Presentationsindico.ictp.it/event/a05188/material/3/0.pdf · block algebras of finite groups and closely connect with Broué’s abelian defect conjecture. For general finite dimensional

SMR1735

Advanced School and Conference on Representation Theory and Related Topics

(9 - 27 January 2006)

Presentations by

Participants

Tate Cohomology for Triangulated Categories

Javad Asadollahi

Motivated by the classical structure of Tate cohomology, we develop and study a

Tate cohomology theory in a triangulated category C. Let E be a proper class of

triangles. By using E-projective, as well as E-injective objects, we give two alter-

native approaches to this theory that, in general, are not equivalent. So, we study

triangulated categories in which these two theories are equivalent. This leads us

to study the categories in which all objects have finite E-Gprojective as well as

finite E-Ginjective dimension. These categories will be called E-Gorenstein trian-

gulated categories. We give a characterization of these categories in terms of the

finiteness of two invariants: E-silpC, the supremum of the E-injective dimension of

E-projective objects of C and E-spliC, the supremum of the E-projective dimension

of E-injective objects of C, where finiteness of each of these invariants for a cate-

gory implies the finiteness of the other. The talk is based on a joint work with Sh.

Salarian.

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GEOMETRY OF REGULAR MODULES OVERCANONICAL ALGEBRAS

GRZEGORZ BOBIŃSKI

Canonical algebras introduced by Ringel play an important role inthe representation theory of finite dimensional algebras. The modulevarieties over canonical algebras are also of interest – some results inthis direction were obtained for example by Barot and Schroer [1]. Aspecial role in the module categories over canonical algebra is playedby so-called regular modules. The closure of the set of regular modules(if non-empty) form an irreducible component of a module variety. Itwas not clear when a given module variety coincides with this closure,although this problem is connected with questions concerning rings ofsemi-invariants studied by Domokos and Lenzing [3,4] and Skowrońskiand Weyman [5]. The aim of the talk is to give a classification of thecanonical algebras such that the above mentioned equality holds forall dimension vectors of regular modules. In particular, it occurs thatirreducibility is equivalent to normality in the considered cases. Finally,main tools used in the proof are planned to be presented.

References

[1] M. Barot and J. Schroer, Module varieties over canonical algebras, J. Algebra246 (2001), no. 1, 175–192.

[2] G. Bobiński, Geometry of regular modules over canonical algebras, Tran. Amer.Math. Soc., to appear.

[3] M. Domokos and H. Lenzing, Invariant theory of canonical algebras, J. Algebra228 (2000), no. 2, 738–762.

[4] , Moduli spaces for representations of concealed-canonical algebras, J.Algebra 251 (2002), no. 1, 371–394.

[5] A. Skowroński and J. Weyman, Semi-invariants of canonical algebras,Manuscripta Math. 100 (1999), no. 3, 391–403.

The Gabriel-Roiter measure for representationdirected algebra

Bo Chen, Bielefeld

January 17, 2006

Abstract: Fix a basic representation directed algebra Λ over an alge-braically closed field k.

For each indecomposable Λ module M which is not simple, by using theGabriel-Roiter measure, we get a short exact sequence 0 → T → M →M/T → 0 with T and M/T indecomposable. In fact we choose a Gabriel-Roiter submodule T of M . We want to present the properties of M/T , theGabriel-Roiter factor module and the Hom space Hom(T, M/T ). We willalso talk the motivations to these questions and some connection with otherproblems.

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Erik Darpo, Uppsala

Restricted quivers in the classification theory of real divi-sion algebras

The category of finite-dimensional real division algebras (not necessarily asso-ciative) is denoted by D. Several important subcategories of D have been shownto be equivalent to representation categories of restricted quivers (Q, R) — thatis to certain subcategories of repR(Q) — for different quivers Q and sets ofrestrictions R.

We give some examples, and treat a subcategory of repR(Q), Q being thesingle loop quiver, from which a classification of all finite-dimensional real flex-ible quadratic division algebras is obtained.

