# Seminars in Algebra

## Fall 2007

## December 14th 2007, 13:15-14:15 in Rom 656

**Title:** Generalized flag varieties and their Euler characteristics

**Speaker:** Jeanne Scott

## December 5th 2007, 13:15 - 14:15 in Rom 656

**Title:** The cluster category of a canonical algebra

**Speaker:** Dirk Kussin

**Abstract:** (Joint work with Michael Barot and Helmut Lenzing.) We study the cluster category of a canonical algebra. By a result of B. Keller it is a triangulated category. We focus on the description of its Grothendieck group and its group of (isoclasses of) exact automorphisms. Of particular interest is the tubular case.

## November 28th 2007, 13:15 - 14:15 in Rom 656

**Title:** Localizations of triangulated categories

**Speaker:** Jan Stovicek

**Abstract:** I will explain the idea of a smashing localization of a compactly generated triangulated category and the Telescope Conjecture in this setting. The two main examples of compactly generated triangulated categories related to representation theory are stable module categories of self-injective artin algebras and (unbounded) derived categories of rings. I will show that the Telescope conjecture, although generally disproved, holds for some simple but non-trivial cases, namely for stable module categories over standard domestic self-injective algebras and derived categories of (non-commutative) noetherian hereditary rings. This allows to demonstrate the concept of a smashing localization on some examples.

## November 21st 2007, 13:15 - 14:15 in Rom 656

**Title:** AR-sequences for modules with a Verma flag

**Speaker:** Dag Madsen

**Abstract:** The category of modules orthogonal to a given tilting module has Auslander-Reiten sequences. In this talk I will illustrate a method for computing these sequences using so(4)-modules with a Verma flag as an example.

## November 14th 2007, 13:15 - 14:15 in Rom 656

**Title:** Invariant subspaces of nilpotent operators

**Speaker:** Claus M. Ringel

## November 7th 2007, 15:00 - 16:00 in Rom 656

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

**Speaker:** Stelios Charalambides

## October 31st 2007, 13:15 - 14:15 in Rom 656

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

**Speaker:** Helge Maakestad

**Abstract:** I will discuss a construction of characteristic classes with values in grothendieck groups of connections using cohomology operations on the grothendieck group functor. Using tensor operations on the category of connections on a p-Lie algebra, I will construct a class of cohomology operations. I will define characteristic classes using gamma operations and prove basic properties of these classes.

## October 17th 2007, 13:15 - 14:15 in Rom 656

**Title:** Towards the semicanonical basis of cluster algebras over affine quivers

**Speaker:** Grégoire Dupont

**Abstract:** For an acyclic quiver Q, the Caldero-Chapoton map gives an explicit application from the cluster category to the cluster algebra. Using this map, Caldero and Keller obtained a Z-basis of the cluster algebra when Q is a quiver of finite representation type. We will propose a generalization of this basis in the case when Q is a quiver of affine type. If we have enough time, we will also compare the semicanonical basis with the already known Sherman-Zelevinsky's canonical basis and Caldero-Zelevinsky's basis for the cluster algebra over the Kronecker quiver.

## October 10th 2007, 13:15 - 14:15 in Rom 656

**Title:** Wild algebras have one-point extensions of representation dimension at least four

**Speaker:** Steffen Oppermann

**Abstract:** The representation dimension of a finite dimensional algebra was introduced by Auslander in order to measure how far an algebra is from having finite representation type. Therefore one may ask for a connection to tame- or wildness of the algebra. In this talk I will indicate how one can for any wild algebra find a one point extension of representation dimension at least four.

## October 3rd 2007, 13:15 - 14:15 in Rom 656

**Title:** Some partial orders on the space of representations of a fixed dimension over a finitely generated algebra

**Speaker:** Sverre O. Smalø

**Abstract:** For a finitely generated algebra A over an algebraically closed field k and a natural number d, there is the space of all algebra homomorphisms from A to M_g(K). This space is in a natural way an affine variety. On this space there are three, or maybe four partial orders appearing naturally. These orders have a natural generalization to the situation when one drops the requirement that k is algebraically closed. In this talk I will take a look at when these partial orders are the same, and indicate that that is rear.

