Generalization of the linear r-matrix formulation through Loop coproducts

Generalizati on of th e linear r − matrix form ulation through Lo op cop ro ducts F abio Musso Departamento de F ´ ısica, Universidad de Burgos, E-090 0 1 Burgos, Spain E-mail: fmusso @ubu. es Abstract. A new method for the construction of clas s ical in tegrable systems, that we call lo op copr o duct formulation, is prese nted. W e show that the linear r − matrix formulation, the Sklyanin algebras and the reflection alg e br as can be obtained as particular sub cases of this framework. W e comment o n the p o ssible genera lizations of the r − matr ix formalism introduced through this approach. P ACS num bers: 02.30.Ik ,45.20 .J j AMS classification sc heme n um ber s: 37J3 5,70H06 1. In tro duction One of the main to ols for the construction and study of classical integrable Hamiltonian systems is t he Lax formalism [1]. In a noteworth y pap er [2], Bab elon a nd Viallet sho w ed that an y in tegrable Hamiltonian sy stem with a finite n um b er of degrees of freedom admits a Lax pair, that is tw o N × N matrices L ( λ ) , M ( λ ), whose en tries ar e functions on the phase space ( λ being the sp ectral parameter ‡ ) and suc h that the Hamiltonian ev olution of the matrix L ( λ ) is of the form dL ( λ ) dt = [ L ( λ ) , M ( λ )] . (1) Equation ( 1 ) implies that the sp ectral in v ariants of the Lax matrix L ( λ ) are conserv ed quan tities under the Hamiltonian ev olution. On the other hand Liouville in tegrability requires the conserv ed quan tities to b e in inv olution among them. In [2], Bab elon and Viallet prov ed that the in volution prop ert y for the sp ectral inv ariants of L ( λ ) is equiv alen t to the existence of a linear r − matrix form ulat io n { L ( λ ) ⊗ 1 , 1 ⊗ L ( µ ) } = [ r 12 ( λ, µ ) , L ( λ ) ⊗ 1 ] − [ r 21 ( µ, λ ) , 1 ⊗ L ( µ )] , (2) ‡ Actually , in the or ig inal pa pe r [2], the spectral parameter does not appear at all, being completely unessential. Ho wev er, since in many cases the natural La x pair formulation involv es a spectr al parameter, w e decided to present the results of [2 ] in this ca se. Gener alization of the line ar r − matrix formulation thr ough L o op c opr o ducts 2 where L ( λ ) is an N × N matrix, r 12 ( λ, µ ) is an N 2 × N 2 matrix, 1 is the N × N iden tity matrix and r 21 ( λ, µ ) = Π r 12 ( λ, µ )Π , with Π b eing the p erm utatio n op erator Π( x ⊗ y ) = y ⊗ x, ∀ x, y ∈ C N . Moreo v er, they sho w ed that if the La x matrix satisfies the equation (2), then the equation of motions asso ciated with any of the sp ectral inv arian ts of L ( λ ) admits a Lax represen tation of the form (1 ) . In this pap er w e prese n t a generalization (in a sense tha t w e will make more precise later) of the linear r − matrix formalism that we call lo op copro duct fo rm ulatio n. A preliminary vers ion of this approac h has b een presen ted in [3], a s a generalization of b oth the Gaudin algebras (see [4] a nd references therein) and the copro duct metho d [5], (see [6 ] for a recen t review on the sub ject). Here w e presen t a more general form ula tion encompassing the linear r − matrix formulation [2], the Skly a nin algebras [7] and also the reflection equation algebra [8]. While all these cases emerge as examples of the lo op copro duct form ulation applied to t he Lie–P o isson a lgebra F ( g l ( N ) ∗ ), the lo op copro duct formalism is defined for a r bit r a ry P oisson algebras. W e think that the implemen tation of this approach to non- linear P oisson algebras could lead to new examples of in tegrable systems and r epresen t s an interes ting op en problem in the field of classical finite- dimensional in tegrable systems . 2. Lo op copro duct formu lation In this section w e presen t o ur main theorem, whic h generalize the result presen ted in [3]. Let us consider a generic P oisson algebra A of dimension M with r Casimir functions C j , j = 1 , . . . , r and let us denote with { y α } M α =1 a set of generators for A with P oisson brac ke ts: { y α , y β } A = F αβ ( ~ y ) ~ y = ( y 1 , . . . , y M ) . Let B b e another Poiss o n algebra and let us denote with ~ Z a set of its generators. Theorem 1 L et us c o n sider a set of m map s dep en d ing on a p ar ameter λ ∆ ( k ) λ : A → B , k = 1 , . . . , m (3) and such that the images of the A gen e r ators satisfy the fol lowing Poisson br acke ts in B : { ∆ ( i ) λ ( y α ) , ∆ ( k ) µ ( y β ) } B = f β γ ( i, k, λ, µ, ~ Z ) F αγ (∆ ( i ) λ ( ~ y )) k > i (4) { ∆ ( k ) λ ( y α ) , ∆ ( k ) µ ( y β ) } B = g β γ ( k , λ, µ, ~ Z ) F αγ (∆ ( k ) λ ( ~ y )) + h α γ ( k , λ, µ, ~ Z ) F γ β (∆ ( k ) µ ( ~ y )) (5) for c ertain functions f β γ ( i, k, λ, µ, ~ Z ) , g β γ ( k , λ, µ, ~ Z ) , h α γ ( k , λ, µ, ~ Z ) . If the map ∆ ( i ) λ is define d on any smo oth function of the gener ators G ∈ A as: ∆ ( k ) λ ( G )( y 1 , . . . , y M )) = G (∆ ( k ) λ ( y 1 ) , . . . , ∆ ( k ) λ ( y M )) , (6) Gener alization of the line ar r − matrix formulation thr ough L o op c opr o ducts 3 then: { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( y β ) } B = 0 k > i (7) { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( C l ) } B = 0 . (8) Pro of: Let us pro ve fir st the equation (7). W e fix an arbitra ry Casimir C j of A . Since C j is a Casimir function, for any β w e m ust ha v e:  C j , y β  A = M X α =1 ∂ C j ∂ y α  y α , y β  A = M X α =1 ∂ C j ∂ y α F αβ ( ~ y ) = 0 . (9) No w w e use this equation together with (4) to prov e equation (7): { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( y β ) } B = M X α =1 ∂ C j  ∆ ( i ) λ ( ~ y )  ∂ ∆ ( i ) λ ( y α ) n ∆ ( i ) λ ( y α ) , ∆ ( k ) µ ( y β ) o B = = M X γ =1 f β γ ( i, k, λ, µ, ~ Z ) M X α =1 ∂ C j  ∆ ( i ) λ ( ~ y )  ∂ ∆ ( i ) λ ( y α ) F αγ (∆ ( i ) λ ( ~ y )) = 0 k > i. F ro m equation { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( y α ) } B = 0 k > i, it follows that { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( C l ) } B = 0 i 6 = k , so that o nly the case k = i remains to b e pro v en. In this case: { ∆ ( i ) λ ( C j ) , ∆ ( i ) µ ( C l ) } B = M X α,β =1 ∂ C j  ∆ ( i ) λ ( ~ y )  ∂ ∆ ( i ) λ ( y α ) ∂ C l  ∆ ( i ) µ ( ~ y )  ∂ ∆ ( i ) µ ( y β ) n ∆ ( i ) λ ( y α ) , ∆ ( i ) µ ( y β ) o B . By using equation (5) a nd (9), w e hav e: { ∆ ( i ) λ ( C j ) , ∆ ( i ) µ ( C l ) } B = = X β ,γ ∂ C l  ∆ ( i ) µ ( ~ y )  ∂ ∆ ( i ) µ ( y β ) g β γ ( k , λ, µ, ~ Z )   M X α =1 ∂ C j  ∆ ( i ) λ ( ~ y )  ∂ ∆ ( i ) λ ( y α ) F αγ (∆ ( i ) λ ( ~ y ))   + + X α,γ ∂ C j  ∆ ( i ) λ ( ~ y )  ∂ ∆ ( i ) λ ( y α ) h α γ ( k , λ, µ, ~ Z )   M X β =1 ∂ C l  ∆ ( i ) µ ( ~ y )  ∂ ∆ ( i ) µ ( y β ) F γ β (∆ ( i ) µ ( ~ y ))   = 0 since the t erms in paren theses v anish, whic h completes the pro of.  W e will call the maps (3) satisfying equations (4) and (5) “lo op copro ducts” and w e will sa y that the inte grable systems tha t can b e obta ined from equations (7 ) ,( 8) “admit a lo op copro duct form ula tion”. Gener alization of the line ar r − matrix formulation thr ough L o op c opr o ducts 4 3. P articular cases Let us now sho w that t he linear r − matrix form ulation (2) is a subcase of the lo op copro duct for mulation. In order to prov e this result, let us take A a s the Lie-P oisson algebra F ( g l ( N ) ∗ ), with the standard set of generators { e ij } , i, j = 1 , . . . , N : { e ij , e k l } = δ j k e il − δ il e k j . (10) The P oisson alg ebra B will b e the one on