Our article introduces and studies a new class of categories associated with quivers (directed graphs), called cluster categories. A tilting theory is developed in such categories. The article was motivated by Fomin-Zelevinsky cluster algebras, (Fomin S and Zelevinsky A, Journal of the American Mathematical Society 15[2]: 497-529, 2002) and the first connections between representation theory and cluster algebras (Marsh R, Reineke M, and Zelevinsky A, Trans. Amer. Math. Soc. 355[10]: 4171-86, 2003).

Left to right, top to bottom: Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, Gordana Todorov “Our article introduces a new kind of tilting theory, known as cluster-tilting theory, which provides a new tool for studying algebras.”

The article is of interest because of its location at the intersection of tilting theory and cluster algebras, leading to applications in both fields. It introduced novel techniques for working with algebras which have been used by articles citing it and it has motivated a variety of new developments:

• The study of cluster-tilted algebras, a family of algebras arising from cluster categories with interesting properties.

• The study of categories of modules over preprojective algebras from a similar perspective, giving new insights into quantum groups and the canonical basis of Lusztig and Kashiwara.

• New insights into the theory of Calabi-Yau categories from a tilting-theoretic perspective (since the cluster categories themselves are Calabi-Yau of dimension 2).

In another paper (Caldero P, Chapoton F, and Schiffler R, Transactions of the American Mathematical Society 358: 1347-1364, 2006), cluster categories and cluster-tilted algebras of type A were independently introduced via the combinatorial geometry of polygons. Together with recent ideas linking Riemann surfaces with cluster algebras, this has motivated geometric descriptions of cluster categories and related algebras in other cases as well.

The article introduces the cluster categories mentioned above, with interesting properties and connections in various directions, while at the same time providing a new approach to the study of relationships between algebras and a new approach to the study of cluster algebras, with implications for quantum groups.

The article grew out of a synthesis of ideas from cluster algebras and representation theory: cluster categories give models for cluster algebras using representation theory, while at the same time providing insights into representation theory.

Associative algebras are spaces of vectors together with an associative multiplication or composition. Examples include spaces of matrices (arrays of numbers) or operators, group algebras and polynomials. Algebras arise in various areas of mathematics, such as algebraic geometry, analysis, and logic, as well as in more diverse fields such as statistical mechanics and mathematical physics.

One way to study algebras is via their representations, in which each element of the algebra is replaced by a matrix of fixed size. Classical Morita theory and tilting theory provide methods for comparing two algebras and have been used to great effect in developing the theory of algebras. Our article introduces a new kind of tilting theory, known as cluster-tilting theory, which provides a new tool for studying algebras. This tool is now being applied to representation theory by a number of authors.

At the same time, cluster categories and research motivated by this article have given insight into the canonical basis for a quantum group: understanding this basis is one of the major open problems in Lie theory, an area of mathematics which arose from the study of continuous symmetries of mathematical objects (Lie groups).

The article is a collaboration between mathematicians with backgrounds in the fields of quantum groups, cluster algebras, and the representation theory of finite dimensional algebras. A project involving such a large number of collaborators is unusual in pure mathematics, but, as a result, benefited from expertise in a number of areas. The members happened to meet at conferences, which led to more intense discussions and visits.

We expect that research inspired by this article will continue for a long time to come, in many different directions. Some directions currently being explored include Calabi-Yau categories, including higher cluster categories, applications to cluster algebras, investigation of cluster-tilted algebras and their properties, new kinds of semi-invariants, constructions of categories arising in representation theory via Riemann surfaces, novel tilting theories, as well as work on understanding the motivating problem of understanding the canonical basis of a quantum group.

A number of conferences are being planned around these topics, many papers are being written, and students are working on doctoral theses in the field, indicating that the research is continuing.

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 total of human knowledge and understanding by providing us with insight into some of the beauties of pure mathematics.

Dr Aslak Bakke Buan
Department for Mathematical Sciences
Norwegian University of Science and Technology (NTNU)
Trondheim, Norway

Dr Robert Marsh
Department of Pure Mathematics
University of Leeds
United Kingdom

Professor Doctor Markus Reineke
Fachbereich C - Mathematics and Natural Sciences
Bergische Universitaet
Wuppertal, Germany

Professor Idun Reiten
Department for Mathematical Sciences
Norwegian University of Science and Technology (NTNU)
Trondheim, Norway.

Professor Gordana Todorov
Department of Mathematics
Northeastern University
Boston, MA, USA