# Talks in 2004

• Dienstag 21.12.2004, 16:15 Uhr in J2.331, Dieter Vossieck (Mexico)
Schemes of representations of artin algebras, IV
Abstract (in PS-format)
• Freitag 10.12.2004, 16:15 in D1.320, Hagen Meltzer (Stettin)
Bernstein-Gelfand-Gelfand correspondence and classical vector bundles
Abstract: We describe the Horrocks-Mumford bundle, null-correlation bundles and Tango bundles in terms of graded modules over the exterior algebra via the Bernstein-Gelfand-Gelfand correspondence. Furthermore we show that properties as indecomposability and stability of these bundles can be proved purely algebraically.
• Freitag 10.12.2004, 14:30 in D1.320, Dan Zacharia (Syracuse)
Linear modules over the exterior algebra and coherent sheaves over the projective n-space
Abstract: Let R be the exterior algebra in n+1 variables over a field K. We study the Auslander-Reiten quiver of the category of linear R-modules, and of certain subcategories of the category of coherent sheaves over projective n-space. If n>1, we prove that up to shift, all but one of the connected components of the A-R quivers, are translation subquivers of a ZA_\infty-type quiver. We also study locally free sheaves over the projective n-space for n>1 and we show that each connected component contains at most three indecomposable locally free sheaves of rank less than n. Finally, using results from the theory of finite dimensional algebras, we construct a family of indecomposable locally free sheaves of arbitrary large ranks, where the ranks can be computed using the Chebycheff polynomials of the second kind.
• Mittwoch 02.12.2004, 16:45 Uhr, Teimuraz Pirashvili (Bielefeld,Tbilisi)
Strict polynomial functors and coherent functors
Abstract: The relation between the representation theory of symmetric groups and polynomial functors goes back to the classical work of Schur, Weyl and Green. Let P_n be the category of homogeneous polynomial functors of degree n. There is an exact functor from P_n to S_n-Rep, which is an equivalence of categories in characteristic zero. In positive characteristic it is only a part of a recollement situation. We will construct another exact functor from C_n to P_n which is also a part of a recollement situation. Here C_n is the category of coherent functors associated with the symmetric group S_n. The composite functor from C_n to S_n-Rep, is the classical functor of Auslander. We will show how to use homological properties of coherent functors to make explicit computation in the category P_n.
• Mittwoch 02.12.2004, 15:15 Uhr, Dieter Vossieck (Mexico)
Schemes of representations of artin algebras, III
Abstract (in PS-format)
• 25.11.2004, 16:15 Uhr, Dieter Vossieck (Mexico)
Schemes of representations of artin algebras, II
Abstract (in PS-format)
• Mittwoch 24.11.2004, 16:15 Uhr in D1.328, Thomas Hüttemann (Göttingen)
From Polytopes to K-Theory
Abstract: A well-known result of Quillen asserts that the algebraic K-theory of n-dimensional projective space over a commutative ring R splits into n+1 copies of the K-theory of R. I will explain how to understand this with combinatorial methods: The splitting is controlled by the behaviour of a certain polynomial p associated to a standard simplex, with the summands of the splitting corresponding to integral roots of p. From this point of view one obtains a generalised (weaker) splitting result for arbitrary projective toric varieties (with the standard simplex replaced by any polytope with integral vertices). I will also demonstrate that the combinatorial approach can be used to treat "non-linear" versions of toric varieties which can be considered as examples for "brave new algebraic geometry" (algebraic geometry over ring spectra).
• 18.11.2004, 16:45 Uhr, Michael Barot (UNAM Mexico)
Cluster algebras of finite type
Abstract: This is an account of a joint work with C. Geiss (UNAM, Mexico) and A. Zelevinsky (Northeastern University, USA). After a brief exposition of the previously known characterizations of cluster algebras of finite type, we will discuss a new one, which has two advantages: it clarifies the connection to the Cartan-Killing classification and it is easier to check.
• 18.11.2004, 15:15 Uhr, Dieter Vossieck (Mexico)
Schemes of representations of artin algebras, I
Abstract (in PS-format)
• 11.11.2004, 16:45 Uhr, Changchang Xi (Beijing Normal University)
Representation dimensions of artin algebras
Abstract: The notion of representation dimension was introduced by Auslander in order to measure homologically how far an algebra is being away from representation-finite. Unfortunately, until five years ago, there was not much progress on the subject. Recently, the subject became vivid. In this talk, I will report some new developments on the subject since 1998.
• 11.11.2004, 15:15 Uhr, Kristian Brüning (Paderborn)
Homological algebra and stable homotopy theory, II
Abstract: See the abstract for part I.
• 04.11.2004, 16:15 Uhr, Kristian Brüning (Paderborn)
Homological algebra and stable homotopy theory, I
Abstract: This talk is based on the article "HZ-algebra spectra are differential graded algebras" of Brooke Shipley. We will describe how homological algebra (i.e. the derived category of a ring R) embeds in stable homotopy theory (i.e. in the stable homotopy category). In fact, we will define a Quillen equivalence between the category of chain complexes over a ring R and the category of modules over the Eilenberg-MacLane-spectrum HR. This implies that the derived category of the ring R is equivalent to the homotopy category of HR-module spectra.
• 28.10.2004, 16:45 Uhr, Ed Green (Virginia Tech)
Polynomial solutions to a reverse engineering problem
Abstract: Given time series data, biologists would like to find a good model for a gene network. That is, given activity levels of genes at a number of diffenent times, a biologist would like to find out which genes affect which other genes. I will discuss the mathematical model and polynomial solutions to the problem. In particular, I will show that the mathematician cannot give reasonable answers to such a problem without more information from the biologist.
• 28.10.2004, 15:15 Uhr, Changchang Xi (Beijing Normal University)
On finitistic dimensions of algebras
Abstract: Suppose we are given an algebra homomorphism from an artin algebra B to another artin algebra A. For example, the inclusion map of a subalgebra B of A into A. We want to transfer, with certain conditions, the finiteness of the finitistic dimension of one algebra to that of the other. In this direction, let us just mention the following two results: (1) Suppose B is a subalgebra of A such that rad(B) is a left ideal in A. If A is representation-finite, then the finitistic dimension of B is finite.
(2) Suppose I and J are two ideals of A with IJ=0. If A/I and A/J are representation-finite, then A has finite finitistic dimension.
As a consequence of (2), we have the following well-known result of Green-Zimmermann-Huisgen: For an algebra A, if the cube of its radical vanishes, then the finitistic dimension of A is finite. Also we may apply the results to pullback algebras, and many other classes of algebras.
• 14.10.2004, 16:45 Uhr, Ed Green (Virginia Tech)
Resolutions over Koszul algebras and Hochschild cohomology, I
Abstract: I will present a number results of work with the following coauthors, Ragnar Buchweitz, Gregory Hartman, Eduardo Marcos, Dag Madsen, Nicole Snashall and Oeyvind Solberg (in various combinations). For a Koszul algebra, there is a comultiplicative' structure to a minimal resoluton of the semisimple part of the algebra. The coefficients of the comultiplicative structure have truly wonderful properties which I will try to describe. As an application, I will give a negative answer to Happel's question: If R is a finite dimensional algebra of infinite global dimension, then must the Hochschild cohomology groups, HH^n(R), be different from 0 for an infinite number if n>=0 ?
• 14.10.2004, 15:15 Uhr, Jue Le (Paderborn, Beijing Normal University)
The Auslander-Reiten formula for complexes of modules
Abstract: In this talk we discuss an Auslander-Reiten formula for complexes of modules, which yields as a special case the classical Auslander Reiten formula for modules relating Ext^1 and stable Hom.
• 29.09.2004, 15:15 Uhr, Henning Krause (Paderborn)
Approximations and virtually Gorenstein algebras
Abstract: Auslander and Buchweitz introduced maximal Cohen-Macaulay approximations in the mid eighties for Gorenstein algebras. In this talk, I will present an alternative construction of such approximations. This is motivated by the relation with an interesting class of algebras (virtually Gorenstein algebras) which have been introduced and studied in recent work of Beligiannis and Reiten.
• 22.07.2004, 16:45 Uhr, Thorsten Holm (Leeds)
Derived equivalences for selfinjective tame algebras
Abstract: Selfinjective algebras of finite representation type have been classified up to Morita equivalence by Riedtmann in the 1980s. A derived equivalence classification was given more recently by Asashiba. The representation theory of selfinjective algebras of tame representation type is far less well understood and still only emerging. In the talk we will present derived equivalence classifications for some classes of selfinjective algebras of domestic representation type. More precisely, we will speak about weakly symmetric algebras of Euclidean type and about selfinjective one-parametric algebras. (This is joint work with R. Bocian and A. Skowronski.)
