$R$-matrices and Hamiltonian Structures for Certain Lax Equations

R -matrices and Hamiltonian Structures for Certain Lax Equations Chao-Z h o ng W u ∗ Marie Curie fello w of th e Istituto Nazionale di Alta Matematica SISSA, via Bonomea 265, 34136 T rieste, Italy Abstract In this pap er a list of R -matrices on a certain coupled Lie algebra is obtained. With one of these R -matrices, we c onstruct infi nitely man y bi-Hamiltonian structures for eac h of the t w o-comp onen t BKP and the T o da lattice hierarc hies. W e also sho w that, w hen suc h t w o h ierarc h ies are reduced to their sub hierar- c hies, these bi-Hamiltonian structures are reduced corresp ond in gly . Key words : R -matrix, Hamiltonian stru ctur e, t wo-co mp on ent BKP hierarch y , T oda lattice hierarch y 1 In tro d uction The existence of Hamiltonian (or P oisson) structures rev eals very import a n t prop- ert y o f a nonlinear ev olutionary equation (see, fo r example, [9]). F or an ev olutionary equation written in Lax form, an efficien t wa y to endo w it with Hamiltonian struc- tures is the so- called classical R -matrix formalism. The classical R - matrix formalism w as prop osed by Semeno v-T y an-Shanskii [21] to construct P oisson brac k ets on a L ie algebra of an asso ciative algebra where the La x equations are defined. Generally , Semeno v-Ty an-Shanskii’s metho d yields t w o compatible Poiss on brac k ets, i.e., an y linear com bination of them is still a P o isson brac k et. Consequen tly one obta ins a bi-Hamiltonian structure of the Lax equation b y p erforming a D irac reduction [1 8] if needed. Suc h a formalism w as first established for anti-sy mmetric R -matr ices satisfy- ing the mo dified Y ang- Baxter equation [2 0]. A s later dev elop ed b y Li a nd P armentier [16], also by Oev el and Ragnisco [19], the R -matr ix formalism b ecomes av ailable for a wider clas s of R -matrices. Moreo v er, this metho d can pro duce three compatible P oisson brac k ets on an a sso ciat ive algebra. ∗ Email addres s : wucz @ sissa.it, T el: + 3 9 040 3787 352, F ax: + 39 040 3787 46 6 1 In t his pap er w e study R -matr ices on a “coupled” Lie algebra o f the form g = G − × G + , whose Lie brac k et is defined diagonally b y the Lie brack ets on G ± (see Section 3 b elow ). By solving the mo dified Y ang-Baxter equation on g , w e will deriv e a class of R -matrices, whic h include the R - matrices used in [2, 1, 13, 2 5] as particular cases. Our goal is to choose an appropriate R -matrix and apply the approac h in [16, 19] to construct Hamiltonian structures for Lax equations defined o n g . Tw o t ypical examples of suc h L a x equations, the tw o-comp onen t BKP hierarc h y [6, 17] and the T o da lattice hierarc h y [24], will b e considered. The t w o-comp onen t BKP hierarc h y , i.e., the t w o-comp onen t Kadomtsev -Pe tviash vili (KP) hierarc h y of t yp e B, w as prop osed by Date, Jim b o, K ashiw ar a and Miw a [6 ] as a bilinear equation in consideration of a neutral fr ee fermions realization of the basic represen tatio n of the Lie algebra D ∞ . This hierarc h y w as found to characterize Prym v arieties in a lg ebra g eometry [22] and D -t yp e top ological Landau-Ginzburg mo dels [23]; it is also kno wn as the univ ersal hierarch y of Drinfeld-Sokolo v hierarc hies of t yp e D, whic h are bi-Hamiltonian equations a sso ciated to unt wisted affine Ka c- Mo o dy al- gebra D (1) n with the zeroth vertex c 0 of its Dynkin diagr a m mark ed [10, 17]. Recen tly w e represen ted the t w o-comp onen t BKP hierarc h y into a La x form [17] (cf. [22]) with tw o t yp es of pseudo-differential op erators. The sets D ± of these tw o types of op erators comp ose a coupled Lie algebra D − × D + of the form g . Based on the Lax represen tatio n, in [25] we derive d a bi-Hamiltonian structure for the t w o-comp onent BKP hierarc h y b y using the R - matrix formalism. Ho w ev er, fro m this bi-Hamiltonian structure w e could not find the corresp onding reductions when the t w o-comp onent BKP hierar ch y is reduced to Drinfeld-So k o lo v hierarc hies men tioned a b o v e. T o re- solv e this pro blem, w e will c ho ose an R -matrix that is differen t from the one in [2 5] on g , then construct a series of bi-Hamiltonian structures for the tw o-comp onent BKP hi- erarc h y . An a dv an tage of these bi- Ha miltonian structures is that, they can b e reduced to the bi-Hamiltonian structures for the corresp onding Drinfeld-Sok olo v hierarc hies of t yp e D. Other reductions of the Hamilto nia n structures for the t w o-comp o nent BKP hierarc h y will also b e studied, see Section 4 b elow . As the second example, the T o