, Karsten Dietrich (Paderborn)
  Higher Auslander-Reiten theory 3
  Abstract: This is the third talk in our series on higher Auslander-Reiten theory. We establish Auslander-Reiten theory and higher Auslander-Reiten theory for the category of maximal Cohen-Macaulay modules over an isolated singularity. It turns out that the same results as for the category of modules over a finite dimensional algebra hold in this framework. We will also explain the Calabi-Yau property of the stable category of maximal Cohen-Macaulay modules under certain additional assumptions.

• 05.11.2009, 14.15, Guodong Zhou (Paderborn)
  Higher Auslander-Reiten theory 2
  Abstract: This is the second talk of our studying group on higher Auslander-Reiten theory. In this talk, working in the setting of Artin algebras, I will present higher Auslander correspondence and introduce $n$-cluster tilting subcategories and $n$-almost split sequences. Some examples will be included as well

• 29.10.2009, 15.45, Dirk Kussin (Bielefeld)
  Categories of vector bundles and invariant subspaces of nilpotent operators
  Abstract: We establish a close link between the category of vector bundles over a weighted projective line of weight type (2,3,p) and the invariant subspace category of linear operators of nilpotency index p studied by Ringel and Schmidmeier. (This is joint work with H. Lenzing and H. Meltzer.)

• 29.10.2009, 14.00, Fei Xu (Paderborn)
  Higher Auslander-Reiten theory 1
  Abstract: This is the first talk of the working seminar on the higher Auslander-Reiten Theory in the sense of Iyama. We will go over the classical Auslander-Reiten Theory, including the Auslander algebras, Auslander correspondence and almost split sequences.

• 26.10.2009, 17.15 in E2.310, Valery Lunts (Indiana University)
  Im Rahmen des IRTG Research Seminars
  Categorical resolution of singularities
  Abstract: Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then  propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasi-coherent sheaves on a scheme. We propose two kinds of such resolutions.
  
  

• 27.08.2009, 15.00 im Raum E2.316, Julian Külshammer (Bonn)
  An Alternative Characterization of Biserial Algebras
  Abstract: Semisimple algebras can be characterized as algebras, which do not admit an indecomposable module of length two. A natural question is, if such a description by the non-existence of certain modules exists for other classes of algebras, e.g. biserial algebras. In my diploma thesis I proved such a theorem in the case of a basic algebra, using a result of R. Vila-Freyer and W. Crawley-Boevey, which states, that such algebras can be described in terms of quivers with certain relations.

• 15.07.2009, 14.30 in E2.304, Jue Le (Paderborn)
  Construction of a superdecomposable pure-injective module over a string algebra
  Abstract: I will introduce the recent work of Gena Puninski, who constructed an element m in a direct product M of finite dimensional modules over a string algebra, such that the smallest direct summand of M containing m is a superdecomposable module.

• 09.07.2009, 15.45, Markus Perling (Bochum)
  Exceptional sequences of invertible sheaves on rational surfaces
  Abstract: The explicit construction of tilting sheaves is an unsolved problem in general. We report about structural results for the case of of tilting sheaves on rational surfaces which decompose into direct sums of invertible sheaves.

• 09.07.2009, 14.15, Martin Langer (Bonn)
  The notion of order in the stable module category
  Abstract: Let k be a field of characteristic p, and let G be a finite group. In 1986, Carlson proved that if p is odd then for every class x of even degree in the (Tate) cohomology of G, x annihilates the cohomology of k/x. Here k/x denotes some choice of cone of multiplication by x on k. For p=2 this statement is not always true, so we can ask why the prime 2 is so special in this situation. We have a similar phenomenon in stable homotopy theory: multiplication by p on the mod-p Moore spectrum S/p vanishes if and only if p>=3. Motivated by the proof of his Rigidity Theorem, Schwede introduced a notion of "order" on triangulated categories which to a certain amount explains this phenomenon. I will carry over this notion to the stable module category and give a generalized version of Carlson's result in which the prime 2 is not special any longer.
  
  

• 25.06.2009, 15.45, Fei Xu (Nantes)
  Tensor product on kC-mod and cohomology rings
  Abstract: Let k be a field and C a finite category. We study the category algebra kC and the monoidal category structure on kC-mod. We show kC is comparable with a cocommutative bialgebra, and hence share many homological properties in common. Based on the tensor product on kC-mod, we can introduce and investigate the ordinary cohomology ring of C, from which we may continue to understand the structure of the Hochschild cohomology ring of kC.

• 25.06.2009, 14.15, Nan Gao (Shanghai / Bielefeld)
  Gorenstein derived categories and  Gorenstein tilting modules
  Abstract: This is a report on joint work with Professor Pu Zhang. We introduce Gorenstein derived categories, and then give the relation with the usual derived categories. We describe explicitly the bounded Gorenstein derived categories of Gorenstein rings and of finite-dimensional algebras via the homotopy categories of Gorenstein-projective modules, and obtain some applications. We also discuss Gorenstein derived equivalences between CM-finite Gorenstein algebras and between finite-dimensional algebras. We introduce Gorenstein tilting modules, and show that the bounded Gorenstein derived categories of Gorenstein artin algebras are triangle-equivalent to the homotopy categories of Gorenstein tilting modules.

• 18.06.2009, 16.45, Karsten Dietrich (Paderborn)
  An upper bound for the finitistic dimension of an EI category algebra
  Abstract: EI categories can be thought of as amalgams of finite posets and finite groups and therefore the associated algebras are build up from incidence algebras and group algebras of finite groups. After recalling the definition and basic facts about representations of this particular class of algebras we will construct an upper bound for the finitistic dimension.

• 18.06.2009, 15.30, Klaus Bongartz (Wuppertal)
  Indecomposables live in all smaller lengths
  Abstract: Let A be an associative algebra of finite dimension over an algebraically closed field. Then there is no gap in the lengths of the indecomposable modules. The proof depends on the theory of representation-finite algebras.

• 18.06.2009, 14.00, Otto Kerner (Düsseldorf)
  Thick subcategories for hereditary algebras
  Abstract: Let A be a finite dimensional algebra. A full subcategory T of A-mod is called a thick subcategory, if T is closed under extensions, direct summands, cokernels of monomorphisms and kernels of epimorphism. For hereditray algebras one can apply Ringels concept of simplification and gets the following theorem: Let H be a finite dimensional hereditary algebra and T a full subcategory of H-mod or of H-reg. Then T is a thick subcategory, if and only if there exists a family M of pairwise Hom-orthogonal bricks in H-mod or H-reg, such that T coincides with the class of modules having a filtration in M.

• 04.06.2009, 15.45, Lars Winther Christensen (Texas Tech University)
  Totally reflexive modules and ADE singularities
  Abstract: In the 1980s, work by Buchweitz, Greuel, and Schreyer, Knörrer, Yoshino and others established remarkable connections between the module theory of a complete Gorenstein local ring and the character of its singularity.  In this talk I will report on recent result that avoids the a priori Gorenstein hypothesis; this is joint work with Piepmeyer, Striuli, and Takahashi.
  Let R be a commutative noetherian local ring. Consider the set of isomorphism  classes of indecomposable totally reflexive R-modules. We prove that if this set  is finite, then either it has exactly one element, represented by the rank 1 free  module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple ADE singularity). The crux of the proof is a result of independent interest: If the residue field of R has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.

• 04.06.2009, 14.15, Jan Stovicek (Trondheim)
  A counterexample to the telescope conjecture of global dimension 2
  Abstract: I will explain how to construct a ring of global dimension two so that the telescope conjecture fails for its derived category. To prove the latter fact, I will use a criterion due to Keller. It may be of interest that although the ring is not hereditary, in which case the telescope conjecture could not fail, it is in many respects rather close to being hereditary.

• 22.05.2009, 14.15 im Raum E2.304, Estanislao Herscovich (Buenos Aires)
  Representation theory and homology of Yang-Mills algebras
  Abstract: Yang-mills algebras have been defined by Alain Connes and Michel Dubois-Violette in connection to some problems arising from physics, string theory and noncommutative quantum field theory.
  The aim of this talk is threefold. In the first place, we shall state the general definitions and recall the main properties for
  these algebras. Then, we will focus ourselves on showing certain families of representations fine enough to separate elements for the Yang-Mills
  algebras.
  Finally, we shall also present several computations in relation to homological properties for these algebras, in particular, the Hochschild
  and cyclic homology.
  These results are part of my PhD thesis with Andrea Solotar.

• 18.05.2009, 17.15 im Raum E2.145, Marcel Wiedemann (Paderborn)
  Ringel-Hall algebras, quantum loop algebras and weighted projective lines
  Im Rahmen des IRTG-Research Seminars

• 14.05.2009, 15.15, Xiao-Wu Chen (Paderborn)
  One-point extension of abelian categories
  Abstract: In this talk, I will introduce the notion of one-point extension of abelian categories and related notions. The relation with Lenzing's p-cycle construction will be discussed; as a consequence of a comparison theorem, we prove a uniqueness theorem for certain one-point extensions. This is based on a on-going project with Henning Krause.

