# Seminars in Algebra

## Fall 2009

## December 3rd 2009, 13:15 - 15:00 in 734

**Title:** Approximation and Duality

**Speaker:** Alex Martsinkovsky

**Abstract:** This is joint work with Roberto Martínez Villa. Given an equivalence (or duality) between categories, we want to extend it to categories that can be approximated (NOT in the sense of homologically finite subcategories) by the original categories. In this lecture I will show how this can be done if one starts with Koszul duality. The main result will establish a triangulated duality between the stable (projective) homotopy theory of modules over a finite-dimensional Koszul algebra with a noetherian Koszul dual and the category of tails over that dual algebra. All terms will be defined and explained.

## November 26th 2009, 13:15 - 14:15 in 734

**Title:** Geometric descriptions of m-cluster categories with applications

**Speaker:** Hermund Andrè Torkildsen

**Abstract:** I will give a short overview over the various geometric descriptions of m-cluster categories. I will consider coloured quiver mutation and use the geometric description to count the number of coloured quivers in the m-mutation class in the A_n case. This is related to the cell-growth problem in combinatorics.

## November 19th 2009, 13:15 - 14:15 in 656

**Title:** Higher cluster categories

**Speaker:** Anette Wraalsen

**Abstract:** There will be a short survey on m-cluster categories, and I will talk about a technique I used to show certain fundamental properties of m-cluster categories.

## November 12th 2009, 13:15 - 14:15 in 656

**Title:** From m-clusters to m-Hom-configurations via exceptional sequences

**Speaker:** Aslak Buan

**Abstract:** This is joint work with Idun Reiten and Hugh Thomas.

We show a bijection with the set of silting objects and the set of Hom-configurations in the derived category of an hereditary algebra. The bijection specializes to a bijection between m-clusters and so called m-Hom-configurations (which will be defined).

The idea of the proof of this bijection will be explained, it involves mutation of exceptional sequences in the derived category.

This result has a combinatorial interpretation in terms of comparing m-non-crossing partitions and m-clusters. If time allows, this will also be explained.

## November 5th 2009, 13:15 - 14:15 in 656

**Title:** The stable AR-quiver of a quantum complete intersection

**Speaker:** Petter Andreas Bergh

**Abstract:** This is joint work with Karin Erdmann. We completely describe the tree classes of the components of the stable AR-quiver of a quantum complete intersection. In particular, we show that the tree class is always A_{infty} whenever the algebra is of wild representation type. Moreover, in the tame case, there is one component of tree class \tilde{A}_{12}, whereas all the others are of tree class A_{infty}.

## October 29th 2009, 13:15 - 14:15 in 656

**Title:** Preprojective algebras, approximations, and c-sortable words

**Speaker:** Gordana Todorov

## October 26th 2009, 10:15 - 11:15 in 734

**Title:** Hochschild (co-)homology and Hodge numbers of varieties with a tilting object

**Speaker:** Ragnar Buchweitz

## October 21st 2009, 10:15 - 11:15 in B333

**Title:** On algebras of finite CM-type

**Speaker:** Apostolos Beligiannis

## October 20th 2009, 16:15 - 17:15 in 734

**Title:** Representation dimension of iterated tilted algebras

**Speaker:** Dieter Happel

## October 15th 2009, 13:15 - 14:15 in 656

**Title:** Preprojective algebras and c-sortable words

**Speaker:** Claire Amiot

**Abstract:** This is a joint work with Iyama, Reiten and Todorov.

Let Q be a finite quiver without oriented cycles. To any element w of the Coxeter group of Q, Buan, Iyama Reiten and Scott have associated a certain subcategory Sub(Lambda_w) of finite length modules over the preprojective algebra Lambda associated to Q. These categories with nice properties ( Frobenius stably 2-Calabi-Yau, with cluster-tilting objects) play an important role in the "categorification" of cluster algebras. Any reduced expression of w gives a nice filtration of \(\Lambda_w\).

In this talk I will explain how to link this filtration with tilting modules over the path algebra kQ when the word w is c-sortable.

## October 12th 2009, 13:15 - 14:00 in EL1

**Title:** Komplette snitt

**Speaker:** Petter Andreas Bergh

**Abstract:** Komplette snitt er en type kommutative lokale ringer med opphav i algebraisk geometri. De svarer til affine varieteter som kan uttrykkes ved så få ligninger som mulig, og som derfor er komplette snitt av hyperflater. Nylig har man også begynt å studere disse ringene innenfor algebraisk topologi, særlig i forbindelse med rasjonal homotopiteori og kommutative ringspektra.

I denne forelesningen skal vi se på de homologiske egenskapene til komplette snitt. Til enhver modul kan vi assosiere en affin varietet, og de geometriske og topologiske egenskapene til denne reflekterer modulens homologiske egenskaper. Videre skal vi se på den deriverte kategorien til komplette snitt. Dette er en triangulert kategori, og det viser seg at dens dimensjon er begrenset av kodimensjonen til det komplette snittet.

## October 8th 2009, 13:15 - 14:15 in 656

**Title:** Constructing tilted algebras from cluster-tilted algebras

**Speaker:** Marco Angel Bertani-Økland

**Abstract:** Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given a cluster-tilting object in a cluster category C and its distribution over the AR quiver of C, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object. This is joint work with Steffen and Anette.

## October 1st 2009, 13:15 - 14:15 in 656

**Title:** Dimensions of thick subcategories of the derived bounded category

**Speaker:** Jan Stovicek

**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 dimension of T is infinite, or T = D^b(A).

## September 24th 2009, 13:15 - 14:15 in 656

**Title:** Total positivity and cluster algebras

**Speaker:** Robert Marsh

**Abstract:** One of the original motivations for Fomin and Zelevinsky's definition of cluster algebras was the study of totally positive matrices: matrices all of whose minors are positive. The clusters correspond to so-called total positivity criteria, i.e. collections of minors whose positivity guarantees that all minors are positive. I will give an introduction to this topic from a cluster algebra perspective via the example of upper unitriangular matrices over the real numbers.

## September 17th 2009, 13:15 - 14:15 in 656

**Title:** Partial Orders on Representations of Algebras

**Speaker:** Tore Alexander Forbregd

**Abstract:** This is joint work with S.O. Smalø and N.M. Nornes.

Let k be a commutative artin ring and let A be an artin k-algebra. For each natural number d let rep_d A be the set of isomorphism classes of A-modules with k-length equal to d. For each natural number n, an (nxn)-matrix with entries in A, can be considered as a k-endomorphism of M^n , where M^n denotes the direct sum of n copies of the A-module M. The quasiorder >n on rep_d A is defined by M >n N if for every (nxn)-matrix B, with entries in A, we have that l_k (M^n/B(M^n)) > l_k (N^n/B(N^n)).

We show that the quasiorder >n is a partial order on rep_d A for n>d^3 (here > includes equality).

## September 10th 2009, 13:15 - 14:15 in 656

**Title:** Tiling bijections between paths and Brauer diagrams

**Speaker:** Robert Marsh

**Abstract:** Joint work with Paul Martin (Leeds). There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.

## September 3rd 2009, 13:15 - 14:15 in 656

**Title:** Quiver Representations and Irreducible Null Cones

**Speaker:** Stéphane Materna

**Abstract:** Given an algebraically closed field k, a quiver Q and a dimension vector d for Q, we obtain an action of the algebraic group Gl(d) (product of linear groups) on the vector space rep(Q, d) (consisting of all representations of Q with dimension vector d). This action induces an action of Gl(d) on the algebra of polynomial functions on rep(Q, d). The polynomial functions which are invariant under the action of the subgroup Sl(d) (product of special linear groups) are called semi-invariants. The aim of the talk is to describe null cones which arise naturally as the common zero sets of the semi-invariants without constant term.

## August 27th 2009, 13:15 - 14:15 in 656

**Title:** Iterated tilted algebras and cluster tilted algebras

**Speaker:** María Inés Platzeck

**Abstract:** The connection of cluster tilted algebras with tilted algebras was studied by Assem, Brüstle and Schiffler. They prove that an algebra is cluster tilted if and only if it is isomorphic to the relation extension R(A) of a tilted algebra A. I will explain the connection between cluster tilted algebras and iterated tilted algebras of global dimension at most 2. Given an iterated tilted algebra B, then B is the endomorphism ring of a tilting complex T in the derived category of a hereditary algebra H. When gl.dim.B < 3, then T defines a cluster tilting object in the cluster category C(H), whose endomorphism ring C is a cluster tilted algebra. Moreover, there is a sequence of algebra homomorphisms B –>C –> R(B) –> B whose composition is the identity of B, and such that the kernel of C–> R(B) is contained in rad^2 C. In particular, C and R(B) have the same quivers. Then I will consider quotients of admissible cuts of cluster tilted algebras of finite representation type, and discuss some applications. (jt work with M.Barot, E. Fernández, I. Pratti and S. Trepode). Some of these results have also been obtained independently by O. Iyama and C. Amiot.

## August 20th 2009, 14:30 - 15:30 in 734

**Title:** n-representation-finiteness and fractional Calabi-Yau properties

**Speaker:** Osamu Iyama

**Abstract:** (joint with Martin Herschend) We study \(n\)-representation-finite algebras from the viewpoint of fractionally Calabi-Yau algebras. We shall show that all \(n\)-representation-finite algebras are twisted fractionally Calabi-Yau. We also show that twisted \(n(i-1)/i\)-Calabi-Yau algebras of global dimension \(n\) are \(n\)-representation-finite for any non-zero \(i\). As an application, we give a construction of \(n\)-representation-finite algebras using the tensor product. If time permits, I will introduce Minamoto's Fano algebras as a possible candidate of '\(n\)-representation-infinite' algebras.

