Symmetries in Connection Preserving Deformations

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 049, 13 pages Symmetries in Connectio n Preserving Deformations ⋆ Christoph er M. ORMER OD L a T r ob e University, Dep artment of Mathema tics and Sta tistics, Bundo or a VIC 3086, A ustr alia E-mail: C.Ormer o d@latr ob e.e du.au Received January 31, 2011, in f inal form May 18, 2 011; P ublished online May 2 4, 2 011 doi:10.38 42/SIGMA.20 11.049 Abstract. W e wish to show that the ro ot lattice of B¨ acklund tr ansformations of the q - analogue of the thir d and fourth Painlev´ e equations, which is of type ( A 2 + A 1 ) (1) , may be expressed as a quo tient of the lattice o f co nnection prese r ving deformatio ns. F urthermore, we will show v arious directions in the lattice o f connection preserv ing deformations present equiv a lent evolution e quations under suita ble transforma tions. These transforma tions cor- resp ond to the Dynkin diagr am automor phisms. Key wor ds: q -Painlev´ e; Lax pair s; q -Schlesinger transformations; connection; isomono dromy 2010 Mathematics Subje ct Classific ation: 34M55; 39A13 1 In tro duction and outline Discrete P ainlev ´ e equ ations are non-autonomous second order dif feren ce equations admitting the Pa inlev ´ e equatio ns as con tinuum limits [25]. These equations arise as con tiguit y relations for the Pa inlev ´ e equations [10], from discrete systems arising in q u an tum gravit y [9], redu ctions of d iscretizations of classically in tegrable soliton equations [8] an d v arious recurr en ce relations for orthogonal p olynomials [6]. There are many w a ys in whic h they ma y b e considered inte grable suc h as the singularit y conf inement prop erty [7], solv ability via asso ciated linear p r oblems [24] and alg ebraic en trop y [2]. A fund amental r esult of O k amoto is that the group of B¨ ac klu nd transformations of the P ainlev ´ e equations are of af f ine W eyl t yp e [19, 20, 18, 21]. Th e w ork of the Kob e group has remark ably shown th e discrete Pa inlev ´ e equations admit s im ilar r epresen tations of af f ine W eyl groups [17 ]. This unders tanding w as enhan ced by the pioneering r esu lt of Sak ai [26], who extended the work of Ok amot o on the asso ciated surface of initial conditions to the discrete P ainlev ´ e equatio ns [26]. A v aluable insight of th is work is that the B¨ ac klund transformations and the discrete P ainlev ´ e equations should be considered to b e elemen ts of th e same group. It has b een w ell established that the Pai nlev ´ e equations admit Lax r epresen tations. The underlying theory b eh in d these L ax rep r esen tations is the theory of isomono dromy [12 , 10, 11]. The f irst evidence that the q -d if ference P ainlev ´ e equations adm itted Lax r ep resen tations can b e fou n d in the work of Papag eorgiou et al. [24] wh ere it w as sho wn that a q -discrete v ers ion of the third P ainlev ´ e equation arises as the compatibilit y condition of t w o systems of q -dif ference equations, written as t w o n × n matrix equations of the form Y ( q x ) = A ( x ) Y ( x ) , (1a) ˜ Y ( x ) = R ( x ) Y ( x ) , (1b) where A ( x ) is some n × n rational matrix in x and q ∈ C is a f ixed constant such that | q | 6 = 1 and the ev olution d enoted b y tildes coincides with the evolutio n of the discrete P ainlev ´ e equa- ⋆ This p ap er is a con tribution to the Pro ceedings of th e Conference “Sym met ries and Integrabilit y of Dif ference Eq u ations (SIDE-9)” (June 14–18, 2010, V arna, Bulgaria). The full collection is av ailable at http://w ww.emis.de/j ournals/SIGMA/SIDE-9.html 2 C.M. Ormero d tion. If (1a) and (1b) form a Lax represen tatio n of a Pa inlev ´ e equ ation, th en the co mpatibilit y condition determines the ev olution of that discrete P ainlev ´ e equation [24]. The q -dif ference analogue of the concept of an isomono dr omic d eformation was pr op osed later on by Jimbo and S ak ai [13]. Th e sys tem admits t w o symbolic solutions, Y 0 ( x ) = ˆ Y 0 ( x ) D 0 ( x ) , Y ∞ ( x ) = ˆ Y ∞ ( x ) D ∞ ( x ) , where ˆ Y 0 and ˆ Y ∞ are series expansions aroun d 0 and ∞ resp ective ly and D 0 and D ∞ are comp osed of q -exp onentia l functions [27]. Sin ce Y 0 and Y ∞ are b oth fund amen tal