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GROTHENDIECK GROUP AND GENERALIZED MUTATION RULE FOR 2CALABI–YAU TRIANGULATED CATEGORIES
"... We compute the Grothendieck group of certain 2Calabi–Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin–Zelevinsky cluster algebras. In this setup, we also prove a generalization of the Fomin–Zelevinsky mutation rule. ..."
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We compute the Grothendieck group of certain 2Calabi–Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin–Zelevinsky cluster algebras. In this setup, we also prove a generalization of the Fomin–Zelevinsky mutation rule.
Tilting theory and cluster combinatorics
 572–618. EQUIVALENCE AND GRADED DERIVED EQUIVALENCE 43
"... of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsk ..."
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of a finitedimensional hereditary algebra H over a field. We show that, in the simplylaced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky
Ptolemy relations for punctured discs
"... Abstract. We construct frieze patterns of type DN with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type DN, we show that the numbers in the pattern ..."
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in the pattern can be interpreted as specialisations of cluster variables in the corresponding FominZelevinsky cluster algebra. 1.
FRIEZE PATTERNS FOR PUNCTURED DISCS
"... Abstract. We construct frieze patterns of type DN with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type DN, we show that the numbers in the patter ..."
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in the pattern can be interpreted as specialisations of cluster variables in the corresponding FominZelevinsky cluster algebra. This is generalised to arbitrary triangulations in an appendix by Hugh Thomas. 1.
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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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
THE ALGEBRA OF MIRKOVIĆVILONEN CYCLES IN TYPE A
, 2005
"... Abstract. Let Gr be the affine Grassmannian for a connected complex reductive group G. Let CG be the complex vector space spanned by (equivalence classes of) MirkovićVilonen cycles in Gr. The BeilinsonDrinfeld Grassmannian can be used to define a convolution product on MVcycles, making CG into a ..."
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commutative algebra. We show, in type A, that CG is isomorphic to C[N], the algebra of functions on the unipotent radical N of a Borel subgroup of G; then each MVcycle defines a polynomial in C[N], which we call an MVpolynomial. We conjecture that those MVpolynomials which are cluster monomials for a FominZelevinsky
nREPRESENTATIONFINITE ALGEBRAS AND FRACTIONALLY CALABIYAU ALGEBRAS
, 908
"... Abstract. In this short paper, we study nrepresentationfinite algebras from the viewpoint of fractionally CalabiYau algebras. We shall show that all nrepresentationfinite algebras are twisted fractionally CalabiYau. We also show that twisted n(ℓ−1)CalabiYau algebras of global dimension n are ..."
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in the representation theory of algebras, especially in the categorification program of FominZelevinsky cluster algebras by cluster tilting theory (e.g. [Am, BIRS, BMRRT, GLS1, GLS2, IR, Ke2, Ke3, KR]). The derived categories of finite dimensional nonsemisimple algebras are never CY, but they are often fractionally
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"... summary) [Cluster algebras and applications (after FominZelevinsky,...)] ..."
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summary) [Cluster algebras and applications (after FominZelevinsky,...)]
Previous Up Next Article Citations From References: 1 From Reviews: 0
"... summary) [Cluster algebras and applications (after FominZelevinsky,...)] ..."
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summary) [Cluster algebras and applications (after FominZelevinsky,...)]
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