The Poisson bracket compatible with the classical reflection equation algebra

The P oisson brac k et compatible with the classi cal reflectio n equation algebra A. V. Tsigano v St.P etersburg State Unive rsity , St.P etersburg, Russia e–mail: tsiganov @m ph.phys.spbu.ru W e introduce a family of compatible P oisson brack ets on the space of 2 × 2 p olyn omial matrices, whic h conta ins the reflection equation algebra b rac ket. Then we use it to derive a m ulti-H amiltonian structure for a set of integrable systems that includes t he X X X Heisen b erg magnet with b ou n dary conditions, the generalized T oda lattices and th e Kow alevski top. 1 In tro d uction In this pa pe r we study a class of finite-dimensional L io uville in teg r able sy stems describ ed b y the repres ent a tions of the quadratic r -matrix Poisson algebr a: { 1 T ( λ ) , 2 T ( µ ) } 0 = [ r ( λ − µ ) , 1 T ( λ ) 2 T ( µ ) ] (1.1) + 1 T ( λ ) r ( λ + µ ) 2 T ( µ ) − 2 T ( µ ) r ( λ + µ ) 1 T ( λ ) , where 1 T ( λ ) = T ( λ ) ⊗ I , 2 T ( µ ) = I ⊗ T ( µ ) and r ( λ, µ ) is a clas sical r -matr ix. The reflection equation a lgebra (1.1) a ppe a red in the quantum in verse scattering metho d [13]. Its represen ta tions play an imp orta nt r ole in the cla ssification and studies of classica l int eg rable systems (see, fo r instance, [4, 7, 8, 13] and referenc e s therein). The main result of this pap er is construction o f the Poisson brackets { ., . } 1 compatible with the bracket { ., . } 0 (1.1) in the s imples t case of the 4 × 4 r ational r -matrix r ( λ − µ ) = − η λ − µ Π , Π =     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     , η ∈ C , (1.2) and 2 × 2 matrix T ( λ ), which dep ends p olyno mially on the par ameter λ T ( λ ) =  A ( λ ) B ( λ ) C ( λ ) A ( − λ )  , deg T ( λ ) =  2 n + 1 2 n + 1 2 n − 1 2 n + 1  . (1.3) Co efficients of the entries A ( λ ) = α λ 2 n +1 + A 2 n λ 2 n + A 2 n − 1 λ 2 n − 1 . . . + A 0 , B ( λ ) = λ 2 n +1 + B n λ 2 n − 1 + B n − 1 λ 2 n − 3 . . . + B 1 λ, C ( λ ) = C n λ 2 n − 1 + . . . + C 2 λ 3 + C 1 λ, (1.4) are gener ators o f the qua dratic Poisson alge br a (1 .1 ). The leading coefficient α a nd 2 n + 1 co efficients of the det T ( λ ) d ( λ ) = det T ( λ ) = Q 2 n λ 4 n + Q 2 n − 1 λ 4 n − 2 + · · · + Q 0 . (1.5) 1 are Casimir s of the br ack et (1.1). Therefore, we hav e a 4 n + 1-dimensional space of the co effi- cients A 0 , . . . , A 2 n , B 1 , . . . , B n , C 1 , . . . , C n (1.6) with 2 n + 1 Casimir op era tors Q i , leaving us with n degrees of freedom. The use of the algebra (1.1) for the theo ry of the in teg rable systems is based on the following construction of commutativ e s ubalgebras [13, 1 4]. Let us in tro duce b o undary matrix K ( λ ) =  A ( λ ) 0 C ( λ ) D ( λ )  (1.7) where entries A ( λ ) , D ( λ ) are polynomia ls with n umerica l co efficients and C ( λ ) is arbitrary po lynomial on λ . If the p olynomia l τ ( λ ) = tr K ( λ ) T ( λ ) = X τ k λ k has n indep endent dynamical co efficients H 1 , . . . , H n only , then { τ ( λ ) , τ ( µ ) } 0 = 0 , ⇒ { H i , H j } 0 = 0 , i, j = 1 , . . . , n . These Poisson in volutive in tegra ls of motion H i define the Lio uville in tegr a ble systems, which are our generic mo de ls for the whole pa pe r . 