## September 28th 2007, 09:00 - 10:00 in Rom 734

**Title:** Auslander algebras and Artin-Schelter regular algebras

**Speaker:** Øyvind Solberg

**Abstract:** To any artin algebra of finite representation type one associates the Auslander algebra. We first discuss how these algebras are related to Artin-Schelter regular algebras. It is known that an artin algebra \(\Lambda\) is of finite representation type if and only if all the objects in the category \(\mathcal{C}\) of finitely presented functors from \((mod\Lambda)^op\) to \(Ab\) have finite length. Moreover, every \(\Lambda\)-module is a direct sum of finitely generated indecomposable \(\Lambda\)-modules if and only if all objects in \(\mathcal{C}\) are noetherian. If time permits, we discuss similar noetherianity conditions on categories we associate to components of the Auslander-Reiten quiver of an artin algebra.

## September 19th 2007, 13:15 - 14:15 in Rom 656

**Title:** From quasitilted algebras to almost laura algebras : the history of a decade

**Speaker:** David Smith

**Abstract:** In 1996, Happel, Reiten and Smalo introduced the quasitilted algebras and characterised them by means of the so-called left and right parts of the module category. In this survey talk, I will discuss various generalisations of quasitilted algebras arising from the study of the left and the right parts, namely the shod (1999), weakly shod (2003), laura (2004) and almost laura algebras (2007).

## September 12th 2007, 13:15 - 14:15 in Rom 656

**Title:** Generic modules and multiplicities

**Speaker:** Dirk Kussin

## September 5th 2007, 13:15 - 14:15 in Rom 656

**Title:** Analysing a pseudo-random bit generator based on elliptic curve arithmetic

**Speaker:** Kristian Gjøsteen

**Abstract:** Generation of pseudo-random bits is an important topic in cryptography. Recently, a generator based on elliptic curve arithmetic was standardized in the US. The generator consists of two parts: (i) generating a pseudo-random sequence of points on an elliptic curve, and (ii) extracting a bit string from the point sequence. We show that while (i) is sound, (ii) is insufficient, and the complete generator is insecure. The talk will focus on concepts and techniques used in modern cryptography.

## August 29th 2007, 14:30 - 15:30 in Rom 656

**Title:** Inductive construction of algebras with n-cluster tilting objects

**Speaker:** Osamu Iyama

## Spring 2007

## June 27th 2007, 14:30-15:30 in Rom 656

**Title:** Triangular matrices and Koszul algebras

**Speaker:** Roberto Martínez-Villa

**Abstract:** Since there is no general algorithm when a given algebra is Koszul, it is of interest to construct new Koszul algebras from given ones, in the talk I will discuss the problem of when given two K-algebras R and T and a R-T bimodule M, the usual triangular matrix ring obtained is also Koszul, as an example, I will describe the bounded graded complexes Koszul of R-modules, when R is Koszul.

## June 13th 2007, 14:30-15:30 in Rom 656

**Title:** H. Cartan's model for equivariant cohomology vs the cubic Dirac operator

**Speaker:** Victor Kac

## June 6th 2007, 14:30-15:30 in Rom 656

**Title:** Localization in algebra and topology

**Speaker:** Halvard Fausk

**Abstract:** We review basic properties of localization functors on categories and consider examples from algebra, algebraic geometry and topology. We emphasize tensor triangulated categories. Here is the definition of the structure we consider. A localization on a category D consist of a functor L : D —> D and a natural transformation e : 1 —> L such that L e_X = e_LX : L X —> L^2 X, and this map is an isomorphism, for all X in D. By letting m : L^2 —> L be the inverse of Le we get that L, e, m is an (idempotent) monad on D. The category of algebras for this monad is equivalent to the full subcategory LD of D consisting of objects X such that e_X is an isomorphism. Hence a localization can always be described as the unit map 1 —> il of an adjunction l : D ←–> LD : i where the right adjoint i : LD —> D is an inclusion of a full subcategory and the left adjoint l : D —>LD is such that il = L. This alternative description of localization, in terms of a left adjoint functor of a full inclusion, is perhaps more familiar. A basic example of a localization is abelianization of discrete groups.

