Talks in 2006


  • 21.12.2006, 16.15 Uhr, Henning Krause (Paderborn)
    Thick subcategories and support for modules over commutative rings
    Abstract:  Given a complex of modules over a commutative noetherian ring, one can ask how its support is related to the support of its cohomology. The same question makes sense in other settings, for instance for representations of finite groups. Joint work with Benson and Iyengar provides an answer to this question. An upshot of this project is a classification of all thick subcategories of modules over a commutative noetherian rings, correcting a result of Hovey from 2001.
  • 14.12.2006, 14.15 Uhr, Dirk Kussin (Paderborn)
    Cluster-Tilted Algebras are Gorenstein and stably Calabi-Yau, II (Report on work of Keller-Reiten)
    Abstract: This is the second part of my report on work of B. Keller and I. Reiten. In the mentioned paper -- like in a recent paper of König-Zhu -- cluster tilting is studied in a general framework. In this talk I concentrate on the proofs of the results that a cluster-tilted algebra is Gorenstein and that its stable Cohen-Macaulay category is Calabi-Yau of CY-dimension 3.
  • 07.12.2006, 16.15 Uhr, Dirk Kussin (Paderborn)
    Cluster-Tilted Algebras are Gorenstein and stably Calabi-Yau (Report on work of Keller-Reiten)
    Abstract: In this talk I report on work of B. Keller and I. Reiten. In the mentioned paper -- like in a recent paper of König-Zhu -- cluster tilting is studied in a general framework. In this talk I concentrate on the proofs of the results that a cluster-tilted algebra is Gorenstein and that its stable Cohen-Macaulay category is Calabi-Yau of CY-dimension 3.
  • 16.11.2006, 16.15 Uhr, Christof Geiss (UNAM Mexico)
    Rigid modules over preprojective algebras - the Kac Moody case
    Abstract: This is a prelimary report on joint work with B. Leclerc and J. Schroeer. Let Q be a quiver without oriented loops and A the correponding preprojective algebra. Thus we have for nilpotent A-modules a functorial isomorphism from Ext^1_A(X,Y) to the the dual of Ext^1_A(Y,X). Thus we may assign to each nilpotent A-module X a function f_M in the dual of the envelopping algebra U(n) of the nilpotent Lie algebra of type |Q| with the following property: If dim Ext^1_A(x,y)=1, whith 0 -> y -> e' -> x -> 0 and 0 -> x -> e" -> y -> 0 the corresponding non-split exact sequences, then f_x f_y = f_e' + f_e". So one wonders if our categorification program of cluster algebra structures on C[N] carries over from the Dynkin case to the general Kac-Moody case. For the moment we rather construct nice subcategories of A-mod which should categorify the cluster algebra structor on coordinate rings of certain open subsets of Schubert cells in the corresponding flag varieties: Let S be a tilting module over Q such that the corresponding torsion category Gen(S) contains only finitely many indecomposables then the subcategory C_S of nilpotent A-modules such that their restriction to Q belongs to Gen(S) is a Frobenius category with remarkable properties.

  • 09.11.2006, 16.15 Uhr, Bernhard Keller (Paris VII) (talk shifted to 10.11.2006, 9.10 Uhr, Hörsaal D2)
    On the Grothendieck group of a resolvable 2-Calabi-Yau category, after Y. Palu
    The Grothendieck group of the cluster category associated with a hereditary category admitting a tilting object was recently computed by Barot-Kussin-Lenzing. In this talk, we will sketch a new proof and a generalization of their result. The main tool will be the link between Calabi-Yau categories of dimensions 2 and 3 studied by Tabuada and used in joint work by the speaker with Idun Reiten. As a  by-product, we will find that any two cluster-tilting objects in an (algebraic) 2-Calabi-Yau category have the same number of pairwise non isomorphic indecomposable factors.