solutions, they should b e expressible in terms of eac h other, which giv es rise to the connection matrix, C ( x ), sp ecif ied b y Y 0 ( x ) = Y ∞ ( x ) C ( x ) . (2) F or regular s y s tems of linear q -d if ference equations, such as the linear problem in [13], the solutions co n v erge under more o r le ss general conditions specif ied in [3]. Ho wev er, in the irregular case Y 0 ( x ) and Y ∞ ( x ) do not necessarily def ine holomorphic fun ctions, hence, we take this def inition is take n to b e sym b olic h ere. F or irr egular systems of q -dif ference equations w e ha v e that there exists at least one co n v ergen t solution, how ever, in general, to describ e th e solutions one w ould b e requ ired to incorp orate a q -analogue of the Stok es p henomena [28]. In th is study , we tak e a connection p reserving d eformation to b e charact erized by a transfor- mation of the form (1b). In fact, the original Lax repr esen tation p rop osed in [24] also d ef ines a connecti on preserving deformation in the sense of Jim b o and Sak a i [13]. In previous stud ies, we considered a lattice of connection preserving deformations, whic h is a lattice of sh ifts of the c haracteristic d ata which d ef ines the connection matrix [23, 22]. Ho w ev er, w e struggled to in corp orate h o w the v ario us Dynkin diagram automorphism s [17] manifested themselve s in the theory of the connection p reserving deformations. W e f in d that man y connection p r eserving d eformations are copies of the same evo lution equations. This induces a natur al automo rphism on the leve l of the evol ution equations. These automorphisms corresp ond to Dynkin diagram automorphisms. W e demonstrate the r ole of the Dynkin diagram automorph isms in a s tu dy of the asso ciated linear pr ob lem f or q - P II I and q - P VI , which p ossesses a group of B¨ ac k lu nd transformations of t yp e W (( A 2 + A 1 ) (1) ), whic h is the case of P ( A (1) 5 ) in Sak ai’s notatio n. This is an interesting case as the symmetries ha v e b een n icely studied b efore [15] and that the t w o equations p ossess the same surface of initial conditions [26] and the same asso ciated linear prob lem where th e t w o systems are dif ferent directions on th e lattice of connection preservin g deformations [23]. This lattice of connection preserving deformations admits a natural group of automorphisms of order six, corresp onding natur ally to the s ub-group of B¨ ac klu nd transformations generated b y the Dynkin diagram automorphisms. T his adv ances our understandin g of the relation b et w een the associated linear problems for q -dif f erence equatio ns and their symmetries. 2 Birkhof f theory for irregular dif ference equations Before sp ecifying the form of the solutions, let u s f ix some notation required to sp ecify the solutions. W e sp ecify some of the building blo c ks used in the Galois theory approac h to the study o f systems of linear q -dif ference equatio ns [27, 29]. Let us consider the q -P o c hhammer Symmetries in Connection Preserving Deformati ons 3 sym b ol, giv en by ( a ; q ) k =              k − 1 Q n =0 (1 − aq n ) if 0 < k < ∞ , 1 if k = 0 , ∞ Q n =0 (1 − aq n ) if k = ∞ , where w e will also u se th e notati on ( a 1 , . . . , a n ; q ) k = n Y i =1 ( a i ; q ) k . W e sp ecify th e Jacobi theta function as θ q ( x ) = X n ∈ Z q ( n 2 ) x n , whic h sati sf ies xθ q ( q x ) = θ q ( x ) . It is also useful to def ine the q -charact er as e q ,c ( x ) = θ q ( x ) θ q ( c ) θ q ( xc ) , whic h co nsequently satisf ies the pair of equations e q ,c ( q x ) = ce q ,c ( x ) , e q ,q c ( x ) = xe q ,c ( x ) . Using the ab ov e building blo c ks, it is an elemen tary task to construct solutions to any f irst order linear q -dif ference equation, hence, w e ma y transform an y system o f q -dif ference equations of the form (1a), where A ( x ) is r ational, to a system of q -dif ference equations where A ( x ) is polynomial. Hence, without loss of generalit y , w e may let A ( x ) = A 0 + A 1 x + · · · + A m x m . W e isolate the set of p oin ts in which A ( x ) is singular, i.e., where d et A ( x ) = 0, b y f ixing the determinan t as det A ( x ) = κx L ( x − a 1 ) · · · ( x − a r ) . W e will call a problem of the form (1a) r