Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras

Mutation-P erio dic Quiv ers, In tegrable Maps and Asso ciated P oisson Algebras ∗ Allan P . F ordy , Sc ho ol of M athematics, Univ ersit y of Leeds. Leeds LS2 9JT, UK. e-mail: am t6apf@leeds.ac.uk 23rd Octob er, 2010 Abstract W e consider a class of map, recently deriv ed in the context of cluster mutatio n. In this pap er w e start with a brief review of the q uiv er con- text, but then mo ve on to a discussion of a related P oisson brack et, along with t he Po isson algebra of a special famil y of functions as so ciated with these maps. A bi-Hamiltonian structure is d erive d and used to construct a seq uence of Poiss on commuting fun ctions and hence show complete in - tegrabilit y . Canonical coordinates are derived, with the map now b eing a canonical transformation with a sequence of comm uting inv ariant func- tions. C ompatibility of a pair of these functions give s rise to Liouville ’s equation and the map plays the role of a B¨ ac klund transformation. Keywor ds : Poisson algebr a, bi-Hamiltonian, int egr a ble map, B¨ acklund tr ans- formation, Laur ent prop erty , cluster alg ebra, quiver g auge theory . 1 In tro duction Robin Bullough’s famous diagra m repre sents a v a st array of areas in Mathe- matics a nd Mathema tica l Physics, together with a “ neural netw ork” o f connec- tions (solid lines when e s tablished, dotted when exp ected) betw een them. This “Grand Syn thesis of So liton Theory” sho ws that some rema rk a ble connectio ns betw een seemingly dis pa rate sub jects hav e come a bo ut throug h the develop- men ts of Integrable Systems, whic h hav e taken place in the last 40 years or so. Of course , the diag ram p erp etually evolv es, as dotted connections b ecome solid and as new sub ject a reas (with co rresp onding links) a r e added to the array . In this paper , I present some connections with sub jects that didn’t even exist un til recently! ∗ This article i s dedica ted to the memory of Robin Bullough. 1 Spec ific a lly , the pres ent pap er is concerne d with inte gr able maps which arise in the c o ntext of clus t er m utations (see F omin and Zelevinsk y (20 02b)). This gives a connection to maps with the Laur ent pro pe r ty , with the a r chet ypical example b eing the Somos 4 iteration (see “The On-Line Ency c lop edia o f In teger Sequences” at Sloane (200 9)), which a rises in the context of elliptic divisibility sequences in num ber theory . In F ordy and Marsh (2009) w e c onsidered a clas s of quiver which had a cer tain p erio dicity pro p er ty under “quiver mutation”. The corresp onding “cluster exc hange relations” then giv e rise to sequences with the Laure nt prope r ty , which gener alise many of the w ell known exa mples. In this pap e r I firs t explain some of this background, but the main emphas is will b e on some asso ciated Poisson alg ebras (with r e s pe ct to an inv ariant Pois- son br a ck et of log- canonical type). In ter ms of c anonic al v aria bles we o btain Hamiltonians (inv ariant under the a ction of the map) in exp onential form. The compatibility of one particula r pair o f inv ariant Hamiltonians lea ds to Lio u- ville’s eq uation fo r each q i , with the map now playing the ro le o f a B¨ acklund transformatio n, bringing us back to one of the or iginal ideas in Soliton Theory! 2 The Laurent Prop ert y The Somos 4 sequence is generated by the iteration o n the rea l line x n x n +4 = x n +1 x n +3 + x 2 n +2 , with x 0 = x 1 = x 2 = x 3 = 1 , (1) giving 1 , 1 , 1 , 1 , 2 , 3 , 7 , 23 , 59 , 314 , 1529 , 8209 , . . . Since we must divide by x n at each step, it is not o bvious that we gener ate int eger s fro m x 8 = 59 onwards. Even more, starting with initial conditio ns x 0 = s, x 1 = t, x 2 = u, x 3 = v , w e find that eac h ter m x n is a L aur ent p olynomial in these initial v a lues (ie, a p olynomia l in s ± 1 , t ± 1 , u ± 1 , v ± 1 ). Int egr a lity o f the ab ov e numerical se q uence then follows by setting r = s = u = v = 1. Considering an obvious gener alisation o f (1) (called the Somos N s e quence) x n x n + N = [ N/ 2] X r =1 x n + r x n + N − r , it is found that it to o has the Laurent proper t y for N = 4 , 5 , 6 , 7, but fails at N = 8 . F ailur e is rather simple to prov e, since non-integer (rational) elements o ccur fairly so on in the sequence. Ad-ho c proo fs exist for v a r ious sequences (see Gale (19 91) for a pr o of in the case of So mo s 4 ), a nd can often b e a dapted to other s e q uences. How ever, a remar k able (but complicated!) pro o f for a very broad class o f iteration was given in F omin and Ze le vinsky (2002a). A t ab out the sa me time, clus ter algebr as were in tro duced in F o min and Zelevinsky (2002b), and it w as shown that an y map whic h aros e as a clust er exchange r ela- tion necessarily had the Laurent prop erty . Cluster algebr as a re a n abstraction of structures which arise in the study of total positivity of matrices and in the 2 canonical basis of a quantum group. Howev er, for this pap er we need none of this background. Neither do we need the full definition of a cluster algebra. The most impo rtant as pe c t for us is the asso cia tio n with quivers a nd quiver mutation . 2.1 Quiv er Mutation A quiver is a dir e cte d gr aph , consisting of N no des with direc ted edges be t ween them. There may b e several ar r ows b etw een a g iven pair of vertices, but for cluster algebras ther e should b e no 1-cycles (an arrow whic h star ts and ends at the same no de) or 2-cyc le s (an a rrow from no de a to no de b , follow ed by one from no de b to no de a ). A quiv er Q , with N no des , can b e ident ified with the unique skew-symmetric N × N matrix B Q with ( B Q ) ij given by the num ber of arrows from i to j minus the n umber of arrows from j to i . An impo r tant quiv er for o ur discussion is the one corr esp onding to the Somo s 4 sequence (b oth quiv er and matr ix in Figur e 1). 3 2 1 4 B =     0 − 1 2 − 1 1 0 − 3 2 − 2 3 0 − 1 1 − 2 1 0     Figure 1: The Somos 4 quiver S 4 and its matr ix. Definition 2. 1 (Quiv er Mu tatio n) Given a quiver Q we c an mut ate at any of its no des. The mutation of Q at no de k , denote d by µ k Q , is c onstructe d (fr om Q ) as fol lows: 1. R everse a l l arr ows which either origi nate or terminate at no de k . 2. Supp ose that ther e ar e p arr ows fr om no de i to no de k and q arr ows fr om no de k to no de j (in Q ). A dd pq arr ows going fr om no de i to no de j to any arr ows alr e ady ther e. 3. R emove (b oth arr ows of ) any two-cycles cr e ate d in the pr evious steps. Note that in Step 2 , pq is just the num b er of paths of length 2 b etw een no des i and j which pass thr o ugh no de k . Remark 2. 2 (Matrix Mutation) L et B and ˜ B b e the skew-symmetric ma- tric es c orr esp onding to the quivers Q and ˜ Q = µ k Q . L et b ij and ˜ b ij b e the 3 c orr esp onding matrix en tries. Then quiver mutation amounts to the fol lowing formula ˜ b ij =  − b ij if i = k or j = k , b ij + 1 2 ( | b ik | b kj + b ik | b kj | ) otherw ise. (2) It is an ex ercise to show that with these definitions, the Somos 4 quiver and matrix are tra nsformed to those of Fig ure 2, if w e mutate at node 1. 1 4 3 2 ˜ B =     0 1 − 2 1 − 1 0 − 1 2 2 1 0 − 3 − 1 − 2 3 0     Figure 2: Mutation ˜ S 4 = µ 1 S 4 of the quiv er S 4 at no de 1 a nd its matrix . 2.2 Cluster E xc hange Relations Given a quiver (with N no des), we a ttach a v ariable at each no de, lab elled ( x 1 , · · · , x N ). When we mutate the quiver we change the a sso ciated matrix according to for mula (2) and, in addition , we transform the cluster v aria bles ( x 1 , · · · , x N ) 7→ ( x 1 , · · · , ˜ x k , · · · , x N ), where x k ˜ x k = Y b ik > 0 x b ik i + Y b ik < 0 x − b ik i , ˜ x i = x i for i 6 = k . (3) If o ne of thes e products is empt y (whic h occurs when all b ik hav e the same sign) then it is r eplaced by the num ber 1. This formula is called the (cluster) exchange r elation . Notice that it just dep ends up on the k th column of the matrix. Since the matrix is skew-symmetric, the v aria ble x k do es no t o ccur o n the right side of (3). After this pro cess we hav e a new quiver ˜ Q , with a new matrix ˜ B . This new quiver has c luster v ariables ( ˜ x 1 , · · · , ˜ x N ). How ever, since the exchange relation (3 ) ac ts a s the identit y o n all except one v ariable, we write these new cluster v a riables as ( x 1 , · · · , ˜ x k , · · · , x N ). W e can now rep eat this proc ess and mutate ˜ Q a t no de ℓ a nd pro duce a third quiver ˜ ˜ Q , with cluster v ariables ( x 1 , · · · , ˜ x k , · · · , ˜ x ℓ , · · · , x N ), with ˜ x ℓ being given b y an ana logous formula (3). Remark 2. 3 (In v o lutiv e Prop erty of the Exc hange Relation) If ℓ = k , then ˜ ˜ Q = Q , so we insist that ℓ 6 = k . 