# Seminars in Algebra

## Fall 2010

## December 7th 2010, 11:00-12:00 in 656

**Title:** A frieze pattern determinant

**Speaker:** Karin Baur

**Abstract:** Broline, Crow and Isaacs have computed the determinant of a matrix associated to a frieze pattern. In joint work with R. Marsh we generalise their result to the corresponding frieze pattern of cluster variables arising from a cluster algebra of type A. We give a representation-theoretic interpretation of this result in terms of configurations of indecomposable objects in the root category of type A.

## December 2nd 2010, 13:15-14:15 in 656

**Title:** Mutation of Auslander generators

**Speaker:** Magdalini Lada

**Abstract:** Let Λ be an artin algebra with representation dimension equal to three and M an Auslander generator of Λ. We show how, under certain assumptions, we can mutate M to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of M . We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.

## November 25th 2010, 13:15-14:15 in 656

**Title:** Consequences of fractional Calabi-Yau properties for derived categories

**Speaker:** Steffen Oppermann

## November 18th 2010, 13:15-14:15 in 656

**Title:** m-cluster tilting objects and m-Hom configurations

**Speaker:** Hugh Thomas

**Abstract:** Exceptional collections are a generalization of tilting objects, which were originally studied in algebraic geometry, and were introduced into the setting of hereditary algebras by Crawley-Boevey and Ringel in the early '90s. I will review this theory, including some more recent additions to it (culminating in work of Igusa-Schiffler), and then explain two special kinds of exceptional collections, m-cluster tilting objects and m-Hom configurations, and describe a bijection between them.

If I have enough time, I will say a little bit about the Dynkin case in particular. Here, the m-cluster tilting objects and the m-Hom configurations correspond to combinatorially defined objects called, respectively, m-clusters and m-noncrossing partitions. It was known that, for a given Q, there were the same (finite) number of m-clusters and m-noncrossing partitions. Our representation-theoretic bijection therefore solves the problem (which was open for m>1) of providing a bijection between m-clusters and m-noncrossing partitions. This is joint work with Aslak Bakke Buan and Idun Reiten, and is drawn from arXiv:1007.0928.

## November 11th 2010, 13:15-14:15 in 656

**Title:** A geometric description of $m$-cluster categories of type $\tilde{A}$

**Speaker:** Hermund Andrè Torkildsen

**Abstract:** A cluster algebra is a commutative algebra together with a set of generators, called cluster variables, which are grouped together in clusters of a given size. Last time Hugh Thomas discussed the Ptolemy cluster algebra, which is a cluster algebra of type A. This cluster algebra can be described using diagonals and triangulations of a regular polygon, where the cluster variables are in bijection with the diagonals, and the clusters are in bijection with the triangulations. Also, the cluster category of type A was constructed using this geometric model. We will briefly recall some of Hugh Thomas' talk last week. There are also some geometric descriptions of cluster categories of other types, for example for type D and \tilde{A}.

In this talk we will consider m-cluster categories. Baur and Marsh generalized the geometric descriptions of cluster categories of type A and D to m-cluster categories. In this talk we will describe the m-cluster category of type \tilde{A} geometrically, by using an annulus. We will also give a bijection between coloured quivers of type \tilde{A} and an equivalence class of (m+2)-angulations of the annulus.

## November 4th 2010, 13:15-14:15 in 656

**Title:** Cyclic polytopes and higher dimensional analogues of cluster categories

**Speaker:** Hugh Thomas

**Abstract:** The simplest cluster algebras, those of type A, can be understood both through a simple combinatorial model in terms of triangulations of a polygon, and also via the now-classical cluster category of Buan-Marsh-Reineke-Reiten-Todorov. I will discuss work with Steffen Oppermann, in which we replace the polygon by a 2d-dimensional cyclic polytope, and the classical cluster category by a certain 2d-Calabi Yau category which we construct in the context of d-representation finite algebras. Various aspects of the d=1 case generalize to the higher-dimensional setting. In particular, we show how mutations coincide in the geometric and representation-theoretic settings, and we give a geometrical interpretation to the exchange sequences which arise from the representation theory. This talk will be mainly based on arXiv:1001.5437.