egular if A 0 and A m are diag onalizable and in vertible a nd irregular if it is not regular. The form al exp ansion for regular pr oblems ma y b e computed under certain non-resonance conditions sp ecif ied by the w ork of Birkhof f [3]. Ho w ev er, to p ass to the irregular theory , we m ust refer to the work of Adams [1], whic h wa s subsequ ently redisco v ered b y Birkh of f and Guenther [4]. Theorem 2.1. The line ar pr oblem (1a) admits two fundamental series solutions, Y 0 ( x ) and Y ∞ ( x ) , sp e cifie d by the exp ansions Y 0 ( x ) =  Y 0 + Y 1 x + Y 2 x 2 + · · ·  diag  e q ,λ i ( x ) θ ( x ) l i  , (3a) Y ∞ ( x ) =  Y ′ 0 + Y − 1 x + Y − 2 x 2 + · · ·  diag  e q ,κ i ( x ) θ ( x ) k i  , (3b) wher e κ i , λ i , l i and k i must satisfy the c ondition that if l i = l j ( k i = k j ) then λ i 6 = λ j q p ( κ i 6 = κ j q p ) for any inte ger p 6 = 0 . 4 C.M. Ormero d In general, we are r equired to choose Y 0 and Y ′ 0 to d iagonalize A 0 and A m resp ectiv ely . More details on the p ro cess of f ind ing these solutions may b e found in the work of Adams [1]. In this irregular setting th e solutions found do not necessarily def ine holomorphic fu n ctions [5]. Once these solutions are def ined, we ma y sp ecify the connection matrix via (2). Ho w ev er, w e are required to tak e (2) to b e a s ym b olic def inition at this p oint . T o characte rize the set of deformations considered , w e introdu ce the charact eristic d ata [23, 22]. Th e c h aracteristic data consists of the v ariables d ef ining th e asymp totic b eha vior of the solutions at x = 0 and x = ∞ , def ined b y (3a) and (3b), and the ro ots and p oles of A ( x ). W e denote this data b y M =  κ 1 , . . . , κ n a 1 , . . . , a r λ 1 , . . . , λ n  . Ho w ev er, wh ile this study extends previous w orks [23, 22], a task remains to fully describ e the set of solutions b y incorp orating a q -analogue of the Stok es p henomenon [28]. That is, in order to f ully mirror the theory of mono dromy , we are also required to include data that enco des the Stok es p henomenon for systems of linear q -dif ference equations [28]. W e no w tak e a deformation of (1a) to b e def ined b y (1b) , where it is easy to see that ˜ Y ( x ) m ust s atisfy ˜ Y ( q x ) =  R ( q x ) A ( x ) R ( x ) − 1  ˜ Y ( x ) = ˜ A ( x ) ˜ Y ( x ) , (4) whic h def ines ˜ A ( x ). W e tak e this new linear pr oblem to b e associated with a new connectio n matrix, ˜ C ( x ), and a new set of characte ristic data, denoted ˜ M . If the fund amen tal solutions satisfy the conditions ˜ Y 0 ( x ) = R ( x ) Y 0 ( x ) , (5a) ˜ Y ∞ ( x ) = R ( x ) Y ∞ ( x ) , (5b) then it is clear from (2) that (5a) and (5b) implies that ˜ C ( x ) = C ( x ). In the regular case, a sp ecif ic case of the co n v erse implication wa s present ed in the w ork Jim b o a nd Sak ai’s work [1 3], ho w ev er, f or the case of irregular systems of q -dif ference equations, where the fundamental solutions are n ot necessa rily holomorphic functions, the analog ous co n v erse implicatio n is n ot so clea r. If w e do tak e R ( x ) to b e rational, as rep orted in a previous wo rk [23], w e can sa y that ˜ M has an alt ered set of c haracteristic constan ts. On the lev el of th e series solutions of Ad ams [1], Birkhof f and Guenther [4], a transformation of the f orm (1 b) ma y • c hange the asymptotic b ehavi or of the fundamental solutions at x = ∞ b y letting κ i → q n κ i ; • c hange the asymptotic b eha vior of the fundament al sol utions at x = 0 b y letting λ i → q n λ i ; • c hange the p osition of a root of the determinant by letting a i → q n a i . W e ca nnot deform these v ariables arbitrarily , as there is one constraint w e must satisfy; if we consider th e d eterminan t of the left and right hand side of (1a) for the solution Y = Y 0 at x = 0, it is easily sho wn that Y κ i Y ( − a i ) = Y λ i . (6) This m ust also hold true for ˜ M . This is a constrain t on ho w we may deform the c haracteristic data. Symmetries in Connection Preserving Deformati ons 5 Con v ersely , let us supp ose that λ i and κ i are c hanged by a multiplica tion by some p ow er of q , then ˜ D 0 D − 1 0 and ˜ D ∞ D − 1 ∞ are b oth r ational functions, meaning that the expansions R ( x ) = ˜ Y 0 ( x ) Y 0 ( x ) − 1 , R ( x ) = ˜ Y ∞ ( x ) Y ∞ ( x ) − 1 , (7) present tw o dif f eren t expansions around x = 0 and x = ∞ resp ectiv ely . F urthermore, taking a determinan t of (4) as it def in es ˜ A ( x ) sp ecif ies that R ( x ) satisf ies the equatio n det R ( q x ) det R ( x ) = det ˜ A ( x ) det A ( x ) . An alternativ e c haracte rization of (4 ) is giv en b y th e w a y in wh ic h w e ha ve t wo w a ys to calcu- late ˜ Y ( q x ): ˜ Y ( q x ) = ˜ A ( x ) R ( x ) Y ( x ) , ˜ Y ( q x ) = R ( q x ) A ( x ) Y ( x ) , leading to the compatibilit y condition ˜ A ( x ) R ( x ) = R ( q x ) A ( x ) . This t yp e of co ndition app ears throughout the in teg rable literature [24, 13]. Giv en a deformation of the charact eristic constan ts, ˜ M , the ab ov e co nstitutes enough inform ation to determine R ( x ) and also determine the tr an s formation R ( x ) induces [23]. A t this p oin t, we w ish to outline the idea of th e symmetries of the asso ciated linear p roblem as one f inds a h ost of deformations of the charac teristic constan ts, whic h w e call the lattic e of connection preserving deformations, p resen ted in a previous w ork [23]. T his is an idea distinct from the w ork of J im b o and Sak ai [13] in that there is no one canonical deform ation, but rather a family of them. If one consid ers the set of all p ossible mov es, one ma y end o w this w ith a lattice structure of dimension r + 2 n − 1 . In addition to the ab o v e translations one exp ects a certain n um b er of sym metries to b e natural in this setting. O ne of the most natural groups that should b e allo w ed to act on the set of c haracteristic d ata should b e the group of p erm utations on the r ro ots [23]. W e wish to examine an add itional structure th at ma y b e obtained from the lattice. W e consider tw o systems to b e equiv alent if there exists a transform ation that maps th e ev olution of one system to the other and vice ve rsa. A classical example would b e P 34 and P II , wh ic h are related via a Miur a trans formation. W e will f in d that the transf orm ations induced by man y of the directions in the lattic e of conn ection pr eserving deformations are equiv ale n t, splitting the group in to classes of equiv alen t systems, r elated to eac h other by a transformation. Conceptually , one ma y think of these transformations that p ermute the directions of the lattice as r otations on the lattice, and that the fact that the lattice admits these rotations as b eing a prop ert y of the latti ce. 3 The asso ciated linear p roblem W e will b egin b y sp ecifying th e pr op erties of the linear system: • The matrix A ( x ) is c h osen so that det A ( x ) = κ 1 κ 2 x ( x − a 1 )( x − a 2 ) . • The solutio n at x = 0 is giv en b y Y 0 ( x ) =  Y 0 + Y 1 x + Y 2 x 2 + · · ·   e q ,λ 1 ( x ) 0 0 e q ,λ 2 ( x ) /θ q ( x )  . (8) 6 C.M. Ormero d • The solutio n at x = ∞ is giv en b y Y ∞ ( x ) =  I + Y − 1 x + Y − 2 x 2 + · · ·   e q ,κ 1 ( x ) /θ q ( x ) 2 0 0 e q ,κ 2 ( x ) /θ q ( x )  . (9) The determinan t and the v arious other asymptotics sp ecify that A ( x ) = A 0 + A 1 x + A 2 x 2 . This wo uld ordinarily lea v e u s to def ine tw elve v ariables, ho w ev er, the n orm alizatio n of the solution at x = ∞ , the determinan tal constraint and the solutions at 0, in addition to the relation κ 1 κ 2 a 1 a 2 = λ 1 λ 2 lea v e three v ariables to b e c hosen. If one c ho ose the v ariables arbitrarily , the ev olution equations are n ot guarante ed to b e n ice. So the question remains, ho w d o es one c hoose these v aria bles so that the resulting evo lution equations app ear in some sort of canonical manner? It is here that the con tin uous case lends incredible insigh t: if one refers to the w ork of Jimb o, Miw a and Ueno [12, 10, 11], it is apparent that the relev ant v ariables, almost in v ariably are c hosen so that one v ariable is the ze ro of th e upp er righ t en try of the matrix, one v ariable encapsulates the gauge freedom and one more v ariable is asso ciated with the ev aluation of the diagonal elemen ts at the ro ot of th e upp er righ t en try . That is to sa y that if w e def in e a notation for the individual entries of A ( x ) as A ( x ) = ( a i,j ( x )), then a 1 , 2 ( y ) = 0. F urth ermore, the gauge freedom