• 22.07.2004, 15:15 Uhr, Anne Henke (Leicester)
Repeating patterns for Schur algebras
Abstract: The modular representation theory of symmetric groups or of Schur algebras is still very little understood; decomposition numbers, for example, are only completely known in case of partitions with very few parts; in the case of two-part partitions they can be described by a fractal structure, a so-called Sierpinski gasket. This suggests the existence of an underlying fractal structure for Schur algebras and symmetric groups in general. It is the aim of the talk to discuss repetitions of different numerical patterns among decomposition numbers and their representation theoretical interpretations.
• 15.07.2004, 16:15 Uhr, Robert Marsh (Leicester)
Cluster-tilting theory
Coauthors: Aslak Buan (Trondheim and Leicester) Markus Reineke (Muenster) Idun Reiten (Trondheim) Gordana Todorov (Northeastern) Abstract: Let Q be a Dynkin quiver, k an algebraically closed field, and kQ the corresponding finite-dimensional path algebra. If i is a vertex of Q, then there is a corresponding APR-tilting module T and tilting theory describes a close connection between the module categories of kQ and kQ', where Q' is the quiver Q with all of the arrows incident with i reversed; in particular, they have the same number of indecomposable objects. We show how such tilting can be generalised to arbitrary vertices of Q (in the general finite-dimesional hereditary situation) and in fact to a whole family of algebras with interesting properties (I will say something about the algebras that can occur). The construction uses a new category, which we call the cluster category, obtained as a quotient of the bounded derived category of the module category of a finite-dimensional hereditary algebra. This category has close connections with the recent theory of cluster algebras (defined by Fomin-Zelevinsky).
• 08.07.2004, 16:15 Uhr, Helmut Lenzing (Paderborn)
Cluster categories II
Abstract: We discuss the basic setting of cluster categories (of hereditary categories), in particular the AR-structure, the concept of tilting objects in cluster categories and the arising cluster tilted algebras.

• 01.07.2004, 16:45 Uhr, Helmut Lenzing (Paderborn)
Cluster categories
• 01.07.2004, 15:15 Uhr, Andrew Hubery (Paderborn)
The centre of the Hall algebra of a cyclic quiver
Abstract: It is known from work of Schiffmann that the Hall algebra of a cyclic quiver is generated by the composition algebra together with a central polynomial ring on infinitely many variables. We provide explicit generators for this polynomial ring and study some of their properties. In particular we show that they generate the whole of the centre of the Hall algebra and consider the coproduct and the values taken under Green's symmetric bilinear form. As a consequence, we deduce that there is a natural isomorphism from Macdonald's ring of symmetric functions to the centre.
• 03.06.2004, Srikanth Iyengar (Lincoln, Nebraska)
Finiteness in derived categories of local rings
Abstract: In this talk I will present aspects of recent work with Dwyer and Greenlees concerned with some new finiteness conditions on complexes of modules over noetherian local rings. These notions, and the techniques we use to study them, are inspired by homotopy theory. They lead to new results on ascent and descent of ring theoretic properties along homomorphisms. I will also attempt to describe some of these.
• 27.05.2004, 16:45 Uhr, Markus Perling (Kaiserslautern)
On Tilting Sheaves on Toric Varieties
Abstract: A by now classical result of Beilinson states that the derived category of bounded complexes of sheaves over the n-dimensional projective space is equivalent to the bounded category of complexes over the exterior algebra in n variables. Generalizations of this result state that for an algebraic variety there exists an equivalence between its derived category and the derived category of the endomorphism algebra of a so-called tilting bundle associated to the variety. A general problem is to prove the existence of such a tilting bundle over a given variety and to determine the precise structure of the endomorphism algebra. We present the case of toric varieties and give an overview of known results and conjectures and some explicit examples.
• 27.05.2004, 15:15 Uhr, Kristian Brüning (Paderborn)
Deriving DG categories, IV
Abstract: This is the continuation of the series of lectures on the paper of Keller. The talk wil be about a Morita Theorem for derived categories.
• 13.05.2004, 16:45 Uhr, Idun Reiten (Trondheim)
Cluster categories
Abstract: We discuss some aspects of the theory of cluster categories.