2 The compatible brac k et The Poisson brack ets { ., . } 0 and { ., . } 1 are compa tible if every linea r combination o f them is still a Poisson brack et [3, 11]. Prop ositio n 1 The br acket (1.1) b elongs to t he fol lowing family of c omp atible Poisson br ack- ets: { B ( λ ) , B ( µ ) } k = 0 , k = 0 , 1 , { A ( λ ) , A ( µ ) } k = η λ + µ  µ 2 k B ( λ ) C ( µ ) − λ 2 k B ( µ ) C ( λ )  , { C ( λ ) , C ( µ ) } k = 2 η  ρ k ( − µ ) A ( µ ) − ρ k ( µ ) A ( − µ )  C ( λ ) − 2 η  ρ k ( − λ ) A ( λ ) − ρ k ( λ ) A ( − λ )  C ( µ ) , { A ( λ ) , B ( µ ) } k = η λ − µ  λ 2 k A ( λ ) B ( µ ) − µ 2 k A ( µ ) B ( λ )  + η λ + µ  λ 2 k A ( λ ) B ( µ ) + µ 2 k A ( − µ ) B ( λ )  − 2 η ρ k ( λ ) B ( λ ) B ( µ ) , { A ( λ ) , C ( µ ) } k = − η λ − µ  µ 2 k A ( λ ) C ( µ ) − λ 2 k A ( µ ) C ( λ )  − 2 η k λA ( λ ) C ( µ ) (2.1) − η λ + µ  µ 2 k A ( λ ) C ( µ ) + λ 2 k A ( − µ ) C ( λ )  + 2 η ρ k ( λ ) B ( λ ) C ( µ ) , { B ( λ ) , C ( µ ) } k = ηλ 2 k λ − µ  A ( − λ ) A ( µ ) − A ( λ ) A ( − µ )  + ηλ 2 k λ + µ  A ( λ ) A ( µ ) − A ( − λ ) A ( − µ )  + 2 η  ρ k ( µ ) A ( − µ ) − ρ k ( − µ ) A ( µ )  B ( λ ) . 2 Her e k = 0 , 1 and ρ k = h λ 2 k − 1 A ( λ ) B ( λ ) i is the quotient of p olynomials in variable λ over a field, such that ρ 0 = 0 , and ρ 1 = αλ + A 2 n . Proof: I t is sufficient to c heck the statement on an op en dense s ubset of the reflection equation algebra defined by the as sumption that A ( λ ) and B ( λ ) are co - prime and all double ro ots of B ( λ ) are distinct. This ass umption allows us to construct a s e paration re presentation for the r eflection equa- tion algebra (1.1). In this sp ecial repr esentation one has n pa irs of Darb oux v a riables, λ i , µ i , i = 1 , . . . , n , having the standard Poisson brack ets, { λ i , λ j } 0 = { µ i , µ j } 0 = 0 , { λ i , µ j } 0 = δ ij , (2.2) with the λ -v aria bles b eing n zeros of the p olyno mial B ( λ ) and the µ -v ar iables b eing v alues of the p olynomial A ( λ ) at those zero s, B ( ± λ i ) = 0 , µ i = η − 1 ln A ( λ i ) , i = 1 , . . . , n. (2.3) The interpola tion data (2.3) plus n + 2 identities A ( λ i ) A ( − λ i ) = − d ( λ i ) , A (0) = p Q 0 , A = αλ 2 n +1 + . . . , allow us to construct the needed s e pa ration representation for the whole algebr a: B ( λ ) = λ n Y k =1 ( λ 2 − λ 2 k ) , (2.4) A ( λ ) =  αλ + ( − 1) n √ Q 0 Q λ 2 k  n Y k =1 ( λ 2 − λ 2 k ) + n X k =1  λ ( λ − λ k )e ηµ k 2 λ 2 k + λ ( λ + λ k )e − ηµ k 2 λ 2 k  C ( λ ) = A ( λ ) A ( − λ ) − d ( λ ) B ( λ ) . Using this repr esentation we c an eas y calcula te the brack et { ., . } 1 (2.1) in ( λ, µ )-v ariables { λ i , λ j } 1 = { µ i , µ j } 1 = 0 , { λ i , µ j } 1 = λ 2 i δ ij , (2.5) In order to complete the pro of we have to check that brac kets (2.5 ) is compatible with the canonical brack e ts (2.2). The compatibility of the brack ets (2.2 ),(2.5) implies the compatibilit y of the br ack ets (1 .1),(2.1) and v ic e versa. This completes the pro o f. Remark 1 The coefficients of th e determinant d ( λ ) (1.5) are the Casimir functions fo r the bo th brackets { ., . } 0 and { ., . } 1 . It mea ns that the Poisson bracket { ., . } 1 has the sa me foliation by symplectic leav es as { ., . } 0 . Prop ositio n 2 The br ackets ( 2.1 ) may b e r ewritten in t he fol lowing r -matrix form { 1 T ( λ ) , 2 T ( µ ) } k = r ( k ) 12 ( λ, µ ) 1 T ( λ ) 2 T ( µ ) − 1 T ( λ ) 2 T ( µ ) r ( k ) 21 ( λ, µ ) (2.6) + 1 T ( λ ) s ( k ) 12 ( λ, µ ) 2 T ( µ ) − 2 T ( µ ) s ( k ) 21 ( λ, µ ) 1 T ( λ ) , 3 wher e r ( k ) 12 ( λ, µ ) = − η      λ 2 k +1 + µ 2 k +1 λ 2 − µ 2 0 0 0 0 0 µ 2 k λ − µ 0 0 λ 2 k λ − µ 0 0 0 2 ρ k ( − µ ) − 2 ρ k ( − λ ) λ 2 k +1 + µ 2 k +1 λ 2 − µ 2      r ( k ) 21 ( λ, µ ) = − η      λ 2 k +1 + µ 2 k +1 λ 2 − µ 2 0 0 0 − 2 ρ k ( λ ) 0 µ 2 k λ − µ 0 2 ρ k ( µ ) λ 2 k λ − µ 0 0 0 0 0 λ 2 k +1 + µ 2 k +1 λ 2 − µ 2      (2.7) s ( k ) 12 ( λ, µ ) = − η      λ 2 k +1 − µ 2 k +1 λ 2 − µ 2 0 0 0 − 2 ρ k ( λ ) 0 µ 2 k λ + µ 0 0 λ 2 k λ + µ 0 0 0 − 2 ρ k ( − µ ) 0 λ 2 k +1 − µ 2 k +1 λ 2 − µ 2      and s ( k ) 21 ( λ, µ ) = Π s ( k ) 12 ( µ, λ )Π . The pro o f co nsists of the stra ig htf o rward calculations . Using the separated representation (2.4 ) we can rewrite the higher or der Poisson brack ets { λ i , λ j } k = { µ i , µ j } k = 0 , { λ i , µ j } k = λ 2 k i δ ij , k = 0 , . . . , n (2.8) at the r -matr ix form (2.6). As a r e sult we o btain a family of the P o isson brackets compatible with the brack et (1.1). F or the Sklyanin r -matr ix algebra such family of the brack ets has be e n constructed in [1 7]. Prop ositio n 3 Inte gr als of motion H i fr om tr K ( λ ) T ( λ ) ar e in the bi-involution { H i , H j } 0 = { H i , H j } 1 = 0 , with r esp e ct t o the br ackets (2.1) or (2.6). Proof: Acco rding to [14] v ariables λ i , µ i (2.4) a r e the sepa rated co or dinates for the coefficients H i of the p oly no mial τ =tr K ( λ ) T ( λ ) and the sepa rated relatio ns look like tr K ( λ i ) T ( λ i ) = X τ k λ k i = A ( λ i ) e ηµ i + D ( λ i ) e − ηµ i , i = 1 , . . . , n. Remind that A ( λ ) , D ( λ ) are numerical p oly nomials and among all the co efficients τ k we hav e only n integrals of motion H i . On the other hand fro m (2.5) follows that λ i , µ i are the Darb oux-Nijenhuis v ariables fo r the brackets { ., . } 0 , 1 (2.1). So , integrals of motion H i are in the bi-inv olution with resp ect to the brackets (2.1) or (2.6) a ccording to the Theor em 3.2 from [3]. This completes the pro of. Summing up, we ha ve pro ved a bi-in volution of the integrals of motion H i using the Darb oux-Nijenhuis v ariables λ i , µ i and the sepa r ation r epresentation (2.4) for the reflection equation algebr a. 3 The Heisen b erg magnet Another imp ortant repr esentations of the quadratic Poisson algebr a with the gener ators A i , B i , C i comes a s a consequence of the co-multiplication prop erty of the reflection equa tion alg e br a (1.1). E s sentially , it means that the matrix T ( λ ) (1.3) can be factor ized in to a pro duct of elementary matrices, each con taining only one degr ee of freedom [13]. In this pictur e, our ma in 4 mo del turns o ut to b e an n -site Heisen b erg magnet with boundary conditions, which is an int eg rable lattice of n sl(2) spins with near est neighbor interaction. According to [13] the 2 × 2 Lax matrix for the g eneralized Heisenber g mag net acquir es the form T = K + ( λ ) T ( λ ) , where K + =  b 1 λ + b 0 0 λ − b 1 λ + b 0  , (3.1) and matrix T ( λ ) = n Y m =1 L m ( λ ) ! K − ( λ ) n Y m =1 L m ( − λ ) ! − 1 , (3.2) with L m ( λ ) = λ − s ( m ) 3 s ( m ) 1 + i s ( m ) 2 s ( m ) 1 − i s ( m ) 2 λ + s ( m ) 3 ! , K − ( λ ) =  a 1 λ + a 0 λ 0 − a 1 λ + a 0  , satisfies to the re fle c tion e q uation algebra at α = a 1 and η = − i. Here a 0 , 1 , b 0 , 1 and c m are arbitrar y n umbers, i 2 = − 1 and s ( m ) 3 are dynamical v ar iables on the dir e c t sum of sl(2) n s ( m ) i , s ( m ) j o 0 = ε ij k