Ralf Schiffler

Bases for cluster algebras from surfaces

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, such as principal coefficients .

FRIEZES, STRINGS AND CLUSTER VARIABLES

To any walk in a quiver, we associate a Laurent polynomial. When the walk is the string of a string module over a 2-Calabi-Yau tilted algebra, we prove that this Laurent polynomial coincides with the corresponding cluster character of the string module, up to an explicit normalising monomial factor.

Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables

We prove a conjecture of Kontsevich, which asserts that the iterations of the noncommutative rational map

CLUSTER AUTOMORPHISMS AND COMPATIBILITY OF CLUSTER VARIABLES

F_r:(x,y)-->(xyx^{-1},(1+y^r)x^{-1})

THE FIRST HOCHSCHILD COHOMOLOGY GROUP OF A CLUSTER TILTED ALGEBRA REVISITED

are given by noncommutative Laurent polynomials with nonnegative integer coefficients.

Algebras and Modules

In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural, then prove that if

Rational smoothness of varieties of representations for quivers of type

\mathcal{A}

Injective Presentations of Induced Modules over Cluster-Tilted Algebras

is unistructural or of Euclidean type, then

Positivity for Cluster Algebras of Rank 3

f: \mathcal{A}\to \mathcal{A}

Modules over cluster-tilted algebras that do not lie on local slices

is a cluster automorphism if and only if