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Cluster algebras, quiver representations and triangulated categories
, 2009
"... This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In additi ..."
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Cited by 106 (6 self)
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This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
Quiver varieties and cluster algebras
, 2009
"... Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a FominZelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric KacMoody Lie algebra, studied earlier by the author via perverse sheaves on ..."
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Cited by 49 (0 self)
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Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a FominZelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric KacMoody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties [49]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [21] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize
Positivity for cluster algebras from surfaces
, 2009
"... We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our for ..."
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Cited by 43 (11 self)
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We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
The periodicity conjecture for pairs of Dynkin diagrams
, 2010
"... We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories. ..."
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Cited by 36 (0 self)
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We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.
Beyond KirillovReshetikhin modules
"... we shall be concerned with the category of finite–dimensional representations of the untwisted quantum affine algebra when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and ..."
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Cited by 33 (12 self)
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we shall be concerned with the category of finite–dimensional representations of the untwisted quantum affine algebra when the quantum parameter q is not a root of unity. We review the foundational results of the subject, including the Drinfeld presentation, the classification of simple modules and qcharacters. We then concentrate on particular families of irreducible representations whose structure has recently been understood:
QUANTUM UNIPOTENT SUBGROUP AND DUAL CANONICAL BASIS
, 2010
"... ... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture w ..."
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Cited by 18 (2 self)
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... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B up. In particular, we prove that the quantum analogue Oq[N(w)] of C[N(w)] has the induced basis from B up, which contains quantum flag minors and satisfies a factorization property with respect to the ‘qcenter’ of Oq[N(w)]. This generalizes Caldero’s results [7, 8, 9] from ADE cases to an arbitary symmetrizable KacMoody Lie algebra.
GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY
"... Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same i ..."
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Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss–Leclerc–Schröer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock–Goncharov concerning the parametrization of a basis of the upper cluster
Simple tensor products
"... Abstract. Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S1 ⊗ · · · ⊗SN of simple objects of F is simple if and only if for any i < j, Si ⊗ Sj is simple. Contents ..."
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Cited by 14 (4 self)
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Abstract. Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S1 ⊗ · · · ⊗SN of simple objects of F is simple if and only if for any i < j, Si ⊗ Sj is simple. Contents