da lattice hierarc h y [24] has a Lax represen ta tion on a coupled Lie algebra of the form g consisting of shift op erators. F or this hierar- c h y , there a re three compatible Hamilto nia n structures found b y Carlet [2], but the reduction prop erty of their Hamilto nian structures has not b een considered b efore. By using a n R -matrix in tro duced in the presen t pa p er, w e will show that the T o da lattice hierarc hy p ossesses infinitely many bi-Ha miltonian structures lab eled by arbi- trary p ositiv e-integer pairs ( N , M ). Particularly , suc h Hamiltonian structures in the case N = M = 1 ar e similar, but not the same, with those giv en in [2]. F urthermore, w e will sho w that the bi-Hamiltonian structure lab eled by ( N , M ) f o r the T o da lattice hierarc h y is reduced to that for the extended ( N , M )-bigraded T o da hierarch y [3 , 5] under suitable constrain t. The bi-Hamiltonia n structures to b e obtained f or the tw o- comp onen t BK P and the T o da lattice hierarchie s, in comparison with those in [2, 25], ha v e mor e a dv an tages. First, the densities of Hamiltonian functionals satisfy the tau-symmetry condition [14], hence they define a tau function o f the hierarc hy . Second, these bi-Hamilto nian struc- 2 tures ha v e imp ortant application in the study of F rob enius manifolds. In the finite- dimensional case, F rob enius manifold w as prop osed b y Dubrov in [11] as a co ordinate- free description o f the WDVV equation in 2 D top ological field t heory . Generally sp eaking, asso ciated to eve ry F rob enius manifold there is a bi-Hamiltonia n struc- ture of h ydro dynamic t yp e, which links integrable hierarc hies with relev ant researc h branc hes of mathematical ph ysics. The t heory of F rob enius manifold w as extended to the infinite-dimensional case b y Carlet, Dubrovin and Mertens [4] in consideration of a bi- Hamiltonian structure for the disp ersionless T o da lattice hierar ch y , whic h coin- cides with the disp ersionless limit of the bi-Hamiltonia n structure with N = M = 1 in Section 5 b elo w. F ollowing the approach of [4], w e constructed a class of infinite- dimensional F rob enius manifolds underlying the bi- Ha miltonian structures obtained in the pap er for the disp ersionless t w o-comp onen t BKP hierarch y [26]. F urt hermore, these infinite-dimensional F rob enius manifolds contain finite-dimensional F rob enius submanifolds corresp onding to the reductions of bi-Hamiltonian structures. This pap er is arranged as follo ws. In next se ction we review the R -matrix formalism for the construction of P oisson brack ets. In Section 3 w e deriv e a list of R -matrices on the Lie a lgebra g = G − × G + , whic h are classified according to the action of some simple in tertw ining in v olutions. With one of these R -matrices, w e apply the R -matrix formalism to the t w o-comp onent BKP and the T o da lattice hierarc hies in Section 4 and Section 5 resp ectiv ely . The reduction prop erty of these Hamiltonian structures will b e clarified. The final section is dev oted to the conclusion. 2 Classical R - matrices Let us recall some prop erties of classical R -matrices, and how to emplo y them to construct P oisson brack et on a Lie algebra. 2.1 The R -matrix formalism All con ten ts in this subsection can b e found in [15, 16, 19, 21]. Let g b e a complex Lie algebra. Definition 2.1 A linear tr a nsformation R : g → g is called an R - matrix if it defines a Lie bra c k et as [ X , Y ] R = [ R ( X ) , Y ] + [ X , R ( Y )] , X , Y ∈ g . (2.1) A sufficien t conditio n for a linear transformation R b eing a n R -matr ix is that it solv es the follo wing mo dified Y ang-Baxter equation [ R ( X ) , R ( Y )] − R ([ X , Y ] R ) = − [ X , Y ] (2.2) for an y X , Y ∈ g . 3 Assume g to b e an a ssociat ive algebra, on whic h a Lie brac k et is defined natura lly b y the commutator. W e a lso assume that there is a function h i : g → C , and it defines a non-degenerate symmetric in v ariant bilinear f o rm (inner pro duct) h , i b y h X , Y i = h X Y i = h Y X i , X , Y ∈ g . Via this inner pro duct g can b e identifie d with its dual space g ∗ . Let T g and T ∗ g denote the tangen t and the cotangent bundles of g , whose fib ers at an y p oin t A are T A g = g and T ∗ A g = g ∗ resp ectiv ely . Giv en an R -ma t r ix R on g , let R ∗ b e the adjoin t transformation of R with resp ect to the a b o v e inner pro duct. The an ti-symmetric pa rt of R reads R a = 1 2 ( R − R ∗ ) . F or an y smo oth functions f , g ∈ C ∞ ( g ), there are three brac k ets: { f , g } 1 ( A ) = 1 2  h [ A, d f ] , R (d g ) i − h [ A, d g ] , R ( d f ) i  , (2.3) { f , g } 2 ( A ) = 