


Representations of Algebras
Org: Ibrahim Assem, Thomas Brustle and Shiping Liu (Sherbrooke) [PDF]
 RAYMUNDO BAUTISTA, UNAM, Instituto de Matemáticas, Campus Morelia, Apartado
Postal 613 (Xangari), CP 58089 Morelia, Michoacan, México
Generic complexes and derived representation type for Artin
algebras
[PDF] 
Let A be an Artin algebra and D(A) its bounded derived category.
We recall that D(A) is called discrete and A derived discrete if
for any sequence h = (h_{i})_{i Î Z} of nonnegative
integers with almost all the h_{i}=0, there are only finitely many
isoclasses of indecomposable objects X Î D(A) with length of H^{i}(X) = h_{i} for all i Î Z.
We prove the following:
Theorem
The Artin algebra A is not derived discrete if and only if there is
a bounded complex of projective Amodules X = (X^{i},d_{X}^{i}) with the
following properties:
(i) for all i the image of d_{X}^{i} is in the radical of
X^{i+1};
(ii) X is indecomposable in the homotopy category of
complexes;
(iii) there is some j such that H^{j} (X) has not finite
length;
(iv) for all i, H^{i} (X) has finite length as left
Emodule, where E is the endomorphism ring of X in the
homotopy category of complexes.
A complex as before is called generic complex. In case A is a
finitedimensional algebra over an algebraically close field, we also
consider the tame representation type in terms of generic complexes.
 FRAUKE BLEHER, University of Iowa, Department of Mathematics, 14 MLH, Iowa
City, IA 522421419, USA
Universal deformation rings and dihedral defect groups
[PDF] 
Let k be an algebraically closed field of characteristic 2, and
let W be the ring of infinite Witt vectors over k. Suppose G is
a finite group, and B is a block of kG with dihedral defect
group D which is Morita equivalent to the principal 2modular block
of a finite simple group. We determine the universal deformation ring
R(G,V) for every kGmodule V which belongs to B and has stable
endomorphism ring k. It follows that R(G,V) is always isomorphic
to a subquotient ring of WD. Moreover, we obtain an infinite series
of examples of universal deformation rings which are not complete
intersections.
 WALTER BURGESS, Department of Mathematics and Statistics, University of
Ottawa, Ottawa, ON, K1N 6N5
On the quasistratified algebras of Liu and Paquette
[PDF] 
Quasistratified algebras are a generalization of standardly
stratified algebras for which both the Cartan Determinant Conjecture
and its converse hold. This talk will relate quasistratified
algebras to other classes of algebras: left serial, the Yamagata
algebras and gentle algebras.
Joint work with Ahmad Mojiri.
 XUEQING CHEN, University of WisconsinWhitewater
Root Vectors, PBW and Canonical Bases of RingelHall
Algebras and Quantum Groups
[PDF] 
Let g = g(C) be the KacMoody Lie algebra
associated to a Cartan matrix C and U = U_{v}(g) its quantum group. A key feature in quantum groups is
the presence of several natural bases (like the PBWbasis and the
canonical basis). There are different approaches to the construction
of the canonical basis: algebraic approach (Lusztig, Kashiwara,
BeckChariPressley, BeckNakajima), geometric approach (Lusztig)
and RingelHall algebra approach (Ringel, LinXiaoZhang). In this
talk, we will recall algebraic and RingelHall algebra approaches to
a PBW basis and a canonical basis of U when C is finite
or affine. Meanwhile, the root vectors in RingelHall algebras will
be discussed.
 FLAVIO ULHOA COELHO, University of São Paulo, Departamento de
MatemáticaIME, Rua do Matão, 1010, Cidade
Universitária/CEP, 05508090 São Paulo, Brazil
On the composition of irreducible morphisms
[PDF] 
Let A be an artin algebra. Using the socalled AuslanderReiten
theory, one can assign to A a quiver G_{A} called the
AuslanderReiten quiver of A which "represents" the
indecomposable finitely generated Amodules together with some
morphisms between them called irreducible. Unfortunately, G_{A}
does not give all the informations on the category mod A of the
finitely generated Amodules one could expect because not all
morphisms can be reconstructed from the irreducible ones. However,
(sum of) compositions of irreducible morphisms can give important
informations on mod A.
