Bernhard Keller & Idun
Reiten talk with ScienceWatch.com and answer a few
questions about this month's Fast Breaking Paper in the
field of Mathematics.
Article Title: Cluster-tilted algebras are
Gorenstein and stably Calabi-Yau
Journal: ADVAN MATH
Year: MAY 1 2007
* Univ Paris 07, CNRS, UMR 7586, UFR Math, Case 7012,2 Pl
Jussieu, F-75251 Paris 05, France.
* Univ Paris 07, CNRS, UMR 7586, UFR Math, F-75251 Paris
(addresses have been truncated)
Why do you think your paper is highly
Our paper is related to cluster algebras, a subject introduced by Sergey
Fomin and Andrei Zelevinsky in 2002, and which, in recent years, has
witnessed spectacular growth. Inside this subject, our paper has unified
the two major existing approaches to categorification: categorification via
cluster categories and categorification via stable categories of
Cluster categories were first introduced and studied by
Buan-Marsh-Reineke-Reiten-Todorov (BMRRT) and in the A_n-case also, by
Caldero-Chapoton-Schiffler, and shown to be triangulated by Keller. Many of
their important properties were proved by Buan-Marsh-Reiten.
Categorification via stable categories is due to Geiss-Leclerc-Schroer.
Does it describe a new discovery, methodology or
synthesis of knowledge?
Our paper contains all three:
1) A new discovery, namely the discovery of remarkable new homological
properties of cluster-tilted algebras: as announced in the title, they turn
out to be Gorenstein and stably 3-Calabi-Yau.
2) A new methodology, namely the axiomatic method based on triangulated
2-Calabi-Yau categories applied to the investigation of endomorphism
algebras of cluster-tilting objects in cluster-categories and in stable
categories of preprojective algebras.
3) A synthesis of knowledge: the axiomatic framework we set up synthesizes
results obtained previously in different settings and allows us to give
How would you summarize the significance of your
paper in layman's terms?
Our paper contributes to the study of certain systems, called 2- and
3-Calabi-Yau categories, which are of two and three dimensions,
respectively. These categories have turned out to be important in a branch
of theoretical physics called string theory, where the elementary particles
are modeled by small strings. In this theory, in addition to the four
dimensions of classical physics—time plus three spatial
dimensions—one has to deal with six real or equivalently three
complex dimensions, whose behavior is expected to be controlled by certain
How did you become involved in this research and
were there any problems along the way?
This research started in informal conversations between the two authors at
a conference at the Banff International Research Station (near Calgary,
Canada). After the first version of the paper was posted on the
arXive, Osamu Iyama of the Graduate School of Mathematics, Nagoya
University, kindly pointed out to the authors that it contained an error in
the part dealing with higher dimensions. This was corrected in the printed
Despite the best efforts of the authors, another minor mistake—kindly
pointed out to the authors by Xiaowu Chen of the Shanghai Institute for
Advanced Studies—unfortunately did make it into the printed version.
It has been corrected in the latest version posted on the arXive.
Where do you see your research leading in the
Our research has already led to a more systematic investigation of 2- and
3-Calabi-Yau categories and to a deeper understanding of the relationship
between cluster algebras and these categories. We expect that, in the near
future, it will play a role in the solution of outstanding problems like
the so-called "periodicity conjecture," which asserts the periodicity of
all the solutions to the so-called Y-systems, and in the ongoing
investigation of higher cluster categories.
Are there any social or political implications for
The research is in the field of algebra in pure mathematics and, as such,
does not have any political implications. From a social perspective, it can
be regarded as adding to the totality of human knowledge and understanding
by providing us with insights into some of the beauties of pure
Professor Bernhard Keller
Université Paris Diderot - Paris 7
Professor Idun Reiten
Norwegian University for Science and Technology (NTNU)
Related: Aslak Bakke Buan, Robert Marsh, Markus Reineke,
Idun Reiten, Gordana Todorov answers a few questions
about this month's (October 2007) fast breaking paper in the field of
Keywords: cluster-tiled algebras, categorification, cluster
categories, stable categories, Gorenstein, 2- and 3-Calabi-Yau
categories, preprojective algebras, theoretical physics, string