## October 28th 2010, 13:15-14:15 in 656

**Title:** Higher Auslander-Reiten theory and precluster tilting categories

**Speaker:** Øyvind Solberg

**Abstract:** This is joint work with Osamu Iyama.

We will briefly review some of the theory of \(n\)-almost split sequences by Iyama, and see how one can arrive at a generalization of this theory. This involves a generalization of \(n\)-cluster tilting categories and connection to Gorenstein artin algebras \(\Lambda\) with \(domdim \Lambda \geq n+1 \geq id \Lambda\).

## October 21st 2010, 13:15-14:15 in 656

**Title:** The representation dimension of Hecke algebras and symmetric groups

**Speaker:** Petter Andreas Bergh

**Abstract:** [This is joint work with Karin Erdmann] In the recent years, there has been quite a lot of focus on finding lower bounds for the representation dimension of various algebras. However, there does not exist a method for computing a good upper bound. The best upper bound available so far was proved by Auslander: the representation dimension of a selfinjective algebra is at most its Loewy length. For some selfinjective algebras, this bound equals the representation dimension, but there also exist algebras for which the difference between this bound and the precise value is arbitrarily large. In this talk, we provide both an upper and a lower bound for the representation dimension of the Hecke algebra of a symmetric group. The proof also carries over to group algebra of certain symmetric groups.

## October 14th 2010, 13:15-14:15 in 656

**Title:** Degeneration in derived categories

**Speaker:** Nils M. Nornes

**Abstract:** Degeneration of modules comes from geometry, but (as seen in Marco's trial lecture) it can also be defined by the existence of certain exact sequences. Replacing sequences with triangles we get a similar relation in the derived category. We will show that this is also a partial order, and discuss some differences from the module case.

The talk is based on the following articles:

"Degenerations for derived categories", B. T. Jensen, X. Su, A. Zimmermann (2005); "Degeneration-like orders in triangulated categories", B. T. Jensen, X. Su, A. Zimmermann (2005).

## October 7th 2010, 13:15-14:15 in 656

**Title:** Connectedness of the tilting graph

**Speaker:** Yvonne Grimeland

## September 30th 2010, 14:30 - 15:30 in 656

**Title:** Classification of tilting modules

**Speaker:** Lidia Angeleri-Hügel

## September 30th 2010, 13:15-14:15 in 656

**Title:** Existence of superdecomposable pure injective modules over some nonpolynomial growth algebras

**Speaker:** Grzegorz Pastuszak

## September 23rd 2010, 13:15-14:15 in 656

**Title:** On selfinjective artin algebras with generalized standard Auslander-Reiten components

**Speaker:** Maciej Karpicz

## September 9th 2010, 13:15-14:15 in 656

**Title:** The gentle algebras derived equivalent to the cluster tilted algebras

**Speaker:** Grzegorz Bobinski

## September 2nd 2010, 13:15-14:15 in B333

**Title:** The almost split triangles for perfect complexes over gentle algebras

**Speaker:** Grzegorz Bobinski

## Spring 2010

## April 29th 2010, 14:15-15:15 in Rom 734

**Title:** Mutating loops and 2-cycles in 2-CY triangulated categories

**Speaker:** Marco Angel Bertani-Økland

**Abstract:** This is joint work with Steffen Oppermann. We derive an algorithm for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras of finite type, which are a class of algebras that satisfy the setup for our mutation algorithm.

## April 22nd 2010, 14:15-15:15 in Rom 734

**Title:** Formal Verification of Reductions in Cryptography

**Speaker:** George Petrides

**Abstract:** A major research direction in cryptography is the design of provably secure cryptographic protocols. Whereas this property gives some assurance about the security of the protocol, at the same time it often leads to lengthy and complicated security proofs that are error-prone and put potential proof-readers off. In this talk I will give a quick and informal introduction to the relevant cryptographic notions before introducing our proposed framework for automated computer verification of security proofs. This is joint work with K. Gjøsteen and A. Steine.

## April 15th 2010, 14:15-15:15 in Rom 734

**Title:** On F(Delta) for quasihereditary algebras

**Speaker:** Karin Erdmann

**Abstract:** (joint with D. Madsen and V. Miemietz) Let \(A\) be a quasi-hereditary algebra, and let \(\mathcal{F}(Delta)\) be the category of modules which have a filtration by standard modules. Then \(\mathcal{F}(Delta)\) is closed under direct summands, and it has relative Auslander-Reiten sequences. It is of finite type if and only if its relative Auslander-Reiten quiver has a finite component. Using this, we classify \(Delta\)-finiteness for families of algebras which include Schur algebras \(S(2,r)\).

