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Cluster Algebras
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- ASLAK BAKKE BUAN, Norwegian University of Science and Technology
Cluster-tilted algebras and quiver mutation
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Quiver mutation is a special case of cluster mutation, as defined by
Fomin and Zelevinsky. Cluster-tilted algebras are finite dimensional
algebras whose representation theory is closely related to the
representation theory of hereditary algebras. We discuss connections,
including some recent developments.
The talk will partially be based on joint work with Robert Marsh and
Idun Reiten.
- SERGEY FOMIN, University of Michigan, Ann Arbor, MI
Cluster complexes of bordered surfaces
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To an oriented surface with boundary and finitely many marked points
one can associate a pure simplicial complex, the cluster complex of
the corresponding cluster algebra. In a joint work with Michael
Shapiro and Dylan Thurston, we provide an explicit combinatorial
description of this complex, and determine its homotopy type and its
growth rate.
- RALF SCHIFFLER, UMASS Amherst, Lederle Graduate Research Tower, Amherst, MA
01003-9305, USA
m-replicated algebras and m-cluster categories
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Let A be a hereditary algebra. Its cluster category as well as its
m-cluster category are by definition quotients of the derived
category Db (mod A). One can construct a fundamental
domain for these categories as the left part, respectively the
m-left part, of the m-replicated algebra of A.
In this talk, we will define the m-replicated algebra, its
(m-)left part and study the projective dimension of its
indecomposable modules, which is crucial for the construction of the
fundamental domain mentioned before.
- HUGH THOMAS, Department of Mathematics and Statistics, University of New
Brunswick, Fredericton, NB
Noncrossing partitions via representations of quivers
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We show how the combinatorics of clusters (viewed as tilting objects
in the cluster category) can be related to the combinatorics of the
noncrossing partitions of the associated Coxeter group. It is known
that, for an arbitrary quiver Q, the tilting objects in the cluster
category for Q are in bijection with partial tilting objects in rep
Q which are tilting on their support. We show that these are also
in bijection with the exact abelian extension-closed subcategories of
rep Q. Further, if Q is of finite or affine type, these are also
in bijection with the noncrossing partitions of the reflection group
associated to Q. When Q is of finite type, we recover Reading's
bijection between clusters and noncrossing partitions. This
perspective also provides a new proof that, in finite type, the
noncrossing partitions form a lattice, which was first given a
type-free proof by Brady and Watt in 2005.
This is joint work with Colin Ingalls (UNB).
- GORDANA TODOROV, Northeastern University, Boston, Massachusetts, USA
m-cluster categories and m-replicated algebras
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Let A be a hereditary algebra over an algebraically closed field.
We prove that an exact fundamental domain for the m-cluster category
of A is the m-left part of the m-replicated algebra A(m) of
A. Moreover, we obtain a one-to-one correspondence between the
tilting objects in the m-cluster category (that is, the
m-clusters) and those tilting A(m)-modules for which all non
projective-injective direct summands lie in the m-left part of
A(m).
Joint work with I. Assem, T. Brüstle and R. Schiffler.
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