is encapsulated b y letting a 1 , 2 ( x ) ∝ w ( x − y ) . There remains one more v ariable to c ho ose, and this is done so that A ( y ) =  z 1 0 ∗ z 2  , where there is a determinanta l constrain t linking z 1 to z 2 . In the con tin uous case, it is typical that this constrain t is that A ( x ) is traceless at x = y , giving z 1 = − z 2 . In our case, we will require that d et( A ( y )) = z 1 z 2 , whic h sp ecif ies one degree of freedom, chosen to b e represente d b y a v ariable, z . Remark ably , this c hoice seems to b e canonical in that the resulting evolutio n equations, after one p erform s the appropriate connection p reserving deformations, app ear to b e in a symplectic form. This completel y sp ecif ies a matrix parameterization of the form A ( x ) = κ 1 (( x − y )( x − α ) + z 1 κ 2 w ( x − y ) κ 1 w ( γ x + δ ) κ 2 ( x − y + z 2 ) ! , where α , γ and δ are fun ctions of y and z sp ecif ied by the constraints. As it is quite easily deriv able from th e ab o v e constraints, and v ariants hav e app eared b efore [16 , 23, 22], w e simply list the parameters as α = − z 1 κ 1 + ( y − z 2 ) κ 2 + λ 1 y κ 1 , γ = − 2 y − α + a 1 + a 2 + z 2 , δ = ( y α + z 1 ) ( y − z 2 ) y , z 1 = y ( y − a 1 ) z , z 2 = z ( y − a 2 ) . Symmetries in Connection Preserving Deformati ons 7 No w that we hav e a parameterization for A ( x ), to f in d the relev ant connection p reserving d e- formations using (7), w e require kn o wledge of the fun damen tal solutions. T o do th is, we simply substitute (8) and (9) int o (1a) and solv e for Y i . It is suf f icien t to compute th e f irs t few term s for compu tation of the conn ection preserving deformations sp ecif ied later on. W e will start with the expansion around x = ∞ . Let us f irs t sp ecify notation as Y k =  y ( k ) i,j  , then if we consider 0 = Y ∞ ( q x ) − A ( x ) Y ∞ , at order x − 3 w e ha ve 0 =       xκ 1  q ( y + α ) − ( q − 1) y ( − 1) 1 , 1  q x  − w κ 2 − κ 1 y ( − 1) 1 , 2  xκ 1  wy ( − 1) 2 , 1 − q γ  q w wκ 2  q y − qz 2 − ( q − 1) y ( − 1) 2 , 2  − q γ κ 1 y ( − 1) 1 , 2 q w       sp ecifying Y − 1 completely . W e could go on to calculate Y − 2 and Y − 3 , ho w ev er, in the int erest of k eeping this accoun t elegan t, w e shall stop. How ever, it suf f ices to sa y that the series solution ma y b e found to an y order. T o sp ecify the solution around x = 0, w e f irst need to def ine Y 0 . Of cour se, th er e is a non- uniqueness in ho w w e def ine Y 0 , as giv en an y matrix that diag onalizes A 0 , multiplica tion on the righ t b y an y in v ertible diagonal matrix results in another matrix th at d iagonalizes A 0 . W e c ho ose a relativ ely simple matrix to simp lify some of our cal culations, namely w e c ho ose Y 0 to b e Y 0 =  wy wy κ 2 − z y + y + z a 2 ( − z y + y + z a 2 ) κ 2 + λ 1  . Although it is q u ite simple to compute Y 1 , Y 2 and so on, the computatio ns b ecome increasingly v erb ose, hence, we will only list the ab o v e terms. 4 The symmetries of the asso ciated linear problem and the B¨ ac klund transformations No w that ev erything has b een def ined, w e ma y study the lattice of connection preserving de- formations. W e ha v e six v ariables in to tal that w e ma y deform and one constraint , sp ecif ied b y (6) . This giv es us a f iv e basis elemen ts to f in d. T o reduce some of the workload, let us consider the m ost basic of transformation: notice that if w e m ultiply the f u ndamenta l solutions b y R ( x ) = xI , where I is simply the iden tit y matrix, w e ma y absorb this into the factors, D 0 and D ∞ , b y letting the κ i → ˜ κ i = q κ i and λ i → ˜ λ i = q λ i . S ince this do es not c hange th e relativ e gauge freedom, this d o es not c hange the def in ition of y , hence, ˜ y = y , ˜ z = z and ˜ w = ˜ w . W e f ind it con v enien t to introd u ce the notation T κ 1 ,κ 2 ,λ 1 ,λ 2 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  q κ 1 q κ 2 a 1 q λ 1 q λ 2 a 2 : ˜ w , ˜ y , ˜ z  , where ˜ w = w , ˜ y = y an d ˜ z = z in accordance with the transformation. W e need to choose four separate generators. T o r educe workload once more, it is clear, in acc ordance with the parameterization, there is a symmetry we may exploit. As m en tioned ab o v e, we exp ect to b e able to p erm ute the ro ots in a natural manner, if we app r opriately def ine ˜ y , ˜ z and ˜ w . W e n ote that ˜ A ( x ) = A ( x ) if we let S a 1 ,a 2 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  κ 1 κ 2 a 2 λ 1 λ 2 a 1 : w , y , z y − a 2 y − a 1  . 