Classification of compatible Lie-Poisson brackets on the manifold e*(3)

Journal of Mathematical Sciences, Vol. 143, No. 1, 2007

CLASSIFICATION OF COMPATIBLE LIE–POISSON BRACKETS ON THE MANIFOLD e ∗(3) A. V. Tsiganov∗

UDC 517.9

A classification of compatible Lie–Poisson brackets on e∗ (3) is constructed. The corresponding bi-Hamiltonian systems are the classical integrable cases of the Euler–Poisson and Kirchhoff equations which describe the motion of a solid body. Bibliography: 12 titles.

1. Introduction Let us recall that any solution r(λ, µ) of the classical Yang–Baxter equation gives rise to a hierarchy of compatible Li–Poisson brackets (see [9]). On the other hand, any pair of linear compatible brackets gives rise to one or more solutions r(λ, µ) of the classical Yang–Baxter equation. Poisson brackets are called compatible if every their linear combination is a Poisson bracket as well, i.e., if any of their linear combinations satisfies the Jacobi identity. In the generic case, description of all compatible linear brackets on a manifold V is a complicated problem. Solution of this problem allows us to construct a sufficiently large class of solutions of the Yang–Baxter equation. For low-dimensional manifolds V , this problem can be solved by using modern systems of symbolic calculations, which can efficiently study the following highly overdetermined system of equations: [[P, P ]] = [[P, P  ]] = [[P  , P  ]] = 0,

(1.1)

where [[., .]] is the Schouten bracket (see [6]) and P and P  are Poisson tensors, which determine a pair of Poisson brackets {., .} and {., .} on the manifold V with local coordinates z = (z1 , . . . , zm ) as follows: {f(z), g(z)} =
df, P dg =




Pik (z)

i,k

∂f(z) ∂g(z) . ∂zi ∂zk

(1.2)

In (1.2), df is the covector with entries ∂f/∂zi and
., . is the standard scalar product. The Jacobi identity for functions f, g, and h ∈ C ∞ (V ) reads as follows: {f, {g, h}} + {g, {h, f}} + {h, {f, g}} =


∂f(z) ∂g(z) ∂h(z) [[P, P ]]ijk . ∂zi ∂zj ∂zk

(1.3)

ijk

Thus, Eq. (1.1) is a condition under which the linear combination P +λP  is a Poisson pencil, i.e., this combination is a Poisson bivector for each λ ∈ C, and, therefore, the corresponding bracket {., .}+λ{., .} is a pencil of Poisson brackets on V . If we consider a family of linear Poisson tensors, Pij (z) =




ckij zk ,

(1.4)

k

then Eqs. (1.1) are algebraic equations on structure constants ckij and c  ij of the Poisson tensors P and P  . Any solution of Eqs. (1.1) can be associated with a dynamical system on V for which the corresponding integrals Hi of motion satisfy the Lenard–Magri recurrence relations [6], i.e., these integrals are solutions of the following complementary system of equations: k

P dH0 = 0, ∗ St.Petersburg

XHi = P dHi = P  dHi−1,

P  dHn = 0.

(1.5)

State University, St.Petersburg, Russia, email: [email protected].

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 231–245. Original article submitted May 10, 2006. c 2007 Springer Science+Business Media, Inc. 1072-3374/07/1431-2831 

2831

Such dynamical systems are called Gelfand–Zakharevich systems (see [2]) since Eqs. (1.5) can be rewritten in the standard form: Pλ dH(λ) = 0, λ ∈ R. (1.6) In (1.6), Pλ = P + λP  is a Poisson pencil, and the function H(λ) = Hn λn + · · · + H1 λ + H0 is the so-called polynomial Casimir function of the Poisson pencil Pλ (see [6]). Condition (1.6) indicates that the functions H0 , H1 , . . . Hn are in bi-involution with respect to both Poisson brackets: {Hi, Hk } = {Hi, Hk } = 0. (1.7) In the present paper, we consider nontrivial solutions of Eqs. (1.1) and (1.6) for which the polynomial Casimir function H(λ) of the Poisson pencil Pλ gives rise to an integrable system on V . This means that the number of functionally independent coefficients Hi of the polynomial H(λ) is enough for Liouville integrability. 2. Poisson pencils on e∗ (3) In this section, we describe all nontrivial solutions of Eqs. (1.1) and (1.6) in the set of linear functions on the manifold V = e∗ (3), which is dual to the Euclidean algebra e(3). The main result is the following classification. Proposition 1. Up to isomorphisms, there are six linear Poisson pencils P + λP  with a canonical tensor P on the Poisson manifold V  e∗ (3). Recall that, up to isomorphisms, we have a complete classification of all reductive (semisimple) Lie algebras g, i.e., we have a complete classification of linear Poisson tensors (1.4) on dual manifolds. This means that on g∗ we can single out some standard (or canonical) tensor P , which is associated, for example, with the Cartan–Weil basis, and any other linear tensor P on g∗ is reducible to standard ones by a linear transformation. The Lie algebra e(3) = so(3) ⊕ R3 is a semidirect sum of the rotation algebra so(3) and the Abel subalgebra of translations in R3 . Thus, on the Poisson manifold V = e∗ (3), it is usual to use coordinates z = (x, J), where x = (x1 , x2 , x3 ) ∈ R3

