17,803 research outputs found
Caldero-Chapoton algebras
Motivated by the representation theory of quivers with potentials introduced
by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave
explicit formulae for the cluster variables of Dynkin quivers, we associate a
Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra.
By definition, the Caldero-Chapoton algebra is (as a vector space) generated by
the Caldero-Chapoton functions of the decorated representations of the basic
algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a
quiver with potential is closely related to the cluster algebra and the upper
cluster algebra of the quiver. The set of generic Caldero-Chapoton functions,
which conjecturally forms a basis of the Caldero-Chapoton algebra) is
parametrized by the strongly reduced components of the varieties of
representations of the Jacobian algebra and was introduced by Geiss, Leclerc
and Schr\"oer. Plamondon parametrized the strongly reduced components for
finite-dimensional basic algebras. We generalize this to arbitrary basic
algebras. Furthermore, we prove a decomposition theorem for strongly reduced
components. Thanks to the decomposition theorem, all generic Caldero-Chapoton
functions can be seen as generalized cluster monomials. As another application,
we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton
algebras lead to several general conjectures on cluster algebras.Comment: 35 pages. v2: Corrected the definition of a generic E-invariant and
also a short list of minor inaccuracies and typos. Final version to appear in
Transactions AM
Generic cluster characters
Let \CC be a Hom-finite triangulated 2-Calabi-Yau category with a
cluster-tilting object . Under a constructibility condition we prove the
existence of a set \mathcal G^T(\CC) of generic values of the cluster
character associated to . If \CC has a cluster structure in the sense of
Buan-Iyama-Reiten-Scott, \mathcal G^T(\CC) contains the set of cluster
monomials of the corresponding cluster algebra. Moreover, these sets coincide
if has finitely many indecomposable objects.
When \CC is the cluster category of an acyclic quiver and is the
canonical cluster-tilting object, this set coincides with the set of generic
variables previously introduced by the author in the context of acyclic cluster
algebras. In particular, it allows to construct -linear bases in acyclic
cluster algebras.Comment: 24 pages. Final Version. In particular, a new section studying an
explicit example was adde
Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II
It is an important aspect of cluster theory that cluster categories are
"categorifications" of cluster algebras. This is expressed formally by the
(original) Caldero-Chapoton map X which sends certain objects of cluster
categories to elements of cluster algebras.
Let \tau c --> b --> c be an Auslander-Reiten triangle. The map X has the
salient property that X(\tau c)X(c) - X(b) = 1. This is part of the definition
of a so-called frieze.
The construction of X depends on a cluster tilting object. In a previous
paper, we introduced a modified Caldero-Chapoton map \rho depending on a rigid
object; these are more general than cluster tilting objects. The map \rho sends
objects of sufficiently nice triangulated categories to integers and has the
key property that \rho(\tau c)\rho(c) - \rho(b) is 0 or 1. This is part of the
definition of what we call a generalised frieze.
Here we develop the theory further by constructing a modified
Caldero-Chapoton map, still depending on a rigid object, which sends objects of
sufficiently nice triangulated categories to elements of a commutative ring A.
We derive conditions under which the map is a generalised frieze, and show how
the conditions can be satisfied if A is a Laurent polynomial ring over the
integers.
The new map is a proper generalisation of the maps X and \rho.Comment: 16 pages; final accepted version to appear in Bulletin des Sciences
Math\'ematique
Cluster algebras as Hall algebras of quiver representations
Recent articles have shown the connection between representation theory of
quivers and the theory of cluster algebras. In this article, we prove that some
cluster algebras of type ADE can be recovered from the data of the
corresponding quiver representation category. This also provides some explicit
formulas for cluster variables.Comment: 17 pages ; 2 figures ; the title has changed ! some other minor
modification
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