# Seminars in Algebra

## Fall 2008

## November 20th 2008, 13:00 - 14:00 in room 656

**Title:** Cluster structures from 2-CY categories with loops

**Abstract:** Inspired by the cluster-type behaviour of maximal rigid objects in certain 2-Calabi-Yau categories, Buan, Iyama, Reiten and Scott provided an axiomatic framework for this phenomenon. However, for these axioms to make sense, one must require that the endomorphism rings of maximal rigid objects do not have loops or 2-cycles. In this talk we will see how one can relax the requirements so the definition also incorporates objects with loops in the quivers of the endomorphism rings. The main example will be maximal rigid objects in the cluster category of a tube. This is joint work with Aslak Buan and Robert Marsh.

## November 13th 2008, 13:00 - 14:00 in room 656

**Title:** Representation dimension of quasi-tilted algebras

**Speaker:** Steffen Oppermann

**Abstract:** I will show that the representation dimension of a quasi-tilted algebra is always at most three.

## November 6th 2008, 13:00 - 14:00 in room 656

**Title:** Semiinvariants on presentation spaces

**Speaker:** Gordana Todorov

## October 30th 2008, 13:00 - 14:00 in room 656

**Title:** Coordinate systems on the Teichmüller and decorated Teichmüller spaces

**Speaker:** Pierre Will

**Abstract:** I will give an introduction to Teichmüller space and decorated Teichmüller space of a Riemann surface \(Sigma\). I will present Fock and Goncharov coordinates on the Teichmüller space, which can be thought of as a toy model of their higher Teichmüller spaces, and try to give some indications about their coordinates in the general case. If I have enough time, I will present Penner's coordinates on the decorated Teichmüller space and try to show the connection with cluster algebras.

## October 23rd 2008, 13:00 - 14:00 in room 656

**Title:** Partial orders on isomorphism classes of modules

**Speaker:** Nils M. Nornes

**Abstract:** We will look at some partial orders on the set of isomorphism classes of d-dimensional A-modules, where A is a finitely generated algebra over a field k. In particular we will look at the relation <_n (n a natural number), defined in the following way: For a matrix B in M_n(A) and a module M, let B_M be the k-endomorphism on M^n induced by B. Then we write M <_n N if dim coker B_M =< dim coker B_N for all B in M_n(A). This relation is not in general a partial order, but when n is large enough it is. We will discuss how large n must be, particularly for the case A=kQ, Q a Dynkin quiver.

## October 9th 2008, 13:00 - 14:00 in room 656

**Title:** The 4 Subspace Problem

**Speaker:** Tore Alexander Forbregd

**Abstract:** We will present a complete solution to the 4 subspace problem in the generality of an algebraically closed field. We do this by means of Auslander-Reiten theory and give the Auslander-Reiten quiver of the extended D4 Dynkin diagram. We also give a geometric interpretation when two configurations of four lines through the origin in the real plane are equivalent.

## October 2nd 2008, 13:00 - 14:00 in room 656

**Title:** Ext-orthogonal pairs for hereditary rings

**Speaker:** Jan Stovicek

## September 23rd 2008, 14:15 - 15:15 in room 656

**Title:** Tilting and cluster tilting for quotient singularities

**Speaker:** Osamu Iyama

**Abstract:** This is a joint work with Ryo Takahashi. Let \(k\) be a field of characteristic zero and \(S=k[x_1,cdots,x_d]\) the polynomial ring of \(d\) variables. For a finite subgroup \(G\) of \({ m SL}_d(k)\) acting on \(k^dackslash{0}\) freely, it is known that the completion \(widehat{R}\) of the quotient singularity \(R:=S^G\) has a \((d-1)\)-cluster tilting object in the category of maximal Cohen-Macaulay \(widehat{R}\)-modules. Using this, we will construct a tilting object in the category of graded maximal Cohen-Macaulay \(R\)-modules. Consequently its stable category is triangle equivalent to the derived category of modules over a finite dimensional \(k\)-algebra.

## September 18th 2008, 14:00 - 15:00 in room 656

**Title:** Cluster categories for algebras of global dimension 2

**Speaker:** Claire Amiot

**Abstract:** Buan, Marsh, Reineke, Reiten and Todorov have associated to an acyclic quiver Q a category C_Q, called the cluster category, in order to understand combinatorial aspects of certain cluster algebras. Other categories coming from representation of preprojective algebras also categorify some cluster algebras. In this talk, I will explain how it is possible to generalize the notion of cluster category, replacing the path algebra of an acyclic quiver with a finite dimensional algebra of global dimension 2. Then I will explain the link between these new categories and the categories coming from preprojective algebras.