s ( m ) k , (3.3) where ε ij k is the totally skew-symmetric tensor. Substituting matrix T ( λ ) (3 .2) in to the brackets { ., . } k (2.1) one get the ov er de ter mined system of a lgebraic equa tions on the elements of the Poisson bivector P 1 asso ciated with the Poisson br ack ets { ., . } 1 : { f ( z ) , g ( z ) } 1 = h d f , P 1 dg i = X i,k P ik 1 ( z ) ∂ f ( z ) ∂ z i ∂ g ( z ) ∂ z k , (3.4) where z = ( z i , . . . , z m ) a re co or dinates on th e Poisson manifold. In our ca se z consist of co ordinates s ( m ) i on n copies of sl(2) (3.3). So lving this system of equations w e obtain the second cubic br ack et { ., . } 1 compatible with (3.3). As an example the loca l brack ets lo ok like n s ( m ) i , s ( m ) j o 1 = ε ij k s ( m ) k  ( s ( m ) 3 + c m ) 2 − 2 a 1 ( s ( m ) 1 + i s ( m ) 2 )( s ( m ) 3 + c m ) − 2 a 0 ( s ( m ) 1 + i s ( m ) 2 )  . F or the sake of br e vity w e omit the explicit form of the nonlo cal cubic br ack ets n s ( m ) i , s ( ℓ ) j o 1 in this pap er. 4 The generalized T o da lattices The T o da lattices a ppea r a s ano ther specia liz ation of our basic mo del. Let us consider gener- alized op en T o da la ttices with the Hamiltonians H g = n X i =1 p i 2 + 2 n − 1 X i =1 e q i − q i +1 + V g ( q ) , where            V B = 2 a 0 e q n V C = a 2 1 e 2 q n V D = 2 a 2 2 e q n − 1 + q n . (4.1) These T o da lattices are asso ciated with the ro ot systems B n , C n and D n [1]. According to [13, 14] the 2 × 2 Lax ma trix for the genera lized op en T o da la ttice acquires the form T open = K + ( λ ) T ( λ ) , where K + =  0 0 1 0  , ( 4 .2) 5 and matrix T ( λ ) = n Y k =1 L k ( λ ) ! K − ( λ ) n Y k =1 L k ( − λ ) ! − 1 , (4.3) with L i =  λ − p i − e q i e − q i 0  , K − ( λ ) =  2 a 2 λ 2 − ia 1 λ + a 0 (4 a 2 e q n + 1) λ 0 2 a 2 λ 2 + ia 1 λ + a 0  , satisfies to the reflection eq ua tion algebra at α = 0 and η = 1. Here p i , q i are dynamical v a riables and a k are para meters. The p olynomia l tr T open = λ 2 n +1 + n X i =1 H i λ 2( n − i ) is a generating function o f indep endent int eg rals of motion H i . The first integral H 1 coincides with one of the the Hamiltonians (4.1), but fo r the T o da la ttices of C n and D n t y p e we hav e to change v ar ia bles p n → p n − ia 1 e q n . (4.4) and p n → − p n cosh( q n ) + 1 sinh( q n ) , q n → − ln( − 2 a 2 cosh  q n + ln( − a 2 )  + 1) . (4.5) resp ectively . Substituting matr ix T ( λ ) (4.3) into the brackets { ., . } k (2.1) one ca n rewr ite the Poisson brack ets { ., . } k in ( p, q ) v ar iables. O f course, at k = 0 w e o btain canonica l brack et { q i , q j } 0 = { p i , p j } 0 = 0 , { q i , p j } 0 = δ ij . F or the T o da la ttice asso ciated with B C n ro ot system after canonical transfor mation (4.4 ) one get the following non-zero brackets i < j { q i , q j } 1 = 2 p i , i = 1 , . . . , n − 1 { p i , p i +1 } 1 = − ( p i + p i +1 ) e q i − q i +1 , { q i +1 , p i } 1 = e q i − q i +1 , { q i , p i } 1 = p 2 i + 2 e q i − q i +1 , i = 1 , . . . , n − 2 { q i , p i +1 } 1 = 2 e q i +1 − q i +2 − e q i − q i +1 , { p n , q i } 1 = 2 e q n − 1 − q n − 2 a 0 e q n − 2 a 2 1 e 2 q n , i = 3 , . . . , n − 1 { p i , q j } 1 = 2 e q i − 1 − q i − 2 e q i − q i +1 , 1 ≤ j ≤ i − 2 , (4.6) and { q n , p n } 1 = p 2 n + 2 a 0 e q n + a 2 1 e 2 q n , { q n − 1 , p n } 1 = 2 a 0 e q n + 2 a 2 1 e 2 q n − e q n − 1 − q n . F or the