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Hall algebras for incidence algebras of posets

Justyna Kosakowska

Let I = (I,�) be a finite poset (partially ordered set) and let K bea finite field. We denote by KI the incidence K-algebra of the poset I and bymod(KI) the category of finite dimensional right KI-modules. Consider thefull subcategory prin(KI) of the category mod(KI) consisting of prinjectiveKI-modules in the sense of the following definition. A KI-module X is saidto be prinjective if there exists a short exact sequence

0 → P1 → P0 → X → 0,

where P0, P1 are projective KI-modules and P1 is, in addition, semisimple.We call the poset I of finite prinjective type if there exist only finitely manyisomorphism classes of indecomposable prinjective KI-modules.

The main aim of our talk is to present results concerning the existence ofHall polynomial for prinjective KI-modules in the case I is of finite prinjec-tive type.

Moreover, in this case, we describe the associated Hall algebra by gener-ators and relations.

Dirk Kussin (Paderborn)

The structure of one-parameter families

By Drozd’s Tame-/Wild-Theorem the one-parameter families for a tame al-gebra over an algebraically closed field are rational, that is, are up to finitelymany points given by the projective line. Over an arbitrary field there isstill no extension of Drozd’s theorem and the general structure of the one-parameter families for tame algebras is still unknown. It is widely acceptedthat such families will be those induced by the one-parameter families asso-ciated with tame bimodules.

We present a structure theorem for the one-parameter families associatedwith a tame bimodule: Each such family can be naturally identified withthe projective prime spectrum of a (not necessarily commutative) gradedfactorial domain. As an application commutativity of the endomorphismring of the generic module (the “function field”) is characterized in terms ofthe so-called multiplicities.

ON DERIVED EQUIVALENCES OF CATEGORIES OF SHEAVESOVER FINITE POSETS

SEFI LADKANI

Throughout this note, the term poset will mean a finite partially ordered set.Given a poset (X,≤), one can define on X a structure of a T0 topological space bysaying that a subset Y ⊆ X is closed if whenever y ∈ Y and y′ ≤ y we have y′ ∈ Y .In this way, posets are identified with finite T0 topological spaces. Such spaces havebeen studied in the past ([2], [5]) and have been shown to be quite general in termsof their homotopy and homology properties.

The Hasse diagram GX of X is a directed graph whose set of vertices is X,with edges x → y whenever x < y and there is no z ∈ X with x < z < y. Givenan abelian category A, the category of sheaves on X (considered as a topologicalspace) with values in A will be denoted by AX . It can be shown that the categoryAX is equivalent to the category of (commutative) diagrams over A with shape GX

([3], [4]), with the global sections functor corresponding to the limit functor.Fix a field k and let A be the category of finite dimensional vector spaces over

k. In this case AX is equivalent to the category of finite dimensional right modulesover the incidence algebra of X, which is a finite dimensional algebra over k. Onecan also think of AX as the category of representations of the quiver GX with allthe commutativity relations.

Given a category AX with A as before, the graph GX , hence X, can be recoveredup to isomorphism by reconstructing the vertices as the simple objects of AX , andthe edges by considering Ext1 between the simple objects.

However, the situation is different for Db(X), the bounded derived category ofAX , as there are equivalences as triangulated categories Db(X) � Db(Y ) (derivedequivalences) for non-isomorphic posets X, Y . Two well-known examples are theBGP reflection functors [1] and the equivalence between the derived categories ofdiagrams over

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This leads to the natural question:

• When Db(X) � Db(Y ) for two posets X, Y ?

We present a general construction that gives many derived equivalences betweenposets, and generalizes the above examples.

Let S be a poset, and let X = {Xs}s∈S be a collection of posets indexed by theelements of S. The lexicographic sum of the Xs along S, denoted ⊕SX, is the poset(X,≤) where X =

∐s∈S Xs is the disjoint union of the Xs and for x ∈ Xs, y ∈ Xt

we have x ≤ y if either s < t (in S) or s = t and x ≤ y (in Xs).1

2 SEFI LADKANI

A poset S is called a bipartite graph if its Hasse diagram is a bipartite graph.This is equivalent to the condition that one can write S = S0

∐S1 as a disjoint

union with S0, S1 discrete (anti-chains) such that s < s′ in S implies s ∈ S0, s′ ∈ S1.Given a poset S, we denote by Sop the opposite poset, with Sop = S and s ≤ s′

in Sop if and only if s ≥ s′ in S.