## September 4th 2008, 13:00 - 14:00 in room 656

**Title:** Dual Garside structures for finite-type Artin groups via quiver representations

**Speaker:** Hugh Thomas

**Abstract:** The Artin groups of type A are the braid groups; for any Coxeter group, there is an associated Artin group, which is called finite-type if the Coxeter group is finite. There is a standard presentation for Artin group analogous to the standard presentation for Coxeter groups. A useful property of the standard presentation for Artin groups of finite type is that there is an associated Garside structure. This gives, for example, an algorithm for computing a normal form for elements of the Artin group. Bessis introduced a dual presentation for finite-type Artin groups which also has this Garside property, and which is, in some respects, computationally preferable. The proofs of Bessis's results make use of type-by-type arguments and computer checks for the exceptional types. I will explain an alternative approach to Bessis's presentation (in crystallographic cases only) using the representation theory of Dynkin quivers in which the proofs are carried out in a uniform way. I will also discuss conjectural applications to non-finite-type Artin groups. No knowledge of Artin groups or Garside structures will be assumed.

## August 28th 2008, 13:00 - 14:00 in room 656

**Title:** Coloured quiver mutation for m-cluster categories

**Speaker:** Hugh Thomas

**Abstract:** In a cluster category, if T is a basic cluster tilting object, then the quiver for End(T) is the crucial combinatorial datum associated to T. One important property of this quiver is that if one knows the quiver for End(T), then one can determine the quiver for the endomorphism ring of a mutation of T, by applying Fomin-Zelevinsky mutation. Simple examples show that when we move to m-cluster categoriesfor m>1 (that is to say, orbit categories of the form C_m=D^b(H)/[m]\tau^{-1}), the endomorphism ring of an m-cluster-tilting object does not have this property, because it does not contain enough information about the m-cluster-tilting object. We show how to define a quiver associated to an m-cluster-tilting object T in C_m whose edges are coloured with one of m+1 colours, which includes the quiver of End(T) as the 0-coloured arrows, and which has the property that, given the coloured quiver associated to T, one can determine the coloured quiver associated to any mutation of T, via a mutation procedure for coloured quivers which generalizes Fomin-Zelevinsky mutation. This is joint work with Aslak Bakke Buan.

## Spring 2008

## May 15th 2008, 10:15 - 11:15 in Room 656

**Title:** Jet bundles on grassmannians and flag varieties

**Speaker:** Helge Maakestad

**Abstract:** Let V be a complex finite dimensional vector space and let G=SL(V) be the special linear group on V. Let F be a flag in V and let P be the subgroup of G fixing the flag F. The quotient G/P is a smooth and projective complex variety parametrizing flags of type F in V. There is an equivalence of categories between the category of locally free finite rank sheaves on G/B with an SL(V)-linearization and the category of rational finite dimensional complex representations of P. The jet bundle J(E) of a locally free sheaf E with an SL(V)-linearization has a canonical SL(V)-linearization and I will discuss its corresponding representation of p=Lie(P) and p_ss=Lie(P_ss) where P_ss is the semi-simplification of the parabolic subgroup P. I will classify the representation as p_ss-module using well known formulas from the representation theory of semi simple Lie algebras.

## May 8th 2008, 10:15 - 11:15 in Room 656

**Title:** On cohomology operations and characteristic classes

**Speaker:** Helge Maakestad

**Abstract:** In this talk I will use tensor operations on exact categories to define cohomology operations and characteristic classes with values in Grothendieck groups of fields, schemes and topological spaces. I will also prove elementary properties of the characteristic classes. I will calculate some examples and show that the characteristic classes of connections with values in Grothendieck groups of fields refine characteristic classes of connections with values in Chevalley-Hochschild cohomology. I will furthermore construct a ramification class in the Grothendieck group of an infinite dimensional Lie algebra associated to an arbitrary finite morphism of projective curves. In special cases we get a characteristic class with values in the Grothendieck group of left modules on an Ore extension of the function field of the base curve.

## April 24th 2008, 10:15 - 11:15 in Room 656

**Title:** Relative support varieties

**Speaker:** Petter Andreas Bergh

**Abstract:** We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.

## April 17th 2008, 10:15 - 11:15 in Room 656

**Title:** Two constructions for representations of Lie algebras

**Speaker:** Alexei Rudakov

**Abstract:** I plan to discuss two constructions of representations that are essentially classical and known to specialists. One is the realization of a representation as a subspace or factor space of algebraic functions on the orbit for the Lie group. The other is making the so called generalized Verma module. Some applications of these constructions in modern research shall be mentioned.

## April 10th 2008, 10:15 - 11:15 in Room 656

**Title:** Injective modules relative to a torsion theory

**Speaker:** Stelios Charalambides

**Abstract:** In this talk I will investigate various types of injectivity relative to a torsion theory. In particular I will introduce \(\tau\)-M-injective and s-\(\tau\)-M-injective modules and look at how they are related with the known concepts of \(\tau\)-injective and \(\tau\)-quasi-injective modules which are torsion theoretic versions of injective and quasi-injective modules. I will also present a torsion theoretic version of Fuch's Criterion which is similar to Baer's Criterion and is used to characterize quasi-injective modules. Time permitting I will talk about the existence and uniqueness up to isomorphism of the \(tau\)-M-injective hull and a generalized version of Azumaya's Lemma characterizing \(tau\)-direct sum M_i-injective modules.

