cluster algebras and cluster categories
نویسندگان
چکیده
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 cluster algebras. from quiver representations, we proceed to the cluster category, which yields a complete categorification of the cluster algebra and its combinatorial underpinnings.
منابع مشابه
CLUSTER ALGEBRAS AND CLUSTER CATEGORIES
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...
متن کاملCluster Algebras and Cluster Categories
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 c...
متن کاملCluster algebras and derived categories
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coeff...
متن کاملGeometric Construction of Cluster Algebras and Cluster Categories
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of [FST06]. Furthermore, we give a description of m-cluster categories via diagonals (arcs) in (punctured) polygons and of m-cluster categories via powers of translation quivers as given in joint work with R. Marsh ([BM08a], [BM07]).
متن کاملCluster Algebras, Quiver Representations and Triangulated Categories
This is an introduction to some aspects of Fomin-Zelevinsky’s 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 fo...
متن کاملM-cluster Categories and M-replicated Algebras
Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category Cm(A) of A is the m-left part Lm(A (m)) of the m-replicated algebra of A. Moreover, we obtain a one-toone correspondence between the tilting objects in Cm(A) (that is, the m-clusters) and those tilting modules in modA(m) for which all non projective-injective di...
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عنوان ژورنال:
bulletin of the iranian mathematical societyجلد ۳۷، شماره No. ۲، صفحات ۱۸۷-۲۳۴
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