1 4  h [ A, d f ] , R ( A · d g + d g · A ) i − h [ A, d g ] , R ( A · d f + d f · A ) i  , (2.4) { f , g } 3 ( A ) = 1 2  h [ A, d f ] , R ( A · d g · A ) i − h [ A, d g ] , R ( A · d f · A ) i  , (2.5) where d f , d g ∈ T ∗ A g a re the gradients of f , g at A ∈ g resp ectiv ely . The brac k ets (2.3)–(2.5) are called the linear, the quadratic and the cubic brac k ets resp ectiv ely . Theorem 2.2 ([16, 19]) The f o l lowing statements hold true: (1) for an y R -matrix R , the line ar br acket is a Poisson br ac k e t; (2) if b oth R a n d its anti-symmetric p art R a solve the mo difi e d Y ang- B axter e q uation (2.2) , then the quadr atic br acket is a Poisson br acke t; (3) if R satisfies the m o difie d Y ang-Baxter e quation (2.2) then the cubic br ac k et is a Poisson b r acket. Mor e over, these thr e e Poisson br ackets ar e c omp atible wh e never al l the ab ov e c ondi- tions ar e fulfil le d. In case the asso ciativ e a lgebra g is non-comm utativ e, then the R -matrix formalism giv es no more Poisson brack ets of order higher tha n 3. Ho w ev er, if g is a comm utativ e asso ciativ e algebra, then one can ha v e P oisson brack ets with order b eing any p o sitiv e in tegers due to the followin g theorem. Theorem 2.3 ([15]) L et g b e a Poiss o n algebr a , that is , a Lie a l g ebr a of a c o mmu- tative asso ciative algebr a with Lie b r acket satisfying [ X , Y Z ] = [ X , Y ] Z + Y [ X, Z ] for al l X , Y , Z ∈ g , and endow g with a n a d -invariant inn e r pr o duct h , i b eing inva riant 4 with r esp e ct to the multiplic ation, i.e., h X Y , Z i = h X , Y Z i . I f R is an R -matrix on g , then ther e exists a se rie s of c om p atible Poisson br ack e ts defi n e d as fol lows : { f , g } r ( A ) = 1 2  h [ A, d f ] , R ( A r − 1 d g ) i − h [ A, d g ] , R ( A r − 1 d f ) i  (2.6) with f , g ∈ C ∞ ( g ) and arbitr ary p ositive in te gers r . 2.2 In tert wining op erators A linear op erator σ : g → g is said to b e inte rtw ining if it satisfies σ [ X , Y ] = [ σ X , Y ] = [ X , σ Y ] , X , Y ∈ g . (2.7) Prop osition 2.4 ([20]) If R is an R -matrix and σ is an in tertwining op er ator, then R ◦ σ is also an R - m atrix. Note that all in tertw ining op erat o rs comp ose a linear family . Hence the R - ma t rices R ◦ σ , with R fixed and σ b eing in tert wining op erators, induce a family of compatible P oisson brac k ets. Definition 2.5 A linear op erator σ : g → g is called an in tertw ining in v olution if it satisfies (2.7) and σ ◦ σ = id . Prop osition 2.6 L et R b e a sol ution of the mo difie d Y ang-Baxter e quation (2 .2) and σ b e an intertwining involution, then R ◦ σ also s o lves e quation (2.2) . Pr o of. This prop osition fo llo ws from a simple calculation: [ R ◦ σ X , R ◦ σ Y ] − R ◦ σ ( [ R ◦ σ X , Y ] + [ X, R ◦ σ Y ]) =[ R ◦ σ X , R ◦ σ Y ] − R ([ R ◦ σ X , σ Y ] + [ σX , R ◦ σ Y ]) = − [ σX , σ Y ] = − σ ◦ σ [ X , Y ] = − [ X , Y ] .  3 R -matrices on a coupl ed Lie algeb ra Let G b e a complex linear space that con tains tw o subspaces G − and G + . On eac h G ± there is a Lie brack et, and these tw o Lie brac k ets coincide with eac h o t her when re- stricted on G − ∩ G + . Moreov er, w e assume that G ± admit the following decomp ositions of Lie subalgebras: G − = ( G − ) − ⊕ ( G − ) + , G + = ( G + ) − ⊕ ( G + ) + , (3.1) 5 and ( G − ) + ⊂ ( G + ) + , ( G + ) − ⊂ ( G − ) − . In tro duce a coupled Lie algebra g = G − × G + (3.2) whose Lie brac k et is defined diagonally by the brac k ets o n G ± as [( X , ˆ X ) , ( Y , ˆ Y )] = ([ X , Y ] , [ ˆ X , ˆ Y ]) , ( X, ˆ X ) , ( Y , ˆ Y ) ∈ g . Consider linear transformations R : g → g of the form R ( X , ˆ X ) = ( a X + + b X − + c ˆ X − , d ˆ X + + e ˆ X − + f X + ) (3.3) with a, b, c, d, e, f ∈ C . Here and b elow w e use the subscripts “ ± ” to denote the pro jections onto the Lie subalgebras ( G − ) ± or ( G + ) ± resp ectiv ely . Prop osition 3.1 The tr a n sformation R i n (3.3) solves the mo difie d Y ang-Baxter e quation (2.2) if and only if ( a, b, c, d, e, f ) is one of the fol lowing : ± (1 , − 1 , − 2 , 1 , − 1 , 2 ) , ± (1 , − 1 , 2 , − 1 , 1 , 2) , (3.4) ± (1 , − 1 , 0 , 1 , 1 , 2) , ± (1 , − 1 , 0 , − 1 , − 1 , 2) , (3.5) ± (1 , 1 , − 2 , 1 , − 1 , 0) , ± (1 , 1 , 2 , − 1 , 1 , 0) , (3.6) ( ± 1 , ± 1 , 0 , ± 1 , ± 1 , 0) . (3.7) Each of these solutions gives an R -matrix on the Lie algebr a g . Pr o of. F or any X = ( X, ˆ X ) , Y = ( Y , ˆ Y ) ∈ g and R in (3.3), the mo dified Y ang-Baxter equation [ R ( X ) , R ( Y )] − R ([ X , Y ] R ) = − [ X , Y ] is expanded to − ( a 2 [ X + , Y + ] + a 2 [ X + , Y − ] + a 2 [ X − , Y + ] + b 2 [ X − , Y − ] − ( a 2 − b 2 )([ X + , Y − ] − + [ X − , Y + ] − ) + c ( − a + b + f )([ X + , ˆ Y − ] − + [ ˆ X − , Y + ] − ) − c ( d + e )[ ˆ X , ˆ Y ] − + c ( e − d − c )[ ˆ X − , ˆ Y − ] , d 2 [ ˆ X + , ˆ Y + ] + d 2 [ ˆ X + , ˆ Y − ] + d 2 [ ˆ X − , ˆ Y + ] + e 2 [ ˆ X − , ˆ Y − ] − ( d 2 − e 2 )([ ˆ X + , ˆ Y − ] − + [ ˆ X − , ˆ Y + ] − ) + f ( d − e + c )([ X + , ˆ Y − ] + + [ ˆ X − , Y + ] + ) + f ( a + b )[ X, Y ] + + f ( a − b − f )[ X + , Y + ]) = − ([ X + + X − , Y + + Y − ] , [ ˆ X + + ˆ X − , ˆ Y + + ˆ Y − ]) . Comparing the co efficien ts on b oth sides, w e ha v e a 2 = b 2 = d 2 = e 2 = 1 , (3.8) c ( − a + b + f ) = 0 , c ( e − d − c ) = 0 , c ( d + e ) = 0 , (3.9) 6 f ( a − b − f ) = 0 , f ( d − e + c ) = 0 , f ( a + b ) = 0 . (3.10) All solutions of these equations are listed in (3.4)–(3.7). Th us the prop osition is pro v ed.  There are t w o in tertw ining in v olutions σ 1 and σ 2 on g defined naturally b y σ 1 ( X , ˆ X ) = ( − X , ˆ X ) , σ 2 ( X , ˆ X ) = ( X , − ˆ X ) . (3.11) They g enerate a group G = { id , σ 1 , σ 2 , σ 1 ◦ σ 2 } of in tert wining in v olutions. Up to the action of G (see Prop osition 2.6), the solutions in eac h line of ( 3 .4)–(3.6) are equiv a lent, while the solutions in line (3.7) are divided to four equiv alence classes. Among the R -matrices giv en in Prop o sition 3.1, f rom now on we fix an R as R ( X , ˆ X ) = ( X + − X − − 2 ˆ X − , ˆ X + − ˆ X − + 2 X + ) . (3.12) This is the R -mat rix that will b e applied to construct Ha milto nian structures fo r in tegrable hierar c hies whose Lax represen tation is defined on a coupled Lie a lg ebra of the form (3.2). Before carrying out the construction, let us give some remarks at t he end of this section. Remark 3.2 The R - matrix corresp onding to the solution (1 , − 1 , 2 , − 1 , 1 , 2) in (3.4), denoted as ˜ R , w a s in tro duced b y Carlet [2] on a coupled Lie algebra of shift op erators (see Section 5 b elow). This is the R -matrix used in [2, 25] to construct Hamilto nian structures for the T o da lattice and the tw o- comp onen t BKP hierarchies . Observ e that ˜ R = R ◦ σ 2 , where R is giv en in (3.12). Remark 3.3 Ev ery solution in line (3.7) splits into R -matrices on G − or G + ; a s a corollary of Prop osition 3.1, they are the only solutions of the fo rm aX + + bX − or d ˆ X + + e ˆ X − to the mo dified Y a ng-Baxeter equation. Remark 3.4 When G − = G + , an R - matrix o f the form (3.12) w as used in [1] to pro v e the Liouville in tegrabilit y of the T o da lattice defined on semi-simple Lie algebras. In this case, G − × G − is called the classical double of the Lie algebra G − . On suc h a classical double, Dubrov in and Skrypny k [13] studied the “Adler-Kostan t-Symes” R - op erators/matrices of the form ˜ R a nd the corresp onding comm utativ e hamiltonian flo ws. It w ould b e in teresting to consider the action of in tert wining inv olutions on suc h R -op erators. 4 Hamiltonian struct ures for the t w o-comp onent BKP hierarc h y Let us emplo y the R -matr ix (3.12) to construct Hamiltonian structures for the tw o- comp onen t BKP hierarc h y . What is more, t he reduction pro p ert y of these Hamiltonian structures will b e studied. 7 4.1 Pseudo-differen tial op erators and Lax represen tation W e first review the definition of the tw o-comp onent BKP hierarc h y a nd some necessary notations. Let A b e an algebra of smo oth functions of x ∈ S 1 , o n whic h there is a deriv ation D = d / d x . Assume A to b e graded as A = Q i ≥ 0 A i suc h t ha t A i · A j ⊂ A i + j and D ( A i ) ⊂ A i +1 . Denote D =  P i ∈ Z f i D i | f i ∈ A  and consider its t w o subspaces D − = ( X i< ∞ f i D i | f i ∈ A ) , (4.1) D + =    X i ∈ Z X j ≥ m ax { 0 ,m − i } a i,j D i | a i,j ∈ A j , m ∈ Z    . (4.2) The subspaces D − and D + , equipped with a pro duct defined b y f D i · g D j = X r ≥ 0  i r  f D r ( g ) D i + j − r , f , g ∈ A , (4.3) are called the algebras of pseudo-differen tial op erator s of t he first t yp e and the second t yp e resp ectiv ely [17]. Observ e that eve ry op erator in D − , of no difference from a usual pseudo-differen tial op erator [9], has an upp er b ound for the p ow ers in D . F or an op erat o r in D + , there may b e neither an upp er nor a low er b ound for the p ow ers in D , but as t he p o w er in D decreases the degree of the co efficien t m ust increase sim ult a neously , whic h mak es D + b e closed f or the pro duct (4.3). Giv en a pseudo-differen tial o p erator A = P i ∈ Z f i D i ∈ D ± , its p ositiv e part, nega- tiv e part, residue and adjoint op erato r are defined resp ectiv ely by A + = X i ≥ 0 f i D i , A − = X i< 0 f i D i , (4.4) res A = f − 1 , A ∗ = X i ∈ Z ( − D ) i · f i . (4.5) The pro jections giv en in (4.4) induce the following decomp ositions o f subalgebras: D ± = ( D ± ) + ⊕ ( D ± ) − . (4.6) Clearly ( D − ) + ⊂ ( D + ) + and ( D + ) − ⊂ ( D − ) − . Assume { u 1 , u 3 , u 5 , . . . , ˆ u − 1 , ˆ u 1 , ˆ u 3 , . . . } to b e a set of indep enden t functions in A 0 ⊂ A ; particularly , w e assume ˆ u − 1 6 = 0. In tro duce t w o pseudo-differential op erat o rs P = D + X i ≥ 1 u i D − i , ˆ P = D − 1 ˆ u − 1 + X i ≥ 1 ˆ u i D i (4.7) suc h that P ∗ = − D P D − 1 , ˆ P ∗ = − D ˆ P D − 1 . Note that P ∈ D − , and ˆ P ∈ D + for D − 1 ˆ u − 1 = P i ≥ 0 ( − D ) i ( ˆ u − 1 ) D − i − 1 . 8 Definition 4.1 The t w o-comp onent BKP hierarc h y is defined by the following La x equations [17]: ∂ P ∂ t k = [( P k ) + , P ] , ∂ ˆ P ∂ t k = [( P k ) + , ˆ P ] , (4.8) ∂ P ∂ ˆ t k = [ − ( ˆ P k ) − , P ] , ∂ ˆ P ∂ ˆ t k = [ − ( ˆ P k ) − , ˆ P ] (4.9) with k ∈ Z odd + . The name of this hierarch y is from its equiv a lent vers ion o f bilinear equation con- structed b y Date, Jim b o, Kashiwara and Miw a [6], see equation (4 .6 8) b elo w. T o study Hamiltonian structures for the tw o-comp o nent BKP hierarc hy , w e need further preparation. By a lo cal functional on A w e mean an elemen t of the quotient space A /D ( A ), written formally as R f d x with f ∈ A . In tro duce a map h i : D → A / D ( A ) , A 7→ h A i = Z res A d x. (4.10) It can b e chec k ed that this map induces an inner pro duct on eac h of D ± b y h A, B i = h AB i = h B A i . (4.11) F or a n y subspace S ⊂ D ± , let S ∗ denote its dual space with resp ect to the inner pro duct (4.11). F or example, one has ( D ± ) ∗ = D ± ,  ( D ± ) ±  ∗ = ( D ± ) ∓ . (4.12) The spaces D ± can b e decomp osed as D ± = D ± 0 ⊕ D ± 1 , D ± ν =  A ∈ D ± | A ∗ = ( − 1) ν A  . (4.13) Note that the dual spaces of D ± ν are ( D ± ν ) ∗ = D ± 1 − ν . Ev ery elemen t of D ± ν can b e expresse d in the form X i ∈ Z  a i D 2 i + ν + D 2 i + ν a i  , a i ∈ A , then for an y l ∈ Z w e ha v e the following decomp ositions o f subspaces: D ± ν = ( D ± ν ) ≥ l ⊕ ( D ± ν ) 1 , how ev er, up to no w w e o nly obtain the bi-Hamilto nian structure (4.66)–(4 .67) for the disp ersionless Lax equations. The difficult y in studying the disp ersiv e case is the lack of a clear description for the manifold comp osed by op erators o f the form ( ˆ P 2 n ) − with ˆ P g iv en in (4.7). Remark 4.14 In [26] w e a sso ciated eac h bi-Ha milto nian structure in Theorem 4.12 to an infinite-dimensional F rob enius manifo ld M m,n consisting of La uren t series of the form ( p ( z ) 2 m , ˆ p ( z ) 2 n ). W e also sho w ed that M m,n con tains an ( m + n )-dimensional F rob enius submanifold M m,n for the Coxc eter group B m + n , whic h is asso ciated with the bi-Hamilto nian structure (4.66) –(4.67), as well as the principal hierarch y (4.65). F rom this p oin t of view, the reductions of such bi-Hamiltonian structures can b e in terpreted b y F r o b enius submanifolds. 5 Hamiltonian structures for the T o da lattice hi- erarc h y Let us apply t he R -matrix formalism to the T o da lattice hierarc hy . 19 5.1 T o da lattice hierarc h y Assume A to b e the set of discrete functions with compact supp ort o n Z , and Λ b e a shift op erator on A suc h that Λ( f ( n )) = f ( n + 1). D enote E = ( X i ∈ Z f i Λ i | f i ∈ A ) . F or A = P i ∈ Z f i Λ i ∈ E one has the follo wing notations: A ≥ k = A >k − 1 = X i ≥ k f i Λ i , A −∞ f i Λ i | f i ∈ A ) . (5.3) On eac h of E ± one in tro duces a pro duct defined b y f ( m )Λ i · g ( n )Λ j = f ( m ) g ( n + i )Λ i + j , then they b ecome asso ciativ e a lgebras, and they are Lie algebras with Lie brac k et giv en b y the commutator. It can b e c hec k ed that h [ A, B ] i = 0 f or any A, B in E − or E + , hence on each of E ± there is an in v arian t inner pro duct h A, B i = h A B i = h B A i . (5.4) In tro duce L = Λ + X i ≤ 0 u i Λ i ∈ E − , ˆ L = X i ≥− 1 ˆ u i Λ i ∈ E + (5.5) with unkno wn functions u i and ˆ u i lying in A . Definition 5.1 The T o da lattice hierarc h y is defined as [24] ∂ L ∂ t k = [( L k ) ≥ 0 , L ] , ∂ ˆ L ∂ t k = [( L k ) ≥ 0 , ˆ L ] , (5.6) ∂ L ∂ ˆ t k = [ − ( ˆ L k ) < 0 , L ] , ∂ ˆ L ∂ ˆ t k = [ − ( ˆ L k ) < 0 , ˆ L ] , (5.7) where k runs o v er all p ositive integers. Recall that the subscripts “ ≥ 0” and “ < 0” mean t he pro jections induced b y the follo wing decomp ositions of Lie subalgebras: E ± = ( E ± ) ≥ 0 ⊕ ( E ± ) < 0 . 20 5.2 Hamiltonian structures The coupled Lie algebra g in Section 3 can b e realized as E = E − × E + , (5.8) whose Lie bra ck et is defined diag o nally . On E there is an inner pro duct induced by (5.4) in the same wa y as (4.16). Consider functionals of the form F ( A ) = P n ∈ Z f ( n ), where f is a discrete function dep ending on A ∈ E . Here w e restrict o urselv es to t he functionals whose gradient lies in E , that is, there exists X ∈ E suc h that δ F ( A ) = h δ A , X i ; in this case w e write δ F ( A ) /δ A = X . The R -matrix (3.1 2) on the Lie algebra E is R ( X , ˆ X ) = ( X ≥ 0 − X < 0 − 2 ˆ X < 0 , ˆ X ≥ 0 − ˆ X < 0 + 2 X ≥ 0 ) . (5.9) Its adjoin t transformation R ∗ , satisfying h R ( X, ˆ X ) , ( Y , ˆ Y ) i = h ( X , ˆ X ) , R ∗ ( Y , ˆ Y ) i , reads R ∗ ( X , ˆ X ) = − R ( X, ˆ X ) + 2 R 0 ( X , ˆ X ) , where R 0 ( X , ˆ X ) = ( R es( X + ˆ X ) , Res( X + ˆ X )) . Th us the an ti-symmetric part of R is R a ( X , ˆ X ) = 1 2 ( R ( X , ˆ X ) − R ∗ ( X , ˆ X )) = R ( X, ˆ X ) − R 0 ( X , ˆ X ) . (5.10) Claim The transformat io n R a satisfies the mo dified Y ang -Baxter equation ( 2.2) on E . Pr o of. Sine R is a solution of the mo dified Y ang-Baxter equation, then fo r any X = ( X , ˆ X ) , Y = ( Y , ˆ Y ) ∈ E , we hav e [ R a ( X ) , R a ( Y )] − R a ([ R a ( X ) , Y ] + [ X , R a ( Y )]) + [ X , Y ] = − [ R 0 ( X ) , R ( Y ) ] − [ R ( X ) , R 0 ( Y )] + R ([ R 0 ( X ) , Y ] + [ X , R 0 ( Y )]) + R 0 ([ R ( X ) , Y ] + [ X , R ( Y )]) − R 0 ([ R 0 ( X ) , Y ] + [ X , R 0 ( Y )]) . (5.11) On the righ t-hand side, the first three terms cancel b y using [ R 0 ( X ) , R ( Y ) ] = R ([ R 0 ( X ) , Y ]) , the fourth term is equal to ( f , f ) with f =Res([ X ≥ 0 − X < 0 − 2 ˆ X < 0 , Y ] + [ X , Y ≥ 0 − Y < 0 − 2 ˆ Y < 0 ]) + Res([ ˆ X ≥ 0 − ˆ X < 0 + 2 X ≥ 0 , ˆ Y ] + [ ˆ X , ˆ Y ≥ 0 − ˆ Y < 0 + 2 Y ≥ 0 ]) =2 Res([ X ≥ 0 − ˆ X < 0 , Y ] + [ X, − Y < 0 − ˆ Y < 0 ] + [ ˆ X ≥ 0 + X ≥ 0 , ˆ Y ] + [ ˆ X , − ˆ Y < 0 + Y ≥ 0 ]) 21 =2 Res([ X , Y ≤ 0 ] − [ ˆ X , Y > 0 ] + [ X, − Y < 0 − ˆ Y < 0 ] + [ ˆ X + X, ˆ Y ≤ 0 ] + [ ˆ X , − ˆ Y < 0 + Y ≥ 0 ]) =Res([ X + ˆ X , Res( Y + ˆ Y )]) = 0 , and clearly the last term v anishes. Thus the claim is v erified.  According to Theorem 2.2, w e ha v e the following lemma. Lemma 5.2 F or arbitr ary functiona l s F an d H on E , ther e ex ist thr e e c omp atible Poisson b r ackets: { F , H } ν ( A ) =  δ F δ A , P ν  δ H δ A  , ν = 1 , 2 , 3 , (5.12) wher e A ∈ E an d the Poisson tensors P ν : T D ∗ → T D r e ad P 1 ( X , ˆ X ) =  [ − X < 0 − ˆ X < 0 , A ] + [ X, A ] ≤ 0 + [ ˆ X , ˆ A ] ≤ 0 , [ X ≥ 0 + ˆ X ≥ 0 , ˆ A ] − [ X, A ] > 0 − [ ˆ X , ˆ A ] > 0  , (5.13) P 2 ( X , ˆ X ) = 1 2  [ − ( AX + X A ) < 0 − ( ˆ A ˆ X + ˆ X ˆ A ) < 0 , A ] + A ([ X, A ] ≤ 0 + [ ˆ X , ˆ A ] ≤ 0 ) + ([ X, A ] ≤ 0 + [ ˆ X , ˆ A ] ≤ 0 ) A, [( AX + X A ) ≥ 0 + ( ˆ A ˆ X + ˆ X ˆ A ) ≥ 0 , ˆ A ] − ˆ A ([ X , A ] > 0 + [ ˆ X , ˆ A ] > 0 ) − ([ X, A ] > 0 + [ ˆ X , ˆ A ] > 0 ) ˆ A  . (5.14) P 3 ( X , ˆ X ) =  [ − ( AX A + ˆ A ˆ X ˆ A ) < 0 , A ] + A ([ X, A ] ≤ 0 + [ ˆ X , ˆ A ] ≤ 0 ) A, [( AX A + ˆ A ˆ X ˆ A ) ≥ 0 , ˆ A ] − ˆ A ([ X , A ] > 0 + [ ˆ X , ˆ A ] > 0 ) ˆ A  . (5.15) W e w ant to reduce these P oisson structures to some appropriate subsets of E on whic h the Lax equations (5.6)–(5.7) are defined. Giv en t w o arbitrary p ositiv e in tegers N and M , with L and ˆ L introduced in (5.5) w e let A = ( A, ˆ A ) = ( L N , ˆ L M ) . (5.16) All suc h op erators form a coset of E : U N ,M = (Λ N , 0) + ( E − ) − N × ( E + ) ≤ M . Lemma 5.3 On the c o set U N ,M ther e ar e two c omp atible Poisson structur es P red ν : T ∗ U N ,M → T U N ,M , ν = 1 , 2 (5.17) define d as P red 1 ( X , ˆ X ) = P 1 ( X , ˆ X ) , (5.18) 22 P red 2 ( X , ˆ X ) = P 2 ( X , ˆ X ) − ([ f , A ] , [ f , ˆ A ]) . (5.19) wher e ( X , ˆ X ) ∈ T ∗ A U N ,M and f = 1 2 (1 + Λ N )(1 − Λ N ) − 1 (Res([ X , A ] + [ ˆ X , ˆ A )) with (1 − Λ N ) − 1 = 1 + Λ N + Λ 2 N + · · · . Pr o of. W e need to perfo rm a Dir ac reduction fo r the P o isson structures P ν in Lemma 5 .2 from E to the coset U N ,M . Let us sk etch the main steps (cf. [2]). F irst, at a n y p o in t A ∈ U N ,M w e ha v e the decomp ositions o f subspaces E = T A U N ,M ⊕ V N ,M = T ∗ A U N ,M ⊕ V ∗ N ,M , where V N ,M = ( E − ) ≥ N × ( E + ) < − M , V ∗ N ,M = ( E − ) ≤− N × ( E + ) >M . Then, the P oisson tensors P ν =   P U U ν P U V ν P V U ν P V V ν   : T ∗ A U N ,M ⊕ V ∗ N ,M → T A U N ,M ⊕ V N ,M are reduced to U N ,M as P red ν = P U U ν − P U V ν ◦  P V V ν  − 1 ◦ P V U ν . After a lo ng but straightforw a rd calculation, w e conclude that, the first tensor P 1 can b e restricted to the coset directly , for P 2 one needs a correction term giv en in (5 .19) (the reduction of P 3 is not clear except for N = M = 1, see b elo w). The lemma is pro v ed.  Let { , } N ,M ν b e the Poisson brack ets o n the coset U N ,M giv en by the tensors P red ν in the ab ov e lemma. The follo wing result can b e v erified with the same metho d a s for Theorem 4.6. Theorem 5.4 Given any p ositive inte gers N and M , the T o da lattic e hier ar c h y (5.6 ) – (5.7) h a s the fol lowing bi-Hamiltonian r epr esentation: for k > 0 , ∂ F ∂ t k = { F , H k + N } N ,M 1 = { F , H k } N ,M 2 , ( 5 .20) ∂ F ∂ ˆ t k = { F , ˆ H k + M } N ,M 1 = { F , ˆ H k } N ,M 2 (5.21) with arbitr ary functional F an d Hamiltonians H k = N k h L k i , ˆ H k = M k h ˆ L k i . (5.22) 23 Remark 5.5 The densities of the Hamiltonian functionals (5.22) ar e tau-symmetric, hence they define a tau function of the T o da L a ttice hierarch y in a similar w a y a s for the t w o-comp onen t BKP hierarc h y . When N = M = 1, o n the coset U 1 , 1 there exists another P oisson structure P red 3 that is compatible with P red 1 and P red 2 . More precisely , P red 3 ( X , ˆ X ) = P 3 ( X , ˆ X ) − ([ Z, A ] , [ Z , ˆ A ]) , ( X, ˆ X ) ∈ T ∗ A U N ,M (5.23) where Z = ( A ( g Λ − 1 + h Λ − 2 ) A ) ≥ 0 with functions g and h determined b y (1 − Λ)( g ) = Res([ X , A ] + [ ˆ X , ˆ A ]) , (1 − Λ)( h ) − g (1 − Λ − 1 )(Res A ) = Res([ X, A ]Λ + [ ˆ X , ˆ A ]Λ) . In this case, the deriv ativ es ∂ /∂ t k and ∂ /∂ ˆ t k with k ≥ 2 of the T o da la ttice hierarc hy can also b e r epresen ted in to Hamiltonian flows of P red 3 . In the case N = M = 1, Carlet [2] deriv ed three Hamiltonian structures for the T o da lattice hierarc h y . Note the difference b etw een them and P red ν in (5.18), (5.1 9) and (5.23). 5.3 Hamiltonian structures for the extended bigraded T o da hierarc hies Supp ose the T o da lattice hierarc h y is constrained by L N = ˆ L M = L , (5.2 4 ) where L has t he fo rm L = Λ N + v N − 1 Λ N − 1 + v N − 2 Λ N − 2 + · · · v − M Λ − M . (5.25) In the same w a y as b efore, under the constrain t (5.24) the P o isson brac k ets in Theo- rem 5.4 are reduced to: { F X ( L ) , F Y ( L ) } 1 = h X , [ Y ≥ 0 , L ] ≤ 0 − [ Y < 0 , L ] > 0 i , (5.26) { F X ( L ) , F Y ( L ) } 2 =  X , [ − ( L Y + Y L ) < 0 , L ] + 1 2 L [ Y , L ] ≤ 0 + 1 2 [ Y , L ] ≤ 0 L  −  X , 1 2 [(1 + Λ N )(1 − Λ N ) − 1 (Res [ Y , L ])  . (5.27) These fo r mulae hav e the same expression as the bi-Hamilto nian structure for the extended ( M , N )-big raded T o da hierarc hy [5, 3]. W e recall briefly the construction of the extended bigr a ded T o da hierarch y . First, a contin uatio n needs to b e p erfo r med, that is, A m ust b e replaced by an algebra o f analytic functions of a spacial v ariable x , mean while the shift op erator Λ replaced by 24 e ǫD with D = d / d x and a small constant ǫ . Second, in order to o bta in a complete inte - grable hierarc hy star t ing f rom L in (5.25), one needs not only the ro ots L = L 1 / N and ˆ L = L 1 / M of the form (5.5), but also a logarithm op erator Log L defined in a dressing w a y , see [5 , 3] for details. Thus up to a scalar transformation of the time v ariables, the extended ( N , M )-bigraded T o da hierarc h y is comp osed of the Hamiltonian flo ws giv en b y the P oisson brack ets (5 .26)–(5.27) together with Hamiltonians (5.22) and H L k = 2 ( k − 1)!  L k − 1  Log L − 1 2  1 M + 1 N  c k − 1  , k ≥ 1 , (5.28) where c 0 = 0 and c k = 1 + 1 / 2 + · · · + 1 / k f or k ≥ 1. One can also consider Hamiltonian structures o f the disp ersionless T o da hierar- c h y . Ho we v er, in con trast to the previous section, nothing is obtained b eyond the disp ersionless limit o f the ab ov e result. Note that the disp ersionless limit of P red 2 for N = M = 1 w as written dow n in [4] (see Prop osition 3.3 there) fo r the purp ose of constructing an infinite-dimensional F rob enius manifold. It is in teresting t o generalize their construction to the case of arbitrary p ositiv e in tegers N and M . 6 Conclus ion On a Lie alg ebra g of t he fo rm (3.2 ), w e ha v e obtained a list of R -mat r ices that satisfy the mo dified Y a ng -Baxter equation. Among t hese R -matrices, the one (3.12) is selected to construct Ha milto nian structures fo r Lax equations defined on g , and these Hamiltonian structures ha v e natural but imp ortant reductions. In the examples of the t w o-comp onen t BK P and the T o da lattice hierarc hies, the bi-Ha milto nian structures and their reductions a r e closely related t o the theory of F rob enius manif o ld. In follo w- up work w e will try to apply the R -matrix f o rmalism to study other hierarc hies like m ulticomp onen t T o da lattice hierarc hies [24] and their generalizations. Ac kno wledgmen t. The author is gra teful to Boris Dubro vin, Si-Qi Liu and Y ouj in Zhang for helpful discussions and commen ts. The researc h leading the this work has re- ceiv ed specific funding under the “Y oung SISSA Scien tists’ Researc h Pro jects” sch eme 2012–201 3, pro moted by the In ternational Sc ho ol for Adv a nced Studies (SISSA), T ri- este, Italy . References [1] Ben Ab d eljelil, K. L ’int ´ egrabilit ´ e du r´ eseau d e 2-T o da sur sl n ( C ) et sa g ´ en´ eralisatio n aux alg ` ebr es de Lie semi-simples. C. R. Math. Acad. S ci. Pa ris 347 (2009), n o. 15–16, 943–9 46. [2] Carlet, G. The Hamiltonian structur es of the tw o-dimensional T o da lattice and R - matrices. Lett. Math. Phys. 71 (2005), n o. 3, 209–22 6. [3] Carlet, G. Th e extended bigraded T o d a hierarc h y . J . Phys. A: Math. Gen. 39 (2006), no. 30, 9411–9 435. 25 [4] Carlet, G.; Dubrovin, B.; Mertens, L. P . Infin ite-dimensional F rob eniu s manifolds for 2+1 in tegrable sys tems. Math. Ann . 349 (2011) , no. 1, 75–115. [5] Carlet, G.; Dubr o vin , B.; Zhang, Y. The extended T o da hierarc hy . Mosc. Math. J. 4 (2004 ), no. 2, 313–332 , 534. [6] Date, E.; Jim b o, M.; Kashiwa ra, M.; Miwa, T. T ransform ation grou p s for soliton equations. IV. A n ew hierarc hy of soliton equations of KP-t yp e. Phys. D 4 (1981/82), no. 3, 343–3 65. [7] Date, E.; Jim b o, M.; Kashiwa ra, M.; Miwa, T. T ransform ation grou p s for soliton equations. Euclidean Lie algebras and red u ction of the KP hierarc h y . P u bl. R es. In st. Math. Sci. 18 (1982), no. 3, 1077–11 10. [8] Date, E.; Kashiw ara, M.; Jimbo, M.; Miw a, T. T rans f ormation groups for soliton equations. Nonlinear in tegrable sys tems—classical th eory and quantum theory (Ky oto, 1981) , 39–11 9, W orld Sci. Publishing, Singap ore, 1983. [9] Dic key , L. A. Soliton equations and Hamiltonian s y s tems. S econd ed ition. Adv anced Series in Mathematic al Physics, 26. W orld Scienti fic Publishin g Co., Inc., Riv er Ed ge, NJ, 2003. [10] Drinfeld, V. G.; Sok olo v, V. V. Lie algebras and equations of Kortew eg-de V r ies type. (Russian) Cu rren t prob lems in m athematics, V ol. 24, 81–180, Itogi Nauki i T ekhniki, Ak ad. Nauk SS SR, Vsesoyuz. Inst. Nauc h n. i T ekhn. Inform., Mosco w, 1984. [11] Dubrovin, B. Geometry of 2D top ological field th eories. Inte grable systems and quan - tum groups (Mont ecatini T erme, 1993), 120–348, Lecture Notes in Math., 1620, Springer, Berlin, 1996. [12] Dubrovin, B.; Liu, S.-Q.; Zhang, Y. F rob enius m anifolds and cent ral inv arian ts for th e Drinfeld-Sok olo v b iHamiltonian structures. Adv. Math. 219 (2008), no. 3, 780– 837. [13] Dubrovin, B.; Skrypnyk, T. Classical double, R -op erators and negativ e flows of int e- grable hierarc hies. Pr eprin t arXiv: nlin.S I/1011 .4894. [14] Dubrovin, B.; Zhang, Y. Normal f orms of integrable PDEs, F rob enius manifolds and Gromo v-Witten inv arian ts. Preprint arXiv: math.DG/010 8160 . [15] Li, L. C. C lassical r -matrices and compatible P oisson stru ctures for Lax equations on P oisson algebras. Comm. Math. Ph ys. 203 (1999 ), n o. 3, 573– 592. [16] Li, L . C.; P armentie r, S. Nonlinear P oisson structur es and r -matrices. Comm. Math. Ph ys. 125 (1989), n o. 4, 545–563. [17] Liu, S.-Q.; W u , C.-Z.; Zhang, Y. On the Drinfeld-Soko lo v h ierarchies of D type. Intern. Math. Res. Notices, 2011, no. 8, 1952–199 6. [18] Marsden, J. E.; Ratiu, T. Reduction of Poi sson manifolds. Lett. Math. Phys. 11 (1986), 161–1 69. [19] Oev el, W.; Ragnisco, O. R -matrices and higher Poisson br ack ets for inte grable sys tems. Ph ys. A 161 (1989), no. 2, 181– 220. [20] Reyman, A. G.; Semeno v-T y an-Sh an s kii, M. A. Compatible Po isson structures for Lax equations: an r -matrix app r oac h. Ph ys. Lett. A 130 (1988), n o. 8–9 , 456–4 60. [21] Semenov- Ty an-Sh an s kii, M. A. What a classical r -matrix is. (Russian) F u nktsional. Anal. i Prilozhen. 17 (1983), no. 4, 17–33. English translation: F un ctional Anal. Appl. 17 (1983 ), 259–272 . 26 [22] Shiota, T. Pry m v arieties and soliton equations. In finite-dimensional L ie algebras and groups (Lu min y-Marseille, 1988), 407–44 8, Adv. Ser. Math. P h ys., 7, W orld Sci. Publ., T eanec k, NJ, 1989. [23] T ak asaki, K . Int egrable h ierarc h y underlying top ological Landau-Ginzburg mo d els of D -t yp e. Lett. Math. Phys. 29 (1993 ), no. 2, 111–121 . [24] Ueno, K .; T ak asaki, K. T o da lattice h ierarc h y . Group r epresen tations and systems of differen tial equ ations (T okyo , 1982), 1–95, Adv. S tud. Pure Math., 4, North-Holland, Amsterdam, 1984. [25] W u, C.-Z.; Xu, D. Bihamiltonian structure of the t wo -comp onent Kad omtsev- P etviash vili h ierarc hy of t yp e B. J . Math. Ph ys. 51 (2010), no. 6, 0635 04, 15 p p. [26] W u, C.-Z.; Xu , D. A class of in finite-dimensional F r ob enius manifolds and their sub- manifolds. In t. Math. Res. Not. IMRN 2012, no. 19, 4520– 4562. 27