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49
Quivers with potentials and their representations I: Mutations
, 2007
"... We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Pono ..."
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Cited by 178 (3 self)
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We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
Cluster algebras and quantum affine algebras
, 2009
"... Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(̂g) of simply-laced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C ..."
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Cited by 71 (10 self)
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Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(̂g) of simply-laced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C
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.
Quantum cluster variables via Serre polynomials
, 2010
"... Abstract. For skew-symmetric acyclic quantum cluster algebras, we express the quan-tum F-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the po ..."
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Cited by 34 (3 self)
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Abstract. For skew-symmetric acyclic quantum cluster algebras, we express the quan-tum F-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.
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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Cited by 15 (0 self)
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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