  • 02.11.2006, 16.15 Uhr, Helmut Lenzing (Paderborn)
    Extended canonical algebras and triangulated categories of singularities
    Abstract: The talk is on joint work with J.A. de la Pena and deals with the shape of the bounded derived category D(B) of an extended canonical algebra B, arising as the one-point extension of a canonical algebra A by an indecomposable projective or injective module. The categories D(B) occur in three different types, depending on the sign of the Euler characteristic of the weighted projective line X associated to A.  For negative Euler characteristic this relates to the triangulated category of the graded surface singularity associated to X. This is connected to recent results by Orlov (2005) and Saito-Takahashi (unpublished).
  • 26.10.2006, 16.15 Uhr, Steffen Oppermann (Köln)
    A lower bound for the representation dimension of elementary abelian groups
    Abstract: The representation dimension of a finite dimensional algebra has been introduced by Auslander in order to measure how far an algebra is from having finite representation type. In my talk I will recall the definition and some basic properties of the representation dimension and the dimension of a triangulated category (due to Rouquier), and point out how they are related. Then I will sketch how to obtain a lower bound in the case of group algebras of elementary abelian groups. Finally we will see what this means for the representation dimension and the ring structure of arbitrary group algebras.
  • 19.10.2006, 16.15 Uhr, Bernard Leclerc (Caen)
    Preprojective algebras and Schubert cells
    Abstract : Let G be a simply connected semisimple complex algebraic group of type A,D,E, and let X = G/B denote its flag variety. For each element w of the Weyl group, we describe a subcategory C(w) of the module category of the preprojective algebra attached to G, which "categorifies" the cluster algebra structure of the coordinate ring of the Schubert cell of X labelled by w. (This cluster algebra structure was introduced by Berenstein-Fomin-Zelevinsky). In particular, taking for w the square of a Coxeter element, one gets a representation-theoretic construction of every cluster algebra of type A,D,E. If time allows, we shall discuss the generalization of these results to the Kac-Moody case. This is a joint work with Christof Geiss and Jan Schroer.
  • 16.10.2006, 17.15 Uhr (im Rahmen des IRTG-Seminars) Lutz Hille (Bielefeld)
    Actions of parabolic subgroups
  • 27.09.2006, 16.15 Uhr, Martin Hamm (Hamburg)
    Torische Beschreibung des versellen Totalraums einer zweidimensionalen zyklischen Quotientensingularität
    Abstrakt: Ausgangspunkt meines Vortrags sind die zweidimensionalen zyklischen Quotientensingularitäten. Mit verwurzelten Bäumen wird eine neue Konstruktionsmethode der versellen Deformation (in der Form von Jürgen Arndt) angegeben. Nach einem endlichen Basiswechsel dieser Deformation sind die Komponenten des Totalraums affine torische Varietäten. Die zugehörigen Kegel ergeben sich aus einem größeren Kegel durch Schnitte mit linearen Teilräumen, wobei diese Teilräume mit einem konvex-geometrischen Kriterium berechnet werden können. Anschließend wird eine konvex-geometrische Beschreibung des gesamten Totalraums angegeben.
  • 24.08.2006, 16.15 Uhr, Bin Zhu (Beijing)
    Generalized cluster complexes via quiver representations
    Abstract: We give a quiver representations' interpretation of generalized cluster complexes defined by Fomin and Reading. By using d-cluster categories, we define a d-compatibility degree on any pair of "colored'' almost positive Schur roots and call that two such roots are compatible provided the d-compatibility degree of them is zero. Associated to the root system corresponding to the valued quiver, by using this compatibility relation, we define a simplicial complex which has colored almost positive Schur roots as vertices and d-compatible subsets as simplexes. If the valued quiver is an alternative quiver of a Dynkin diagram, this complex is the generalized cluster complex defined by Fomin and Reading.
  • 13.07.2006, 14.15 Uhr, Andrew Hubery (Paderborn)
    Some results concerning cluster algebras and cluster categories
    Abstract: In the cluster-multiplication theorem of Caldero-Keller, one needs to calculate the Euler characteristic of the constructible sets Hom_C(M,N)_E, the set of morphisms in the cluster category C whose cone is isomorphic to E. However, in the Ringel-Hall approach, only those triangles in the image of the natural functor from D to C are needed. We give a simple proof of the fact that all other triangles form a set with Euler charcteristic zero.
