Let C be the category of finite-dimensional representations of a quantum affine algebra Uq(ĝ) of simply-laced type. We introduce certain monoidal subcategories Cl (l ∈ N) of C and we study their Grothendieck rings using cluster algebras.

We describe the c-vectors and g-vectors of the Markov cluster algebra in terms of a special family of triples of rational numbers, which we call the Farey triples.

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.

in this paper, we introduce the notion of multiplier in -algebra and study relationships between multipliers and some special mappings, likeness closure operators, homomorphisms and ( -derivations in -algebras. we introduce the concept of idempotent multipliers in bl-algebra and weak congruence and obtain an interconnection between idempotent multipliers and weak congruences. also, we introduce...

We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra...

After introducing double derivations of $n$-Lie algebra $L$ we‎ ‎describe the relationship between the algebra $mathcal D(L)$ of double derivations and the usual‎ ‎derivation Lie algebra $mathcal Der(L)$‎. ‎In particular‎, ‎we prove that the inner derivation algebra $ad(L)$‎ ‎is an ideal of the double derivation algebra $mathcal D(L)$; we also show that if $L$ is a perfect $n$-Lie algebra‎ ‎wit...

The polynomial ring Z[x11, . . . , x33] has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group Uq(sl3(C)) [8] [5]. On the other hand, Z[x11, . . . , x33] inherits a basis from the cluster monomial basis of a geometric model of the type D4 cluster algebra [3] [4]. We prove that these two bases are equal. This extends work of S...

Berenstein and Zelevinsky introduced quantum cluster algebras [3] the triangular bases [4]. The support conjecture in [12] asserts that of a basis element for rank-2 algebra is bounded by an explicitly described region possibly concave. In this paper, we prove all skew-symmetric algebras.

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of ...

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 can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster ...