Donnerstag 15.12.2005, 16:45 Uhr, Alexander Zimmermann (Amiens)
Derived invariance of Külshammer's ideals
Abstract: For symmetric finite dimensional k-algebras A over an algebraically closed field k, Külshammer defined in the early 1980's a descending series of ideals of the center of A. We show that given an equivalence between the derived categories of A and another algebra B, there is an isomorphism between the centres of A and B mapping the corresponding ideals to each other. In joint work with Thorsten Holm we apply this to tame blocks of group rings with 2 simple modules and show that it is possible to detect parameters of certain of these algebras in the derived category which were previously not known to be visible.
Donnerstag 15.12.2005, 15:15 Uhr, Bin Zhu (Tsinghua University)
Applications of BGP-reflection functors to cluster algebras
Abstract: For any sink or source in a valued quiver, the associated Bernstein-Gelfand-Ponomarev reflection functor (BGP-reflection functor for short) induces a triangle equivalence between the corresponding cluster categories. In this talk, we will speak about their applications to cluster algebras, including the bijection from the set of indecomposable objects in the cluster category of Dynkin type to the set of cluster variables of corresponding cluster algebras which sends the shift by 1 of an indecomposable projective module to u_i and sends tilting objects to clusters, and the denominator theorem in the Dynkin case.
Dienstag 13.12.2005, 17:45 Uhr in P1.4.17 (im Rahmen des Fakultätskolloquiums), Markus Reineke (Münster)
Simultane Konjugation von Matrizen
Abstract: Es ist ein klassisches ungelöstes Problem der Linearen Algebra, eine Normalform für ein Tupel von Matrizen unter simultaner Konjugation´zu finden. Einige neuere Ansätze zum qualitativen Verständnis dieses Problems mit geometrischen Methoden sollen vorgestellt werden.
Donnerstag 8.12.2005, 16:45 Uhr, Martin Langer (Bonn)
Realizability of Modules over Tate Cohomology - some examples
Abstract: Let G be a group and k be a field. Given any kG-module M, the Tate cohomology of M is a graded module over the Tate cohomology ring of G. This leads us to the following question: Given a graded module over the Tate cohomology ring, how can we decide whether it is the Tate cohomology of some kG-module M? After a short introduction to Tate cohomology, a result due to Benson, Krause and Schwede giving a partial answer to this question will be explained. Then we will discuss several examples G, in particular abelian 2-groups and Q_{8}, the quaternion group with 8 elements.
Donnerstag 8.12.2005, 15:00 Uhr, Birgit Huber (Paderborn)
Realizability and Localization
Abstract: Let A be a differential graded algebra such that its cohomology ring H*(A) is graded commutative and coherent. A graded module over H*(A) is called realizable if it is (up to direct summands) of the form H*(M) for some differential graded A-module M. Benson, Krause and Schwede have given a local and a global obstruction for the realizability. We show that a finitely presented graded module X is realizable if and only if X localized at p is realizable for all primes p of the graded spectrum of H*A. The global obstruction is given by the secondary multiplication of the A_{\infty}-algebra H*(A). We define a localization A_p of the dga A at a prime p of H*(A) and show that the secondary multiplications of H*(A) and H*(A_p) are related in a nice way. Further we discuss whether the global obstruction being trivial is a local property.
Freitag 2.12.2005, 17:15 Uhr im Raum D1.328, Mechthild Koreuber (FU Berlin)
Die Entstehung und Entwicklung einer mathematischen Tatsache: Das Hasse-Brauer-Noether-Theorem – Historische und wissenschaftstheoretische Anmerkungen
Abstrakt: Zwischen Emmy Noether und Helmut Hasse gab es in den zwanziger und Anfang dreißiger Jahren einen regen Briefwechsel, der einen Einblick in den Entstehungsprozess des Hasse-Brauer-Noether-Theorems gestattet. Noethers Briefe lassen sich wie Labortagebücher lesen, in denen die Begeisterung für Mathematik, die Aufregung über den Entstehungsprozess und die Freude über den gelungenen Beweis zu finden sind. Im Vortrag werden auf der Grundlage dieser Briefe die Entstehung eines bedeutenden algebraischen Theorems im Kontext der der damaligen mathematischen Forschungslandschaft und seine weiteren Entwicklungen dargestellt. Im Anschluss an den Vortrag werden ergänzend Bilder von Emmy Noether gezeigt.