## April 3rd 2008, 10:15 - 11:15 in Room 656

**Title:** Relative homology and maximal n-orthogonal modules

**Speaker:** Magdalini Lada

## March 27th 2008, 10:15 - 11:15 in Room 656

**Title:** Cohomology of twisted tensor products

**Speaker:** Petter Andreas Bergh

**Abstract:** This is joint work with Steffen. It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. As an application, we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the Hochschild cohomology ring. Moreover, both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra.

## March 13th 2008, 10:15 - 11:15 in Room 656

**Title:** Tilting abelian categories

**Speaker:** Jan Stovicek

**Abstract:** I will show a general argument for proving derived equivalence of abelian categories through calculus of fractions and relative derived categories. This covers various versions of tilting modules and objects from literature. Needless to say that a big part of the ideas is available in literature, but they are spread over several not very closely related papers.

## March 6th 2008, 10:15 - 11:15 in Room 656

**Title:** t-structures on the bounded derived category of a commutative Noetherian ring

**Speaker:** Manolo Saorín

## February 28th 2008, 10:15 - 11:15 in Rom 656

**Title:** n-APR tilting and n-cluster tilting objects

**Speaker:** Steffen Oppermann

**Abstract:** By a result of Iyama the module category of the Auslander algebra of linear oriented A_n contains a 2-cluster tilting object. We will see that there is a certain tilting, which we call 2-APR tilting, such that the endomorphism algebra of the tilting object also has a 2-cluster tilting object in its module category. This construction can be iterated in order to get more algebras of global dimension 2 with 2-cluster tilting objects. This is joint work in progress with Osamu Iyama.

## February 21st 2008, 10:15 - 11:15 in Rom 656

**Title:** Counting cluster-tilted algebras of finite representation type

**Speaker:** Hermund Andrè Torkildsen

**Abstract:** I will give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type \(A_n\), by counting the mutation class of any quiver with underlying graph \(A_n\). It will also follow that if \(T\) and \(T'\) are cluster-tilting objects in a cluster category \(\mathcal{C}\), then \(End_{\mathcal{C}}(T)\) is isomorphic to \(End_{\mathcal{C}}(T')\) if and only if \(T=\tau^i T'\). I will also look at the D_n case.

## February 15th 2008, 10:15-11:15 in Rom 656

**Title:** Accessible algebras and spectral analysis

**Speaker:** José Antonio de la Peña

## February 14th 2008, 10:15-11:15 in Rom 656

**Title:** Remarks on the representation dimension

**Speaker:** Osamu Iyama

## February 7th 2008, 10:15 - 11:15 in Rom 656

**Title:** On finite generation for Hochschild cohomology of some Koszul algebras

**Speaker:** Karin Erdmann

**Abstract:** (Work in progress, joint with O. Solberg). Let \(A\) be a finite-dimensional self-injective algebra. The basic problem is to determine whether or not \(HH^*(A)\) satisfies suitable finiteness conditions which ensure that \(A\)-modules have support varieties, with good properties. Motivated by some tame Hecke algebras, this has led us to a criterion which applies to finite-dimensional Koszul algebras. This is based on work of [Buchweitz-Green-Solberg-Snashall].

## January 31st 2008, 10:15 - 11:15 in Rom 656

**Title:** Denominators of cluster variables

**Speaker:** Aslak Buan

**Abstract:** There is a 1-1 correspondence between cluster variables for an acyclic cluster algebra and indecomposable exceptional objects in the corresponding cluster category. The cluster variables are certain Laurent polynomials, and our aim is to describe the denominators of the cluster variables in terms of the cluster category. One obtains particularly nice results in the case of Dynkin or Extended Dynkin type. This is based on joint work with Marsh and Reiten.

## January 24th 2008, 10:15 - 11:15 in Rom 656

**Title:** Tilting theory and finitistic dimension

**Speaker:** Soud Khalifa Mohammed

**Abstract:** We give a sufficient condition for the category of modules of finite projective dimension over an artin algebra to be contravariantly finite in the category of all finitely generated modules over the artin algebra. This is a sufficient condition for the finitistic dimension of the artin algebra to be finite [Ausander-Reiten].

## January 17th 2008, 10:15 - 11:15 in Room 656

**Title:** Hochschild homology and finite global dimension

**Speaker:** Dag Madsen

**Abstract:** Does finite Hochschild homology imply finite global dimension? Let A be a finite dimensional algebra. For A commutative it is known that A has finite global dimension if and only if the Hochschild cohomology groups HH^n(A) vanish when n is large enough. For noncommutative algebras this does not hold, there are examples where A has infinite global dimension, but HH^n(A) is non-zero only for n=0,1,2. For A commutative the corresponding result also holds in Hochschild homology. But the statement known as Han's conjecture is still open for A non-commutative. Han's conjecture: Let A be a finite dimensional associative algebra. Then A has finite global dimension if and only if the Hochschild homology groups HH_n(A) vanish when n is large enough. In this talk I will report on some work done together with Petter Bergh on this problem.