• 12.05.2009, 16.15 in E2.145, Andrew Hubery (Leeds)
  The simplicial complex of tilting modules
  Abstract: In relating acyclic cluster algebras to the tilting theory of hereditary algebras, there are several important ingredients: showing that the simplicial complex of tilting modules is connected, the Caldero-Chapoton map, and the Caldero-Keller multiplication theorem. These have all been done for quivers (or the skew-symmetric case), and it is natural to ask for analogues for species (or the skew-symmetrisable case). We prove the first of these - that the simplicial complex of tilting modules is connected.

• 30.04.2009, 14.15, Marco Angel Bertani-Økland and Anette Wrålsen (Trondheim)
  Constructing tilted algebras from cluster-tilted algebras
  Abstract: Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given a cluster-tilting object, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object. Furthermore, in the Dynkin case, we provide a technique to find a cluster-tilting object, having a given cluster-tilted algebra as endomorphism ring.

• 16.04.2009, 15.45, Stephen Doty (Chicago)
  Rational Schur algebras
  Abstract:  This is joint work with R. Dipper and F. Stoll. Rational Schur algebras are a new family of quasihereditary algebras depending on 3 parameters; when one of the parameters is set to zero the classical Schur algebras are recovered as a special case. Rational Schur algebras control the rational representation theory of general linear groups over an infinite field, and there is a Schur-Weyl duality involving "mixed" tensor powers of the natural representation and its dual. The centralizer algebra is a nice subalgebra of the Brauer algebra described by certain "walled" Brauer diagrams. Of course, there are q-analogues of everything.

• 16.04.2009, 14.15, Jan Stovicek (Trondheim)
  Dimensions of thick subcategories of the derived bounded category
  Abstract: This is a joint work with Steffen Oppermann. Given an artin algebra A and a bounded complex X of finitely generated A-modules, we study the dimension of the thick subcategory T of D^b(A) generated by X. We show that if T contains A, then the dimension is given by the nilpotency degree of an ideal of the category of perfect complexes. As a consequence, either the dimenstion of T is infinite, or T = D^b(A).

• 14.04.2009, 17.15 in A 2 337, Dieter Vossieck (Bielefeld)
  The representation type of finite dimensional Lie algebras
  Abstract: With a finite dimensional complex Lie algebra one can associate a quiver such that the finite-dimensional representations of the Lie algebra correspond to certain finite-dimensional nilpotent representations of the quiver.  The calculation of the quiver is indicated in a not so well-known article by Michèle Loupias from 1972. We will explain some of her results, putting special emphasis on explicit examples.
  

• 14.04.2009, 16.00 in A 2 337, Dan Zacharia (Syracuse University)
  Auslander-Reiten theory for modules of finite complexity over selfinjective algebras
  Abstract: Let R be an artin algebra, and let M be an indecomposable R-module. The complexity of M measures the growth of its minimal projective resolution. For instance complexity zero, means that the projective dimension is finite, and complexity one is the same as saying that the terms of the minimal projective resolution have bounded dimension. We will look at modules of finite complexity over a selfinjective artin algebra using the shape of the Auslander-Reiten sequences ending at them.

• 26.03.2009, 16.15, Thorsten Weist (Wuppertal)
  Localization in Kronecker moduli spaces and tree modules
  Abstract: Torus fixed points of Kronecker moduli spaces are given by stable bipartite quivers coming along with a certain colouring. By use of the glueing method it is possible to construct a huge class of such quivers implying a lower bound for the Euler characteristic. Stable torus fixed points are also often indecomposable tree modules. Modifying these fixed points it is possible to construct an indecomposable tree module for every root of the Kronecker quiver.

• 24.03.2009, 17.00, Helmut Lenzing (Paderborn)
  Examples, illustrating Orlov's theorem
  Abstract: We are going to illustrate the trichotomy expressed by Orlov's theorem by a couple of examples. The most impressing examples we are going to discuss arise from weighted projective lines, whose Euler characteristic plays a key role for the Orlov context.

• 24.03.2009, 15.30, Xia-Wu Chen (Paderborn)
  Orlov's Theorem II
  Abstract: I shall discuss the proof of Orlov's Theorem and some of its corollaries.

• 24.03.2009, 14.15, Marcel Wiedemann (Paderborn)
  Orlov's Theorem I
  Abstract: I shall present all the definitions and background results necessary to prove Orlov's Theorem, such as semiorthogonal decompositions, triangulated categories of singularities and quotient categories of graded modules.