Donnerstag 1.12.2005, 16:45 Uhr, Bo Chen (Bielefeld)
The Gabriel-Roiter measure for representation directed algebras
Abstract: Let A be a finite dimensional directed algebra over an algebraically closed field k and M be a finitely generated A-module. Can we write M as an extension of an orthogonal pair? With the question, I start with the Gabriel-Roiter measure of an A-module M which was defined to be a subset of the natural numbers which corresponds to certain filtration of M with indeocmposable terms. I want to present some properties of the GR measure and the connection with our question.
Donnerstag 1.12.2005, 15:15 Uhr, Carl Frederik Berg (Trondheim)
Hereditary categories
Abstract: In the first part I will introduce and speak about ray quviers, an invention of Ringel which is very handy to construct hereditary categories of different types. This first part would be interesting for people who work with hereditary categories and want examples to work on. In the second part I will give some results about hereditary Ext-finite abelian categories with almost split sequences. Parts of those results are from work in progress with Van Roosmalen from Hasselet University in Belgium.
Dienstag 29.11.2005, 17:45 Uhr in P1.4.17 (im Rahmen des Fakultätskolloquiums), Steffen König (Köln)
Relative Äquivalenzen - Darstellungstheorie mit und ohne Orientierung
Abstrakt: Wählt man eine Kategorie von Darstellungen etwa für Liealgebren oder algebraische Gruppen, so bieten sich zwei Prinzipien an: Eine symmetrische Situation erreicht man oft durch Verwendung symmetrischer diskreter Objekte wie Weylgruppen. Eine orientierte Situation erreicht man dagegen durch Graduierungen, etwa durch höchste Gewichte. Beides zugleich geht nur in einfachen oder halbeinfachen Situationen. Relative homologische Algebra erlaubt es aber manchmal, orientierte Situationen mit symmetrischen zu vergleichen, ohne die Orientierung zu verlieren. Dies soll an Beispielen erklärt werden.
Donnerstag 24.11.2005, 16:45 Uhr, Øyvind Solberg (Trondheim)
Support varieties over complete intersections
Abstract: Let (R,m) be a local complete intersection with maximal ideal m. The support of a perfect complex over R is defined in terms of the prime ideal spectrum of R. The support of a finitely generated R-module is defined in terms of the prime spectrum of the ring R/m[x_1,...,x_t] (here t = codim R). The aim of this talk is to show that these two support theories can both be derived from a theory of support varieties for the derived category of bounded complexes of finitely generated R-modules. This talk is based on joint work with: A. Buan, H. Krause and N. Snashall.
Donnerstag 24.11.2005, 15:15 Uhr, Michael Barot (UNAM Mexico)
Lie algebras associated to positive definite unit forms
Abstract: Serre's Theorem gives an explicit description (in terms of a given Dynkin diagram D) of a finite set of relations defining a semisimple Lie algebra A(D), whose Coxeter diagram is D. The talk will show how this set of relations can be expanded in a nice way if the initial data is the bigraph B(q) associated to a positive definite unit form q (whose Dynkin type is D). The presented results are part of the Ph.D thesis of Daniel Rivera.
Donnerstag 17.11.2005, 16:15 Uhr in D1.320, Andrew Hubery (Paderborn)
From cluster categories to cluster algebras, IV
Abstract: In the last of this series of talks we complete the proof of the cluster multiplication theorem by Caldero and Keller. This will involve understanding the triangles in the cluster category of the form N->E->M->N[1] when N and M are modules such that there are no extensions of M by N in the module category.
Donnerstag 10.11.2005, 16:15 Uhr in D1.320, Bin Zhu (Tsinghua University)
From cluster categories to cluster algebras, III
Abstract: This is the third part of a series of talks. We will present a part of the proof of Caldero-Keller's multiplication theorem by using Green's Theorem (which was stated in the talk by A.Hubery).
Donnerstag 03.11.2005, 15:00 Uhr in D1.320, Dirk Kussin (Paderborn)
From cluster categories to cluster algebras, II
Abstract: This is the second part of a series of talks. We will present the proofs of the results of P. Caldero and F. Chapoton which were stated in the first talk (see talk by A. Hubery on 13th of October in this seminar).
Donnerstag 27.10.2005, 16:15 Uhr, Ragnar-Olaf Buchweitz (Toronto)
Hochschild Cohomology and the Centre of the Derived Category
Abstract: We will discuss the nature of the characteristic homomorphism from Hochschild cohomology to the centre of the derived category, presenting the little we know about the latter in examples. Among the tools employed to bound kernel and image of the map are Grothendieck residues, the theory of levels of objects in a triangulated category with respect to a subcategory, and some old fashioned spectral sequences.