## May 16th 2007, 14:30-15:30 in Rom 656

**Title:** Mutations for quivers with potentials and their representations II

**Speaker:** Andrei Zelevinsky

## April 25th 2007, 14:45-15:45 in Rom 656

**Title:** Rime R-matrices

**Speaker:** Oleg Ogievetsky

## April 25th 2007, 13:30-14:30 in Rom 656

**Title:** Connections in characteristic p

**Speaker:** Helge Maakestad

**Abstract:** Let K be a field of characteristic p>0 and let be L a p-Lie algebra. Ill define and prove basic properties of some characteristic classes for modules on p-Lie algebras with values in Lie algebra cohomology and K-theory. Ill show using explicit examples that the invariants in K-theory refine the invariants in Lie algebra cohomology. Hence the Chern-character from K-theory to Lie algebra cohomology is seldom an isomorphism. Ill also show how a theorem of Cartier/Katz can be used to calculate some grothendieck groups of p-Lie algebras.

## April 18th 2007, 14:30-15:30 in Rom 656

**Title:** Monoid curves and surfaces

**Speaker:** Pål Hermunn Johansen

**Abstract:** A monoid hypersurface is an irreducible hypersurface of degree d which has a singular point of multiplicity d-1. Any monoid hypersurface admits a rational parameterization, and is hence of potential interest in computer aided geometric design. Properties of monoids in general and of monoid surfaces in particular will be discussed. The main results include a description of the possible real forms of the singularities on a monoid surface other than the point of multiplicity d-1. These results are applied to the classification of singularities on quartic monoid surfaces, complementing earlier work on the subject. Secondly (if time allows it), the classification of the quartic monoids is extended to the stratification of the space of quartic monoids. The dimension of each stratum is computed, and for the strata where the tangent cone of the monoid is not smooth, the number of components can be counted. Additionally, each component is rational and the parameterizations (of the components of the strata) are closely related to the method of classifying the monoids.

## March 28th 2007, 13:15-14:15 in Rom 656

**Title:** Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

**Speaker:** Xiuping Su

**Abstract:** Let g be a reductive Lie algebra over an algebraically closed field. A Lie subalgebra q of g is called a seaweed Lie algebra, if there exists a pair of parabolic subalgebras (p,p') such that pcap p'=q and p+p'=g. Let G be the adjoint group of a simple Lie algebra g, P and P' the parabolic subgroup G associated to p and p', respectively, and let Q=Pcap P'. There is a question, proposed independently by Duflo and Panyushev, on the existence of an open Q-orbit in the nilpotent radical of q. The elements with an open Q-orbit are called Richardson elements. In the case where q is just a parabolic, the answer is yes and the result is commonly called Richardson's Theorem. In this talk I will show the existence and explain how to construct Richardson elements for seaweed Lie algebras of type A, using representations of quivers.

## March 28th 2007, 14:45-15:45 in Rom 656

**Title:** Cluster tilting objects for one-dimensional hypersurface singularities

**Speaker:** Osamu Iyama

**Abstract:** This is a part of joint work with I. Burban, B. Keller and I. Reiten. Again we consider certain 2-Calabi-Yau triangulated categories having cluster tilting objects. Let R=kx,y/(f) (fin (x,y)) be a one-dimensional reduced hypersurface singularity over an algebraically closed field of characteristic zero. Then the stable category C:=ul{CM}R of maximal Cohen-Macaulay R-modules forms a 2-Calabi-Yau triangulated category. We show that, if f is a product f_1f_2…f_n of power series satisfying f_iin(x,y)ackslash (x,y)^2, then C has exactly (n!) cluster tilting objects and exactly (2^n-2) indecomposable rigid objects. By a result in birational geometry, the converse is also true, i.e. C contains a cluster tilting object if and only if f has the above form.