    The second result is an explicit formula for the rank two cluster variables, which is a possible first step in proving the positivity conjecture in this case.
  • 06.07.2006, 15.15 Uhr, Gus Lehrer (Sydney)
    Endomorphism algebras of tensor powers
    Abstract: Let g be a complex semisimple Lie algebra and U_q its Drinfeld-Jimbo quantisation over the field C(q) of rational functions in the indeterminate q. If V and V_q are corresponding irreducible modules for g and U_q, it is known that in several cases, the endomorphism algebra of a tensor power (V_q)^r is a deformation of the endomorphism algebra of V^r, and both algebras have a cellular structure, which in principle permits one to study non-semisimple deformations of either.  We present a framework ("strongly multiplicity free" modules) where the endomorphism algebras are "generic" in the sense that in the classical (unquantised) case, they are quotients of Kohno's infinitesimal braid algebra T_r, while in the quantum case, they are quotients of the group ring C(q)B_r of the r-string braid group B_r. In addition to the well known cases above, these include two new cases: the irreducible 7 dimensional module in type G_2 and arbitrary irreducibles for sl_2.  This is joint work with Ruibin Zhang of Sydney University.
  • 03.07.2006, 17.15 Uhr (im Rahmen des IRTG-Seminars) Maurizio Martino (Köln)
    Symplectic leaves for representations of preprojective algebras and deformed symplectic quotients.
    Abstract: For any complex affine algebraic Poisson variety one can consider its stratification into symplectic leaves. In this talk I will consider the symplectic leaves of certain varieties arising in noncommutative algebra. Firstly I will define the (deformed) preprojective algebras, which are quotients of the path algebra of a quiver. The representation spaces of these are certain symplectic reductions of the representation space of the corresponding quiver, and I will show that the symplectic leaves of these has a nice representation theoretic description. I will also relate this result to the deformations of symplectic quotient singularities defined by Etingof and Ginzburg.
  • 22.06.2006, 16.15 Uhr, Srikanth Iyengar (Lincoln)
    Hochschild cohomological criteria for the Gorenstein property for commutative algebras
    Abstract: A classical result of Hochschild, Kostant, and Rosenberg characterizes smoothness of commutative algebras essentially of finite type over a field in terms of its Hochschild cohomology. I will discuss similar characterizations of the Gorenstein property. This is joint work with L. L. Avramov.

  • 01.06.2006, 16.15 Uhr, Karsten Schmidt (Paderborn)
    A topological space with infinitely many AR-components
    Abstract: This is a short informal report on an example of an AR-quiver of a topological space (in the sense of Peter Joergensen) with infinitely many components.
  • 29.05.2006, 17.15 Uhr (im Rahmen des IRTG-Seminars) Lidia Angeleri Hügel (Varese)
    A solution to the Baer splitting problem
    Abstract: Let R be a commutative domain. I will present a proof, obtained in joint work with S. Bazzoni and D. Herbera, of the fact that an R-module B is projective if and only if Ext^1(B,T)=0 for any torsion module T. This answers in the affirmative a question raised by Kaplansky in 1962.
  • 24.05.2006, 17.30 Uhr, Hörsaal A3, Lidia Angeleri Hügel (Varese)
    On the telescope conjecture for module categories
    Abstract: In a paper of Krause and Solberg of 2003, the Telescope Conjecture was formulated for the module category ModR of an artin algebra R as follows: "If (A,B) is a complete hereditary cotorsion pair in ModR with A and B closed under direct limits, then A coincides with the limit closure of its finitely presented objects." My talk will be devoted to recent results concerning this conjecture over an arbitrary ring R. In particular, we will see that the conjecture holds true for the cotorsion pairs that arise in tilting theory.
