


Representations of Associative Algebras and Related Topics / Représentations d'algèbres associatives et sujets connexes Org: Vlastimil Dlab (Carleton) and/et Shiping Liu (Sherbrooke)
 YURI BAHTURIN, Department of Mathematics and Statistics, Memorial University
of Newfoundland, St. John's, Newfoundland A1C 5S7
Gradings on finitedimensional algebras

In this talk we are going to discuss various aspects of gradings on
finitedimensional algebras and their representations. Among topics
covered are the description of gradings by finite groups on simple
finitedimensional algebras, the structure of not necessarily simple
finitedimansional algebras, graded modules over graded algebras, the
connection between the properties of the identity component of the
grading and those of the whole of algebra. The talk will be based on
joint results with V. Drensky, S. Montgomery, S. Sehgal and
M. Zaicev.
 RAYMUNDO BAUTISTA, Universidad Nacional Autonoma de Mexico, Instituto de Matematicas
Unidad Morelia, Morelia, Mich. Mexico
The category of morphisms between projective modules

Let A be an artin algebra over a local commutative artinian ring
k. We consider the category p(A) whose objects are morphisms
f: P® Q with P and Q finitely generated left
Amodules. We introduce an exact structure on p(A). We prove a
relation between the Homfunctor and the Extfunctor of the exact
structure. From here we can prove the existence of almost split
sequences in p(A) and its relation with the almost split sequences
for finitely generated Amodules.
 FRAUKE BLEHER, University of Iowa, USA
Actions of finite groups on homogeneous coordinate rings

Let k be a field of positive characteristic p, let G be a finite
group and let P be a Sylow psubgroup of G. Suppose X is a
nonsingular projective variety over k with a faithful linear action
of G. We discuss the question of whether X has an embedding into
projective Nspace P^{N} for some N such that the
corresponding coordinate ring S(X) of X can be written as a
direct sum of indecomposable kGmodules which lie in finitely many
kGisomorphism classes. This was shown to be the case by Symonds and
Karagueuzian in case k is finite and X=P^{N} (embedding into
itself). We show that this is also true if either (i) X is a
projective curve, or (ii) X is a projective surface,
H^{1}(X,O_{X})=0 = H^{2}(X,O_{X}) and X^{P} ¹ Æ.
 THOMAS BRUESTLE, Sherbrooke University
Elliptic Lie algebras and tubular algebras

It is wellknown that the positive part g^{+} of a
KacMoody algebra g is isomorphic to Ringel's composition
algebra c(Q), defined by a quiver Q with the same root
system as g. Using vertex operator algebras, one can
construct a Lie algebra g(L) for every generalized root
lattice L. If L is elliptic and simply laced, Saito and Yoshii
gave a description of the Lie algebras g(L) by generators
and relations. Our aim is to compare in some cases g(L)
with the composition algebra c(A) of certain tubular
algebra A=kQ/I. Here the quiver Q encodes the Serre relations of
g(L) and the ideal I of the path algebra kQ encodes
the additional relations described by Saito and Yoshii. These
additional relations (which Saito and Yoshii consider to be strange,
since they involve more than two simples) are very natural from the
point of view of finitedimensional algebras.
 RAGNAR BUCHWEITZ, Department of Mathematics, University of Toronto, 100 St. George
Street, Toronto, Ontario M5S 3G3
The derived EckmannHilton argument and commutativity
of Hochschild cohomology

Surely, one of the deepest results in homological algebra is
Gerstenhaber's theorem that Hochschild cohomology of any associative
algebra is commutative with respect to the Yoneda product.
It has been observed by several authors recently that for flat algebras
the result follows very elegantly from the EckmannHilton argument
originally used to prove commutativity of higher homotopy groups.
Here, we investigate the scope of this method and show that various
incarnations of Hochschild cohomology, such as selfextensions of the
identity functor on the category of endofunctors, are amenable to the
same treatment.
We prove further that flatness can be replaced by the weaker hypothesis
that the algebra K® A satisfies Tor_{+}^{K}(A,A) = 0.
 XUEQING CHEN, University of Ottawa, Ottawa, Ontario K1N 6N5
Root vectors and integral PBW basis of the composition
algebra of the Kronecker algebra

