Cluster algebras introduced by fomin and zelevinsky in 2001 are commutative rings with unit and no zero divisors equipped with a distinguished family of generators cluster variables grouped in overlapping subsets clusters of the same cardinality the rank of the cluster algebra connected by exchange relations. Cluster algebras and poisson geometry about this title michael gekhtman university of notre dame notre dame in michael shapiro michigan state university east lansing mi and alek vainshtein university of haifa haifa mount carmel israel publication mathematical surveys and monographs publication year 2010 volume 167. This book is the first devoted to cluster algebras after presenting the necessary introductory material about poisson geometry and schubert varieties in the first two chapters the authors introduce cluster algebras and prove their main properties in chapter 3 this chapter can be viewed as a primer on the theory of cluster algebras in the . Cluster algebras and poisson geometry michael gekhtman michael shapiro alek vainshtein p cm mathematical surveys and monographs v 167 includes bibliographical references and index isbn 978 0 8218 4972 9 alk paper 1 cluster algebras 2 poisson algebras i shapiro michael 1963 ii vainshtein alek 1958 iii title qa2513g45 2010 512 44 dc22 2010029529 copying . Surveys and monographs volume 167 clusteralgebras and poisson geometry michael gekhtman michael shapiro alekvainshtein american mathematical society providencerhode island contents preface ix chapter 1 preliminaries 1 11 flag manifolds grassmannians pliicker coordinates andpliicker relations 1 12 simple lie algebras and groups 3 13 poisson lie groups 8 bibliographical notes 13
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