The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface Σ. This is a noncommutative algebra AΣ generated by “noncommutative geodesics” between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plücker relations. It turns out that the algebra AΣ exhibits a noncommutative Laurent Phenomenon with respect to any tria...

The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface Σ. This is a noncommutative algebra AΣ generated by “noncommutative geodesics” between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Plücker relations. It turns out that the algebra AΣ exhibits a noncommutative Laurent Phenomenon with respect to any tria...

We show that in a triangulated category, the existence of a cluster tilting object often implies that the homomorphism groups are bounded in size. This holds for the stable module category of a selfinjective algebra, and as a corollary we recover a theorem of Erdmann and Holm. We then apply our result to Calabi-Yau triangulated categories, in particular stable categories of maximal Cohen-Macaul...

The A∞ T-system, also called the octahedron recurrence, is a dynamical recurrence relation. It can be realized as mutation in a coefficient-free cluster algebra (Kedem 2008, Di Francesco and Kedem 2009). We define T-systems with principal coefficients from cluster algebra aspect, and give combinatorial solutions with respect to any valid initial condition in terms of partition functions of perf...

A lot of recent activity has been directed toward various constructions of “natural” bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig’s construction of a canonical basis, and the pioneering construction of the Kazhdan–Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig’s Lemma that we apply to a...

Nakajima (2003) introduced a t-deformation of q-characters, (q, t)-characters for short, and their twisted multiplication through the geometry of quiver varieties. The Nakajima (q, t)-characters of Kirillov-Reshetikhin modules satisfy a t-deformed T -system. The T -system is a discrete dynamical system that can be interpreted as a mutation relation in a cluster algebra in two different ways, de...

Let G=SL(2,5) be the special linear group of 2×2-matrices with coefficients in field 5 elements. We show that principal block over a splitting K characteristic two algebra KG has 3-cluster tilting module. This gives first example representation-infinite having cluster module and answers question by Erdmann Holm.

We describe and investigate a connection between the topology of isolated singularities plane curves mutation equivalence, in sense cluster algebra theory, quivers associated with their morsifications.

We recover the Jones polynomials of knots and links from K-theory a cluster C*-algebra sphere with two cusps. In particular, an interplay between Chebyshev is studied.