21 - 22.10.2005, 6th NWDR Represenation Theory Workshop
Donnerstag 13.10.2005, 16:15 Uhr in D1.320, Andrew Hubery (Paderborn)
From cluster categories to cluster algebras
Abstract: This is a report on recent work by P. Caldero, F. Chapoton and B. Keller concerning cluster categories. The first article shows how one can realise the cluster variables in a cluster algebra of finite type via quiver representations. The second article then interprets the multiplication in the cluster algebra via the triangulated structure of the cluster category.
P. Caldero and F. Chapoton, `Cluster algebras as Hall algebras of quiver representations' math.RT/040187
P. Caldero and B. Keller, `From triangulated categories to cluster algebras' math.RT/0506018
Donnerstag 29.09.2005, 15:15 Uhr in D1.320, Karsten Schmidt (Paderborn)
The Auslander-Reiten quiver of a Poincare duality space
Abstract: This is the continuation of a talk given on the 22nd of September on two articles of Peter Jorgensen. Let k be a field and X a simply connected Poincare duality space with HX being finite dimensional. Last time I explained that D_{dg}(C^*(X;k))^c has Auslander-Reiten triangles. This time I will be concerned with the structure of the corresponding Auslander-Reiten quiver. I will present a result of Jorgensen saying that all components are of the form ZA_\infty.
Donnerstag 22.09.2005, 15:45 Uhr in D1.338, Lutz Hille (Bielefeld)
Parabolic group actions and tilting modules
Abstract: Actions of parabolic groups can be understood using good modules over certain quasi-hereditary algebras. In particular, the classification of all pairs (P,n) (consisting of a parabolic subgroup in GL and a P-ideal n in the Lie algebra of its unipotent radical), where P acts with a dense orbit on n, is equivalent to the classification of good modules without selfextensions over a corresponding quasi-hereditary algebra. We first explain the classical results (the action of P on the Lie algebra of its unipotent radical) due to Richardson for the parabolic group actions and Br"ustle, H., Ringel and R"ohrle for the tilting modules. In a second part we explain more recent results obtained by my diploma student M. Goller.
Donnerstag 22.09.2005, 14:15 Uhr in D1.338, Markus Perling (Paderborn)
On Tilting Sheaves on Toric Varieties II
Abstract: I report about progress made in the search for tilting sheaves on toric varieties. I will present a variety of new examples and some speculations/conjectures.
Donnerstag 22.09.2005, 10:15 Uhr in D1.338, Karsten Schmidt (Paderborn)
Auslander-Reiten theory over topological spaces
Abstract: I will give a report on two articles of Peter Jorgensen: [J1] Auslander-Reiten theory over topological spaces, and [J2] The Auslander-Reiten quiver of a Poincare duality space. Given a topological space X and a field k one considers the triangulated subcategory of compact objects in the derived category of the singular cochain differential graded algebra C^*(X;k) of X. Jorgensen shows that this category has Auslander-Reiten triangles if and only if the space X has Poincare duality. Furthemore, he shows - if AR triangles exist - that the components of the corresponding Auslander-Reiten quiver are of the form ZA_\infty. The latter will be presented in a second talk (probably next week).
Donnerstag 21.07.2005, 16:15 Uhr, Steffen Sagave (Bonn)
Universal Toda brackets and realizability obstructions
Abstract: Given a homological functor from a triangulated category to an abelian category, one can ask whether an object or a morphism lies in the image of this functor. I will explain what Massey products and Toda brackets are and how they are related to this problem. As applications, I will discuss the homology functor on the derived category of a differential graded algebra and the homotopy groups functor on the homotopy category of the module category of a ring spectrum.
Donnerstag 14.07.2005, 15:15 Uhr, Andrew Hubery (Bielefeld)
Stratifying Systems and Quantum Schur Algebras
Abstract: We take a fresh look at stratifying systems as introduced by Erdmann and Saenz. In particular we obtain a stronger result about existence of relative injectives. This can then be applied to quantum Schur algebras in the same spirit as Erdmann for the Schur algebras. In particular, the indecomposable relative injectives correspond to the Young modules.
Donnerstag 07.07.2005, 15:15 Uhr, Christof Geiss (UNAM Mexico)
A bound for rigid modules over preprojective algebras
Abstract: This is joint work with J. Schroer. We show, that the number of non-isomrphic indecomposable summands of a rigid module over a preprojective algebra of Dynkin type Q is bounded by the number of positive roots associated to Q. This bound is actually sharp.
