Seminar
Friday, 19 October 2018
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14:15, Room U2-205
Sarah Scherotzke (Münster): The Chern character and categorification
Abstract: The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives.
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15:30, Room U2-205
Jörg Feldvoss (Mobile, Alabama): Cohomological Vanishing Theorems for Leibniz Algebras
Abstract: Leibniz cohomology was introduced by Bloh and Loday as a non-commutative analogue of Chevalley-Eilenberg cohomology of Lie algebras. It turned out that Leibniz cohomology works more generally for Leibniz algebras which are a non-(anti)commutative version of Lie algebras. Many results for Lie algebras have been proven to hold in this more general context.
In the talk I will start from scratch and define Leibniz algebras, Leibniz (bi)modules, and their cohomology. Then I will explain the Leibniz analogues of vanishing theorems for the Chevalley-Eilenberg cohomology of semisimple and solvable Lie algebras due to Whitehead, Dixmier, and Barnes. In particular, we obtain the second Whitehead lemma for Leibniz algebras and the rigidity of semisimple Leibniz algebras in characteristic zero. The latter results were conjectured to hold for quite some time. Our main tools are the cohomological analogues of two spectral sequencesof Pirashvili for Leibniz homology and a spectral sequence due to Beaudouin.
All this is joint work with Friedrich Wagemann.
For a regular email announcement please contact birep.
Future Talks
Friday, 09 November 2018
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Kevin Coulembier (Sydney): tba
Friday, 16 November 2018
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Lidia Angeleri Hügel (Verona): tba
Friday, 18 January 2019
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Catharina Stroppel (Bonn): tba
Seminar Archive
Friday, 24 August 2018
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14:15, V2-200
Hiroyuki Minamoto (Sakai): Finite dimensional graded Iwanaga-Gorenstein algebras and Happel's functor
Abstract: Let A be a finite dimensional algebra and T(A)= A + D(A) be the trivial extension by the dual bimodule D(A). Happel constructed a functor H from the derived category of A to the stable category of graded T(A)-modules and proved that it is fully faithful and gives an equivalence precisely when A is of finite global dimension. Happel's functor is generalized to a finite dimensional graded Iwanaga-Gorenstein algebra S and gives a functor from the derived category of the Beilinson algebra, a finite dimensional algebra constructed from S, to the stable category of graded Cohen-Macaulay modules. In this talk, first we give two characterizations of S such that Happel's functor is fully faithful or equivalence. Next we show that Happel's functor admits a fully faithful left adjoint which has left adjoint provided that the degree 0 subalgebra of S is of finite global dimension. If time permits, we give several applications. This is a joint work with Kota Yamaura.
Friday, 20 July 2018
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14:15, Room T2-205
Baptiste Rognerud (Bielefeld): The derived category of the Tamari lattice is fractionally Calabi-Yau
Abstract: In this talk, I will introduce an interesting family of indecomposable objects in the bounded derived category of the Tamari lattice. Then, I will give a combinatorial description of the action of the Serre functors on these objects and explain how we can deduce that the bounded derived category is fractionally Calabi-Yau.
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15:30, Room T2-205
Fan Xu (Beijing): Ringel-Hall algebras and categorification
Abstract: The aim of this talk is to generalize Lusztig's construction of quantum groups to Ringel-Hall algebras. We construct the geometric analog of Green's theorem on the comultiplication of a Ringel-Hall algebra. It is an extension version of the comultiplication of a quantum group defined by Lusztig. As an application, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework. This is based on joint work with Jie Xiao and Minghui Zhao.
Friday, 13 July 2018
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15:00, Room T2-205
Louis Rowen (Ramat Gan): Algebraic systems and exterior semi-algebra
Abstract: In this talk, we describe negation maps and `systems', and their application to linear algebra in a rather general framework that includes tropical algebra, hyperfields and fuzzy rings.
The usual definition of Grassmann (exterior) algebras generalizes directly to semi-algebras, and has a built-in negation map for elements of degree > 1, so the theory of systems can be applied directly to Gatto's theory, unifying results of linear algebra from different perspectives including the classical perspective.
This will include joint work with Akian, Gaubert, Gatto, Jun, Knebusch, and Mincheva, and does not require prerequisites.
Friday, 06 July 2018
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14:15, Room T2-205
Ulrich Thiel (Sydney): Finite-dimensional graded algebras with triangular decomposition
Abstract: I will discuss a new approach to the representation theory of self-injective finite-dimensional graded algebras with triangular decomposition (such as restricted enveloping algebras, Lusztig’s small quantum groups, hyperalgebras, finite quantum groups, restricted rational Cherednik algebras, etc). We show that the graded module category of such an algebra is a highest weight category and has a tilting theory in the sense of Ringel. We can then show that the degree zero part of the algebra (the "core") is cellular and can construct a canonical highest weight cover à la Rouquier of it. The core captures essential information of the representation theory of the original algebra, hence we can approach the latter with these additional structures. This is joint work with Gwyn Bellamy (Glasgow).
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15:30, Room T2-205
Yann Palu (Amiens): Non-kissing complex and tau-tilting over gentle algebras
Abstract: This is a report on a joint paper with Vincent Pilaud and Pierre-Guy Plamondon. The non-kissing complex is a simplicial complex introduced by T. McConville who studied some of its lattice theoretic aspects. After explaining the properties of the non-kissing complex that seem the most relevant to representation theory, I will relate it to tau-tilting theory, as defined by Adachi-Iyama-Reiten. This allows to generalise non-kissing to a more general set up by making use of gentle algebras.
