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Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
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Cluster algebras and quantum affine algebras
, 2009
"... Let C be the category of finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced 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 finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C
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
Skewsymmetric cluster algebras of finite mutation type
"... Abstract. In the famous paper [8] Fomin and Zelevinsky obtained CartanKilling type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation ty ..."
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Cited by 44 (6 self)
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Abstract. In the famous paper [8] Fomin and Zelevinsky obtained CartanKilling type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skewsymmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skewsymmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems E6, E7, E8,
Cluster tilting for higher Auslander algebras
, 2008
"... The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category M ..."
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Cited by 30 (9 self)
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The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras.
DILOGARITHM IDENTITIES FOR CONFORMAL FIELD THEORIES AND CLUSTER ALGEBRAS: Simply Laced Case
, 2010
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THE HALL ALGEBRA OF A SPHERICAL OBJECT
"... Abstract. We determine the Hall algebra, in the sense of Toën, of the algebraic triangulated category generated by a spherical object. 1. ..."
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Cited by 19 (2 self)
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Abstract. We determine the Hall algebra, in the sense of Toën, of the algebraic triangulated category generated by a spherical object. 1.