• 20.03.2009, 14.15, Michael Barot (UNAM, Mexico)
  Tubular cluster algebras of type (2,2,2,2)
  Abstract: This is a joint work with Christof Geiss. We show the close relationship between tilting objects in coh X, where X is a weighted projective line of weight type (2,2,2,2) with the tagged triangulations of a sphere with 4 punctures and with the cluster algebra associated to it.

• 26.02.2009, 14.15, Oeyvind Solberg (Trondheim)
  Koszul theory and components of the Auslander-Reiten quiver
  Abstract: Auslander showed that the category of all additive contravariant functors from the category of finitely generated left modules, mod(Lambda), to the category of vectorspaces, for a finite dimensional algebra Lambda, is Noetherian if and only if Lambda is of finite representation type. We study the associated graded category, A_gr(mod(Lambda)), of mod(Lambda), which has the same objects as mod(Lambda) while the morphisms are given by the direct sum of the radical layers in mod(Lambda). We obtain a characterization of when the category of additive graded contravariant functors from A_gr(mod(Lambda)) to graded vectorspaces is Noetherian in terms of the shape to the left stable parts of the components in the Auslander-Reiten quiver of Lambda.
  
  

• 12.02.2009 REPRESENTATION THEORY DAY

• 05.02.2009, 16.45 in D1.320, Guodong Zhou (Koeln)
  On a conjecture of Auslander and Reiten
  Abstract:  A conjecture of Auslander and Reiten says that if two finite dimensional algebras defined over an algebraically closed field have equivalent stable categories, then they have the same number of isoclasses of non-projective simple modules. In this talk, we are interested in this conjecture for a special class of stable equivalences: stable equivlances of Morita type. This kind of stable equivqlences, introduced by Broué, appear naturally in modular representation theory of finite groups. We shall give some equivalent conditions of this conjecture, for example, this conjecture is equivalent to the invariance of the dimension of the degree zero Hochschild homology, and for symmetric algebras, equivalent to the invariance of the dimensions of the centers. We will also present some conditions stronger than the conjecture of Auslander and Reiten.
• 05.02.2009, 15.30, Dirk Kussin (Paderborn)
  Introduction to weighted projective lines IV: Representation type and Euler characteristic
  Abstract: We will explain how the Euler characteristic of a weighted projective line determines the representation type of the category of coherent sheaves and of the associated category of graded Cohen-Macaulay modules, respectively.
• 05.02.2009, 14.15, Claudia Koehler (Paderborn)
  Introduction to weighted projective lines III: Axiomatic approach
  Abstract: We show that each abelian category satisfying certain properties (noetherian, hereditary, no non-zero projectives, existence of a tilting complex) is derived equivalent to the module category of a so-called squid algebra and therefore equivalent to the category of coherent sheaves on the associated weighted projective line.

• 22.01.2009, 15.45, Xiao-Wu Chen (Hefei/Paderborn)
  Vector bundles and Cohen-Macauly modules
  Abstract: In this talk, I will recall the definition of weighted projective lines and sketch the proof of Serre's theorem, and a refinement of Serre's theorem which relates vector bundles with graded maximal Cohen-Macaulay modules. If time permits, to go back to canonical algebras, we will also discuss Serre duality and the canonical tilting sheaves.

• 22.01.2009, 14.15, Karsten Dietrich (Paderborn)
  Constructing weighted projective lines from canonical algebras
   Abstract: After recalling the definition of a canonical algebra and the structure of its module category we will construct the category of coherent sheaves on the associated weighted projective line (inside the bounded derived category of the canonical algebra) and discuss some properties.

• 15.01.2009, 14.15, Marcel Wiedemann (Paderborn)
  Knoerrer's periodicity
  Abstract:  I shall discuss that the notions of simple singularity and finite representation type are equivalent for hypersurfaces.
  
  

• 08.01.2009, 15.45, Jue Le (Hefei/Paderborn)
  Triadic categories
  Abstract: I will talk about some recent work of Rump. Given any additive category A, he introduces on the homotopy category M(A) of two-termed complexes a new structure, which he calls "triad". This seems to be a new tool to study the Auslander-Reiten quiver of A.

• 08.01.2009, 14.15, Marcel Wiedemann (Paderborn)
  Matrix factorizations
  Abstract: In my talk I will discuss Eisenbud's matrix factorization. This plays a key role in the treatment of Cohen-Macaulay modules over hypersurface singularities.