Contemporary Mathematics
Volume 406, 2006
Cluster-tilting theory
Aslak Bakke Buan and Robert Marsh
ABSTRACT.
We aim to present a survey of recent developments concerning con-
nections between cluster algebras, representation theory and the dual canoni-
cal basis of a quantum group. Cluster algebras were introduced by Fomin and
Zelevinsky in order to study the dual canonical basis and total positivity for
algebraic groups; the corresponding cluster category can be obtained as a quo-
tient of the derived category of representations of an appropriate quiver. We
describe links between cluster categories and cluster algebras, and we survey
representation-theoretic applications of cluster categories, in particular how
they provide an extended version of classical tilting theory. We also briefly
discuss a number of interesting new developments linking cluster algebras,
cluster categories, representation theory and the canonical basis.
Introduction
Cluster algebras were introduced by Fomin and Zelevinsky
[FZ3]
in order to
understand the dual canonical basis of the quantised enveloping algebra of a quan-
tum group and total positivity for algebraic groups. Cluster categories are certain
quotients of derived categories of hereditary algebras (or hereditary categories),
and were introduced in
[BMRRT].
A graphical description in type
An
in terms
of triangulations of a polygon was given in
[CCSl].
The aim was to model cluster
algebras using the representation theory of quivers. The definition in
[BMRRT]
arose from work by Marsh, Reineke and Zelevinsky [MRZ]linking cluster algebras
and the representation theory of Dynkin quivers. This was a result of the com-
bination of two approaches to the canonical basis
[Ka, Lul, Lu3]
of a quantised
enveloping algebra: via the representation theory of the preprojective algebra and
via cluster algebras.
Cluster categories have led to new developments in the theory of the canonical
basis and particularly its dual. They are providing insight into cluster algebras
and their related combinatorics, and they have also been used to define a new kind
of tilting theory, known as cluster-tilting theory, which generalises APR-tilting for
2000
Mathematics Subject Classification.
Primary 16G20, 16G70, 16899, 17B37; Secondary
17B20, 52Bll.
Key words and phrases.
Cluster algebras, representations of finite-dimensional algebras,
hereditary algebras, cluster categories, canonical basis, quantised enveloping algebras, tilting the-
ory, cluster-tilting, preprojective algebras.
@2006 American Mathematical Society
http://dx.doi.org/10.1090/conm/406/07651
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