T oda lattice asso c ia ted with D n ro ot system after canonical tra nsformation (4.5) o ne 6 gets the following non-zero brackets i < j { q i , q j } 1 = 2 p i , i = 1 , . . . , n − 2 { p i , p i +1 } 1 = − ( p i + p i +1 ) e q i − q i +1 , { q i +1 , p i } 1 = e q i − q i +1 , { q i , p i } 1 = p 2 i + 2 e q i − q i +1 , { p n , q i } 1 = 2 e q n − 1 − q n − 2 a 2 2 e q n − 1 + q n , i = 1 , . . . , n − 3 { p n − 1 , q i } 1 = 2 e q n − 2 − q n − 1 − 2 e q n − 1 − q n − 2 a 2 2 e q n − 1 + q n , i = 3 , . . . , n − 1 { p i , q j } 1 = 2 e q i − 1 − q i − 2 e q i − q i +1 , 1 ≤ j ≤ i − 2 , (4.7) and { q n , p n } 1 = p 2 n , { q n − 2 , p n − 1 } 1 = − 2 e q n − 2 − q n − 1 + 2 e q n − 1 − q n + 2 a 2 2 e q n − 1 + q n , { q n − 1 , p n − 1 } 1 = p 2 n − 1 + 2 e q n − 1 − q n + 2 a 2 2 e q n − 1 + q n , { q n , p n − 1 } 1 = { p n , q n − 1 } 1 = e q n − 1 − q n − a 2 2 e q n − 1 + q n , { p n − 1 , p n } 1 = − ( p n − 1 + p n ) e q n − 1 − q n + ( p n +1 − p n ) a 2 2 e q n − 1 + q n . So, using br ack et (2.1) at k = 1 w e recov er ed all the known second brack ets for the B C n and D n T o da lattices [2]. Remark 2 Acco rding to [9, 14] if a k 6 = 0 then after mor e complicated canonical transfor mation of p n and q n we can descr ib e generalized T o da la ttice with the following p o tent ia l V g = 2 a 2 2 e q n − 1 + q n + a 1 sinh 2 q n + 2 a 0 sinh 2 ( q n / 2) , which was discovered by Inozem tsev [6]. O f course, the brack et (2.1) gives rise the second Poisson s tructure for this s ystem. Remark 3 Acco rding to [7, 9, 13, 14] if we m ultiply matrix T ( λ ) (4.3) o n the matrix K + ( λ ) =  2 b 2 λ 2 − ib 1 λ + b 0 0 (4 b 2 e − q 1 + 1) λ 2 b 2 λ 2 + ib 1 λ + b 0  the τ ( λ ) = tr K + ( λ ) T ( λ ) ar e genera ting function of int eg rals of motion for the p e r io dic T o da lattices as so ciated with all the classical ro ot sy s tems. The Dar bo ux-Nijenhuis v ariables λ i , µ i (2.4) are the separated coo rdinates for these integrals and, therefore, integrals o f mo tion for the per io dic T o da lattices ar e in bi-inv olution with resp ect to the same bra ck ets (2.1) or (4.6-4.7). All the details may b e found in [18]. 5 The Ko w alevski top on so ∗ (4) Let us consider the Kow alevs ki top on so ∗ (4) with the Hamilton function H 1 = J 2 1 + J 2 2 + 2 J 2 3 − 2 b x 1 , (5.1) and the second integral of motion H 2 = ( J 2 + + 2 b x + − P b 2 )( J 2 − + 2 b x − − P b 2 ) , (5.2) J ± = J 1 ± i J 2 , x ± = x 1 ± i x 2 , (i 2 = − 1 ) , 7 which a re Poisson co mm uting { H 1 , H 2 } 0 = 0 (5.3) with resp ect to the following Poisson brack ets { J i , J k } 0 = ε ikl J l , { J i , x k } 0 = ε ikl x l , { x i , x k } 0 = −P ε ikl J l . (5.4) Here ε ikl is the completely antisymmetric tensor, P is a complex (or r e al) parameter, x = ( x 1 , x 2 , x 3 ) a nd J = ( J 1 , J 2 , J 3 ) a r e co o r dinates on the Poisson manifold so ∗ (4). The Casimir s of the br ack et (5.4) hav e the form C 1 = x 1 J 1 + x 2 J 2 + x 3 J 3 , C 2 = x 2 1 + x 2 2 + x 2 3 − P ( J 2 1 + J 2 2 + J 2 3 ) . (5.5) According to [10] the 2 × 2 Lax matrix for the K ow alevsk i top a cquires the form T = K + ( λ ) T ( λ ) , where K + =  1 0 λ 1  , (5.6) and matrix T ( λ ) is the following repre s ent a tion of the reflection equa tion alg ebra at n = 2, α = 0 a nd η = 2 i: A 4 = − 1 2  X 2 + J 2 3 − P b 2 / 2  , A 3 = − i 2  X 3 + ( J 2 3 + bx 1 − P b 2 ) X + bx 2 J 3  , A 2 = J 2 2  X 2 + J 2 3  + bx 2 J 3 X − bx 1 J 2 3 + b 2 2  P J 2 3 − x 2 3 + C 2 2  , A 0 = − b 2 C 2 1 4 , A 1 = i 2  J 2 X 3 + 2 b x 2 J 3 X 2 +  J 2 3 ( J 2 − bx 1 ) + b 2 ( x 2 2 − x 2 3 − P J 2 1 ) + bJ 1 ( x 1 J 1 + x 2 J 2 )) X + bx 2 J 3 ( J 2 2 + J 2 3 − bx 1 ) + J 1 J 2 J 3 ( bx 1 − P b 2 )  , B 3 = X 2 − J 2 + 2 b x 1 − P b 2 , (5.7) B 1 = − J 2 X 2 − 2 b ( x 2 J 3 − x 3 J 2 ) X − 2 b C 1 J 1 − b 2 ( x 2 2 + x 2 3 − P J 2 1 ) , C 3 = 1 4  X 2 + J 2 3   X 2 + J 2 3 − P b 2  , C 1 = − 1 4  J 2 X 4 + 2 b x 2 J 3 X 3 +  2 J 2 3 ( J 2 − bx 1 + P b 2 / 2) + C 2 b 2 − b 2 (2 x 2 3 + x 2 1 )  X 2 +2 bx 2 J 3 ( J 2 3 − bx 1 ) X + J 4 3 ( J 2 − 2 b x 1 ) − b 2 J 2 3 ( x 2 3 − x 2 1 + P ( J 2 1 + J 2 2 )) + P b 4 x 2 3  , where X = J 1 J 3 + bx 3 J 2 , J 2 = J 2 1 + J 2 2 + J 2 3 . (5.8) In contrast with [10] we use the transp ose d ma tr ices K and T in (5.6). Substituting this re presentation of the reflection eq uation alg ebra into the br ack ets { ., . } 1 (2.1) and solving the resulting system of algebraic equa tions o ne gets the following Poisson 8 brack ets { ., . } 1 in ( x, J ) v ar iables: { J 1 , J 2 } 1 = − J 3 X 2 − b ( x 3 J 1 + x 2 X ) , { J 1 , J 3 } 1 = J 2 X 2 − b ( x 3 J 2 − x 2 J 3 ) X J 2 + bx 2 (2 J 1 J 2 + bx 2 ) J 2 , { J 2 , J 3 } 1 = − J 1 X 2 − b ( x 1 J 1 + x 3 J 3 ) − bx 2 ( J 2 1 − J 2 3 + bx 1 ) J 2 , { x 1 , J 1 } 1 = − 2 Q 1  b ( x 2 J 2 + x 3 J 3 ) + ( J 2 2 + J 2 3 ) J 1  J 2 2 , { x 1 , J 2 } 1 = − x 3 X 2 + b P J 2 X + Q 1 (2 J 2 1 − H 1 ) − bx 2 x 3 J 1 J 2 , { x 1 , J 3 } 1 = x 2 X 2 − b P ( J 1 J 2 + bx 2 ) + Q 1 ( J 1 X + J 2 J 3 ) + bx 2 2 J 1 J 2 , { x 2 , J 1 } 1 = − x 3 J 2 1 − ( bx 2 + J 3 X + J 1 J 2 )  ( b P − 2 x 1 ) J 3 + x 3 J 1  J 2 , { x 2 , J 2 } 1 = − Q 2 ( H 1 − J 2 1 + b 2 P ) + bJ 3 ( P J 2 3 + x 2 3 ) J 2 (5.9) { x 2 , J 3 } 1 = b P ( J 2 1 + J 2 3 ) + Q 2 ( J 2 J 3 + J 1 X ) + ( b P − x 1 )( bx 2 + J 3 X ) J 1 + bx 2 x 3 J 3 J 2 { x 3 , J 1 } 1 = x 2 J 2 1 +  b ( x 2 J 2 + x 3 J 3 ) + ( J 2 2 + J 2 3 ) J 1   b P − 2 x 1 + x 2 J 1 J 2  J 2 , { x 3 , J 2 } 1 = x 1 X 2 − b P ( bx 1 + J 2 1 − J 2 3 ) + Q 3 (2 J 2 1 − H 1 ) + bx 2 ( x 1 J 1 + x 3 J 3 ) J 2 , { x 3 , J 3 } 1 = ( Q 3 − b P J 2 )( J 2 J 3 + J 1 X ) J 2 − bx 2 ( x 1 X + x 2 J 3 ) J 2 and { x 1 , x 2 } 1 = ( Z 1 x 3 − Z 2 J 3 ) − b P J 1 Q 1 J 2 { x 1 , x 3 } 1 = − ( Z 1 x 2 − Z 2 J 2 ) + b P X Q 1 J 2 , { x 2 , x 3 } 1 = ( Z 1 x 1 − Z 2 J 1 ) + b P ( x 1 J 1 − x 3 J 3 ) − b P x 2 ( J 2 1 − J 2 3 ) J 2 . Here Q = x ∧ J = [ x 2 J 3 − x 3 J 2 , x 3 J 1 − x 1 J 3 , x 1 J 2 − J 1 x 2 ] and Z 1 = x 3 J 3 − x 2 J 2 − 2 x 1 J 1 + x 2 ( J 2 1 − 2 J 2 3 ) J 2 +  J 3 ( b P − 2 x 1 ) + x 3 J 1  X J 2 , Z 2 = b 2 P 2 + ( H 1 − J 2 3 ) P . F unctions C 1 , 2 (5.5) a r e Casimirs w ith resp ect to the both Poisson structures { ., . } 0 , 1 (5.4),(5.9). It allo ws as to o bta in the recur s ion opera to r N on the s ymplectic lea ves o f so ∗ (4) and the corres p o nding Darbo ux -Nijenh uis v ar iables λ 1 , 2 . On the other hand w e ca n get these Darbo ux-Nijenhuis v ariables using the cont r ol matrix theory . Remind, that accor ding to [3] the bi-inv o lutivit y o f in tegr als of motion { H 1 , H 2 } 0 = { H 1 , H 2 } 1 = 0 (5.10) 9 is equiv alent to the existence of the non-degenerate control ma trix F , such that P 1 dH i = P 0 2 X j =1 F ij dH j , i = 1 , 2 . (5.11) Here P 0 , 1 are the Poisson bivectors