A morphism f : X ® Y is called irreducible
provided it does not split and whenever f = gh, then either h is a
split monomorphism or g is a split epimorphism. It is not difficult
to see that such an irreducible morphism f belongs to the radical
rad(X,Y) but not to its square rad^{2} (X,Y). Consider now a
nonzero composition g = f_{n} ¼f_{1} : X_{0} ® X_{n}
of n ³ 2 irreducible morphisms f_{i}¢s. It is not always true
that g Î rad^{n} (X_{0}, X_{n}) \rad^{n+1} (X_{0}, X_{n}). In
this talk, we shall discuss some results on the problem of when such a
composition does lie in rad^{n} (X_{0}, X_{n}) \rad^{n+1} (X_{0},X_{n}). The particular cases n=2,3 will be considered in more
details.
Joint work with C. Chaio and S. Trepode (Universidad de Mar del Plata).
 JOSÉ ANTONIO DE LA PEÑA, UNAM, Mexico
Coxeter transformations: from Lie algebras to singularity
theory
[PDF] 
Coxeter transformations play an important role in the theory of Lie
algebras. Namely, the Weyl group is finite (resp. affine, contains a free
subgroup) if the Coxeter elements are periodic (resp. have spectral radius
1, > 1). For a hereditary algebra A = k D associated to a quiver
D without oriented cycles, the Coxeter transformation is induced
from the AuslanderReiten equivalence of the derived category D^{b}(mod A)
to the Grothendieck group of A. The spectral properties of this
transformation are essential to understand the representation theory of
A. For canonical algebras A over the complex numbers, spectral
properties of the Coxeter transformations are related to the
classification of Fuchsian groups and their asociated singularities.
 EDWARD GREEN, Virginia Tech
Noetherianity and Ext
[PDF] 
The importance of the relationship between an algebra and its
Extalgebra is well established. On the other hand, little is known
about which properties of the algebra or its representations imply, or
are implied by the noetherianity of the Extalgebra. The main
thrust of this talk is the study of such properties. Particular
attention is given to the case when the algebra is Koszul. Some of
the results presented are given below.
We prove that if every module in gr(R) has a finitely generated
Extmodule, Å_{n ³ 0} Ext^{n}_{R} (R/J,R/J), where J is
the graded Jacobson radical of a standard graded algebra R, then R
is left noetherian.
We prove that if R is a Koszul algebra of finite global dimension,
then R being left noetherian is equivalent to every module M = Å_{i} M_{i} in grR has the property that for some n, the
module M_{n} ÅM_{n+1} Å¼ is linear.
 MARK KLEINER, Syracuse University, Department of Mathematics, Syracuse, NY
132441150, USA
Reduced words in the Weyl group of a KacMoody algebra and
preprojective representations of valued quivers
[PDF] 
We discuss the interplay between the preprojective representations of
a connected valued quiver, the (+)admissible sequences of vertices,
and the Weyl group. To each preprojective representation corresponds
a canonical (+)admissible sequence. A (+)admissible sequence is
the canonical sequence of some preprojective representation if and
only if the product of simple reflections associated to the vertices
of the sequence is a reduced word in the Weyl group. As a
consequence, for any Coxeter element of the Weyl group associated to
an indecomposable symmetrizable generalized Cartan matrix, the group
is infinite if and only if the powers of the element are reduced
words. The latter strengthens known results of Howlett and
FominZelevinsky.
The talk is based on joint work with Helene R. Tyler and with Allen
Pelley.
 MARCELO LANZILOTTA, Unviersidad de la República, Iguá 1445, CP 11400,
Montevideo, Uruguay
La technique de coupement à deux côtés pour la
conjecture finitistique / Double cut approach for the
finitistic conjecture
[PDF] 
There is a famous conjecture ("Finitistic conjecture") in
Representation theory of artin algebras:
"Given an artin algebra, its finitistic dimension is finite".
