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This is an introduction to some aspects of Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
By using the approach in cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an applicatio
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $mathcal{P}_lambda$ of the quantum group $U_q(mathfrak{sl}_{n+1})$ is closed under tensor products. Our results general
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic qui
In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relat