4 Example 2.4 (The Somo s 4 Quiver S 4 ) Placing x 1 , x 2 , x 3 , x 4 resp ectively at nodes 1 to 4 of quiv er S 4 (of Figure 1) giv es the initial cluster . Along with the quiver mutation (lea ding to µ 1 S 4 of Figure 2), we also hav e the exchange r elation x 1 ˜ x 1 = x 2 x 4 + x 2 3 . (4) This cor resp onds to one arrow coming into no de 1 from ea ch of no des 2 and 4 with 2 ar rows going out to node 3. W e can now consider mutations of q uiver ˜ S 4 = µ 1 S 4 . T o av oid to o many “tildes”, let us write ˜ x 1 = x 5 , s o q uiver ˜ S 4 has x 5 , x 2 , x 3 , x 4 resp ectively at no des 1 to 4. Mutation at no de 1 w ould just ta ke us back to quiver S 4 (as noted in the ab ov e r emark). W e compare the exchange relations w e would obtain by m utating a t nodes 2 or 3. Mutation at no de 2 would lead to exc hange relation x 2 ˜ x 2 = x 3 x 5 + x 2 4 , (5) whilst that at no de 3 would lead to x 3 ˜ x 3 = x 2 x 2 5 + x 3 4 . (6) W e see that the righ t hand sides of formula (4) and (5) are related by a shift , whilst form ula (6) is en tirely differ ent . In fact, it can be seen in Figures 1 and 2 that the configura tion o f arr ows at no de 2 of quiver ˜ S 4 is exactly the same as that at no de 1 o f quiver S 4 , th us giving the same exchange rela tion. In fact, we hav e more. The whole quiver ˜ S 4 is obta ine d from S 4 by just rotating the arro ws , whilst keeping the no des fixe d. It follo ws that mutation of quiver ˜ S 4 at no de 2 just lea ds to a further rotation, with node 3 inheriting this same configuratio n of arrows. If a t each s tep we r e lab el ˜ x n as x n +4 , the n th exchange relation can be wr itten x n x n +4 = x n +1 x n +3 + x 2 n +2 . (7) This rotatio na l property of the quiver has lead to an iteration , which in this case is just Somos 4 . In F ordy and Ma rsh (2009) we intro duce d and studied quiv ers with this type of rotationa l pro p erty . Consider the N × N matrix ρ =       0 · · · · · · 1 1 0 . . . . . . . . . . . . 1 0       . The above rota tio n, whic h w e write S 4 → ˜ S 4 = ρS 4 , is achiev ed in the matrix formulation by ˜ B = ρB ρ − 1 , 5 with N = 4 in this case. Consider a quiver Q = Q (1), with N no des. W e consider a sequence of m utations, starting at no de 1 , follow ed by no de 2, and so o n. Mutation at no de 1 o f a quiver Q (1) will pro duce a second quiv er Q (2). The mutation a t no de 2 will therefore b e of quiver Q (2), giving rise to quiver Q (3) and so on. W e define a p erio d m quiver as follows. Definition 2. 5 A quiver Q has p erio d m if it satisfies Q ( m + 1) = ρ m Q (1) (with m the minimum such inte ger). The mutation se quen c e is depicte d by Q = Q (1) µ 1 − → Q (2) µ 2 − → · · · µ m − 1 − → Q ( m ) µ m − → Q ( m + 1) = ρ m Q (1) , (8) and c al le d the perio dic chain asso ciate d to Q . The c orr esp onding matric es would then satisfy B ( m + 1) = ρ m B (1 ) ρ − m . Remark 2. 6 (The Sequence of Mutations) We mu st p erform the correct se qu en c e of mutations. F or instanc e, if we mu tate µ 1 S 4 at no de 3 , we obtain a quiver which has 5 arr ows fr om no de 4 to no de 1 , whic h cannot b e p ermu tation e quivalent to Q (1) = S 4 . As we pr eviously saw, the c orr esp onding exchange r elation (6) was a lso differ ent . Remark 2. 7 (P e rio dicity and Iterations) Perio d 1 quivers c orr esp ond to iter ations on the r e al line. P erio d m quivers c orr esp ond to iter ations on R m . The formula (3 ) c onsists of only two t erms (additively), c orr esp onding to incoming and outgoing arr ows. Both the Somos 4 and Somos 5 iter ations c an b e built in this wa y, but not Somos 6 or 7 , whi ch c ontain 3 terms. In F ordy a nd Mars h (2009) we giv e a ful l classificatio n of p erio d 1 quivers, a partial c la ssification of p erio d 2 q uivers and ex amples of higher perio d ones. 2.3 Primitiv e Quiv ers In our cla ssification o f p erio d 1 quivers we in tro duced a sp ecial class o f quivers, called primitives . An impo rtant feature o f a primitiv e is that node 1 is a sink , so only step 1 of the m utation (Definition 2 .1) is needed. W e constr ucted a basis of primitives for each N (see Figure 3 for N = 4 , 5 ). The basis consists of P ( r ) N , 4 1 2 3 (a) P (1) 4 3 1 2 4 (b) P (2) 4 5 1 2 3 4 (c) P (1) 5 4 5 1 2 3 (d) P (2) 5 Figure 3: The p erio d 1 pr imitives for 4 a nd 5 no des. 6 for 1 ≤ r ≤ N / 2. In our classificatio n, the pr imitives are the “atoms” out of which we build the gener al p erio d 1 quiver for each N . F or given N , we can start with a n arbitrar y linear c o mbination of P ( r ) N , with integer co efficients Q 0 N = [ N/ 2] X r =1 m r P ( r ) N . If all the m r hav e the same sign, then Q 0 N already has p erio d 1, but other wise, we m ust add “corr ection terms”, which are integer com binations of primitives with N − 2 k no des (1 ≤ k ≤ [ N / 2]). O ur class ification of p erio d 1 quivers gives the formula for these co efficient s in terms o f the origina l co efficie nts m r . F or the Somos 4 quiver, we hav e m 1 = 1 , m 2 = − 2 and our formula req uir es the addition of a further tw o ar rows between no des 3 and 2 (see Figure 4). 4 1 2 3 (a) P (1) 4 3 1 2 4 (b) P (2) 4 3 1 2 4 (c) P (1) 2 3 2 1 4 (d) S 4 Figure 4: One of P (1) 4 min us t w o of P (2) 4 plus tw o of P (1) 2 gives S 4 2.4 Lauren t Prop erty vs Complete Integrabilit y Each iteration w e obtain through our co nstruction is guaran teed to hav e the Laurent pr op erty (b y the r esults of F omin a nd Zelevinsky (2002b)). How ever, only sp ecial cases are exp ected to b e completely in tegra ble in a ny sense (see V eselov (1991) for v arious definitions). F o r insta nce, the most gener al 4 node, per io d 1 quiver corr esp onds to the itera tion x n x n +4 = x r n +1 x r n +3 + x s n +2 . (9) The iterations with r = 1 , s ∈ { 0 , 1 , 2 } are ana ly sed b y Hone (200 7), who shows that these ca ses are Liouville in tegrable (even su p er-int e gr able when s = 0 , 1). W e first write (9) as a map o n the 4 − dimensiona l space with co or dinates x 0 , x 1 , x 2 , x 3 : ϕ ( x 0 , . . . , x 3 ) =  x 1 , x 2 , x 3 , x r 1 x r 3 + x s 2 x 0  . (10) The log -canonical Poisson br ack et { x i , x j } = P ij x i x j (see Gekhtman et al. 7 (2003) for a general discuss ion), where P =     0 r s r (1 + s ) − r 0 r s − s − r 0 r − r (1 + s ) − s − r 0     , (11) is invariant under the ac tio n of the map ϕ . This means tha t, if ˜ x = ϕ ( x ), then { ˜ x i , ˜ x j } = P ij ˜ x i ˜ x j . Remark 2. 8 It is an inter esting fact t hat the matrix P is (u p t o an over al l factor) the inv erse of the B matrix for the c orr esp onding quiver. Th e factor is t he Pfaffian (2 + s ) r 2 − s 2 , which actual ly vanishes in the Somos 4 c ase. Nevertheless, the matrix P , with r = 1 , s = 2 , is stil l invaria nt u nder the map. Liouville in tegra bility is defined in the same wa y as for contin uo us Hamil- tonian s y stems (see V eselov (199 1)). W e must first us e the Casimir functions (when the Poisson matrix is degenerate) to reduce to the sy mplectic leav es, whose dimension is 2 d , where d is the num b er of degrees of free dom. W e then require the existence of d , functionally indep endent Hamiltonians , h 1 , . . . , h d , which should be in involution (so { h i , h j } = 0, for all i, j ). F or the discr ete case, we have the extra requir e ment that the functions h 1 , . . . , h d are invariants of the map. This means that the map has a system of d c o mmu ting c ontinuous symmetries (the Hamiltonian flows). In the contin uous ca se w e can say that the Hamiltonian s ystem is solv able, up to quadr atu r e , but this no tio n is not carried to the discr ete case. In Hone (2007), it is shown tha t for cases r = 1 , s = 0 , 1, there are 3 independent, in v ariant functions , out of which it is po ssible to construct tw o Poisson commuting functions. Suc h systems (with additional first in tegra ls) are known as sup er-inte gr able . In the case r = 1 , s = 2 (Somos 4) the Poisson brack et is degener ate, with tw o Casimir functions, which a re not invariant under the map. Howev er, the action o f the map on these Casimirs is by a 2 − dimensional int egr a ble map (a s pe cial ca se of the symmetric QR T map (see Quisp el et al. (1988))). It is not k nown whether the map (9) is Liouville integrable for other v a lues of r and s , but some of the standard integrabilit y tests (suc h as algebr aic entr opy (see Bellon and Viallet (1999))) indicate non-in tegrabilit y . Remark 2. 9 Isolating and analysing the inte gr able c ases is one of the most inter esting out standing pr oblems. 3 The P (1) N Iteration as a Map The following iteratio n cor resp onds to the p erio d 1 primitive P (1) N with N nodes : x n x n + N = x n +1 x n + N − 1 + 1 , (12) 8 with initial co nditio ns x i = a i for 0 ≤ i ≤ N − 1. In F or dy and Ma rsh (20 09) it was shown that there exis ts a s p ecia l sequence of functions J n = x n + x n +2 x n +1 , satisfying J n + N − 1 = J n . (13) With the giv en initial conditions, w e ha ve { J i = c i : 0 ≤ i ≤ N − 2 } , together with the per io dicity condition, which can also b e written as J n = c n with c n + N − 1 = c n . Theorem 3.1 (Li ne arisation) If t he se quenc e { x n } is given by the iter ation (12), with initial c onditions { x i = a i : 0 ≤ i ≤ N − 1 } , then i t also satisfies x n + x n +2( N − 1) = S N x n + N − 1 , (14) wher e S N is a function of c 0 , · · · , c N − 2 , which is symmetric un der cyclic p er- mutations. Here we restrict to the case of even N . The firs t few S N take the fo rm S 2 = c 0 ; S 4 = c 0 c 1 c 2 − c 0 − c 1 − c 2 ; S 6 = c 0 c 1 c 2 c 3 c 4 − c 0 c 1 c 2 − c 1 c 2 c 3 − c 2 c 3 c 4 − c 3 c 4 c 0 − c 4 c 0 c 1 + c 0 + c 1 + c 2 + c 3 + c 4 . The function S N is a n invariant fun ct ion of the nonlinea r map (12), s o this linearisatio n dep ends upon the particular initial conditions . 3.1 The Log-Canonical Poisson Brack et As in the 4 th order case (10), write the N th order iteratio n (12) as a map of the space with c o ordinates ( x 0 , . . . , x N − 1 ), given b y ϕ ( x 0 , . . . , x N − 1 ) =  x 1 , . . . , x N − 1 , x 1 x N − 1 + 1 x 0  . (15) Again we seek an inv ariant Poisson brack et of log - canonical form: { x i , x j } = P ij x i x j , 0 ≤ i < j ≤ N − 1 , (16) for some constants P ij . W e seek the v alue of these constants for which this Poisson bra ck et is in v aria nt under the action of the map ϕ . W riting ˜ x = ϕ ( x ), we require { ˜ x i , ˜ x j } = P ij ˜ x i ˜ x j . The shift structure of the map (15) implies a banded structure, with P i +1 j +1 = P ij , so the undetermined co nstants are P 0 j , j = 1 , . . . , N − 1. The pr ecise for m of ˜ x N − 1 puts strong constraints o n these, which can b e determined up to an ov erall multiplicativ e constan t. 9 Lemma 3.2 F or a nontrivial Poisson br acket of the form (16) to b e invariant under the map (15), we re quir e N to b e ev en , in which c ase the c o efficients take the form P ij =  1 when i < j and i + j is o dd , 0 when i < j and i + j is even . This Poisso n br acket is non-de gener ate. Remark 3. 3 A gain, it is an inter esting fact that the matrix P is (up to an over al l factor) the in verse of the B matrix for the c orr esp onding quiver. 3.2 The P oisson Algebra of F unctions J m The indep endent functions J 0 , . . . , J N − 2 , written in ter ms of the c o ordinates ( x 0 , . . . , x N − 1 ), are given by J m = x m + x m +2 x m +1 , m = 0 , . . . , N − 3 , J N − 2 = x 0 x N − 2 + x 1 x N − 1 + 1 x 0 x N − 1 . (17) Under the actio n of the map ϕ th ey satisfy the cyclic conditions J n ◦ ϕ = J n +1 , n = 0 , . . . , N − 3 and J N − 2 ◦ ϕ = J 0 . (18) W e just need to calcula te the N − 3 brac kets { J 0 , J n } , n = 1 , . . . , N − 3, since all others fo llow through the a b ove rela tions. These a re easily calc ula ted to b e { J 0 , J 1 } = 2 J 0 J 1 − 2 , { J 0 , J 2 m − 1 } = 2 J 0 J 2 m − 1 , 2 ≤ m ≤ M − 1 , { J 0 , J 2 m } = − 2 J 0 J 2 m , 1 ≤ m ≤ M − 2 , (19) where N = 2 M . The cyclic a ction of ϕ then implies { J m , J m +1 } = 2 J m J m +1 − 2 , 1 ≤ m ≤ N − 3 , { J m , J n } = 2( − 1) m + n − 1 J m J n for 1 ≤ m ≤ n − 2 ≤ N − 4 , { J 0 , J N − 2 } = − 2 J 0 J 1 + 2 , (20) where the r elation for { J 0 , J N − 2 } was obtained from that of { J N − 3 , J N − 2 } through the action of ϕ . By taking cyclic sums of an y function of the J n we ca n build functions which are inv a r iant under the action o f ϕ . Since the Poisson brack et (1 6) (with P ij as g iven in L e mma 3.2) is non-degenera te (on a 2 M − dimensional space), our task is to select M in v a riant functions which ar e in involution . It is, in fact, easier to work with the Poisson bracket relations (19) and (2 0), which de fine a Poisson bracket on the (2 M − 1 ) dimensional J − space. The c o rresp onding Poisson matrix P is the sum o f tw o homog e neous parts: P = P 2 + P 0 , each of which is itself a Poisson matrix. Thes e therefore define a compatible pair of Poisson brack ets: 10 Definition 3. 4 (Compatible Poisson Brac k ets) The matric es P 0 , P 2 and P = P 2 + P 0 define c omp atible Poisson br ackets { f , g } i = ∇ f P i ∇ g , i = 0 , 2 and { f , g } P = ∇ f ( P 2 + P 0 ) ∇ g . W e use these brackets to define a bi-Hamiltonia n ladder (see Magri (1978)), starting with the Ca simir function of P 0 and ending with that of P 2 : ( P 2 + P 0 )( ∇ h M − ∇ h M − 1 + ∇ h M − 2 − · · · + ( − 1) M +1 ∇ h 1 ) = 0 , (21) where h k is a ho mogeneous p olynomial of degree 2 k − 1. The ho mo geneity prop erty o f P 0 , P 2 , h k leads to equation (21) decoupling into a sequence o f M + 1 homogeneous equatio ns (the bi-Hamiltonian ladder): P 0 ∇ h 1 = 0 , P 0 ∇ h k = P 2 ∇ h k − 1 , for 2 ≤ k ≤ M , and P 2 ∇ h M = 0 . (22) The functions h 1 , h M are eas y to find fro m the form of the P oisson matrices : h 1 = 2 M − 1 X k =1 J k , h M = 2 M − 1 Y k =1 J k . (23) The remaining functions, h 2 , . . . , h M − 1 , a re obtained by solving the “central” sequence o f equations (22). Since P 0 has a Casimir function, we need to c heck that the equations are compatible, in that ∇ h 1 P 2 ∇ h k − 1 = 0 . W e use the following result Lemma 3.5 (Bi -Hamiltoni an Relations) With the Poisson br ackets given by Definition 3.4, the functions h 1 , . . . , h M satisfy { h i , h j } 0 = { h i , h j − 1 } 2 and { h i , h j } 2 = { h i +1 , h j } 0 . Pro of: The ladder relations (22) imply { h i , h j } 0 = ∇ h i P 0 ∇ h j = ∇ h i P 2 ∇ h j − 1 = { h i , h j − 1 } 2 , and { h i , h j } 2 = ∇ h i P 2 ∇ h j = −∇ h j P 2 ∇ h i = −∇ h j P 0 ∇ h i +1 = ∇ h i +1 P 0 ∇ h j = { h i +1 , h j } 0 . Lemma 3.6 (Co m patibility of Equations (22)) Equations (22) ar e c om- p atible. Pro of: The compatibility co ndition ∇ h 1 P 2 ∇ h k − 1 = 0 is just { h 1 , h k − 1 } 2 = 0. The first e q uation is just { h 1 , h 1 } 2 = 0 , 11 which is o bviously satisfied, so it is p ossible to solve for h 2 . Now suppos e we hav e functions h 1 , . . . , h k − 1 . The compatibilit y condition is { h 1 , h k − 1 } 2 = { h 2 , h k − 1 } 0 = { h 2 , h k − 2 } 2 = · · · = { h s , h s } ℓ = 0 , for some s, ℓ . T o solve E quations (22) wr ite the equations in ter ms of the co or dinates ( J 0 , . . . , J 2 M − 3 , z 1 ) (with z 1 = h 1 ), after which P 0 has a co mplete row (and column) of zero s, with the non-z ero par t being inv ertible. The ab ov e c o m- patibilit y means that the final en try in the column vector P 2 ∇ h k − 1 is zero. These calculations are s traightforw ar d and give r ise to a sequence of functions of ( J 0 , . . . , J 2 M − 3 , z 1 ). These are only defined up to an additiv e function of z 1 , which can be discarded. Replacing z 1 by h 1 , we obtain the desir ed functions o f ( J 0 , . . . , J 2 M − 3 , J 2 M − 2 ). W e then hav e the following theo rem of Mag ri (1978): Theorem 3.7 (Co mplete In tegrabili ty) The functions h 1 , . . . , h M ar e in in- volution with r esp e ct to b oth of the ab ove Poisson br ackets { h i , h j } 0 = { h i , h j } 2 = 0 , and henc e { h i , h j } P = 0 . It t hen fol lows fr om Liouvil le’s the or em that t he fun ctions h 1 , . . . , h M define a c ompletely inte gr able Hamiltonian system. Pro of: Without loss o f gener ality , choo se i < j . Then { h i , h j } 0 = { h i , h j − 1 } 2 = { h i +1 , h j − 1 } 0 = · · · = { h k , h k } ℓ = 0 , for some k , ℓ . Similarly { h i , h j } 2 = { h i +1 , h j } 0 = { h i +1 , h j − 1 } 2 = · · · = { h k , h k } ℓ = 0 , for some k , ℓ . The Casim i r F unction F or m ula (21) just sta tes that the function C = h M − h M − 1 + h M − 2 − · · · + ( − 1) M +1 h 1 (24) is the Casimir of the P oisson matrix P , so (with resp ect to { , } P ) commutes with each J i and hence with al l functions of J i (not just with h 1 , . . . , h M ). Example 3.8 (The Case N = 4 ) Here we have 3 ba sic functions J 0 , J 1 , J 2 . With, M = 2, we hav e h 1 and h 2 , given by (23). Example 3.9 (The Case N = 6 ) Here we have 5 basic functions J 0 , . . . , J 4 . With, M = 3 we hav e h 1 and h 3 , given by (2 3), and h 2 = J 0 J 1 J 2 + J 1 J 2 J 3 + J 2 J 3 J 4 + J 3 J 4 J 0 + J 4 J 0 J 1 . 12 Example 3.1 0 (The Case N = 8 ) Here we have 7 basic functions J 0 , . . . , J 6 . With M = 4 , we have h 1 and h 4 , given by (2 3), and h 2 = 6 X i =0 J i J i +1 ( J i +2 + J i +4 ) , h 3 = 6 X i =0 J i J i +1 J i +2 J i +3 J i +4 , the indices her e being ta ken mo dulo 6. Remark 3. 11 It c an b e se en fr om the list fol lowing The or em 3.1, that S 4 , S 6 (r eplacing c i by J i ) ar e just C of t he ab ove examples. W e c an use t he line ar differ enc e e quation (14) to define S N , r ep e ate d ly using the formula x n x n + N = x n +1 x n + N − 1 + 1 to re write this as a funct ion of x 0 , . . . , x N − 1 . Conjecture: The Casimir function for g eneral N can also b e written a s C = x 0 + x 2( N − 1) x N − 1 written in terms of x 0 , . . . , x N − 1 . (25) 4 The Maps in Canonical Co ordinates The Poisso n br ack et (16), with the P ij being g iven by Lemma 3 .2, naturally separates the o dd and even num b ered v aria bles, from which we co nstruct re - sp ectively cano nic a l v ariables p i and q i as follows: q i = log( x 2( i − 1) ) , i = 1 , . . . , M , (26) p 1 = 1 2 log( x 1 x N − 1 ) , p i = 1 2 log  x 2 i − 1 x 2 i − 3  , i = 2 , . . . , M , (27) where N = 2 M . Defining π r = r X i =1 p i − M X i = r +1 p i , 0 ≤ r ≤ M − 1 , π M = M X i =1 p i , (so π i = log( x 2 i − 1 )) the in verse of this transformation is written x 2 r = e q r +1 , x 2 r + 1 = e π r +1 , 0 ≤ r ≤ M − 1 , (28) and the functions J k take the form J 2 r = e − π r +1 ( e q r +1 + e q r +2 ) , 0 ≤ r ≤ M − 2 , J 2 r + 1 = e − q r +2 ( e π r +1 + e π r +2 ) , 0 ≤ r ≤ M − 2 , (29) J 2 M − 2 = e − q 1 − π M ( e q 1 + q M + e π 1 + π M + 1) . 13 The map ϕ (see (15 )) is ca nonical, now having the form ˜ q r = π r , 1 ≤ r ≤ M ˜ p 1 = 1 2 ( q 2 − q 1 + Log (1 + e 2 p 1 )) , ˜ p M = 1 2 ( − q 1 − q M + log(1 + e 2 p 1 )) , ˜ p r = 1 2 ( q r +1 − q r ) , 2 ≤ r ≤ M − 1 . (30) The v ar iables π r , trans form a s ˜ π r = q r +1 , 1 ≤ r ≤ M − 1 , ˜ π M = − q 1 + log(1 + e ( π 1 + π M ) ) . (3 1) The functions (29 ) inherit the cyclic b ehaviour (18) under this map. Now consider the function C = M − 1 X i =1 e − π i ( e − q i + e − q i +1 ) + e − π M ( e q 1 + e − q M ) + e π M − q 1 (32) = M − 1 X i =1 e − q i +1 ( e − π i + e − π i +1 ) + e − q 1 ( e − π 1 + e π M ) + e q 1 − π M . The second line is just a re -ordering of the first, but useful. Lemma 4.1 (Symm etry under the map (30)) Under the map (30), the func- tion C is invariant: ˜ C = C . Pro of Using (31) it is e a sy to show that e − π i ( e − q i + e − q i +1 ) → e − q i +1 ( e − π i + e − π i +1 ) , e π M − q 1 → e − q 1 ( e − π 1 + e π M ) , e − π M ( e q 1 + e − q M ) → e q 1 − π M , so the fir st line o f (3 2 ) tra ns forms to the seco nd, giving the result. Theorem 4.2 (Casi mir F unction) The function C is a Casi mir function for the Poisso n algebr a of fun ctions J i . Pro of First note th at  J 0 , e − π i ( e − q i + e − q i +1 )  =  J 0 , e − π M ( e q 1 + e − q M )  =  J 0 , e π M − q 1  = 0 , so { J 0 , C } = 0. Since C is an inv ariant function under the map (30), this implies that { J r , C } = 0, for all r , giving the result. Remark 4. 3 We now have 3 expr essions for t he Casimir function of the J algebr a ((24 ), (25) and (32)), which c oincide on al l known explicit examples. However, I have no pr o of that these ar e the same. 14 5 The B¨ ac klun d T ransformati on for Liouville’s Equation Here we consider the Hamiltonian flows g enerated by the Casimir C and the first Hamiltonian h 1 . Suppos e these flows a re respe c tively par ameterised b y x and t , so w e hav e f x = { f , C } , f t = { f , h 1 } , for any function f ( q 1 , . . . , p M ) . Since these Hamiltonia ns Poisson c ommut e, their respec tive flows commute, so can b e co nsidered as co o rdinate curves on the le vel surface given b y C = c 1 , h 1 = c 2 . Cons ide r the second order par tial deriv ative q ixt = {{ q i , C } , h 1 } = {{ q i , h 1 } , C } . T o c alculate this in g e neral, we need the for m ula { q i , π j } =  1 if i ≤ j, − 1 if i ≥ j + 1 . First consider q 1 . F rom the definitions (2 9), we hav e { q 1 , J 2 r } = − J 2 r , { q 1 , J 2 r + 1 } = J 2 r + 1 , 0 ≤ r ≤ M − 2 { q 1 , J 2 M − 2 } = − J 2 M − 2 + 2 e π 1 − q 1 . (33) Since C commutes with all J k , {{ q 1 , h 1 } , C } = 2 { e π 1 − q 1 , C } . W e ha ve { e π 1 − q 1 , e − π 1 ( e − q 1 + e − q 2 ) } = 2 e − 2 q 1 , { e π 1 − q 1 , e − π i ( e − q i + e − q i +1 ) } = 0 , for i 6 = 1 , (34) { e π 1 − q 1 , e − π M ( e q 1 + e − q M ) } = 0 , { e π 1 − q 1 , e π M − q 1 } = 0 , so q 1 xt = {{ q 1 , h 1 } , C } = 4 e − 2 q 1 . Now act with ϕ on this equation (recalling the formulae (30) and (31)) to obtain Liouville’s equa tio n for each of q i , π i : q ixt = 4 e − 2 q i , π ixt = 4 e − 2 π i , i = 1 , . . . , M . (35) The B¨ ac klund transfor ma tion for this equation is well kno wn (see Rogers and Schief (2002)), but here we show how to construct it from o ur ca nonical transfor mation (30) and (31). 