• 13.05.2004, 15:15 Uhr, Dirk Kussin (Paderborn)
Characterization of commutative exceptional curves
Abstract: Let k be a field. We show that the function field of an exceptional curve X is commutative if and only if X is multiplicity free.
• Mittwoch 12.05.2004, 14:15 Uhr, Raum C5.206, Birgit Huber (Paderborn)
Magic exact sequences
Abstract: Let R be a non-semisimple Frobenius algebra. We call an exact sequence 0->A->B->R->0 of (R,R)-bimodules magic if for each f.g. R-module X the tensored sequence is either split or almost split modulo injectives, and the latter case occurs for at least one module X. We show how to construct the magic sequences over a class of Hopf algebras and characterize the magic sequences in terms of defects. We further show that tensoring the AR-sequence 0->A->B->R->0 of (R,R)-bimodules gives always a split sequence.
• Dienstag 11.05.2004 im Rahmen des Mathematischen Kolloquiums: 17.45 Uhr, Hörsaal D 2
Idun Reiten (Trondheim)
Cluster algebras and tilting theory
Abstract: Cluster algebras were introduced by Fomin and Zelevinsky a few years ago, with motivation from semisimple Lie algebras and algebraic groups. We give some easy examples of cluster algebras, and discuss, through examples, connections with tilting theory for finite dimensional algebras and related categories. This is based upon work by several people.
• 06.05.2004, 16:45 Uhr, Shigeo Koshitani (Chiba University, Japan)
Broué's abelian defect group conjecture in representation theory of finite groups
Abstract: Late eighties in the last century Michel Broué in Paris announced a wonderful conjecture in representation theory of finite groups. It says that two block algebras A and B of group algebras of finite groups corresponding via the Brauer correspondence and having abelian defect groups should be derived equivalent, namely, the bounded derived categories D^b(mod-A) and D^b(mod-B) of the categories mod-A and mod-B of finitely generated right A- and B-modules, respectively, should be equivalent as triangulated categories. The conjecture is called "Broué's abelian defect group conjecture" (Broué's ADGC, for short). We have had only several partial answers, of course, affirmative ones, for Broué's ADGC. For example, N. Kunugi and the speaker in [J. Algebra 248 (2002), 575--604] solve it affirmatively for the case where A is the principal block algebra (and hence so is B) and their defect groups P is elementary abelian of order 9, say C_3 × C_3. Here in this talk we will be discussing on Broué's ADGC for the case where A and B are NOT principal block algebras but having the same defect group P = C_3 × C_3 as above.
• 06.05.2004, 15:15 Uhr, Jue Le (Paderborn)
Deriving DG categories, III
Abstract: This is the continuation of the series of lectures on the paper of Keller. The talk covers Sections 4 of this paper, which includes an derived analogue of Frey'ds characterization of module categories among abelian categories.
• 29.04.2004, Hörsaal D 2, Angela Holtmann (Bielefeld)
The tame dimension vectors for stars and one parameter families of indecomposable representations
Abstract: We classify the tame dimension vectors for stars. A dimension vector d is called tame if there is a one parameter family of indecomposable representations for d and every direct summand in a family of representations with dimension vector d depends on at most one parameter. The tame dimension vectors are special cases of the s-tame dimension vectors for stars. The latter ones can be characterised by their Tits forms and those of their s-decompositions, but it is also possible to construct a complete list of them. Furthermore, when the quiver is of wild type, the s-tame dimension vectors are exactly those, for which every family of subspace representations depends on a single parameter. We show an analogue of the classification theorem of the s-tame dimension vectors for stars. In the last part of the talk we show that all families of indecomposable representations for the tame dimension vectors for stars with any orientation can be derived from the families of indecomposable subspace representations for the s-tame dimension vectors for stars (with subspace orientations) by means of the BGP-reflection functors. Since the explicit construction procedures for the s-tame dimension vectors are known, it is possible to construct all families of indecomposable representations for the tame dimension vectors of stars explicitly.
Deriving DG categories, II
Abstract: This is the continuation of the series of lectures on the paper of Keller. The talk covers Sections 2 and 3 of this paper, with an alternative proof for the existence of projective and injective resolutions. There are also some notes available (in PS Format).
Deriving DG categories, I
Abstract: This is the first in a series of talks on the paper [Keller, Bernhard: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63--102]. The talk gives an introduction into the subject, with some definitions and examples.

