We study the notion of positive and negative complexity of pairs of objects in cluster categories. The first main result shows that the maximal complexity occurring is either one, two or infinite, depending on the representation type of the underlying hereditary algebra. In the second result, we study the bounded derived category of a cluster tilted algebra, and show that the maximal complexity...

Let $A$ be an arbitrary Banach algebra and $varphi$ a homomorphism from $A$ onto $Bbb C$. Our first purpose in this paper is to give some equivalent conditions under which guarantees a $varphi$-mean of norm one. Then we find some conditions under which there exists a $varphi$-mean in the weak$^*$ cluster of ${ain A; |a|=varphi(a)=1}$ in $A^{**}$.

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...

We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in [8][6]. As an application, when Q is a valued quiver of finite type or rank 2, we obtain 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) and various bases of the quantum cluster algebras...

The cluster category was introduced in (BMRRT, 2006) and also in (CCS1, 2006) for type A, as a categorical model to understand better the cluster algebras of Fomin and Zelevinsky (FZ, 2002). It is a quotient of the bounded derived category Db(modA) of the finitely generated modules over a finite dimensional hereditary algebra A. It was then natural to consider the endomorphism algebras of tilti...

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The n-Auslander-Reiten translation functor τn plays an important role in the study of n-cluster tilting subcategories. We study the category Mn of preinjective-like modules obtained by applying τn to injective modules repeatedly. We call a f...

This thesis is concerned with the development and application of categorical tools in the study of the cluster algebras of S. Fomin and A. Zelevinsky. C. Amiot’s generalized cluster category is a triangulated category which has been used, in the case where it is Hom-finite, to categorify a certain class of cluster algebras, using cluster characters in the sense of Y. Palu. In this thesis, we ge...

We construct bar-invariant Z[q 1 2 ]−bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of [14],[4] and [11] respectively. As a byproduct, we prove the positivity of the elements in these bases.