these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

We introduce a new cluster character with coefficients for category $\mathcal{C}$ and rather than using Frobenius $2$-Calabi-Yau realization to incorporate into the representation-theoretic model algebra, as done by Fu Keller, we exploit intrinsic properties of $\mathcal{C}$. For this purpose, define an ice quiver associated each tilting object in In Dynkin case $\mathbb{A}_n$, also prove that ...

We call a 2-Calabi-Yau triangulated category a cluster category if its cluster-tilting subcategories form a cluster structure as defined in [4]. In this paper, we show that the canonical orbit category of the bounded derived category of finite dimensional representations of a quiver without infinite paths of type A∞ or A∞ is a cluster category. Moreover, for a cluster category of type A∞, we gi...