Wednesday, 27 June 2018
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14:15, Room U2-205 (90 minutes)
Giovanni Cerulli Irelli (Rome): Cell decompositions and algebraicity of cohomology for quiver Grassmannians
Abstract: I will report on a joint project with F. Esposito (Padova), H. Franzen (Bochum) and M. Reineke (Bochum), arXiv: 1804.07736. We show that the cohomology ring of a quiver Grassmannian associated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this, we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with a rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.
Friday, 15 June 2018
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14:15, Room U2-113
Jan Geuenich (Bielefeld): Tilting modules for the Auslander algebra of the truncated polynomial ring
Abstract: I present the classification of the tilting modules for the Auslander algebra of the truncated polynomial ring. More precisely, I construct an isomorphism between the tilting poset and a finite interval in the braid group. This extends an isomorphism described by Iyama and Zhang between the classical tilting poset and the symmetric group with the opposite left weak order.
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15:30, Room U2-113
Estanislao Herscovich (Grenoble): The curved A_infinity-coalgebra of the Koszul codual of a filtered dg algebra
Abstract: In this talk I will present a result allowing to compute the coaugmented curved A_infinity-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. This provides a generalisation of a result by B. Keller, which described the A_infinity-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra. As an application, I will show how to compute the coaugmented curved A_infinity-coalgebra structure of the Koszul codual of a PBW deformation of an N-Koszul algebra, extending a previous result by G. Fløystad and J. Vatne.
Friday, 08 June 2018
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14:15, Room T2-205
Mikhail Gorsky (Cologne): Extended Hall algebras
Abstract: Hall algebras play an important role in representation theory and algebraic geometry. The Hall algebra of an exact or a triangulated category captures information about the extensions between objects. It turns out that in some cases twisted and extended Hall algebras of triangulated categories are well-defined even when their non-extended counterparts are not. I will explain how to associate a twisted extended Hall algebra to a triangulated category, when the latter arises as the homotopy category of a hereditary exact model category or as an orbit category of certain kind. I will discuss applications of this constructions to graded quiver varieties and to categorification of modified quantum group.
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15:30, Room T2-205
William Crawley-Boevey (Bielefeld): A new approach to simple modules for preprojective algebras
Abstract: This is joint work with Andrew Hubery. My earlier work on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.
Friday, 25 May 2018
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14:15, Room T2-205
Sophiane Yahiatene (Bielefeld): Thick subcategories in hereditary abelian categories
Abstract: Let H be a connected Ext-finite hereditary abelian k-category with tilting complex. In this talk I present a group-theoretic method to classify the thick subcategories generated by exceptional sequences. For that, we consider the Grothendieck group of H, which can be seen as a root lattice of a generalized root system and define a reflection group acting on it.
As an example we consider the category of coherent sheaves of a weighted projective line of tubular type.
(Joint work with B. Baumeister and P. Wegener)
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15:30, Room T2-205
Henning Krause (Bielefeld): Morphic enrichments of triangulated categories (after Keller)
Abstract: The talk presents some recent work of Bernhard Keller. The notion of a triangulated category suffers from the fact that cones are not functorial. Morphic enrichments provide a concept to overcome this problem, and basically all triangulated categories that arise in nature admit such an enrichment. A morphic enrichment of a triangulated category T is given by a recollement of triangulated categories with T at both ends, and the additive structure of this recollement determines the triangulated structure of T. This idea goes back to work of Keller (Derived categories and universal problems, Comm. Algebra 19, 1991); it turns out to be useful for defining a triangulated structure on the completion of a triangulated category.
Friday, 04 May 2018
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13:15, Room T2-205
Magnus Bakke Botnan (München): Representation Theory in Topological Data Analysis
Abstract: Topological data analysis (TDA) is a relatively recent approach to data analysis in which topological signatures are assigned to data. In this talk I will survey how the theoretical foundations of TDA rest on classical results from the representation theory of quivers. I will also discuss recent results in representation theory inspired by questions in TDA. This is joint work with Ulrich Bauer, Steffen Oppermann and Johan Steen.
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14:30, Room T2-205
Haydee Lindo (Stuttgart): Trace modules, Rigidity and Endomorphism rings
Abstract: I will speak on some recent developments in the theory of trace modules over commutative Noetherian rings. This will include applications of trace modules in understanding endomorphism rings and a discussion of ongoing work examining the relationship between trace modules and modules having no self-extensions.
Friday, 27 April 2018
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14:15, Room T2-205
Thomas Poguntke (Bonn): Higher Segal structures in algebraic K-theory and Hall algebras
Abstract: One of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces concerns the fibrancy properties of Waldhausen's simplicial construction of the algebraic K-theory of an exact category, which are in particular responsible for the associativity of various Hall algebras. We will explain their results, with an emphasis on this latter aspect. Finally, we will introduce a higher dimensional analogue of the construction, where short exact sequences are replaced by longer extensions, whose algebraic ramifications are yet to be clearly understood.
Saturday, 21 April 2018
Friday, 20 April 2018
Friday, 13 April 2018
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14:15, Room T2-205
Minghui Zhao (Beijing): On purity theorem of Lusztig's perverse sheaves
Abstract: Let Q be a finite quiver without loops and U the quantum group corresponding to Q. Lusztig introduced the canonical basis of the positive part of U via some semisimple perverse sheaves (Lusztig's perverse sheaf). When Q is a Dynkin quiver, Lusztig proved that any Lusztig's perverse sheaf L possesses a Weil structure such that the Frobenius eigenvalues on the stalk of the i-th cohomology sheaf H^i(L) at x are equal to q^(i/2) for any k-rational point x, where k is the finite field of q elements. The purpose of this talk is to generalize this result to all finite quiver without loops. As an application, we shall prove the existence of a class of Hall polynomials. This is a joint work with Jie Xiao and Fan Xu.
For information on earlier talks please check the complete seminar archive.
