Mutation-periodic quivers, integrable maps and associated Poisson algebras

Fordy, Allan P.

doi:10.1098/rsta.2010.0318

Cited by 15 publications

(28 citation statements)

References 16 publications

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“…A recent development was the observation in [8] that (with all β i = 0) a combination of these compatible brackets for the dressing chain coordinates J i arises by reduction from the log-canonical Poisson structure for the cluster variables in cluster algebras associated with affine A-type Dynkin quivers. In this context, a further observation was that the linear relations between cluster variables found in [9] (see also [2,7,16]), which are of the form…”

Section: Connection With the Dressing Chainmentioning

confidence: 99%

“…In [7], it was shown that theÃ 1,N −1 exchange relations produce integrable maps for N even; in the non-commutative setting, the exchange relations for N odd were considered in [3]. The simplest case (N = 2) is just the recurrence (1.2), which preserves the Poisson bracket {x n , x n+1 } = x n x n+1 , and has a conserved quantity C (independent of n) that appears as a non-trivial coefficient in the linear relation for the sequence of cluster variables (x n ), that is,…”

Section: Introductionmentioning

confidence: 99%

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A family of linearizable recurrences with the Laurent property

Hone

Ward

2014

Bulletin of the London Mathematical Society

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We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained.

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Section: Connection With the Dressing Chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A family of linearizable recurrences with the Laurent property

Hone

Ward

2014

Bulletin of the London Mathematical Society

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“…The following can be deduced from the quantum counterpart, Lemma 6.5, by taking the semiclassical limit or directly from (22). Parts (i) and (iv) were observed by Fordy in [12]. Lemma 8.2.…”

Section: Commutative Cluster Algebras With Poisson Structurementioning

confidence: 99%

“…, x n ] that are semiclassical limits of the families L q n and C q n . The Poisson algebra that is the semiclassical limit of C q 2 n was introduced by Fordy [12] and this sparked our interest in C q n . In this section we shall present results of an analysis of the Poisson prime spectrum of the semiclassical limits of L q n and C q n .…”

Section: Poisson Structuresmentioning

confidence: 99%

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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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Discrete Integrable Systems and Poisson Algebras From Cluster Maps

Fordy

Hone

2013

Commun. Math. Phys.

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162

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We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised them in terms of the skew-symmetric matrix that defines the quiver. The associated nonlinear recurrences are equivalent to birational maps, and we explain how these maps can be endowed with an invariant Poisson bracket and/or presymplectic structure.Upon applying the algebraic entropy test, we are led to a series of conjectures which imply that the entropy of the cluster maps can be determined from their tropical analogues, which leads to a sharp classification result. Only four special families of these maps should have zero entropy. These families are examined in detail, with many explicit examples given, and we show how they lead to discrete dynamics that is integrable in the Liouville-Arnold sense.

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Mutation-periodic quivers, integrable maps and associated Poisson algebras

Cited by 15 publications

References 16 publications

A family of linearizable recurrences with the Laurent property

A family of linearizable recurrences with the Laurent property

Connected quantized Weyl algebras and quantum cluster algebras

Discrete Integrable Systems and Poisson Algebras From Cluster Maps

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