asso ciated with the bra ck ets { ., . } 0 , 1 . In our c ase F loo ks like F =      H 1 + b 2 P 2 − X 2 − J 2 3 1 4 Z 3 H 1 + b 2 P 2      , where Z 3 = H 2 − 2  b ( b P − 2 x 1 ) − J 2 1  X 2 + 8 b x 2 J 3 X + (2 P J 2 3 − 2 x 2 3 ) b 2 + 4 bQ 2 J 3 + 2( J 2 2 + 2 J 2 1 ) J 2 3 . Then we can prov e that en tr y B ( λ ) of the matrix T ( λ ) (5.7) coincides with the characteristic po lynomial of F and, therefore, with the minimal characteristic p olyno mial of the recursion op erator N on the symplectic leav es of s o ∗ (4). Summing up, w e hav e proved that roots λ 1 , 2 of the p olynomia l B ( λ ) a r e the Dar b o ux- Nijenh uis co or dinates with r esp ect to the Poisson struc tur es (5.4) and (5.9). At P = 0 thes e v a riables λ 1 , 2 coincide with the well-kno wn Kow alev s ki v aria bles [10]. 6 The Ko w alevski-Gory ac hev-Chaplygin gyrostat. Let us consider the Kowalevski-Goryac hev- Chaplygin gyros tat with the fo llowing Hamilton function H 1 = J 2 1 + J 2 2 + 2 J 2 3 + 2 ρJ 3 + c 1 x 1 + c 2 x 2 + c 3 ( x 2 1 − x 2 2 ) + c 4 x 1 x 2 + δ x 2 3 c 1 , c 2 , c 3 , c 4 , ρ, δ ∈ R . (6.1) Here x = ( x 1 , x 2 , x 3 ) a nd J = ( J 1 , J 2 , J 3 ) a re co o rdinates o n the dual Lie algebra e ∗ (3) with the Lie-Poisson brack ets  J i , J j  0 = ε ij k J k ,  J i , x j  0 = ε ij k x k ,  x i , x j  0 = 0 , (6.2) and with the following Casimirs C 1 = x 1 J 1 + x 2 J 2 + x 3 J 3 , C 2 = x 2 1 + x 2 2 + x 2 3 . (6.3) The Ha milton function (6.1) determines dynamical sys tem o n e ∗ (3), which is a n integrable by Liouville at C 1 = 0 o nly . Let us s tart with the 2 × 2 Lax ma tr ix for the symmetr ic Neuma nn system L ( λ ) =     λ 2 − 2 J 3 λ − J 2 1 − J 2 2 − δ x 2 3 λ ( ix 1 + x 2 ) − x 3 ( iJ 1 + J 2 ) λ ( ix 1 − x 2 ) − x 3 ( iJ 1 − J 2 ) x 2 3     , (6.4) which is a repr e sentation o f the Sklyanin algebra on the subset of e ∗ (3) defined b y C 1 = 0 [8, 15]. Using a family of the Poisson brackets compatible with the Sklyanin alg ebra [17] w e can g et compatible bi-hamiltonian str uctures on e ∗ (3) a sso ciated with the matrix L ( λ ). All the detail may b e found in [5, 17]. 10 According to [8 , 1 5], the Lax matrix for the Kow alevski- Goryac hev -Chaplygin gyrostat acquires the form T ( λ ) = K + ( λ ) T ( λ ) , where T ( λ ) = L ( λ − ρ ) K − ( λ )  0 1 − 1 0  L T ( − λ − ρ )  0 1 − 1 0  , ρ ∈ R (6.5) is the repre s entation of the reflection equa tio n a lgebra (1.1) at n = 2, α = 0 and η = 2i. Here L ( λ ) is g iven by (6.4) and K − =  a 1 λ + a 0 λ 0 − a 1 λ + a 0  , K + =  b 1 λ + b 0 0 λ − b 1 λ + b 0  (6.6) are numerical ma trices dep ending o n arbitrary para meters a 0 , 1 and b 0 , 1 . The Hamilton function e H 1 from tr T ( λ ) = λ 6 − 2 f H 1 λ 4 + f H 2 λ 2 + 2 a 0 b 0 ( ρ 2 C 2 − δ ) (6.7) coincides with the previous function H (6.1) after canonical transformation of v ar iables J → J + U x , U =   0 0 i β + 0 0 β − − i β + − β − 0   , β ± = a 1 ± b 1 2 and exchange of parameters a 2 1 =  c 3 + i c 4 2  , b 2 1 =  c 3 − i c 4 2  , a 0 = i c 1 − c 2 2 , b 0 = i c 1 + c 2 2 . Substituting ma trix T ( λ ) (6.5) in to the br ack ets { ., . } k (2.1) one get the second cubic brack et { ., . } 1 compatible with brack et (6 .2). F or brevity we present these brack ets a t c 3 = c 4 = ρ = δ = 0 o nly , i.e. for the Kow ale vski top: { J 1 , J 2 } 1 = − ( J 3 ( J 1 − i J 2 ) + 2 a 0 x 3 )( J 1 + i J 2 ) { J 1 , J 