(The finitistic dimension is the supremum of the projective dimensions
of the modules with finite projective dimension.)
This conjecture has more than 45 years, and is one of the most
interesting problems at this area. In the talk we explain a new
technique to treat the conjecture, using the IgusaTodorov
function. We exhibit a new family of quivers algebras with finite
finitistic dimension.
 AARON LAUVE, UQAM, LaCIM, C.P. 8888, Succ. CentreVille, Montréal,
H3C 3P8
On novel ways to invert a matrix
[PDF] 
Given an n×n matrix M over a (not necessarily commutative)
field F and a candidate inverse M¢, the n^{2} equations M·M¢=I, if solvable, define an inverse for M in End_{F}(F^{n}). For us, it is a small wonder that
(i) the solution is unique, and
(ii) the solution is the same as one would reach in solving
the n^{2} different equations M¢·M=I.
We are led to the following question: from the 2·n^{2} equations
mentioned above, which choices of n^{2} yield a unique solution M¢?
The case n=2 is already interesting, involving a (reducible) Coxeter
group of order eight, a nice lemma of Cohn's on the roots of
noncommutative polynomials, ... .
 ROBERTO MARTINEZVILLA, Instituto de Matemáticas, UNAM, Morelia
ArtinSchelter regular algebras and categories
[PDF] 
Having in mind non connected algebras, like the preprojective algebra,
we introduce a generalization of the notion of a noncommutative
regular algebra given by Artin and Schelter, we obtain some basic
results and apply them to the the polynomial algebra. In order to
include the category of finitely presented functors from the finitely
generated modules over a finite dimensional Kalgebra, to the
category of Kvector spaces, we extend the notion of Artin Schelter
regular to additive categories. Finally, we give an application to
the structure of the Auslander Reiten components.
The results presented here are part of a joint work with Oeyvind
Solberg.
 MARKUS SCHMIDMEIER, Florida Atlantic University
Nilpotent Linear Operators
[PDF] 
In this talk I will present recent results from joint work with Claus
Michael Ringel (Bielefeld) on nilpotent linear operators and their
invariant subspaces.
Let k be a field. We consider triples (V,U,T) where V is a
finite dimensional kspace, U a subspace of V and T : V®V a linear operator with T^{n}=0 for some n, and such that T(U) Í U. Thus, T is a nilpotent linear operator on V and U
is an invariant subspace with respect to T.
If v=dimV and u=dimU then (v,u) is the dimension pair of the
triple (V,U,T). It turns out that whenver the nilpotency index n
is at most 6, then interesting properties about an indecomposable
triple (V,U,T) can be read off from the dimension pair.
 DAVID SMITH, Université de Sherbrooke, 2500 boul. de l'Université,
Sherbrooke (Québec), J1K 2R1
Piecewise hereditary skew group algebras
[PDF] 
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 representationfinite,
being hereditary, being tilted or quasitilted, etc.
In this talk, we study the interplay between the skew group algebras
and the socalled piecewise hereditary algebras, that is algebras A
for which there exist a hereditary abelian category H and
a triangleequivalence 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.
 SONIA TREPODE, Universidad Nacional de Mar del Plata
On AuslanderReiten Components with Bypasses
[PDF] 
We study the AuslanderReiten components which have a sectional
bypass and we characterize them. We show that a bypass defines a new
irreducible morphism.
Joint work with Claudia Chaio and Edson Ribeiro Alvares.
 RITA ZUAZUA, UNAM Campus Morelia
When the graph subrings admit standard Noether
normalizations?
[PDF] 
In Noether normalizations of some subrings of graphs (Comm. Algebra 29(2001), 55255534), Alcántar asked when a
standard Noether normalizations of the monomial subring or edge
subring of a graph exists. In this talk we will give an answer to the
above question.
This is a joint work with Florian Luca.