15 Again, first co nsider q 1 and ˜ q 1 = π 1 . Lo oking at the formulae (34), w e see that q 1 − π 1 commutes with a ll but one ter m in the ex pr ession (32) for C . The remaining term g ives { q 1 − π 1 , C } = { q 1 − π 1 , e − π 1 ( e − q 1 + e − q 2 ) } = − 2 e − q 1 − π 1 . W e no w use (33), tog ether with their consequence under the map { π 1 , J 0 } = J 0 − 2 e q 1 − π 1 , { π 1 , J 2 r } = J 2 r , for r 6 = 0 , { π 1 , J 2 r + 1 } = − J 2 r + 1 , { π 1 , J 2 M − 2 } = J 2 M − 2 , so { q 1 + π 1 , h 1 } = 2( e π 1 − q 1 − e q 1 − π 1 ) . In summary , we have shown q 1 x − ˜ q 1 x = − 2 e − q 1 − ˜ q 1 , q 1 t + ˜ q 1 t = 2 ( e ˜ q 1 − q 1 − e q 1 − ˜ q 1 ) , (36) which is the B¨ ac klund transfor ma tion for Liouville’s equation (35) (for i = 1). Again, act with ϕ on these equations to obtain q ix − π ix = − 2 e − q i − π i , q it + π it = 2 ( e π i − q i − e q i − π i ) , i = 1 , . . . , M , π ix − q i +1 x = − 2 e − π i − q i +1 , i = 1 , . . . , M − 1 , π it + q i +1 t = 2( e q i +1 − π i − e π i − q i +1 ) , i = 1 , . . . , M − 1 . W e can a ct again with ϕ , but the calculatio n is mo re complicated, since it no w inv o lves ˜ π M (see (31 )). W e get a relationship involving deriv atives of 3 v ariables ( π M , q 1 and π 1 ), but use (36) to eliminate der iv atives of π 1 = ˜ q 1 , to obta in π M x + q 1 x = − 2( e q 1 − π M − e π M − q 1 ) , π M t − q 1 t = 2 e − π M − q 1 . Notice that x, t seem to hav e reversed their ro les at this step. How ever, the next a ction of ϕ (aga in req uir ing mor e complica ted ma nipulations) brings us full circle to the o r iginal for mulae (3 6) for q 1 , π 1 . 6 Conclusions In F or dy and Mars h (2 009) a new clas s of quiver, with a certain p erio dicity prop erty , w as intro duced and par tially class ified. The corre sp onding cluster m utation relatio ns give rise to itera tions with the Laure nt pr op erty . An imp or- tant op en question is the classificatio n of the subcla ss of such iterations which define Liouville integrable maps. T he main conten t of this pap e r is the study of one particular fa mily of suc h maps. The question of in tegrability for the g eneral class is co nsidered in F ordy and Hone (20 10). In F ordy and Marsh (2 009) we noted a s urprising co nnection b etw een our examples of p erio dic quiv ers and those which arise in the c o ntext of quiver gauge 16 the ories (see Hanany et al. (2005)). Unfortunately , I had no spa ce to describ e this here, but an ex pla nation of this is also an important o pe n question. This brings us bac k, fina lly , to Robin Bullough’s famous diagr a m. Some new boxes and co nnections are needed to incor p o rate the sub ject of this paper, but that is the nature of this dia gram, whic h will grow indefinitely and become more and more co mplex as new disco veries are ma de . References M. Bellon and C-M. Viallet. Algebra ic entropy . Comm.Math.Phys. , 204:425–3 7 , 1999. S. F omin and A. Zelevinsky . The La urent phenomenon. A dvanc es in Applie d Mathematics , 28:1 19–1 4 4, 2002a. S. F o min and A. Z elevinsky . Cluster algebra s I: F oundations. J. Amer Math So c , 15:497–5 29, 2002b. A.P . F ordy and A.N.W. Ho ne . Integrable maps and Poisson algebr as derived from cluster a lg ebras. 20 1 0. In prepa ration. A.P . F ordy and R.J. Marsh. Cluster mutation-pe r io dic quivers and ass o ciated laurent s equences. 2009. P reprint arXiv:0904.02 00 v 3 [math.CO]. T o a ppea r in the J ournal of Algebra ic Combinatorics. D. Ga le. The strange and surprising saga of the Somos sequences. Math. Intel- ligenc er , 13:40–2, 19 91. M Gek ht man, M Shapiro, and A V ainshtein. Cluster alge br as and Poisson geometry . Mosc ow Math.J , 3 :899– 934, 200 3. A. Hanany , P . Kazakopoulos , and B. W ech t. A new infinite class o f quiver gauge theories. J . High Ener gy Phys. , 08:0 54, 2005 . A.N.W. Hone. Laurent p olynomia ls and s uper integrable maps. SIGMA , 3:022, 18 pages, 2 0 07. F. Magr i. A simple mo del of the integrable Hamiltonia n equa tio n. J.Math.Phys , 19:115 6–11 62, 1978. G.R.W. Quisp el, J.A.G. Rob erts, and C.J. Thompson. Integrable mappings a nd soliton equatio ns. Phys. L ett s . A. , 126:419– 421, 1988. C. Rogers and W.K. Sc hief. B¨ acklund and Darb oux T r ansformations: Ge ometry and Mo dern Applic ations in Soliton Th e ory . CUP , Cam bridge, 2002. N.J.A. Sloane . T he On-Line Encyc lop edia of In teger Sequences. www.r ese ar ch.att.c om/ ∼ njas/se quenc es , 2009. A.P . V eselov. Int egr able maps. Russ. Math. S u rveys , 46, N5:1– 51, 1991 . 17