whic h the r -matrix for m ulation (2) is define d. The map ∆ λ : A → B is defined on the A generators a s ∆ λ ( e ij ) = L ij ( λ ) (11) and is extended to an a rbitrary elemen t of A through equation (6). Let us denote with E ij the canonical g l ( N ) matr ix generators  E ij  k l = δ ik δ j l and let us compute the Poisson brac ket (5) in this case: { ∆ λ ( e ij ) , ∆ λ ( e k l ) } B = { L ij ( λ ) , L k l ( µ ) } = T r  { L ( λ ) ⊗ 1 , 1 ⊗ L ( µ ) } E j i ⊗ E lk  = = T r  { [ r 12 ( λ, µ ) , L ( λ ) ⊗ 1 ] − [ r 21 ( µ, λ ) , 1 ⊗ L ( µ ) ] } E j i ⊗ E lk  = = T r  [ r ab,cd ( λ, µ )( E ab ⊗ E cd ) , L ef ( λ )( E ef ⊗ 1 )] E j i ⊗ E lk  − − T r  [ r ab,cd ( µ, λ )( E cd ⊗ E ab ) , L ef ( µ )( 1 ⊗ E ef )] E j i ⊗ E lk  = = X a r ia,k l ( λ, µ ) L aj ( λ ) − r aj,k l ( λ, µ ) L ia ( λ ) − r k a,ij ( µ, λ ) L al ( µ ) + r al,ij ( µ, λ ) L k a ( µ ) . (12) On the other hand, the righ t hand side of the equation (5), with the c hoices (10 ) and (11), is g iv en b y: X a,b g k l ab ( k , λ, µ, ~ Z ) ( δ j a L ib ( λ ) − δ ib L aj ( λ )) + h ij ab ( k , λ, µ, ~ Z ) ( δ bk L al ( µ ) − δ al L k b ( µ )) (13) and (13) coincides with the last line of (12) when g k l ab ( k , λ, µ, ~ Z ) = − r ba,kl ( λ, µ ) h ij ab ( k , λ, µ, ~ Z ) = − r ba,ij ( µ, λ ) . (14) Let us stress that the r − matrix in equation (14) can b e of dynamical ty p e, since the functions g k l ab and h ij ab are a r bit r a ry functions of the B manifold co ordinates. W e also p oint out that the images of the F ( g l ( N ) ∗ ) Casimirs C i , i = 1 , . . . , N under the lo o p copro duct ( 1 1) coincide with the sp ectral in v ariants of the L ax matrix L ( λ ): ∆ λ ( C i ) = T r( L ( λ ) i ) , i = 1 , . . . , N . When the r -matrix is non-dynamical and unitary ( r 12 ( λ, µ ) = r ( λ − µ ) = − r 21 ( µ, λ )), then equation (2) reduces to { L ( λ ) ⊗ 1 , 1 ⊗ L ( µ ) } = [ r ( λ − µ ) , L ( λ ) ⊗ 1 + 1 ⊗ L ( µ )] Gener alization of the line ar r − matrix formulation thr ough L o op c opr o ducts 5 and defines the so called Gaudin algebras (see [4] and referenc es therein). This case has b een extensiv ely considered, in the framew o rk of the loo p copro duct form ulation, in [3]. F or suc h r − matrices, o ne can also define the Skly anin algebra [7] { L ( λ ) ⊗ 1 , 1 ⊗ L ( µ ) } = [ r ( λ − µ ) , L ( λ ) ⊗ L ( µ )] , (15) whic h is quadrat ic in the Lax matrix en tries. Th e Skly anin algebras are also a sub case of the lo op copro duct form ulation with the same lo o p copro duct as defined b y (11) and the particular c hoice g k l ab ( k , λ, µ, ~ Z ) = − 1 2 X c ( r ba,kc ( λ − µ ) L cl ( µ ) + r ba,cl ( λ − µ ) L k c ( µ )) , h ij ab ( k , λ, µ, ~ Z ) = 1 2 X c ( r ic,ba ( λ − µ ) L cj ( λ ) + r cj,ba ( λ − µ ) L ic ( λ )) . This is a plain conseque nce of the fact that equation (15) can b e reform ulated into the form (2) by in tro ducing the dynamical r − matrix r 12 ( λ, µ ) = 1 2 ( r ( λ − µ )( 1 ⊗ L ( µ )) + ( 1 ⊗ L ( µ )) r ( λ − µ )) . Another example of a quadratic P oisson algebra asso ciated with a unitary non-dynamical r − matrix is the reflection algebra [8]: { L ( λ ) ⊗ 1 , 1 ⊗ L ( µ ) } = [ r ( λ − µ ) , L ( λ ) ⊗ L ( µ )] + + ( L ( λ ) ⊗ 1 ) r ( λ + µ )( 1 ⊗ L ( µ )) − ( 1 ⊗ L ( µ )) r ( λ + µ )( L ( λ ) ⊗ 1 ) . It is immediate to ch ec k that this alg ebra is also a sub case of the lo o p copro duct form ula tion with the same lo op copro duct as defined b y ( 11) provid ed that g k l ab ( k , λ, µ, ~ Z ) = X c ( r ba,kc ( λ + µ ) L bl ( µ ) − r ba,cl ( λ − µ ) L k c ( µ )) , h ij ab ( k , λ, µ, ~ Z ) = X c ( r ic,ba ( λ − µ ) L cj ( λ ) + r ic,ba ( λ + µ ) L cj ( λ )) . 