According to the canonical isomorphism between the positive part
U^{+}_{q}(g) of the DrinfeldJimbo quantum group
U _{q} (g) and the generic composition algebra
C (D) of L, where the KacMoody Lie algebra
g and the finite dimensional hereditary algebra L
have the same diagram, in specially, we get a realization of quantum
root vectors of the generic composition algebra of the Kronecker
algebra by using the RingelHall approach. The commutation relations
among all root vectors are given and an integral PBWbasis of this
algebra is also obtained.
 CHRISTOF GEISS, Instituto de Matematicas, UNAM, Ciudad Universitaria, 04510 Mexico
D.F., Mexico
Semicanonical bases and preprojective algebras

This is a report on joint work with J. Schröer and B. Leclerc.
Let g be a simple Lie algebra of type A,
D, E and n a maximal nilpotent
subalgebra of g. Moreover, let N be a maximal unipotent
subgroup of a simple Lie group with Lie algebra g.
Finally, let P denote the corresponding preprojective algebra.
Lusztig's semicanonical basis B of U(n) is parametrized
by irreducible components of the corresponding preprojective varieties
mod (P,d). The dual B^{*} is a basis of C[N].
We can show:
$\bullet$ For two elements of B^{*} holds
b_{C}·b_{D} Î B^{*} if for the corresponding
irreducible
components holds ext_{P}^{1}(C,D)=0.
$\bullet$ The dual canonical basis B_{q}^{*} of
U_{q}(n) specializes for q=1 to B if and only if P
is representation finite.
This explains the multiplicative properties of the dual canonical basis
observed previously in the cases A_{2,3,4}. On the other
hand it gives us a good control over the dual semicanonical basis in
the cases A_{5} and D_{4}, i.e. when P is
tame, since we have in this case a precise combinatorial description of
the irreducible components of mod(P,d) in terms of
indecomposable components.
 ED GREEN, Virginia Tech, Blacksburg, Virginia 24060, USA
Extalgebras of Koszullike algebras

In this talk I describe some joint work with Eduardo N. Marcos of
University of São Paulo, Brazil. We give a definition of a
Koszullike algebra in terms of graded projective resolutions and
finite generation of the Extalgebra. This class includes Koszul
algebras and DKoszul algebras (introduced by R. Berger and studied
by GreenMarcosMartínezZhang). For such algebras, we find
necessary and sufficient condition on the degrees of generators of the
projective modules in the resolution for the Extalgebra to be finitely
generated. We investigate the finite generation of a restricted class
of bigraded algebras.
Another consequence of our work is studying certain subalgebras of
Extalgebras. In particular, we prove the following proposition. Let
A be a Koszul algebra and E(A) be its Extalgebra å_{k ³ 0}Ext_{A}^{k}(A/J,A/J) where J is the graded Jacobson radical of
A. Then for n ³ 1, the subalgebra sum_{k ³ 0}Ext_{A}^{nk}(A/J,A/J) is always a Koszul algebra.
 ELLEN KIRKMAN, Wake Forest University, WinstonSalem, North Carolina 27109,
USA
Fixed subrings of Noetherian graded regular rings

Let A be a graded algebra having the AuslanderGorenstein property,
and let G be a group of graded automorphisms of A. P. Jørgensen
and J. Zhang and N. Jing and J. Zhang have given conditions when the
fixed subring A^{G} also satisfies the AuslanderGorenstein property;
these conditions involve the "homological determinant" of the
automorphisms in G. We study groups of graded automomorphisms of
downup algebras and generalized Weyl algebras, and compute the
homological determinants of these automorphisms. We use these results
to produce examples of fixed subrings having the AuslanderGorenstein
property.
 SHIPING LIU, University of Sherbrooke, Sherbrooke, Quebec J1K 2R1
The no loop conjecture for special biserial algebras