and J = (J1 , J2 , J3 ) ∈ R3  so(3)     are two three-dimensional vectors. Here and below, we identify the algebra R3 , ∧ with the algebra so(3), [., .] by the following well-known isomorphism of Lie algebras:     z1 0 z3 −z2 0 z1  . (2.1) z =  z2  → zM =  −z3 z3 z2 −z1 0 In these coordinates, the canonical Lie–Poisson tensor (1.4) on V = e∗ (3) reads as 

0 ∗  ∗ P = ∗  ∗ ∗

0 0 ∗ ∗ ∗ ∗

0 0 0 −x3 0 x2 ∗ 0 ∗ ∗ ∗ ∗

x3 0 −x1 J3 0 ∗

 −x2 x1    0  0 ≡ −J2  xM  J1 0

xM JM

.

(2.2)

The corresponding Lie–Poisson brackets (1.2) for linear functions on e∗ (3), i.e., for elements of the Lie algebra e(3), coincide with the Lie bracket on e(3): {Ji, Jj } = εijk Jk ,

{Ji, xj } = εijk xk ,

{xi , xj } = 0,

(2.3)

where the εijk is a totally skew-symmetric tensor. The bracket (2.3) is degenerate and has two Casimir functions: |x|2 ≡

n=3
k=1

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x2k ,


x, J ≡

n=3
k=1

xk Jk ,

P d|x|2 = P d
x, J = 0.

(2.4)

Below, the first tensor P is the canonical Poisson tensor (2.2) on V  e∗ (3) and P  is a solution of Eqs. (1.1). It is obvious that the tensors P  , and P  → P  + αP define the same Poisson pencil. One more class of isomorphisms consists of the following linear transformations: J → J  = U J + w ∧ U x,

x → x  = U x,

U ∈ SO(3),

w ∈ R3 ,

(2.5)

which preserve the canonical tensor P in (2.2). Summing up, in order to describe the Poisson pencil it is enough to present a specific tensor P  that belongs to this pencil and is independent with the canonical tensor. 2.1. Compatible Lie–Poisson tensors with a common Casimir function If we add to Eqs. (1.1) some subsidiary equations, P  d
x, J = 0

or P  d|x|2 = 0,

(2.6)

we get only two nontrivial solutions of (1.1). The corresponding classical r-matrices can be found in [9]. The first Lie–Poisson pencil Let A be a symmetric 3×3 numerical matrix and let a be an arbitrary parameter. Then the second Lie–Poisson tensor satisfying (2.6) is   a JM A xM  , P1 d
x, J = 0. (2.7) P1 =  xM A (A J)M Here A xM is a product of two matrices A and xM , while (AJ)M is a matrix which is the result of application of mapping (2.1) to the vector AJ. The tensor P1 is defined up to canonical transformations (2.5); therefore, we may assume without loss of generality that the matrix A in (2.7) is diagonal. Solutions of the complementary system of Eqs. (1.6) at H0 = |x|2 are integrals of motion for the Clebsch system on e∗ (3) (see [1]): H1 = −a
J, J −
(tr A − A) x, x,

H2 = a
A J, J +
A∨ x, x,

(2.8)

where A∨ = det(A) A−1 is the cofactor matrix. If two parameters ai coincide, for instance, ai = aj , then the  = Jk . In this case, one gets an integrable integrals of motion (2.8) are in involution with the linear function H system, which can be called a generalized Lagrange top in a quadratic potential. The group transformations (2.5) with U = Id and w = 0 reduce the Lie–Poisson tensor P1 (2.7) to the following form:   WxM − xM W JM W − (w ∧ x)M W , P1 = P1 + a  −WJM + W(w ∧ x)M
w, J W where W = wM is an antisymmetric matrix associated with the vector w. The corresponding Hamilton function is as follows:  1 = H1 + 2a
w, x ∧ J − a|w ∧ x|2 . H The second Lie–Poisson tensor Let A be a symmetric 3 × 3 numerical matrix; then the second Lie–Poisson tensor that satisfies the second equation of (2.6) is equal to  P2 = 

axM

AxM

xM A

(C1 J − C2 x)M

where C1 = tr A − 2A and

 ,

P2 d|x|2 = 0,

(2.10)