4. Discussion and p ersp ectiv es While it is true that a n y in tegrable system admits, at least in a neighborho od o f generic p oints of the phase sp ace, an r − matrix formulation [2] (so that, as a conseq uence, w e can state that any integrable system admits a lo op copro duct formulation), t he pro of of the existence of the La x matrix L ( λ ) relies on the kno wledge of the action- angle co ordinates, so that, in general, finding it could b e not easier than solving explicitly t he asso ciated dynamical system. On the other hand, for some Hamiltonian systems , the r − matrix approac h t urns out to b e a precious instrumen t to establish the in tegrability of the syste m, and suc h systems can b e considered to p ossess a “natural” r − matrix form ulation. In the same sense there will b e Hamilto nia n systems that admits a “natural” lo op copro duct formulation. W e claim that the class of systems of the latter kind is p oten tially m uc h larger than those of the former one. Indeed, as w e hav e sho wn, t he linear r − matrix form ulation can b e seen as a sub case of the lo op-copro duct form ula tion when Gener alization of the line ar r − matrix formulation thr ough L o op c opr o ducts 6 (i) there is a unique lo op- copro duct ∆ λ : A → B , (ii) A is the Lie–Poiss on algebra F ( g l ( N ) ∗ ). Hence, w e can generalize the linear r − matrix approa ch b y lo oking for Hamiltonian systems admitting a natural lo op copro duct form ulat io n with (i) k > 1 lo op copro duct maps ∆ ( i ) λ : A → B , i = 1 , . . . , k , (ii) A b eing a non-linear Pois son algebra. Examples of systems of the first kind emerge natura lly in the so called copro duct metho d [5], [6 ]. Namely , in [3] it has b ee n prov en tha t all the in tegrable systems coming from the copro duct metho d admit a natural lo op copro duct fo r mulation, so that one can define k lo op copro duct maps (3) satisfying (4) a nd (5) and the in tegrabilit y of the sys tem can b e deduced fro m equations (7) and (8) . Ty pically , in suc h systems , one can asso ciate with eac h map a linear r − matrix that assures that { ∆ ( i ) λ ( C j ) , ∆ ( i ) µ ( C l ) } B = 0 (16) but notice that the following relations also hold: { ∆ ( i ) λ ( C j ) , ∆ ( k ) µ ( C l ) } B = 0 k 6 = i. (17) W e stress that finding a unique r − matrix form ula tion of the form ( 2 ) accoun ting fo r b oth (16) and (17) seems not to b e an easy task. While this example sho ws that the lo op coproduct formulation should b e indeed regarded as more general of the r − matrix one (in the sense previously explained), the second kind of generalization is, in our opinion, the really in teresting one. Therefore, we suggest that finding a lo op copro duct form ula tion for a non-linear Poiss on algebra A could lead to t he construction of new classes of in tegrable systems and w e prop ose it as an op en researc h line in the field of classical, integrable and finite-dimensional Hamiltonian systems . Ac kno wledgmen t s The author thanks A. Ballesteros, F.J. Herranz, M. P etrera and O. Ragnisco for useful discussions . This w or k w a s partially supp or t ed by the Spanish MICINN under grant MTM2007-67389 (with EU-FEDER supp ort) , b y Junta de Castilla y Le´ on (Pro ject GR224) and b y INFN- CICyT. References [1] L a x P D 19 68 Comm. Pur e Appl. Math. 21 467 [2] B ab elon O and Viallet C M 19 90 Phys. L ett . B 237 411 [3] Mus s o F 2009 Loo p Copro ducts Pr eprint nlin.SI/0907.49 2 7 [4] Sk rypnyk T 2007 J. Phys. A: Math. The or. 40 133 37 [5] B allesteros A and Ra gnisco O 1998 J . Phys. A: Math. Gen. 3 1 3791 [6] B allesteros A et al 2 009 J. Phys.: Conf. Ser. 175 0120 04 [7] F addeev L D a nd T akht a jan L A 1986 Hamiltonians metho ds in the theory of solito ns (Ber lin: Springer) [8] Sk lyanin E K 1988 J. Phys. A: Math. Gen. 21 2 375