Subject: Workshop on Representation Theory of Algebras 2008
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Date: Wed, 13 Aug 2008 12:09:37 +0900

Workshop on Representation Theory of Algebras 2008

Period: September 2 - 4, 2008

Place: Shizuoka University B-301 (Common building)

(http://www.shizuoka.ac.jp/ippan/shizuoka.html)
836 Ohya, Suruga-ku,
422-8529, Shizuoka
Tel.: 054-238-4722

Workshop Dinner: 18:30 - 20:30, September 3

Organizer: Hideto Asashiba (Shizuoka Univ.)

(shasash [at] ipc [dot] shizuoka [dot] ac [dot] jp)

Financial support: Hideto Asashiba; Jun-ichi
Miyachi
Grant-in-Aid for Scientific Research (C), JSPS

Speakers, titles of lectures

Bernhard Keller (Paris 7): Cluster algebras, cluster
categories and periodicity

Henning Krause (Paderborn) (joint with D. Benson and S. Iyengar):
Support varities and stratifications for stable module categories

Mitsuo Hoshino (Tsukuba.) (joint with Hiroki Abe):
Construction of quasi-Frobenius rings

Kiriko Kato (Osaka Prefecture) (joint with Osamu Iyama and
Jun-ichi Miyachi): Recollement of homotopy categories and Cohen-Macaulay
modules

Ryo Takahashi (Shinshu): Resolving subcategories of modules
over commutative rings

Hiroyku Minamoto (Kyoto): A non-commutative algebro-geometric
characterization of representation type of a quiver

Program

Tue. Sept. 2

Wed. Sept. 3

Thu. Sept. 4

09:00 - 10:00

Minamoto

Minamoto

10:15 - 11:15

Takahashi

Takahashi

Krause

11:30 - 12:30

Kato

Kato

Keller

--- Lunch ---

14:00 - 15:00

Krause

Krause

15:25 - 16:25

Hoshino

Hoshino

16:40 - 17:40

Keller

Keller

Abstract

Bernhard Keller
Cluster algebras were invented by Sergey
Fomin and Andrei Zelevinsky at the beginning of this decade. The initial
motivation came from the study of canonical bases in quantum groups and total
positivity in algebraic groups. In recent years, cluster algebras have become
very popular thanks, notably, to the many links to other subjects that were
discovered. Representation theory figures prominently among these since it
allows to categorify the basic constructions of cluster theory. A central role
is played by cluster categories, certain triangulated categories associated
with quivers without oriented cycles. We will present a survey of the links
between cluster algebras and cluster categories and then generalize the
construction of cluster categories to algebras of global dimension two
following recent work by Claire Amiot. As an application, we will sketch a
proof for a conjecture from mathematical physics known as the periodicity
conjecture, which claims that a certain discrete dynamical system associated
with a pair of Dynkin diagrams is periodic.

Henning Krause
Support varieties are used to classify
representations of finite groups, up to some suitable equivalence relation.
The plan is to explain the various ingredients of this joint work with Benson
and Iyengar. We construct support varieties and local cohomology functors for
triangulated categories with respect to the action of a graded commutative
ring. Combining this with some commutative differential graded algebra, one
obtains a stratification of the stable module category of a finite group.

