Cluster mutation-periodic quivers and associated Laurent sequences

Abstract

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting new families of non-linear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations. We extend our construction to include recurrences with parameters, giving an explanation of some observations made by Gale. Finally, we point out a connection between quivers which arise in our classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages, 35 figure

Connected quantized Weyl algebras and quantum cluster algebras

“…We continue to fix an odd integer n ≥ 3 and a scalar q ∈ K * that is not a root of unity. We now aim to explain the connection between the cyclic connected quantized Weyl algebra C q 2 n and the quantum cluster algebra Q q of the quiver denoted P (1) n+1 in the classification of periodic mutation by Fordy and Marsh [11]. We shall express this quantum cluster algebra as a quotient U/∆U of an iterated skew polynomial extension U of K, with ∆ central in U.…”

“…The original motivation can be traced back to the classification of mutation periodic quivers by Fordy and Marsh [11] and to a Poisson algebra P introduced by Fordy [12] in a further study of some such quivers.…”

“…For an even positive integer m we present the quantum cluster algebra for a Dynkin quiver of type A m as a factor of the linear connected quantized Weyl algebra L q m+1 . For an odd integer n ≥ 3 we present the quantum cluster algebra for the periodic quiver denoted P (1) n+1 in [11] as an extension of the cyclic connected quantized Weyl algebra C q 2 n , requiring one further generator, and also as a factor, by the principal ideal generated by a central element, of an iterated skew polynomial ring in n + 2 variables over the base field K.…”

Mutation-periodic quivers, integrable maps and associated Poisson algebras

“…In Fordy & Marsh [3], we introduced and studied quivers with this type of rotational property. Consider the N × N matrix…”

“…This gives a connection to maps with the Laurent property, with the archetypical example being the Somos 4 iteration (see The On-line Encyclopedia of Integer Sequences at Sloane [2]), which arises in the context of elliptic divisibility sequences in number theory. In Fordy & Marsh [3], we considered a class of quiver that had a certain periodicity property under 'quiver mutation'. The corresponding 'cluster exchange relations' then give rise to sequences with the Laurent property, which generalize many of the well-known examples.…”