8 C.M. Ormero d W e choose the follo wing additional elemen ts to rep resen t the f ive dimensional latt ice: T a 1 ,λ 1 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  κ 1 κ 2 q a 1 q λ 1 λ 2 a 2 : ˜ w, ˜ y , ˜ z  , T a 2 ,λ 1 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  κ 1 κ 2 a 1 q λ 1 λ 2 q a 2 : ˜ w, ˜ y , ˜ z  , T κ 1 ,λ 1 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  q κ 1 κ 2 a 1 q λ 1 λ 2 a 2 : ˜ w , ˜ y , ˜ z  , T κ 2 ,λ 2 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  q κ 1 κ 2 a 1 λ 1 q λ 2 a 2 : ˜ w , ˜ y , ˜ z  , where ˜ w , ˜ y and ˜ z is to b e determined in eac h case. T h ese four elemen ts and T κ 1 ,κ 2 ,λ 1 ,λ 2 form a basis for the lattice of connectio n preservin g d eform ations. Theorem 4.1. The tr ansformations T a 1 ,λ 1 , T κ 1 ,λ 1 and T κ 1 ,λ 2 ar e induc e d by tr ansformations of the line a r pr oblem ˜ Y ( x ) = R a 1 ,λ 1 ( x ) Y ( x ) , ˜ Y ( x ) = R κ 1 ,λ 1 ( x ) Y ( x ) , ˜ Y ( x ) = R κ 1 ,λ 2 ( x ) Y ( x ) , r esp e ctively, wher e R a 1 ,λ 1 =         x + q y − q a 1 + q y ( y − a 1 ) z ( a 2 − y ) x − q a 1 q w y ( y − a 1 ) z ( x − q a 1 ) ( y − a 2 ) − q ( y ( z − 1) − z a 2 ) ( y ( z − 1) + a 1 − z a 2 ) wz ( x − q a 1 ) ( y − a 2 ) x + q y  y − a 1 y z − z a 2 − 1  x − q a 1         , (10a) R κ 1 ,λ 1 ( x ) =     x + ( y ( z − 1) − z a 2 ) κ 2 y κ 1 wκ 2 κ 1 y ( z − 1) − z a 2 wy 1     , (10b) R κ 1 ,λ 2 ( x ) =     x + ( y ( z − 1) − z a 2 ) κ 2 − λ 1 y κ 1 wκ 2 κ 1 ( y ( z − 1) − z a 2 ) κ 2 − λ 1 wy κ 2 1     . (10c) Pro of . In considering R a 1 ,λ 1 , since κ 1 and κ 2 are unc hanged, we know that ˜ Y ∞ ( x ) Y ∞ ( x ) − 1 = I + O  1 x  , this, coupled with the determinan tal relation sp ecify that det R a 1 ,λ 1 = x x − a 1 , hence, w e seek a parameterization of the f orm R a 1 ,λ 1 ( x ) = xI + R 0 x − q a 1 . W e consider the compatibilit y in determinin g ˜ Y ( q x ) to b e giv en b y R ( q x ) A ( x ) = ˜ A ( x ) R ( x ). T aking the residu e at x = a 1 giv es us R 0 completely in terms of the un transformed v ariables, namely , we obtain a r epresent ation of R a 1 ,λ 1 giv en by (10a). Symmetries in Connection Preserving Deformati ons 9 T o compute R κ 1 ,λ 1 ( x ), the dif ference in asymptotic b ehavior of Y ∞ ( x ) w ith ˜ Y ∞ ( x ) is u sed as w e notice ˜ Y ∞ ( x ) Y ∞ ( x ) − 1 = ˜ ˆ Y ∞ ( x ) ˜ D ∞ ( x ) D ∞ ( x ) − 1 ˆ Y − 1 ∞ = ˜ ˆ Y ∞ ( x )  x 0 0 1  ˆ Y − 1 ∞ . The leading te rms of ˆ Y ∞ ( x ) and ˜ ˆ Y ∞ ( x ) are b oth I , hence, R κ 1 ,λ 1 is giv en by a formal expansion of the form R κ 1 ,λ 1 ( x ) = x  1 0 0 0  + R 1 + O  1 x  , where w e u s e the fact det R κ 1 ,λ 1 ( x ) = x to b ound the order of the expansion. Where the previous calc ulation allo wed a simple deriv ation of the en tries of R a 1 ,λ 1 ( x ) via the computation of the residue, there is an added dif f icult y in computing R κ 1 ,λ 1 . W e are required to lo ok at v a- rious combinatio ns of the entries of the compatibilit y relation, ˜ A ( x ) R κ 1 ,λ 1 ( x ) = R κ 1 ,λ 1 ( q x ) A ( x ). Ho w ev er, it is not a v ery dif f icult calculation. Th e result is given b y (10b). The deriv ation of R κ 1 ,λ 2 follo ws th e same pattern as the previous case.  While we ha v e mainly used the compatibilit y relation to def ine the entries of the R matrices, it is also p ossible to expand the ˜ Y ∞ ( x )( Y ∞ ( x )) − 1 to higher orders to f ind R . Ho w ev er, ho wev er one ma y determine the en tries, it b ecomes abundantly clear that w e ha v e man y more r elations than we need to def ine the entries of R in eac h case. T he remaining relations ma y b e used to express ˜ y , ˜ z and ˜ w in terms of y , z and w . Theorem 4.2. The effe c t of the tr ansfo rmations, T a 1 ,λ 1 , T κ 1 ,λ 1 and T κ 1 ,λ 2 , ar e sp e cifie d by the r elations T a 1 ,λ 1 : ˜ w = w (1 − ˜ z ) , ˜ y y = ˜ z ( ˜ z a 2 κ 2 + q λ 1 ) q ( ˜ z − 1) κ 1 , ˜ z z = q y ( y − a 1 ) κ 1 ( y − a 2 ) κ 2 , T κ 1 ,λ 1 : ˜ w = w ( ˜ z − 1 ) ( q y κ 1 − z κ 2 ) z κ 1 , ˜ yy = z ( ˜ z a 2 κ 2 + q λ 1 ) q ( ˜ z − 1) κ 1 , ˜ z = q y a 1 κ 1 + q z λ 1 q y 2 κ 1 − y z κ 2 , T κ 1 ,λ 1 : ˜ w = − w ( ˜ z − 1) , y ˜ y = ˜ z ( ˜ z a 2 κ 2 + q λ 1 ) q ( ˜ z − 1) κ 1 , ˜ z z = − q κ 1 ( z a 1 a 2 κ 2 + ( a 1 − y ) λ 1 ) κ 2 ( a 2 ( κ 2 − q y κ 1 ) + λ 1 ) . Note that in v erses of al l these transformations are ea sily computed in the form pr esen ted. W e now state that the lattice of conn ection p reserving deformations as L = h T κ 1 ,κ 2 ,λ 1 ,λ 2 , T a 1 ,λ 1 , T a 2 ,λ 1 , T κ 1 ,λ 1 , T κ 1 ,λ 2 i ∼ = Z 5 , the elemen ts ab ov e commute and form a basis for this lattice. 5 The m ultiple corresp ondences to q - P I I I and q - P IV There are tw o problems w e wish to address in this section th at are not co v ered by previous studies [23]. Firstly , th e d imension of th e lattice of connection pr eserving deformations is f ive, whereas the ro ot lattice of t yp e ( A 2 + A 1 ) (1) is three. W e also wish to look at how the v arious corresp onden ces b et w een the t w o latti ces giv es rise to the Dynkin d iagram automorph ism s. One exp ects that the set of trans lational B¨ ac k lu nd transformations em b ed naturally in this lattice of connection preserving d eformations. W e in tro d uce a similar notati on on the v ariables 10 C.M. Ormero d for the af f ine W eyl group of typ e ( A 2 + A 1 ) (1) , as found in the w ork of Ka j iwara et al. [14]. W e def ine one ref lectio n, s 1 , and one rotatio n, σ 0 , whic h generates a group of t yp e A (1) 2 , giv en by s 0 :  b 0 b 1 b 2 f 0 , f 1 , f 2  →  1 b 0 b 1 b 0 b 2 b 0 f 0 , f 1  b 0 + f 0 1 + b 0 f 0  , f 2  1 + b 0 f 0 b 0 + f 0  , σ 0 :  b 1 b 2 b 0 f 1 , f 2 , f 0  →  b 0 b 1 b 2 f 0 , f 1 , f 2  , where the other ref lections may b e def ined to b e s 1 = σ 0 ◦ s 0 ◦ σ 2 0 and s 2 = σ 2 0 ◦ s 0 ◦ σ 0 . W e also def ine some extra generators, σ 1 and w 0 , to b e σ 1 :  b 0 b 1 b 2 f 0 , f 1 , f 2  →  b 0 b 1 b 2 1 f 0 , 1 f 1 , 1 f 2  , w 0 :  b 0 b 1 b 2 f 0 , f 1 , f 2  →  b 0 b 1 b 2 b 0 b 1 ( b 2 b 0 + b 2 f 0 + f 2 f 0 ) f 2 ( b 0 b 1 + b 0 f 1 + f 0 f 1 ) , b 1 b 2 ( b 0 b 1 + b 0 f 1 + f 0 f 1 ) f 0 ( b 1 b 2 + b 1 f 2 + f 1 f 2 ) , b 1 b 2 ( b 0 b 1 + b 0 f 1 + f 0 f 1 ) f 0 ( b 1 b 2 + b 1 f 2 + f 1 f 2 )  , where we def ine one more operator, w 1 = σ 1 ◦ w 0 ◦ σ 1 . It is a g eneral r esult of [14] that the generators of G = h s 0 , s 1 , s 2 , w 0 , w 1 , σ 0 , σ 1 i satisfy all the relations of the extended af f ine W eyl group of t yp e ( A 2 + A 1 ) (1) . F ollo wing [14], there are four translatio nal comp onen ts, T 0 , . . . , T 3 , where T 0 ◦ T 1 ◦ T 2 = I , whic h ge nerates a three dimensional lattice. These are T 0 = σ 0 ◦ s 2 ◦ s 1 , T 1 = σ 0 ◦ T 1 ◦ σ 2 0 , T 2 = σ 2 0 ◦ T 1 ◦ σ 0 , T 3 = σ 1 ◦ w 0 . W e n ote that if w e def in e 1 q c 2 = f 0 f 1 f 2 and b 0 b 1 b 2 = √ q , then the subgroup, h s 0 , s 1 , s 2 , σ 0 i preserve s c , while T 3 maps c → √ q c . The task remains to mak e a corresp ondence b et w een the root lattice and the lattice of connec- tion preserving deformations. How ever, give n the theory established b y previous authors [14], T 0 , T 1 and T 2 present isomorphic ev ol utions, h en ce, w e sh ould b e ab le to determine six corre- sp ond ences b etw een the connection p r eserving deformations and the lattices. W e seek to reco v er these isomorphisms from the connectio n preserving deformation stand p oint. Let us list explicitly giv e the represent ation of T 0 : T 0 :  b 0 b 1 b 2 f 0 , f 1 , f 2  → ( √ q b 0 b 1 √ q b 2 ˜ f 0 , ˜ f 1 , ˜ f 2 ) , where ˜ f 0 f 0 = q c 2 ( b 1 + f 1 ) f 1 ( b 1 f 1 + 1) , ˜ f 1 f 1 = q c 2  b 0 ˜ f 0 + √ q  ˜ f 0  b 0 + √ q f 0  , and ˜ f 2 is def ined by ˜ f 0 ˜ f 1 ˜ f 2 = q c 2 . With regards to T 0 , we can imm ed iately mak e a corresp on- dence with T − 1 a 1 ,λ 1 as they are of similar forms. By let ting y = − a 1 b 0 f 0 , z = − f 1 b 1 , b 2 0 = a 1 a 2 , b 2 1 = − a 2 κ 2 λ 1 , c 2 = a 1 a 2 κ 1 √ q λ 1 , (11) 1 This def inition is trivially dif ferent to th at of [14], this is t o mak e a full corresp ondence b etw een the connection preserving deformation ab ov e and th e translational components of t he group of B¨ ac klund transformations. Symmetries in Connection Preserving Deformati ons 11 w e obtain a corresp onden ce b et w een the evol ution in y and z and f 0 and f 1 . The corresp ond en ce sough t b et w een T κ 1 ,λ 1 and T 0 ma y b e sp ecif ied b y y = − a 1 b 0 b 1 f 0 f 1 c 2 q 3 / 2 , z = − f 0 b 0 , b 2 0 = − a 2 κ 2 λ 1 , b 2 1 = − q λ 1 a 1 κ 2 , c 2 = a 1 a 2 κ 1 √ q λ 1 , (12) while th e corresp ondence b et w een T − 1 a 2 ,λ 1 and T 0 is sp ecif ied b y y = − a 2 b 1 f 1 , z = q a 1 κ 1 b 0 b 1 f 0 f 1 κ 2 , b 2 0 = − q λ 1 a 1 κ 2 , b 2 1 = a 1 a 2 , c 2 = a 1 a 2 κ 1 √ q λ 1 . (13) Lastly , there are secondary corresp ondences b et w een the giv en lattices. Although (11) giv es one w a y of mapping the evo lution of T − 1 a 1 ,λ 1 to T 0 , w e also ha v e that y = − a 2 b 0 f 0 , z = − 1 b 1 f 1 , b 2 0 = a 1 a 2 , b 2 1 = − a 2 κ 2 λ 1 , c 2 = λ 1 q √ q a 1 a 2 κ 1 (14) giv es another corresp ondence that inv erts the v alue of c . W e h av e similar corresp ondences that in v ert c for T − 1 a 2 ,λ 1 and T κ 1 ,λ 1 . The corresp ondences betw een th e ev olutions is summarized in T able 1. T able 1. W e list the wa y in which the lattice of connectio n preserv ing deformatio ns may a lig n with the ro ot lattice. The f irst three f ix c while the last three inv ert c . T 0 T 1 T 2 T 3 T − 1 a 1 ,λ 1 T κ 1 ,λ 1 T − 1 a 2 ,λ 1 T κ 1 ,λ 2 T κ 1 ,λ 1 T − 1 a 2 ,λ 1 T − 1 a 1 ,λ 1 T κ 1 ,λ 2 T − 1 a 2 ,λ 1 T − 1 a 1 ,λ 1 T κ 1 ,λ 1 T κ 1 ,λ 2 T − 1 a 1 ,λ 1 T κ 1 ,λ 1 T − 1 a 2 ,λ 1 T − 1 κ 1 ,λ 2 T κ 1 ,λ 1 T − 1 a 2 ,λ 1 T − 1 a 1 ,λ 1 T − 1 κ 1 ,λ 2 T − 1 a 2 ,λ 1 T − 1 a 1 ,λ 1 T κ 1 ,λ 1 T − 1 κ 1 ,λ 2 If w e consider the ele men t that c h anges corresp ondences betw een (11) and (12 ): ˜ b 2 0 = a 1 a 2 = q b 2 0 b 2 1 = b 2 2 , ˜ b 2 1 = − a 2 κ 2 λ 1 = b 2 0 , ˜ c 2 = c 2 , similarly ˜ f 0 = a 1 ˜ b 0 y = q c 2 f 0 f 1 = f 2 , ˜ f 1 = − ˜ b 1 z = − b 0 z = f 0 , hence, th e action b i → ˜ b i and f i → ˜ f i is σ 0 . Similarly , if w e consider the elemen t that c h anges corresp onden ces b etw een (11) and (14), we ha v e that ˜ b i = b i and ˜ c = 1 /q c , or m ore p recisely , q ˜ c 2 = 1 /q c 2 . The tr an s formations of the f i are giv en by ˜ f 0 = a 1 ˜ b 0 y = 1 f 0 , ˜ f 1 = − ˜ b 1 z = 1 f 1 . The transformation, b i → ˜ b i and f i → ˜ f i is σ 1 . This adds additional structure to the lattices found in previous studies [23, 22]. An in teresting secondary consequence is that the abov e table also presen ts us with a wa y in whic h w e may reduce the d im en sion of the lattice. If there is to b e a fu ll corresp ondence 12 C.M. Ormero d b et w een the connection preserving deformations and the ro ot latt ice of t yp e ( A 2 + A 1 ) (1) then the comp osition of the conn ection preserving d eformations that represent T 0 ◦ T 1 ◦ T 2 w ould b e a trivial transformation. Indeed w e f ind that T − 1 a 1 ,λ 1 ◦ T κ 1 ,λ 1 ◦ T − 1 a 2 ,λ 1 :  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : w , y , z  →  κ 1 κ 2 a 1 λ 1 λ 2 a 2 : κ 2 w κ 1 , y q , z  , whic h corresp onds to th e iden tit y element in eac h of the ca ses presente d in T able 1. Hence, w e ma y write h T 0 , T 1 , T 2 , T 3 i ∼ = L / h T κ 1 ,κ 2 ,λ 1 ,λ 2 , T − 1 a 1 ,λ 1 ◦ T κ 1 ,λ 1 ◦ T − 1 a 2 ,λ 1 i . This quotien t could b e remo v ed b y app ropriately f ixing or remo ving tw o redundant v ariables, whic h ma y allo w a more exp licit corresp ondence b et w een the v ariables asso ciated w ith the linear problem and the f i ’s and b i ’s. Using (11), w e f ind that th e symmetry , S a 1 ,a 2 is equiv alen t to the symmetry s 1 :  b 0 b 1 b 2 f 0 , f 1 , f 2  →  b 0 b 1 1 b 1 b 2 b 1 f 0 b 1 f 1 + 1 b 1 + f 1 , ˜ f 1  , while u sing (12) w e f ind s 0 and b y using (13) w e f ind s 2 . 6 Conclusion W e ha v e that there is add itional s tructure in the lattice of connection preservin g deformations as v arious dir ections on the lat tice presen t the same ev olution equations. By considering the fact that several of the translations on the lattice of connection pr eservin g deformations present equiv alent ev olution equ ations, w e r eco v er an automorphism group that p ermutes v arious direc- tions on the lattice, whic h corresp onds to the group of Dynkin diagram automo rphism s . These automorphisms could b e incorp orated int o all the lattices found in the p revious study [23]. 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