## March 25th 2010, 15:30-16:30 in Rom 734

**Title:** The multiplicity problem and the isomorphism problem for modules over algebras

**Speaker:** Andrzej Mroz

**Abstract:** I will formulate two simple and natural problems: the multiplicity problem (M.P. - the problem of determining the multiplicities of indecomposables appearing in direct sum decomposition of a given module) and the isomorphism problem (I.P. - how to find out if two given modules are isomorphic). The aim of my talk is to discuss the connections between these two problems, the possible ways of eventual solutions and a nature of difficulties one has to face. I will recall more or less known classical examples of solutions of M.P. and I.P. and also some related theoretical results. I will also propose some theoretical method for solving M.P. which roughly speaking relies on encoding Auslander-Reiten sequences in terms of (infinite) matrices (this approach has surprising connections with some pathological properties of partial ring of infinite matrices). As a summary I will present briefly the solution of both problems for domestic canonical algebras (one of the simplest considered class of algebras).

## March 25th 2010, 14:15-15:15 in Rom 734

**Title:** Graded mutation for cluster categories coming from hereditary categories with a tilting object.

**Speaker:** Anette Wraalsen

**Abstract:** Let H be a hereditary category (without projectives) with a tilting object, such that H is not of finite type. We present a graded mutation rule for its cluster category C_H. To to this we assign to each summand of a cluster-tilting object the property of being either a sink or a source. We do this by extending the definition of this property given by Hübner for coh X for some weighted projective line X, and then lifting it to the cluster category. This is joint work with Marco Bertani-Økland and Steffen Oppermann.

## March 24th 2010, 11:15-12:15 in Rom 734

**Title:** Geometric degenerations of algebras

**Speaker:** Adam Hajduk

**Abstract:** TBA

## March 18th 2010, 15:30-16:30 in Rom 734

**Title:** 2-d Koszul algebras

**Speaker:** Edward L. Green

**Abstract:** Both Koszul and d-Koszul algebras have proven to be interesting classes of algebras and I briefly review the definitions of these algebras. Both types are quotients of path algebras via ideals generated by homogeneous elements of one fixed degree; 2 for Koszul algebras and d for d-Koszul algebras. In this talk I will discuss algebras whose relations can be generated in two degrees: 2 and d. I will describe results obtained jointly with Eduardo Marcos. In particular, we investigate conditions which imply that the Ext-algebra of such an algebra is finitely generated.

## March 18th 2010, 14:15-15:15 in Rom 734

**Title:** Stable categories of higher preprojective algebras

**Speaker:** Osamu Iyama

**Abstract:** This is a joint work with Steffen Oppermann. Let kQ be a Dynkin quiver. It is well-known that the preprojective algebra of Q is selfinjective and stably 2-Calabi-Yau. Moreover its stable category has cluster tilting objects (Geiss-Leclerc-Schroer), and moreover it is triangle equivalent to the generalized cluster category of the stable Auslander algebra of kQ (Amiot). We show that these results are special case of general phenomenon for higher preprojective algebras. A typical case is the following: The (n+1)-preprojective algebras of n-representation-finite algebras are selfinjective, and triangle equivalent to the generalized (n+1)-cluster categories of the stable n-Auslander algebras. Another typical case is: The stable categories of Cohen-Macaulay modules over cluster tilted algebras are triangle equivalent to the generalized 3-cluster categories. As a byproduct of the proof, we show that the stable Auslander algebras of Dynkin quivers with the same underlying graph are derived equivalent. This was independently shown by Ladkani. Also we give a classification of iterated-tilted 2-representation-finite algebras.

## March 11th 2010, 14:15-15:15 in Rom 734

**Title:** Cluster equivalence and graded derived equivalence

**Speaker:** Claire Amiot

**Abstract:** This is a joint work with Steffen Oppermann. We will call two algebras of global dimension at most 2 cluster equivalent if their generalized cluster categories are triangle equivalent. For example, two derived equivalent algebras are cluster equivalent. However, the converse is not true in general. In this talk, I will explain how to put Z-gradings on cluster equivalent algebras \(\Lambda_1\) and \(\Lambda_2\) to get a derived equivalence \(D^b(gr \Lambda_1) \simeq D^b(gr \Lambda_2)\). I will also explain how to detect if two cluster equivalent algebras are derivedequivalent. All results will be illustrated by examples.