Freitag 01.07.2005, NWDR Workshop in Bielefeld
Donnerstag 30.06.2005, 16:15 Uhr, Thomas Brüstle (Sherbrooke)
From cluster tilted algebras to tilted algebras - and back
Abstract: Cluster tilted algebras have been introduced by Buan,Marsh,Reineke,Reiten and Todorov as a tool to better understand the cluster algebras of Fomin and Zelevinsky. The concept of "cluster tilting" is a slight variation of "classical" tilting, and there are many similar results. For instance, the minimal representation-infinite cluster tilted algebras correspond bijectively to the minimal representation-infinite tilted algebras. However, this bijection is so far only obtained by counting the numbers of minimal representation-infinite algebras. The aim of this talk is to provide an actual map from cluster tilted algebras to tilted algebras or the other way round.
Donnerstag 23.06.2005, 14:30 Uhr in Bielefeld (Raum A3-137), Michael Butler (Liverpool)
Abstract: Direct decompositions of infinite rank integral representations of finite groups and orders
The question has recently arisen, both in group theory and in functional analysis, as to whether every action of a finite group, G, on an infinite rank free abelian group decomposes into a direct sum of finite rank representations, that is, ZG-lattices. A necessary condition for this is that the Sylow subgroups of G are cyclic and of cube-free order, and this is now known to be sufficient if G is a p-group. More generally, I shall discuss Wolfgang Rump's remarkable new combinatorial characterisation of separable orders over Dedekind domains with the property that all `infinite rank lattices' are direct sums of lattices.
Donnerstag 09.06.2005, 16:45 Uhr, Igor Burban (MPI Bonn)
On the derived categories of an irreducible projective curve of arithmetic genus one
Abstract: The aim of this talk based on a series of joint articles with Bernd Kreussler is to discuss the common features and principal differences between the derived categories of coherent sheaves of a smooth and a singular irreducible projective curve of arithmetic genus one. The main technical tools are the Fourier-Mukai transforms and Harder-Narasimhan filtrations in triangulated categories. This approach allows to describe the group of exact auto-equivalences of the derived category of a singular Weierstrass curve and obtain a positive answer on a question posed by Polishchuk about a description of spherical objects.
Donnerstag 09.06.2005, 15:15 Uhr, Igor Burban (MPI Bonn)
On the Hall algebra of an elliptic curve
Abstract: In this talk based on a joint work with Olivier Schiffmann I shall describe the Hall algebra H_X of an elliptic curve X defined over a finite field and show that the group SL(2,Z) of exact auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld double DH_X of H_X by algebra automorphisms. Next, I am going to consider a certain natural subalgebra U_X of DH_X for which an explicit presentation by generators and relations will be given. This algebra turns out to be a flat two-parameter deformation of the ring of symmetric Laurent polynomials in two sets of countably many variables under the simultaneous symmetric group action.
Donnerstag 02.06.2005, 16:15 Uhr, Victor Levandovskyy (Kaiserslautern)
Applications of Computer Algebra in Non-commutative PBW Algebras
Abstract: We present several important applications of the non-commutative computer algebra together with their implementation in the Computer Algebra System Singular:Plural. Among others, we show how Groebner bases can be used for computing intersection of modules with subalgebras, kernels of module homomorphisms and preimages of ideals under morphisms of algebras. Some interesting examples, including the decomposition into central characters, will be computed live.
Donnerstag 19.05.2005, 16:15 Uhr, Dan Zacharia (Syracuse)
Koszul algebras and their representations
Abstract: This talk is an elementary introduction to Koszul algebras. I will also talk about some applications to the representation theory of selfinjective algebras and some geometric connections.
Donnerstag 12.05.2005, 16:15 Uhr, Jan Stovicek (Prague/Düsseldorf)
All tilting modules are of countable type
Abstract (in PDF format)
Donnerstag 21.04.2005, 16:15 Uhr, Jose Antonio de la Pena (UNAM, Mexico)
Root-induced integral quadratic forms
Abstract: This is joint work with M. Barot. Given an integral quadratic form q and a tuple of q-roots r, we consider the induced form q_r. Two non-negative unit forms are of the same Dynkin type precisely when they are root induced one from the other. Root-induction yields an interesting partial order on Dynkin types which is studied.
Donnerstag 14.04.2005, 16:15 Uhr, Thorsten Holm (Leeds)
Cartan determinants for gentle algebras
Abstract: The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very useful for derived equivalence classifications. In the talk we show how to compute the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.