Theorem. If S is a bipartite graph and X = {Xs}s∈S is a collection of posets, thenDb(⊕SX) � Db(⊕SopX) as triangulated categories.

In particular, we get that Db(X ⊕ Y ) � Db(Y ⊕X) where ⊕ is the ordinal sum,without any restriction on the posets X, Y .

The construction is based on ideas from sheaf theory. We note that the techniqueused in the proof can also give derived equivalences between categories of sheavesof posets and representations of quivers with additional relations.

By considering the Euler bilinear form over the Grothendieck group of Db(X),questions on derived equivalence can be transformed to questions on congruency ofintegral matrices.

If time permits, the question when Db(X1 ⊕ X2 ⊕ X3) � Db(X2 ⊕ X1 ⊕ X3) forthree posets will be addressed. While there are examples showing that this is nottrue in general, one can still have such derived equivalences for certain classes ofposets.

References

[1] Bernstein I.N., Gelfand I.M., Ponomarev V.A. Coxeter functors, and Gabriel’s theorem.

Uspehi Mat. Nauk 28 (1973), no.2 (170), 19–33.[2] McCord, M.C. Singular homology groups and homotopy groups of finite topological spaces.

Duke Math. J. 33 (1966), 465–474.

[3] Mitchell, B. On the dimension of objects and categories. II. Finite ordered sets. J. Algebra9 (1968), 341–368.

[4] Spears, W.T. Global dimension in categories of diagrams. J. Algebra 22 (1972), 219–222.[5] Stong, R.E. Finite topological spaces. Trans. Amer. Math. Soc. 123 (1966), 325–340.

Stable equivalence of Morita typeYuming Liu

Beijing Normal UniversitySchool of Mathematical Sciences

Beijing, China

It was observed by Rickard that a derived equivalence between two self-injectivealgebras implies a special kind of stable equivalence, which is called a stable equivalenceof Morita type by Broué. Stable equivalences of Morita type occur frequently betweenblock algebras of finite groups and closely connect with Broué’s abelian defectconjecture. For general finite dimensional algebras, this notion is still of particularinterest. In this talk, we shall introduce a construction of stable equivalences of Moritatype for finite dimensional algebras. This is a joint work with Changchang Xi.

Braid action on derived category ofNakayama algebras

Intan Muchtadi-Alamsyah

Abstract

We construct an action of a braid group associated to a complete graphon the derived category of a certain symmetric Nakayama algebra whichis also a Brauer star algebra with no exceptional vertex. We connect thisaction with the affine braid group action on Brauer star algebras definedby Schaps and Zakay-Illouz. We show that for Brauer star algebras withno exceptional vertex, the action is faithful.

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On degenerations between preprojectivemodules over wild quivers

Roland Markus Olbricht

January 17, 2006

We study minimal degenerations between preprojective modules over thealgebras of wild quivers with no relations. Asymptotic properties of suchdegenerations are studied, with respect to codimension and numbers of inde-composable direct summands. We state the existence of families of minimaldisjoint degenerations of arbitrary codimension for almost all wild quivers.It can be proven that no such examples exist for the remaining quivers.

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Towards Hecke Algebras for GLn(Zp)

Amritanshu Prasad

January 2006

For any finite field k, the Bruhat decomposition associates to each doublecoset of GLn(k) with respect to the subgroup of upper triangular matrices,a unique permutation matrix. Therefore the resulting convolution algebras(which are Iwahori-Hecke algebras) can be studied using the combinatoricsof symmetric groups.

We will discuss what aspects of this construction can be generalized togroups such as GLn(Z/pkZ), or to GLn(Zp). In particular, we will show howto associate to each such matrix a family of doubly stochastic matrices, whichremains invariant when the given matrix is multiplied on the left or the rightby an upper triangular matrix. We will discuss some simple properties ofthese invariants.

Some of this is joint work with Uri Onn and Leonid Vaserstein.

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