On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories

By R. Dehy, B. Keller. 2007

112 citations recorded by the source corpus.

Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T′ form a basis of the Grothendieck group of T and that, if T and T′ are related by a mutation, then the indices with respect to T and T′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin-Zelevinsky's g-vectors.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/49fb3274bbd05222cce3f46e59a413decd5944ca

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Citations in the corpus

On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, methodology): “The mutation of ice quivers satisfying conditions 1)-3) is deﬁned via the mutation of the corresponding integer matrices recalled in section 2.1.”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, methodology): “In this section, we study g-vectors [16] (see also [4,32]) and g-matrices of support τ -tilting modules.”
Generic bases for cluster algebras from the cluster category (2011, background): “Each object M of the cluster category has an index indΓM (as defined in [11]) which is an element of K0(addΓ).”
Cluster categories (2010, background): “There are several interesting generalizations beyond the acyclic case, and we refer to ([37],[52],[56],[57],[58],[93],[96],[97]).”
The periodicity conjecture for pairs of Dynkin diagrams (2010, background): “1 of [12], the indecomposable Tk cannot occur in both T 0 L and T 1 L and so m0 = 0 or m1 = 0.”
Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, methodology): “In this section, we study g-vectors [16] (see also [4,32]) and g-matrices of support τ -tilting modules.”
Wall and chamber structure for finite-dimensional algebras (2018, methodology): “We start by recalling the notion of g-vectors first introduced by Dehy and Keller in [14] and later adapted to module categories by Adachi, Iyama and Reiten in [1].”
Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors (2019, background): “8, we need to recall the following results and definition from [31, 75].”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, methodology): “riables. When the cluster algebra admits a categoriﬁcation by a cluster category, this formula (10) can be interpreted as the transformation rule of the indices of object in the cluster category, cf. [DK07][KY11][Pla13]. We refer the reader to [FZ07, Remark 7.15] [Pla13, Section 3.5] [FG09]for more ”
Categorification of Skew-symmetrizable Cluster Algebras (2009, methodology): “In the skew-symmetric case, according to the work of Caldero–Chapoton [7], of Caldero–Keller [9, 10], of Palu [55], of Dehy–Keller [ 15 ] and of Fu–Keller [21] one can assign to every object X of C a Laurent polynomial in the cluster variables of an initial seed of the cluster algebra A categorified”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “idered as an element of K0(T) does not depend on the choice of this triangle. In the situation of the lemma, we deﬁne the index ind(L) of L as the element [T0]−[T1] of K0(T). Theorem 7.9 (Dehy-Keller [38]). a) Two rigid objects are isomorphic iﬀ their indices are equal. b) The indices of the indecom”