41 research outputs found
Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules
Given a tagged triangulation of a once-punctured polygon with
vertices, we associate an ice quiver with potential such that the frozen part
of the associated frozen Jacobian algebra has the structure of a Gorenstein
-order . Then we show that the stable category of the category
of Cohen-Macaulay -modules is equivalent to the cluster category
of type . It gives a natural interpretation of the usual
indexation of cluster tilting objects of by tagged triangulations
of . Moreover, it extends naturally the triangulated categorification by
of the cluster algebra of type to an exact categorification
by adding coefficients corresponding to the sides of . Finally, we lift the
previous equivalence of categories to an equivalence between the stable
category of graded Cohen-Macaulay -modules and the bounded derived
category of modules over a path algebra of type .Comment: 50 pages. Several improvements after refereeing. arXiv admin note:
text overlap with arXiv:1307.067
A survey on maximal green sequences
Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster
algebras. They are useful for computing refined Donaldson-Thomas invariants,
constructing twist automorphisms and proving the existence of theta bases and
generic bases. We survey recent progress on their existence and properties and
give a representation-theoretic proof of Greg Muller's theorem stating that
full subquivers inherit maximal green sequences. In the appendix, Laurent
Demonet describes maximal chains of torsion classes in terms of bricks
generalizing a theorem by Igusa.Comment: 15 pages, submitted to the proceedings of the ICRA 18, Prague,
comments welcome; v2: misquotation in section 6 corrected; v3: minor changes,
final version; v4: reference to Jiarui Fei's work added, post-final version;
v4: formulation of Remark 4.3 corrected; v5: misquotation of Hermes-Igusa's
2019 paper corrected; v5: reference to Kim-Yamazaki's paper adde
-tilting finite algebras, bricks and -vectors
The class of support -tilting modules was introduced to provide a
completion of the class of tilting modules from the point of view of mutations.
In this article we study -tilting finite algebras, i.e. finite
dimensional algebras with finitely many isomorphism classes of
indecomposable -rigid modules. We show that is -tilting finite
if and only if very torsion class in is functorially finite. We
observe that cones generated by -vectors of indecomposable direct summands
of each support -tilting module form a simplicial complex . We
show that if is -tilting finite, then is homeomorphic to
an -dimensional sphere, and moreover the partial order on support
-tilting modules can be recovered from the geometry of .
Finally we give a bijection between indecomposable -rigid -modules and
bricks of satisfying a certain finiteness condition, which is automatic for
-tilting finite algebras.Comment: 29 pages. Changed title. Added Theorem 6.5 and Proposition 6.
Lattice theory of torsion classes
The aim of this paper is to establish a lattice theoretical framework to
study the partially ordered set of torsion
classes over a finite-dimensional algebra . We show that
is a complete lattice which enjoys very strong
properties, as bialgebraicity and complete semidistributivity. Thus its Hasse
quiver carries the important part of its structure, and we introduce the brick
labelling of its Hasse quiver and use it to study lattice congruences of
. In particular, we give a
representation-theoretical interpretation of the so-called forcing order, and
we prove that is completely congruence
uniform. When is a two-sided ideal of , is a lattice quotient of which is
called an algebraic quotient, and the corresponding lattice congruence is
called an algebraic congruence. The second part of this paper consists in
studying algebraic congruences. We characterize the arrows of the Hasse quiver
of that are contracted by an algebraic
congruence in terms of the brick labelling. In the third part, we study in
detail the case of preprojective algebras , for which
is the Weyl group endowed with the weak
order. In particular, we give a new, more representation theoretical proof of
the isomorphism between and the Cambrian
lattice when is a Dynkin quiver. We also prove that, in type , the
algebraic quotients of are exactly its
Hasse-regular lattice quotients.Comment: 65 pages. Many improvements compared to the first version (in
particular, more discussion about complete congruence uniform lattices
Categorification of skew-symmetrizable cluster algebras
We propose a new framework for categorifying skew-symmetrizable cluster
algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with
the action of a finite group G, we construct a G-equivariant mutation on the
set of maximal rigid G-invariant objects of C. Using an appropriate cluster
character, we can then attach to these data an explicit skew-symmetrizable
cluster algebra. As an application we prove the linear independence of the
cluster monomials in this setting. Finally, we illustrate our construction with
examples associated with partial flag varieties and unipotent subgroups of
Kac-Moody groups, generalizing to the non simply-laced case several results of
Gei\ss-Leclerc-Schr\"oer.Comment: 64 page
Cluster algebras and preprojective algebras : the non simply-laced case
We generalize to the non simply-laced case results of Gei\ss, Leclerc and
Schr\"oer about the cluster structure of the coordinate ring of the maximal
unipotent subgroups of simple Lie groups. In this way, cluster structures in
the non simply-laced case can be seen as projections of cluster structures in
the simply-laced case. This allows us to prove that cluster monomials are
linearly independent in the non simply-laced case.Comment: 6 pages, submitted to "comptes-rendus de l'Acad\'emie des Sciences",
french version 4 pages and english abridged version 2 page
Quotients of exact categories by cluster tilting subcategories as module categories
We prove that some subquotient categories of exact categories are abelian.
This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated
categories. As a particular case, if an exact category B with enough
projectives and injectives has a cluster tilting subcategory M, then B/M is
abelian. More precisely, it is equivalent to the category of finitely presented
modules over the stable category of M.Comment: 21 pages. Slight modifications after referring. Accepted for
publication in Journal of Pure and Applied Algebr