From triangulated categories to cluster algebras II

Abstract

In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi-Yau property of the cluster category.

Toda Systems, Cluster Characters, and Spectral Networks

“…There is a general correspondence between quiver Grassmannians and moduli spaces of framed quiver representations [Rei08], and in this section we recall how to rewrite the cluster characters CCpM λ i q in terms of framed quiver moduli. While the perspective of quiver Grassmannians is more natural from the point of view of cluster algebras [CC06,CK06,Pal08], the perspective of framed quiver moduli is more natural from the point of view of framed BPS indices in N " 2 field theory [GMN13a, CDM`13, CN13, Cir13]. Thus our aim here is to clarify how to equate the expressions CCpM λ k q with the types of expressions considered in the literature on line operators in N " 2 theories.…”

“…Additive categorification refers to the study of cluster algebras through the representation theory of associative algebras, in particular Jacobian algebras of quivers with potential [Ami11,Kel12]. A key notion is that of the cluster character (or Caldero-Chapoton function) of a representation of a quiver with potential, a generating function of the Euler characteristics of its quiver Grassmannians [CC06,CK06,Pal08,Pla11]. In particular, while cluster variables are a priori defined in a recursive, combinatorial way, they can in retrospect be described in a nonrecursive, representation-theoretic way as the cluster characters of rigid indecomposable representations.…”

Quantized Chebyshev polynomials and cluster characters with coefficients

“…So we can apply Caldero-Keller's one-dimensional multiplication formula for cluster characters without coefficients [6] to ι(S i+n ) and ι(S (n) i ) in C A . We get : −1 ))).…”

“…Inspired by works on cluster characters for the coefficient-free case [3,6,7,19], Fu and Keller introduced in [13] cluster characters with coefficients in order to realize elements in the cluster algebra A(Q, y, x) from objects in the cluster category C Q . In particular cluster variables in A(Q, y, x) are characters associated to indecomposable rigid objects in the cluster category C Q .…”

On Cluster Algebras Arising from Unpunctured Surfaces

“…In [CCS2], the authors obtained a formula for the denominators of the cluster expansion in types A, D and E, see also [BMR]. In [CC,CK,CK2] an expansion formula was given in the case where the cluster algebra is acyclic and the cluster lies in an acyclic seed. Palu recently generalized this formula to arbitrary clusters in an acyclic cluster algebra [Pa].…”