  • 24.05.2006, 16.15 Uhr, Hörsaal A3, Kristian Brüning (Paderborn)
    Cotorsion pairs and model categories
    Abstract: This is a report on the paper 'Cotorsion pairs, model category structures, and representation theory' by Mark Hovey. We will discuss when a complete cotorsion pair determines a model structure and vice versa. Furthermore, we will explain how methods from model category theory can be used to construct approximations for a not necessarily complete cotorsion pair.
  • 23.05.2006, 17.45 Uhr (im Rahmen des Fakultätskolloquiums) Andrei Zelevinsky (Northeastern University Boston)
    Laurent expansions in cluster algebras via quiver representations
  • 18.05.2006, 16.45 Uhr, Jose Antonio de la Peña (UNAM Mexico)
    Decomposition of modules over rings with several objects.
    Abstract: This reports joint work with Lidia Angeleri. We consider rings R with enough idempotents (in a set E) and which are locally bounded, meaning that for any two idempotents e,f in E the module eRf is of finite length over eRe and for each e there is only a finite number of f in E with eRf not zero or fRe not zero. We study decomposition of right modules over R as direct sums of indecomposable modules, in particular for modules X such that the lenght of Xe is finite over eRe for any e in E. We consider properties of indecomposable modules and decompositions.
  • 18.05.2006, 15.15 Uhr, Csaba Lehel Szanto (Cluj-Napoca)
    Submodules of Kronecker modules via Hall algebras
    Abstract:  Using some results  on the Hall algebra of the Kronecker algebra over a finite field, we provide numerical criteria (depending on Kronecker invariants) for a Kronecker module to be isomorphic with the submodule of an another Kronecker module. The results can be applied in matrix pencil theory to partially solve 'the problem of giving necessary and sufficient conditions for the existence of a matrix pencil with prescribed Kronecker invariants and a prescribed arbitrary subpencil'.
  • 04.05.2006, 16.45 Uhr, Hagen Meltzer (Stettin)
    Omnipresent exceptional modules for hyperelliptic algebras
    Abstract: A hyperelliptic algebra is a canonical algebra in the sense of Ringel of type (2,2, ...,2). Using universal extensions we give an explicit description of all omnipresent exceptional modules of minimal rank over  those algebras. All  these modules are given by matrices involving as coefficients 0,1, x_3,...,x_t provided the  algebra is defined by these parameters.
  • 04.05.2006, 15.15 Uhr, Helmut Lenzing (Paderborn)
    Representing roots by exceptional objects, a theorem of T. Hübner
    Abstract: Let X be a weighted projective line of positive Euler characteristic, equivalently assume that the category coh(X) of coherent sheaves on X has tame domestic representation type. Hübner's theorem (1989) states that the map which sende E to[E] establishes a bijection between  isomorphism classes of exceptional bundles in coh(X) and roots of the Euler form having positive rank. We present a simple conceptual proof of this theorem and discuss extensions to  exceptional curves of positive Euler characteristic and also applications to  module categories  related to these curves.
  • 27.04.2006, 16.15 Uhr, Henning Krause (Paderborn)
    An introduction to virtually Gorenstein algebras
    Abstract: Virtually Gorenstein algebras have been introduced some years ago by Beligiannis and Reiten and generalize the class of Gorenstein algebras. I will explain the definition and will give some motivation for studying these algebras. In addition, an example of an algebra which is not virtually Gorenstein will be presented.
  • 13.04.2006, 14.15 Uhr, Shiping Liu (Sherbrooke)
    The derived category of an algebra with radical squared zero, II
    Abstract: This is the second part of a series of two talks. For an algebra with radical squared zero, we are going to describe the indecomposable objects, the Auslander-Reiten triangles, the shape of Auslander-Reiten components, and the representation type of its derived category. This is joint work with R. Bautista.
