Cluster Algebras and Poisson Geometry

By Michael I. Gekhtman, Michael Shapiro, A. Vainshtein. 2002

438 citations recorded by the source corpus.

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. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. 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 remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.|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. Examples of cluster algebras include coordinate rings of many algebraic varieties that play a prominent role in representation theory, invariant theory, the study of total positivity, etc. The theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links to a wide range of subjects including representation theory, discrete dynamical systems, Teichmuller theory, and commutative and non-commutative algebraic geometry. 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 remaining chapters, the emphasis is made on geometric aspects of the cluster algebra theory, in particular on its relations to Poisson geometry and to the theory of integrable systems.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b188bc567c7a59fb36b880ee2c131fbf138a9c7a

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Citations in the corpus

Matrix formulae and skein relations for cluster algebras from surfaces (2011, positive): “ self-folded triangles. Note that a self-folded triangle consists of an arc ℓ whose endpoints coincide, together with an arc r to an enclosed puncture which we dub a radius. Following the notation of [GSV11], we will refer to an arc ℓ cutting out a once-punctured monogon as a noose. Deﬁnition 2.5 (O”
Canonical bases for cluster algebras (2014, methodology): “ure on CGr(k,n) (the aﬃne cone over Gr(k,n) in its Plu¨cker embedding) every frozen variable has an optimized seed. Proof. As was pointed out to us by Lauren Willams, for Gr(k,n), the initial seed in [GSV], Figure 4.4, is optimized for one frozen variable (the special upper right hand vertex for the”
Locally acyclic cluster algebras (2011, methodology): “ntersection of the Laurent rings of all clusters. From this perspective, X Uis the anization of X; the universal ane scheme with a map from X. 6The subscheme Xis called the cluster manifold; see [GSV03] and [GSV10]. However, this terminology can be misleading, since the smooth locus of X A can be bi”
Higher pentagram maps, weighted directed networks, and cluster dynamics (2011, methodology): “This relation was discovered and explored by Glick [12] who proved that the pentagram map, acting on the quotient space Pn/R, is described by coefficient dynamics [6] (also known as τ -transformations, see Chapter 4 in [8]) for a certain cluster structure.”
Tilting theory and cluster combinatorics (2004, background): “... properties of the dual canonical basis of the quantised enveloping algebra of a semisimple Lie algebra over C, and that it should model the (classical and quantised) coordinate rings of varieties associated to algebraic groups (now shown in several cases — see [BFZ02], [Sco03].), with particular”
Total positivity and cluster algebras (2010, background): “In particular, not covered here are the theory of cluster categories and the various facets of categorification [39, 40, 41, 50]; the connections between cluster algebras and Poisson geometry [32, 33]; closely related work on cluster varieties arising in higher Teichmüller theory [16, 17]; the polyh”
A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules (2013, neutral): “aining the vertex (2,1). For another illustration, with C of type G2, see Figure 3. More examples can be found in the Appendix, §6.5, §6.6, §6.7. 2.2 Cluster algebras We refer the reader to [FZ2] and [GSV] for an introduction to cluster algebras, and for any undeﬁned terminology. 2.2.1 Cluster algeb”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, background): “Cluster algebras quickly gained popularity in various branches of mathematics, for example Poisson geometry (e.g. [16]), higher Teichm¨uller theory (e.g. [8]), combinatorics (e.g. [31]), integrable systems (e.g. [13]), etc.”
CLUSTER MUTATION VIA QUIVER REPRESENTATIONS (2004, background): “Cluster algebras have also been related to Poisson geometry [ GSV1 ], Teichm¨uller spaces [GSV2], positive spaces and stacks [FG], dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representation theory, see amongst others [BMRRT, BMR, CC, CCS”
Categorification of Skew-symmetrizable Cluster Algebras (2009, background): “Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other subjects: combinatorics (see for instance [8, 22]), Poisson geometry [ 35 , 36], integrable systems [25], Teichmuller spaces [20], and, last but not least, representations ”