• 11.03.2004, 14:15 Uhr, Raum D 1.338, Henning Krause (Paderborn)
The stable derived category of a noetherian scheme
Abstract: The topic of this talk will be the unbounded stable derived category for a noetherian scheme. The construction of this stable category leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal Cohen-Macaulay approximations and an extension of the classical Grothendieck duality. The concept of the unbounded stable category is of fairly general nature and covers, for instance, applications in representation theory. There is a preprint available (in PS Format).
• 11.03.2004, 11:15 Uhr, Raum D 1.338, Andrew Hubery (Paderborn)
Triangulated categories and Kac-Moody algebras, III
Abstract: This is the continuation of the series of lectures on the paper of Peng and Xiao. The talk covers the proofs in Sections 6,7 and 8 (based on a new approach by Hubery). There are also some notes available (in PS Format).
• Freitag 20.02.2004, 14:15 Uhr, Raum D 1.303, Bangming Deng (Beijing Normal University)
Triangulated categories and Kac-Moody algebras, IV
Abstract: This is the continuation of the series of lectures on the paper of Peng and Xiao. The talk covers the results of Section 4, showing that for the root category of a species, we obtain the corresponding (derived) Kac-Moody Lie algebra. Maybe also a sketch of some of the ideas in Section 5 (e.g. for the quiver A_2).
• 19.02.2004, Raum D 1.303, Hagen Meltzer (Stettin)
Exceptional modules for canonical algebras
Abstract: We describe exceptional modules over canonical algebras of tubular type. We discuss the part of the proof, which was not presented during the 1.Bielefeld-Paderborn Representation Theory Workshop. We show further how certain exceptional modules over tubular canonical algebras can be described explicitly.
• Dienstag 17.02.2004, 14:15 Uhr, Raum D 1.303, Henning Krause (Paderborn)
Triangulated categories and Kac-Moody algebras, II
Abstract: This is the continuation of the series of lectures on the paper of Peng and Xiao. The talk covers the proofs in Sections 6,7 and 8 (based on a new approach by Hubery). There are also some notes available (in PS-format).
• Dienstag 17.02.2004, 11:15 Uhr,Raum D 1.303, Hideto Asashiba (Osaka)
Triangulated categories and Kac-Moody algebras, I
Abstract: This is the first in a series of talks on the paper [Peng, Liangang; Xiao, Jie: Triangulated categories and Kac-Moody algebras. Invent. Math. 140 (2000), no. 3, 563--603]. The talk provides some background to the paper, including a sketch of what happens in the module category case (as motivation for what appears later) and the statement of the main results in Section 3. There are also some notes available (in PS Format).
• Freitag 13.02.2004, 10:15 Uhr, Bangming Deng (Beijing Normal University)
Frobenius morphisms and representations of algebras
Abstract: By defining the Frobenius morphism F on a finite dimensional algebra A over the algebraic closure of a finite field and the Frobenius twist of an A-module, we establish a relation between the representation theory of A and that of the F-fixed ponit algebra A^F=:B. More precisely, we show that the indecomposable B-modules correspond bijectively to the Frobenius orbits of indecomposable A-modules and that the Auslander-Reiten quiver of B can be obtained by folding' the Auslander-Reiten quiver of A via the Frobenius action.
• 05.02.2004, Kristian Brüning (Bielefeld)
Models of the stable homotopy category and algebraic K-theory
Abstract: The standard model of the stable homotopy category SHC has the disadvantage that the smash product is only commutative, associative and unital up to homotopy. Lydakis introduced the category of simplicial functors as a model of the stable homotopy category that is additionally equipped with a strict product. We compare the algebraic K-theory of the category of simplicial functors with the algebraic K-theory of spectra in the sense of Bousfield and Friedlander, the standard model of SHC.
• 29.01.2004, Igor Burban (Paris VI)
Kleinian singularities
Abstract: In this talk I am going to discuss basic geometric and algebraic properties of the quotient (Kleinian or Du Val) singularities. It will include such topics as the classification of finite subgroups of SU(2,C), geometry of the minimal resolution and McKay correspondence.
• 22.01.2004, Igor Burban (Paris VI)
Weighted projective lines with singularities
Abstract: I am going to introduce a certain generalization of weighted projective lines. Some of such "weighted projective lines with singularities" arise as equivariant Z/2Z quotients of singular projective curves of arithmetic genus 1. We consider the non-commutaive normalization of such curves and describe its derived category of coherent sheaves.