Mitsuo Hoshino

Recall that a ring $A$ is said to be quasi-Frobenius if it is artinian and
selfinjective on both sides (see e.g. \cite{Na1, Na2}). There have been given
several characterizations of quasi-Frobenius rings. For instance, a ring $A$
is quasi-Frobenius if it is noetherian on either side and selfinjective on
either side (see e.g. \cite{Fa} and its references). Also, many attempts have
been made to generalize the notion of quasi-Frobenius rings (see e.g.
\cite{Th}, \cite{Os} and so on). On the other hand, only a few attempts seem
to have been made to construct quasi-Frobenius rings. We provide several ways
to construct quasi-Frobenius rings.
We modify the notion of
Frobenius extensions of rings due to Nakayama-Tsuzuku \cite{NT1, NT2} as
follows. Let $A$ be a ring containing a ring $R$ as a subring. Then $A$ is
said to be a Frobenius extension of $R$ if the following conditions are
satisfied: (1) $A_{R}$ and ${_{R}A}$ are finitely generated projective; and
(2) $A_{A} \cong \mathrm{Hom}_{R}(_{A}A,R)$ and ${_{A}A} \cong
\mathrm{Hom}_{R^{\mathrm{op}}}(A_{A},R)$. Frobenius extensions preserve
various homological properties. For instance, the following hold: $\mathrm{inj
\ dim} \ A_{A} \le \mathrm{inj \ dim} \ R_{R}$ and $\mathrm{inj \ dim} \
{_{A}A} \le \mathrm{inj \ dim} \ {_{R}R}$; if $R$ is a noetherian ring
satisfying the Auslander condition (see \cite{BE}) then so is $A$; and, if $R$
is a quasi-Frobenius ring, i.e., a selfinjective artinian ring then so is
$A$.
Also, we provide several ways to construct weakly symmetric
rings inductively, starting from a given quasi-Frobenius ring local ring. To
do so, we provide several ways to extend a given quasi-Frobenius ring $R$ to a
quasi-Frobenius ring $A$ containing an idempotent $e$ such that $eAe \cong R$.
For instance, we provide a way to construct, up to Morita equivalence, every
Brauer tree algebra over a field $k$ inductively, starting from
$k[t]/(t^{1+m})$, where $m$ is the multiplicity of the exceptional vertex (see
\cite{GR} for Brauer tree algebras).

References

[AH] H. Abe and M. Hoshino, Frobenius extensions and tilting complexes,

Algebras and Representation Theory (to appear).

[AH] H. Abe and M. Hoshino, Constructions of quasi-Frobenius rings,
preprint.

[BE] J. -E. Bj\"{o}rk and E. K. Ekstr\"{o}m, Filtered
Auslander-Gorenstein rings,

Progress in Math. {\bf 92}, 425--447, Birkh\"{a}user,
Boston-Basel-Berlin, 1990.

[Fa] C. Faith, Rings with ascending condition on annihilators,

Nagoya J. Math. {\bf 27} (1966), 179--191.

[GR] P. Gabriel and C. Riedtmann, Group representations without groups,

Comment. Math. Helv. {\bf 54} (1979), 240--287.

[Ho] M. Hoshino, Strongly quasi-Frobenius rings,

Comm. Algebra {\bf 28}(8)(2000), 3585--3599.

[Na1] T. Nakayama, On Frobeniusean algebras I,

Ann. Math. {\bf 40} (1939), 611--633.

[Na2] T. Nakayama, On Frobeniusean algebras II,

Ann. Math. {\bf 42} (1941), 1--21.

[NT1] T. Nakayama and T. Tsuzuku, On Frobenius extensions I,

Nagoya Math. J. {\bf 17}(1960), 89--110.

[NT2] T. Nakayama and T. Tsuzuku, On Frobenius extensions II,

Nagoya Math. J. {\bf 19}(1961), 127--148.

[Os] B. J. Osofsky, A generalization of quasi-Frobenius rings,

J. Algebra {\bf 4} (1966), 373--387; Errata, {\bf 9} (1968), p. 120.

[Th] R. M. Thrall, Some generalizations of quasi-Frobenius algebras,

Trans. Amer. Math. Soc. {\bf 64} (1948), 173--183.

Kiriko Kato
This is joint work with O. Iyama and J.
Miyachi. We study the homotopy category of unbounded complexes with bounded
homologies and its quotient category by the homotopy category of bounded
complexes. We show the existence of a recollement of the above quotient
category and it has the homotopy category of acyclic complxes as a
triangulated subcategory. In the case of the homotopy category of finitely
generated projective modules over a coherent ring R of finite
self-injective dimension both sides, we show that the above quotient category
are triangle equivalent to the stable module category of Cohen-Macaulay
modules over the 2×2 upper triangular matrix ring of R.

Ryo Takahashi
The notion of a resolving subcategory was
introduced in the 1960s by Auslander and Bridger in the study of modules of
Gorenstein dimension zero, which are now also called totally reflexive
modules. We study resolving subcategories of the category of finitely
generated modules over commutative noetherian rings. First, we consider
contravariant finiteness of resolving subcategories, and show a classification
theorem of contravariantly finite resolving subcategories. Second, we consider
constructing a "better" module from each module in a fixed resolving
subcategory, and investigate nonfree loci of resolving subcategories.

Hiroyuki Minamoto
TBA