Cluster Algebras and Poisson Geometry

Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek

doi:10.1090/surv/167

Cited by 283 publications

(495 citation statements)

References 14 publications

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“…This example is directly related to the original motivations for cluster algebras coming from total positivity and canonical bases. The same example was also studied by Gekhtman et al (19) from the point of view of Poisson geometry. PoissonLie structures on a complex Lie group have been classified by Belavin and Drinfeld (61).…”

Section: Webs On Surfaces Rings Of Invariants and Clustersmentioning

confidence: 66%

Cluster algebras

Leclerc¹,

Williams²

2014

Proc. Natl. Acad. Sci. U.S.A.

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What Is a Cluster Algebra?Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2-15; Poisson geometry (16-19); Teichmüller theory (20-24); string theory (25-31); discrete dynamical systems and integrability (6, 32-38); and combinatorics (39-47).Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36,(48)(49)(50)(51)(52)(53) and to the cluster algebras portal (www.math.

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Section: Webs On Surfaces Rings Of Invariants and Clustersmentioning

confidence: 66%

Cluster algebras

Leclerc¹,

Williams²

2014

Proc. Natl. Acad. Sci. U.S.A.

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“…It is known that all quivers with the Dynkin diagram of type A n−1 as underlying graph are mutation equivalent, see [13,Lemma 3.23]. The adjacency matrix is B = (b ij ) where, for 1 ≤ i ≤ j ≤ n − 1, b ij = 1 if j = i + 1 and b ij = 0 if j = i + 1.…”

Section: Quantum Cluster Algebrasmentioning

confidence: 99%

Connected quantized Weyl algebras and quantum cluster algebras

Fish

Jordan

2018

Journal of Pure and Applied Algebra

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Abstract. For an algebraically closed field K, we investigate a class of noncommutative K-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators {x 1 , . . . , x n } such that each pair satisfies a relation of the form x i x j = q ij x j x i + r ij , where q ij ∈ K * and r ij ∈ K, with, in some sense, sufficiently many pairs for which r ij = 0. For such an algebra it turns out that there is a single parameter q such that each q ij = q ±1 . Assuming that q = ±1, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let n ≥ 3 be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type A n−1 as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver P (1) n+1 identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in n + 2 variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of A n−1 and P (1) n+1 are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element. Key words and phrases. skew polynomial ring, quantized Weyl algebra, quantum cluster algebra, Poisson algebra.Some of the results in this paper appear in the University of Sheffield PhD thesis of the first author, supported by the Engineering and Physical Sciences Research Council of the UK. We thank Vladimir Dotsenko for his helpful comments at an early stage of this project, the referee for many helpful suggestions of improvements in the exposition and Bach V Nguyen for pointing out some errors in our original manuscript.

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“…The main feature of the generalized cluster algebras is the appearance of polynomials in the exchange relations of cluster variables and coefficients, instead of binomials in the ordinary case. Generalized cluster algebras naturally appear so far in Poisson dynamics [GSV03], Teichmüller theory [CS14], representation theory [Gle14], exact WKB analysis [IN14], etc. It has been shown in [CS14,Nak14] that essentially all important properties of the ordinary cluster algebras are naturally extended to the generalized ones.…”

Section: Introductionmentioning

confidence: 99%

Quantum generalized cluster algebras and quantum dilogarithms of higher degrees

Наканиши

2015

Theor Math Phys

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Abstract. We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds. IntroductionThe generalized cluster algebras were introduced by Chekhov and Shapiro [CS14]. In this note we demonstrate that the notion of quantum cluster algebras is also extended to the generalized ones. To be more precise, there are two kinds of formulations of quantum cluster algebras, the one quantizing the cluster variables by [BZ05] and the one quantizing the coefficients by [FG09a,FG09b], and it is known that they are closely related to each other. Here, we concentrate on the latter one. As shown by [FG09a,FG09b], in the ordinary case, the quantization of the coefficients is tightly integrated with the quantum dilogarithm [FV93,FK94]. Similarly, in the generalized case, it is tightly integrated with certain generalizations of the quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds, which are also parallel to the ones in the ordinary case.The main message of this note is that the fundamental (and perhaps all) features of the quantum cluster algebras are also extended to the generalized ones.2010 Mathematics Subject Classification. 13F60.

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Cluster Algebras and Poisson Geometry

Cited by 283 publications

References 14 publications

Cluster algebras

Cluster algebras

Connected quantized Weyl algebras and quantum cluster algebras

Quantum generalized cluster algebras and quantum dilogarithms of higher degrees

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