## March 21st 2007, 14:30-15:30 in Rom 656

**Title:** Finding lower bounds for Auslander's representation dimension

**Speaker:** Steffen Oppermann

**Abstract:** The representation dimension of a finite dimesional algebra has been introduced by Auslander in order to measure how far an algebra is from having finite representation type. In my talk I will first recall the definition and some properties of the representation dimesion. Then Rouquier's definition of dimesion of a triangulated category will be extended to module categories by regarding them as subcategories of their derived category. I will point out that this dimension is a lower bound for Auslander's representation dimension. Finally, I will present a criterion which provides lower bounds for the dimension of the module category, and therefore also for the representation dimension, and indicate how it can be applied.

## March 14th 2007, 14:30-15:30 in Rom 656

**Title:** A construction of 2-Calabi-Yau categories with cluster tilting objects

**Speaker:** Osamu Iyama

**Abstract:** This is a part of joint work with A. Buan, I. Reiten and J. Scott. Motivated by study of cluster algebra structure of coordinate rings of unipotent subgroups of Kac-Moody groups, we construct certain family of Frobenius/triangulated categories satisfying 2-Calabi-Yau property Hom(X,Y)simeq DExt^2(Y,X) and having cluster tilting objects. For preprojective algebras Lambda of non-Dynkin diagrams Delta, we construct a family of 2-sided ideals T_w parametrized by elements w in corresponding Coxeter group W of Delta. These T_w are tilting Lambda-modules of projective dimension at most one, and the factor algebra Lambda/T_w is Gorenstein (in the sense of Happel) of dimension at most one. The category CM(Lambda/T_w) of Cohen-Macaulay (Lambda/T_w)-modules is Frobenius 2-Calabi-Yau. For each reduced expression of w, we construct a cluster tilting object in CM(Lambda/T_w). Now it is a quite interesting question whether all cluster tilting objects in CM(Lambda/T_w) is transitive under successive mutations.

## March 7th 2007, 14:30-15:30 in Rom 1329

**Title:** Noetherian hereditary categories with Serre duality over perfect fields

**Abstract:** For an algebraically closed field \(k\), Reiten and Van den Bergh have classified all connected \(Ext\)-finite noetherian hereditary abelian \(k\)-linear categories satisfying Serre duality and containing non-zero projective objects. We will follow the lines of Reiten and Van den Bergh, but instead of working over an algebraically closed field, we will work over a perfect field. The aim is the classification of all such categories in terms of species.

## February 21st 2007, 14:30-15:30 in Rom 656

**Title:** Telescope conjecture for module categories

**Speaker:** Jan Stovicek

**Abstract:** The original Telescope Conjecture (or Generalized Smashing Conjecture) comes from algebraic topology and it is stated for compactly generated triangulated categories with the stable homotopy category in mind. But when the conjecture was investigated for stable module categories over self-injective artin algebras by Krause and Solberg, it was shown to be equivalent to stating that certain cotorsion pairs (with both classes closed under direct limits) in the usual module category are generated by finitely presented modules. While the finite type is still open, the main result here is to prove that such cotorsion pairs are always generated by countably presented modules, and that this holds not only for self-injective algebras, but for any associative ring. Also, several analogies between the settings of module categories and triangulated categories that work far beyond the (stable) modules categories over self-injective artin algebras, where the translation by Krause and Solberg works, will be shown. This is a joint work with Jan Saroch.

## February 14th 2007, 14:30-15:30 in Rom 656

**Title:** Stability of Quiver Representations

**Speaker:** Asgeir Steine

**Abstract:** A stability structure on the category Rep(k, Q) of representations of the quiver Q over the field k gives rise to a family of abelian subcategories (one for each level of the stability structure). The Jordan-Holder filtration in these subcategories, and what is called the Harder-Narisimhan filtration in Rep(k,Q) are tools that enables us to study the representations using the properties of our stability structure. The aim of this talk is to present these basic tools and to (show) that for any finite connected quiver with no oriented cycles there exist atleast one "non-trivial" stability structure.