3 } 1 = −  J 2 1 + J 2 2 + 2 J 2 3 − 2 a 0 ( x 1 − i x 2 )  J 2 { J 2 , J 3 } 1 =  J 2 1 + J 2 2 + 2 J 2 3 − 2 a 0 ( x 1 − i x 2 )  J 1 { x 1 , J 1 } 1 = (2i J 2 2 − 2 a 0 x 2 ) x 3 − 2 x 2 ( J 1 + i J 2 ) J 3 { x 1 , J 2 } 1 = − i( J 2 1 − 2i J 1 J 2 + J 2 2 − 2i a 0 x 2 ) x 3 + 2i x 2 ( J 1 + i J 2 ) J 3 { x 1 , J 3 } 1 = −  J 2 1 + J 2 2 + 4 J 2 3 − 2 a 0 ( x 1 + i x 2 )  x 2 + 2 x 3 J 2 J 3 (6.8) { x 2 , J 1 } 1 = ( J 2 1 + 2i J 1 J 2 + J 2 2 − 2 a 0 x 1 ) x 3 + 2 x 1 ( J 1 + i J 2 ) J 3 , { x 2 , J 2 } 1 = 2i( J 2 1 + a 0 x 1 ) x 3 − 2i x 1 ( J 1 + i J 2 ) J 3 , { x 2 , J 3 } 1 =  J 2 1 + J 2 2 + 4 J 2 3 − 2 a 0 ( x 1 + i x 2 )  x 1 − 2 x 3 J 1 J 3 , { x 3 , J 1 } 1 = x 2 ( J 2 1 + 2i J 1 J 2 + J 2 2 ) − 2i x 1 J 2 2 , { x 3 , J 2 } 1 = − x 1 ( J 2 1 − 2i J 1 J 2 + J 2 2 ) − 2i x 2 J 2 1 , { x 3 , J 3 } 1 = 2( x 2 J 1 − x 1 J 2 ) J 3 and { x i , x j } 1 = 2i ε ij k ( x 1 J 2 − x 2 J 1 − i x 3 J 3 ) x k . 11 A t C 1 = 0 integrals of motio n e H 1 , 2 (6.7) ar e in the bi-inv olutio n with r e s pe c t to the co mpatible brack ets { ., . } 0 , 1 and satisfy to the following relatio ns P 1 d e H i = P 0 2 X j =1 F ij d e H j , i = 1 , 2 , (6.9) where P 0 , 1 are the Poisson biv ec to rs asso ciated with the brack ets { ., . } 0 , 1 . a nd F =   2 J 2 1 + 2 J 2 2 + 4 J 2 3 − 2 a 0 ( x 1 + i x 2 ) 1 2 a 0 (i x 2 J 1 − i x 1 J 2 + x 3 J 3 )( J 1 + i J 2 )1 − ( J 2 1 + J 2 2 ) 2 0   . Of cour s e, ent r y B ( λ ) of the matr ix T ( λ ) (6 .5) coincides with the c har acteristic p olyno mial of F and, therefor e, with the minima l characteristic p olynomial of the r ecursion oper ator N on the symplectic le av es of e ∗ (3). Summing up, we have pro ved that at C 1 = 0 the Kow alevski top has the po lynomial brack ets (6.8) in additional to the rationa l br ack ets (5.9) c o nsidered ab ove. These differen t Poisson structure s ar e related with the different separated v ar iables, which giv e rise to the different r epresentations of the reflection equation a lg ebra. Remark 4 Ther e is another bi-hamiltonian structure for the Kow alevsk i to p on e ∗ (3), asso ci- ated with linear r -matrix alge br a [1 2]. According to [16] the corresp onding Poisson tenso r is the r ational function b P 1 = C − 1 1 P pol , where C 1 is the Casimir function (6.3) and P pol is a cubic po lynomial in v ar ia bles ( x, J ). The Casimir C 2 do es not the Casimir function for the sec o nd Poisson bivector b P 1 from [12, 16], in contrast with the sec o nd Poisson bivectors a sso ciated with the consider ed ab ov e brack ets (5.9) and (6 .8). 7 Conclusion W e present a f a mily of compatible Poisson brac kets (2.1) ,(2 .6), that includes the reflection equation algebr a. The application of the r -matrix formalism is extremely useful here resulting in drastic r eduction of the calcula tio ns for a whole set of integrable s ystems. F or the r ational 4 × 4 matrix r ( λ − µ ) (1.2) the similar constructio n has been applied to other r -matrix a lgebras in [17, 5] and [19]. It will b e int e r esting to cons tr uct the similar families of compa tible Poisson brack ets asso ciated with the 4 × 4 matrices r ( λ − µ ), which ar e trigonometric and elliptic functions on sp ectra l parameter. The research w as partially supp orted b y the RFBR grant 06-0 1-001 40 and grant NSc 5403.2 006.1 . References [1] O.I. 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