(Joint work with JeanPhilippe Morin)
Let A be a finite dimensional algebra over a field given by a quiver
with relations. Let S be a simple Amodule with a nonsplit
selfextension, that is, the quiver has a loop at the corresponding
vertex. The strong no loop conjecture claims that S is of
infinite projective dimension. First we show that Ext^{i}_{A}(S,S), for
all i ³ 1, does not vanish if the selfextension is almost split.
As a byproduct of the proof, we get a new characterization of Nakayama
algebras, strengthening the one given by AuslanderReitenSmalø.
Further, using a result of GreenSolbergZacharia, we deduce that
Ext^{i}_{A}(S,S) does not vanish for all i ³ 1 if no power of the
loop is a component of a polynomial relation. Finally, we show that
Ext^{i}_{A}(S,S) does not vanish for all i ³ 1 if the convex support
of S is a special biserial algebra.
 FRANK MARKO, Pennsylvania State University, Hazleton, Pennsylvania 18202,
USA
Properties of Schur superalgebras in positive characteristics

(Joint work with Alexandr Zubkov)
Classical Schur algebras are known to be quasihereditary and cellular
and the Schur superalgebras in "large" characteristics are
semisimple. In case of "small" characteristics we determine that the
Schur superalgebras are no longer quasihereditary or cellular. In
case when the characteristic divides the degree r we determine all
Schur superalgebras S(mn,r) of finite representation type.
 ALEX MARTSINKOVSKY, Northeastern University, Boston, Massachusetts, USA
Stabilizing the AuslanderReiten formula

As was shown by R. Martinez Villa and A. Martsinkovsky, noncommutative
Serre duality for generalized ArtinSchelter regular algebras can be
deduced from the AuslanderReiten formula for quasiFrobenius (QF)
algebras. This is based on a functorial isomorphism, provided by
Koszul duality, between cohomology of tails and stable cohomology and
on the interpretation of the the AuslanderReiten formula for QF
algebras as a duality in stable cohomology. This result opens a
possibility for establishing Serretype duality formulas for much more
general classes of algebras. Their Koszul duals would not be QF and,
therefore, the original AuslanderReiten formula would have to be
replaced by a hypothetical formula expressing a duality in stable
cohomology. We establish such a formula by subjecting the
AuslanderReiten formula to the process of stabilization, which will be
explained in the lecture. This is joint work with Idun Reiten.
 MARKUS SCHMIDMEIRER, Florida Atlantic University, USA
Birkhoff's problem and poset representations

Let L be an artin algebra. We denote by S(L) the
category of pairs (A,B) where A is a finitely generated
Lmodule and B is a submodule of A; a map f:(A,B)®(A¢,B¢) is just a Llinear map f: B® B¢ such that
f(A) Ì A¢. The case of L = Z/(p^{n}) with p a
prime number and n a positive integer has attracted a lot of interest
since the categories S(Z/(p^{n})) describe the
possible embeddings of a subgroup in a finite abelian p^{n}bounded
group. The category S(Z/(p^{n})) has finitely many
indecomposables for n £ 5, is tame for n=6 and wild for n ³ 7.
For L a commutative local uniserial ring of Loewy length n,
and (A,B) an object in S(L), the radical and socle series of
A and B give rise to a filtration of B; the subsequent factors
form the vector spaces for a subspace representation of a poset. In
case n=6, this poset is tame of tubular type E_{8}; from its
indecomposable representations we can reconstruct parts of the category
S(L).
This is a report about joint work with Claus Michael Ringel.
 RITA ZUAZUA, Instituto de Matematicas, UNAM, Morelia
Almost split sequences for complexes of fixed size

Let A be the category of projective L modules for
L an Artin algebra. In this talk we show the existence of
almost split sequence in the category of projective finite complexes
C(A).