## February 7th 2007, 15:00-16:00 in Rom 656

**Title:** Cluster-tilted algebras of type A_n

**Abstract:** Cluster-tilted algebras were introduced by Buan, Marsh and Reiten a few years ago and have received quite some attention since then. They are defined as endomorphism rings of (cluster-)tilting objects in cluster categories. Homologically, they are very different from hereditary and tilted algebras, since they have in general infinite global dimension. In this talk we will classify the cluster-tilted algebras arising from the cluster category of path algebras of type A_n, by describing the mutation-class of A_n. We will use this description, and tilting theory, to give a necessary and sufficient condition for when such algebras are derived equivalent. The work is joint with Aslak Bakke Buan.

## January 31st 2007, 14:30-15:30 in Rom 1329

**Title:** Piecewise hereditary skew group algebras

**Speaker:** David Smith

**Abstract:** The study of the representation theory of skew group algebras was started in the eighties with the works of de la Peña, and Reiten and Riedtmann. Given an algebra A and a group G acting on A, we define the skew group algebra A[G]. It turns out that A[G] often retains many features from A, such as being representation-finite, being hereditary, being tilted or quasitilted, etc. In this talk, we study the interplay between the skew group algebras and the so-called piecewise hereditary algebras, that is algebras A for which there exist a hereditary abelian category H and a triangle-equivalence between the derived categories of bounded complexes over A and H. Those algebras, first studied by Happel, Rickard and Schofield and later by Happel, Reiten and Smalø, played a decisive role in the classification of selfinjective algebras of finite and tame representation type. We show that, under some assumptions, the skew group algebra A[G] is piecewise hereditary when so is A. The talk is based on joint work in progress with Julie Dionne and Marcelo Lanzilotta.

## January 17th 2007, 14:30-15:30 in Rom 1329

**Title:** Extended Canonical Algebras I

**Speaker:** Helmut Lenzing

**Abstract:** The series of talks is on joint work with J.A. de la Pena and deals with the shape of the bounded derived category D(B) of an extended canonical algebra B, arising as the one-point extension of a canonical algebra A by an indecomposable projective or an indecomposable injective module. These categories D(B) occur in three different types, depending on the sign of the Euler characteristic of the canonical algebra, equivalently of the weighted projective line X associated to A. For positive Euler characteristic, where the canonical algebra is tame domestic, D(B) is triangle-equivalent to the derived category of a Dynkin quiver while for negative Euler characteristic, where the canonical algebra is wild, D(B) relates to the triangulated category of the graded surface singularity associated to X. The case of Euler characteristic zero, where the attached canonical algebra is tubular, has to be dealt with in a slightly different fashion. In all three cases, a spectral analysis of the extended canonical algebra yields additional interesting insight. The results are related to recent results by Orlov (2005), Ueda (2006) and Saito-Takahashi (unpublished); also there is a link to old unpublished work of Buchweitz (1987).

## January 10th 2007, 15:00-16:00 in Rom 734

**Title:** Noncrossing partitions via quiver representations

**Speaker:** Hugh Thomas

**Abstract:** We show how the combinatorics of clusters (viewed as tilting objects in the cluster category) can be related to the combinatorics of the noncrossing partitions of the associated Coxeter group. In order to do this, we show that the tilting objects in the cluster category for Q are in natural bijection with the exact abelian, finitely generated, and extension-closed subcategories of rep Q. As a biproduct of our approach, we show that any such subcategory can be obtained as the set of semistable objects with respect to some stability condition. When Q is of finite type, we recover the bijections between clusters, c-sortable elements, and noncrossing partitions studied by Reading. We extend the bijection between clusters and noncrossing partitions to affine type. This talk is based on the paper math.RT/0612219, which is joint work with Colin Ingalls (UNB).