  • 10.04.2006, 17.15 Uhr (im Rahmen des IRTG-Seminars), Markus Perling (Grenoble)
    Derived categories of toric manifolds and a conjecture of King

  • 06.04.2006, 16.15 Uhr, Shiping Liu (Sherbrooke)
    The derived category of an algebra with radical squared zero, I
    Abstract: For an algebra with radical squared zero, we are going to describe the indecomposable objects, the Auslander-Reiten triangles, the shape of Auslander-Reiten components, and the representation type of its derived category. This is joint work with R. Bautista.
  • 23.03.2006, 14.15 Uhr, Hideto Asahiba (Osaka)
    Realization of simple Lie algebras via Hall algebras of domestic canonical algebras
    Abstract: Let  A  be a domestic canonical algebra of Dynkin type  D over a finite field, and  d  the positive generator of the radical of the Euler form  q  of  A. It is known that if  v  is a positive root of  q, then there is a unique indecomposable module  M(v) having the dimension vector  v  up to isomorphisms. In the degenerate composition Lie algebra  L  of A, we have shown that if we "identify" the isoclasses  [M(v)]  of indecomposables  M(v)  with  [M(v + d)] for all positive root  v  of  q, then we obtain the simple Lie algebra g  of type  D, and the equivalence classes of  [M(v)]  are in 1-1 correspondence with (positive and negative) real roots of  g. But here the way of identification contained an error. It is now clear that we cannot identify as  [M(v)] = [M(v + d)], otherwise the resulting algebra sometimes turns out to be zero. That is, the ideal  I  of  L generated by the set  {[M(v + d)] - [M(v)]| v  positive root of  q} was too big. I will talk how to fix the error in the lecture.  Namely, instead of the ideal  I, we have to use the ideal  I(A) generated by the set {[M(v + d)] - [M(v)]| M(v)  simple}, and then it automatically yields a relation that  [M(v + d)] = r(v)[M(v)]  for some nonzero rational number  r(v)  for each positive root  v  of  q. In the induction step in the proof the notion of Gabriel-Roiter submodules was useful. It was also useful in examining the existence of Hall polynomials that was needed in the proof.
  • 15.02.2006, 14.00 Uhr s.t. in D1.312, Kristian Brüning (Paderborn)
    Coherent and thick subcategories
    Abstract: We will classify the thick subcategories of the bounded derived category of a hereditary category A in terms of coherent subcategories of A. This result can yield a first step towards an understanding of the telescope conjecture for the derived category of a hereditary algebra.
  • 09.02.2006, 16.15 Uhr, Dirk Kussin (Paderborn)
    The Grothendieck group of the cluster category
    Abstract: This is a report on joint work with M. Barot and H. Lenzing. Let A be a hereditary or a canonical algebra. By a result of B. Keller the cluster category of A admits a triangulated structure containing the induced triangles of the derived category. We describe its Grothendieck group explicitly in the cases where A is canonical or the path algebra of a Dynkin quiver.
  • 02.02.2006, 16.15 Uhr, Osamu Iyama (Nagoya)
    Non-commutative crepant resolutions and tilting modules over 3-Calabi-Yau algebra
    Abstract: We study tilting modules over 3-Calabi-Yau algebras. They are closely related to non-commutative crepant resolution introduced by Van den Bergh to study derived equivalence in birational geometry. By definition, reflexive equivalent algebras have the same non-commutative crepant resolutions. We will show that, for 3-Calabi-Yau algberas, non-commutative crepant resolutionsare nothing but tilting modules of projective dimension at most one. As a conclusion, we can construct a certain family of tilting modules over 3-Calabi-Yau algebras. On the other hand, we show some results on the number of indecomposable summands of rigid reflexive modules over a three dimensonal quotient singularities.
  • 19.01.2006, 16.15 Uhr, Henning Krause (Paderborn)
    The Gabriel-Roiter measure of a finite dimensional algebra
    Abstract: This is a report on recent work of Ringel on the combinatorial structure of the category of representations over a finite dimensional algebra. The talk is based on a (new) axiomatic definition of the Gabriel-Roiter measure. Also, I propose a definition for the Gabriel-Roiter measure of the derived category.

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