## August 20th 2009, 13:15 - 14:15 in 734

**Title:** Faithfulness of actions of Artin groups on derived categories

**Speaker:** Hugh Thomas

**Abstract:** (joint work with Chris Brav, University of Toronto) Fix a simply-laced extended Dynkin type. There is a certain triangulated category which can be thought of in a few different ways, either as the bounded derived category of modules over the (completed) preprojective algebra, or as the bounded derived coherent sheaves on the minimal resolution of the corresponding Kleinian singularity. This triangulated category admits an action of the corresponding affine Artin group, either via the twist functors of Seidel-Thomas, or via tensoring by certain tilting modules, as in Iyama-Reiten. One can also consider the action of the corresponding finite-type Artin group contained inside the affine Artin group; Seidel-Thomas showed that this action is faithful in type A. Ishii-Ueda-Uehara later showed that the action of the whole affine Artin group is faithful in type A. We generalize this result to any simply-laced extended Dynkin type. The main new ingredients in our approach are the classical (Deligne, Brieskorn-Saito) Garside structure for finite-type Artin groups, and a presentation of the affine-type Artin group due to Macdonald.

## Spring 2009

## May 15th 2009, 13:15 - 14:15 in 734

**Title:** Jacobian 2-CY tilted algebras and gradings

**Speaker:** Claire Amiot

**Abstract:** In this talk I will explain why stably 2-CY categories with a cluster-tilting object whose endomorphism ring is a Jacobian algebra with a 'nice' grading are equivalent to some cluster categories coming from algebras of global dimension 2.

## May 11th 2009, 13:15 - 14:15 in 734

**Title:** Towards a Sketch based Specification Framework for Software Engineering

**Speaker:** Yngve Lamo

**Abstract:** The talk present the work of the Bergen group in developing a formal Diagrammatic Specification Framework, DPF for Software Engineering. The framework is based on general sketches, a generalization of (categorical) sketches (first Introduced by Ehresman in the late 60's). A sketch can be seen as a graphical specification of categories, where a diagram is interpreted as a commutative diagram, cones and co-cones are interpreted as universal constructions such as limit cones and co-limit cones. A general sketch generalize this approach by allowing arbitrary predicates not only universal constructions. The talk motivates the use of a such graph based approach from a computer science view, especially for model driven engineering, by pointing on some problem with set based approaches such as algebraic specifications.

## May 4th 2009, 13:15 - 14:15 in 734

**Title:** Jet bundles on grassmannians

**Speaker:** Helge Maakestad

**Abstract:** Let V be an arbitrary finite dimensional irreducible SL(E)-module where E is a finite dimensional vector space over a field of characteristic zero. Let v be the corresponding highest weight vector and let g=Lie(SL(E)). There is a filtration U^k(g)v in V by P-modules where P is the subgroup fixing v. There is an equivalence of categories between the category of SL(E)-linearized locally free sheaves on SL(E)/P and the category of P-modules, and the aim of this talk is to relate the filtration U^k(g)v to SL(E)-linearized jet bundles J on the flag variety SL(E)/P.

## March 30th 2009, 13:15 - 14:15 in 734

**Title:** Silting mutation in triangulated categories

**Speaker:** Osamu Iyama

**Abstract:** This is a joint work with K. Yamaura and T. Aihara. Following Keller-Vossieck, we call an object X in a triangulated category T a silting object if thick(X)=T and Hom(X,X[i])=0 for any positive integer i>0. This concept sometimes appears in representation theory, e.g. work by Hoshino-Kato-Miyachi. The point is that one can develop nice mutation theory for silting objects (not for tilting objects!). We introduce mutation and partial order on silting objects, and prove Happel-Unger-type result which asserts that Hasse quiver describes mutation. We observe a few class of algebras A such that successive mutation acts transitively on silting objects in K^b(proj A).

## March 23rd 2009, 13:15 - 14:15 in 734

**Title:** n-representation-finite algebras of type A

**Speaker:** Osamu Iyama

**Abstract:** This is a joint work with S. Oppermann. We call a finite dimensional algebra A of global dimension at most n n-representation-finite if there exists an n-cluster tilting object M in the module category mod A. In this case, M is unique up to multiplicities of direct summands. Gabriel's Theorem asserts that 1-representation-finite algebras are path algebras of Dynkin quivers. In my talk, a class of n-representation-finite algebras of `type A' will be constructed. We also classify n-representation-finite algebras with dominant dimension at least n. Our main tools are n-AR translation functor and n-APR tilting modules. An interesting combinatorics will appear in our consideration as slices in n-cluster tilting subcategories of derived categories.