## March 4th 2010, 14:15-15:15 in Rom 734

**Title:** Hochschild homology and split pairs

**Speaker:** Petter Andreas Bergh

**Abstract:** This is a report on joint work with Dag Madsen. We study the Hochschild homology of algebras related via so-called split pairs. This elementary concept is applied to various kinds of algebras: fibre products, trivial extensions, monomial algebras, graded-commutative algebras and algebras containing oriented cycles of a special form.

## February 25th 2010, 14:15-15:15 in Rom 734

**Title:** Artin Gorenstein algebras with finite dominant dimension

**Speaker:** Øyvind Solberg

**Abstract:** The n-cluster tilting modules correspond bijectively to the n-Auslander algebras up to natural equivalences. Recall that an artin algebra A is called an n-Auslander algebra if the dominant dimension is greater or equal to n+1 and the global dimension of A is less or equal to n+1. In particular, a 1-Auslander algebra is the same as a classical Auslander algebra, that is, an artin algebra A with domdim A >= 2 >= gldim A. A more general class of algebras is artin algebras A with domdim A >= 2 >= injdim_A A = injdim A_A. This class was considered by Auslander-S, and it was shown to correspond to socalled DTr-selfinjective artin algebras. In this talk we will discuss what happens when we replace 2 with n+1, as for going from Auslander algebras to n-Auslander algebras.

## February 18th 2010, 14:15-15:15 in Rom 734

**Title:** On irreducible components of module varieties

**Speaker:** Grzegorz Bobinski

**Abstract:** In the talk I will discuss properties of the irreducible components of the module varieties for the dimension vectors of the indecomposable modules over the tame quasi-tilted algebras. In particular, I will show they are regular in codimension one.

## February 11th 2010, 15:30-16:30 in Rom 734

**Title:** Dual semicanonical and generic bases for cluster algebras asociated to unipotent cells

**Speaker:** Christof Geiss

**Abstract:** In previous work we categorified the canonical cluster algebra structure on the coordinate ring of each unipotent cell by a 2-Calabi-Yau Frobenius subcategory of the modules over the corresponding preprojective algebra. An essential ingredient of this project is our cluster character which actually takes values in the coordinate ring. We show here, that the Fu-Keller cluster character can be used to provide a cluster expansion formula for our cluster character with respect to any cluster. An easy consequence is, that the twist automorphism of a unipotent cell is categorified in a strong sense by the inverse of the Auslander-Reiten translate. As an other application we show, in generalization of a conjecture of Dupont, that a properly defined generic basis of a quiver potential coming from a unipotent cell can be identified with the dual semicanonical basis.

## February 11th 2010, 14:15-15:15 in Rom 734

**Title:** Laurent expansions for twisted Plücker coordinates

**Speaker:** Jeanne Scott

**Abstract:** This talk will begin with a description of of special coordinate systems for the Grassmannian associated to Postnikov diagrams. This is followed by a brief discussion of cross-products and the Grassmannian's twist transform — conjecturally related to an AR-translation via the Geiss-Leclerc-Schröer categorification of the Grassmannian's cluster algebra structure. The talk concludes with a formula — involving perfect matchings in the bipartite dual of a Postnikov diagram — which tabulates Laurent expansions for twisted Plücker coordinates with respect to the coordinates associated to the Postnikov diagram.

## January 28th 2010, 14:15-15:15 in Rom 734

**Title:** Internet voting security and algebraic cryptography

**Speaker:** Kristian Gjøsteen

**Abstract:** With internet voting, a voter uses his pc to send his ballot to a ballot box. After the voting period is over, the ballot box is opened and the votes are counted. Compared to ordinary paper voting, internet voting is susceptible to many new types of fraud. One example: The pc might cheat. How does the voter know that the pc sent the correct ballot to the ballot box? One possible solution involves some surprisingly simple algebra. Given a finite cyclic group of prime order (satisfying certain cryptographic requirements), we show how ballots can be encrypted and how a check-sum of the vote can be computed from the ciphertext _without_ revealing the decryption. The voter can use the check-sum to verify that the ballot box received the correct ballot, which implies that the pc did not cheat.