We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) ...

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.

We extend the definition of a quantum analogue of the CalderoChapoton map defined [17]. When Q is a quiver of finite type, we prove that the algebra A H |k|(Q) generated by all cluster characters (see Definition 1) is exactly the quantum cluster algebra E H |k|(Q).

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson-Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on G corresponds to a cluster structure in O(G). We have shown before that this conjecture holds for any G in the case ...

To David Kazhdan for his 60th birthday " Loxadь sostoit iz trh neravnyh polovin ". 4 The quantum dilogarithm and its properties 26 4.1 The quantum logarithm function and its properties. . 1 " A horse consists of three unequal halves ". cf. A. de Barr, Horse doctor. Moscow 1868. Cluster varieties [FG2] are relatives of cluster algebras [FZI]. Cluster modular groups act by automor-phisms of clust...

This note shows that for any cluster algebra, the exchange graph has non-leaving-face property.

Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph is independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

In this paper, we initiate the study of a new class of algebras, which we call cluster algebras. Before giving precise definitions, we present some of the main features of these algebras. For any positive integer n, a cluster algebra A of rank n is a commutative ring with unit and no zero divisors, equipped with a distinguished family of generators called cluster variables. The set of cluster v...

In this paper, we initiate the study of a new class of algebras, which we call cluster algebras. Before giving precise definitions, we present some of the main features of these algebras. For any positive integer n, a cluster algebra A of rank n is a commutative ring with unit and no zero divisors, equipped with a distinguished family of generators called cluster variables. The set of cluster v...

What Is a Cluster Algebra? Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory o...