## March 16th 2009, 13:15 - 14:15 in 734

**Title:** Solution to the Clebsch-Gordan problem for string algebras

**Speaker:** Martin Herschend

**Abstract:** String algebras are a class of tame algebras whose indecomposable finite dimensional modules are classified by strings and bands. They originate from the classification of Harish-Chandra modules over the Lorentz group by Gelfand and Ponomarev. By definition, any string algebra is presented as the path algebra of a quiver subject to certain monomial relations. Hence its module category is equipped with a tensor product defined point-wise and arrow-wise in terms of the underlying quiver. This leads to the following problem: given two indecomposable modules over a string algebra, decompose their tensor product into a direct sum of indecomposables. In my talk I will present the solution to this problem for all string algebras. Moreover, I will describe the corresponding representation rings.

## March 12th 2009, 10:15 - 11:15 in 656

**Title:** T-Koszul algebras revisited

**Speaker:** Dag Madsen

**Abstract:** In this talk I will give a much simplified definition of T-Koszul algebras and give some slightly surprising examples.

## March 2nd 2009, 13:15 - 14:15 in 734

**Title:** On the classification of periodic binary sequences into nonlinear complexity classes

**Speaker:** George Petrides

**Abstract:** Starting from a naive approach based on the definition of nonlinear complexity, we arrive at some nice recursions with interesting properties that help us obtain the number of periodic binary sequences of a given nonlinear complexity.

## February 23rd 2009, 13:15 - 14:15 in 734

**Title:** Computation of p-integral basis of quartic number fields and applications

**Speaker:** L. Houssain El Fadil

**Abstract:** Based on the Newton polygon, for each prime integer \(p\), a \(p\)-integral basis of §K§ and an explicit decomposition of §pZ_K§, where \(K\) is a quartic number field, defined by an irreducible polynomial \(P(X)=X4 + aX+bin Z[X]\) is given. The discriminant \(d_K\) of \(K\) and an integral basis of \(K\) are then obtained from its \(p\)-integral bases.

## February 16th 2009, 13:15 - 14:15 in 734

**Title:** Computing Hilbert class polynomials

**Speaker:** Øystein Thuen

**Abstract:** An elliptic curve over a field k is an algebraic curve of genus 1. To a curve E, we assign a value j(E) in k, known as the j-invariant. This value uniquely determines the isomorphism class of an elliptic curve. We look at an elliptic curve E over the complex numbers. Since E is an additive group, its endomorphism ring End(E) always contains the integers. If End(E) is strictly larger than the integers, we say that E has complex multiplication. In this case the j-invariant is an algebraic integer, and its minimal monic polynomial is called a Hilbert class polynomial. We will look at methods of computing Hilbert class polynomials and why they are of interest in cryptography.

## February 9th 2009, 13:15 - 14:15 in 734

**Title:** Degenerations of modules over the quantum plane

**Speaker:** Sverre O. Smalø

## February 2nd 2009, 13:15 - 14:15 in room 734

**Title:** Number of elements in the mutation class of a quiver of type D_n

**Speaker:** Hermund Andrè Torkildsen

**Abstract:** Quiver mutation is an important ingredient in the definition of cluster algebras. It is an operation on quivers, which induces an equivalence relation on the set of quivers. The mutation class M of a quiver Q consists of all quivers mutation equivalent to Q. It is known that if Q is a Dynkin quiver, then M is finite. We will use a geometric realization of the cluster category of type D_n to give an explicit formula for the number of quivers in the mutation class of a quiver of type D_n.

## January 26th 2009, 13:15 - 14:15 in room 734

**Title:** Vanishing of homology over complete intersection rings

**Speaker:** Petter Andreas Bergh

**Abstract:** When does the homology of two modules over a commutative Noetherian local ring vanish? This has been one of the major topics in commutative ring theory since 1961, when Auslander studied torsion properties over regular local rings. In this talk, we give a couple of results on homology vanishing patterns for modules over complete intersection rings, that is, certain quotients of regular local rings. The talk is a report on recent joint work with Dave Jorgensen.

## January 15th 2009, 13:00 - 14:00 in room 656

**Title:** Vanishing of Ext_R^*(M,M) and Gorenstein presentations

**Speaker:** David Jorgensen

**Abstract:** The vanishing of Ext_R^i(M,M) for all i>0 happens quite easily for finitely generated R-modules M of either finite projective or finite injective dimension over R. But what if M has neither finite projective nor finite injective dimension over R? In this talk we give a characterization of such modules over a Cohen-Macaulay local ring satisfying Ext_R^i(M,M)=0 for all i>0, in terms of presentations of R by Gorenstein rings. The characterization is in some sense a generalization of a result of I. Reiten. This involves joint work with G. Leuschke and S. Sather-Wagstaff.