  C2 = a−1 tr A∨ − 2A∨ − A2 . 2833

Solutions of the complementary system (1.6) at H0 =
x, J are integrals of motion for the Steklov–Lyapunov system:   2H1 = −a
J, J +
A x, J + a−1
A2 x, x + 2
A∨ x, x ,   (2.11) H2 = a
A J, J − 2
A∨ x, J − a−1
A3 x, x − tr A2
Ax, x . The function H1 is called the Lyapunov Hamiltonian (see [4]), and the function H2 is called the Steklov Hamiltonian (see [11]). If two parameters ai coincide, for instance, ai = aj , then integrals of motion (2.8) are in  = Jk . In this case, one gets an integrable system, which can be called a involution with the linear function H generalized Lagrange top in quadratic potential as above. Using the group transformations (2.5), we can diagonalize the matrix A in P2 (see formulas (2.10)) and obtain the following Poisson tensor:   0 WxM − (Wx)M , P2 = P2 +   xM W − (Wx)M −(C2 x)M where W = wM is an antisymmetric matrix associated with the vector w, and  1 −1 −1 2 2  C2 = a (WA − AW) − a W − tr W . 2 2.2. Compatible Lie–Poisson tensors without common Casimir functions Below, we consider two polynomial Casimir functions H α (λ) and H β (λ) of the Poisson pencil P + λP  that are solutions of differential equations (1.5)–(1.6) with different boundary conditions: H0α = |x|2

and

H0β =
x, J.

By definition (1.5), coefficients of these polynomial Casimir functions H α (λ) and H β (λ) are in bi-involution (1.7). We do not know closed-form expressions for classical r-matrices associated with the Lie–Poisson pencils constructed in this section. The third Lie–Poisson pencil Let A be a symmetric 3×3 numerical matrix and let a be an arbitrary parameter; then the second Lie–Poisson tensor compatible with canonical ones is   (a J + A x)M (AJ)M − AJM  P3 =  ∗ −a−1 (A2 J)M   a−1 (Ax)M A − (A2 x)M − AxM A

  +

0 ∗



−a−2 (A3 x)M + A(Ax)M A − A(A2 x)M



 . − (A2 x)M A

(2.12)

The corresponding first coefficients of the polynomial Casimir functions H α (λ) and H β (λ) have the following form:  tr A α (2.13) H1α = −H1 and H1β = −a−1 H1 + H2 , 2 where H1,2 are the Lyapunov and Steklov Hamiltonians (2.11), respectively. Thus, the Steklov–Lyapunov system is related with two different Lie–Poisson pencils (2.10) and (2.12). We may assume that precisely this fact is related with the existence of two different 2 × 2 Lax representations for this system with rational and elliptic dependence on the spectral parameter (see [9, 12]). Recall that the equations of motion for the Steklov–Lyapunov system are linearized at the Jacobian of a hyperelliptic curve instead of the Prym variety of an elliptic curve as in the Clebsch case. 2834

The fourth Lie–Poisson pencil Let us introduce special diagonal matrices: A1 = diag (0, 0, a1),

A2 = diag (b1 , b1 , b2 ),

B1 = 2A1 − tr A1 ,

and B2 =

A3 = diag (c1 , c1 , −c1 ), 1 tr A2 − A2 , 2

which depend on four parameters (a1 , b1 , b2 , c1 ); then, up to transformation (2.5), the second Lie–Poisson tensor is   A1 xM + xM A1 B1 JM + JM A1 + A2 xM . P4 =  (2.14) A1 JM + JM B1 + xM A2 B2 JM + JM B2 + (A3 x)M The first coefficients of the polynomial Casimir functions H α (λ) and H β (λ) are 1 H1α = 2
A1 x, J +
(A2 − b1 ) x, x and H1β =
A1 J, J −
(A3 − c1 ) x, x. 2  = J3 . Thus, the fourth It is easy to prove that these integrals of motion are in involution with the function H ∗ Poisson pencil can be associated with an integrable system on e (3) which admits a linear integral of motion. The fifth Poisson pencil Let A1 and A2 be two matrices that can be diagonalized simultaneously: A1 = diag (a1 , a2 , a3 )

and A2 = diag (b1 , b2 , b3 );

then the second Poisson tensor is 



0

(A1 x)M

(A1 x)M

(A1 J)M + (A2 x)M

P5 = 

.

(2.15)

The Casimir functions H α (λ) and H β (λ) are completely defined by their first coefficients 1 H1α = −
A1 x, x and H1β = −
A1 x, J −
A2 x, x. 2 We have to underline that there do not exist linear integrals of motion in involution with the Hamiltonian H1β . Thus, the fifth Poisson pencil can be associated with a nontrivial integrable system on e∗ (3) which admits quadratic integrals of motion only. The sixth Lie–Poisson pencil Let A1 and A2 be two matrices that can be diagonalized simultaneously: A1 = diag (a1 , a2 , a3 )

and A2 = diag (b1 , b2 , b3 );

then the second Lie–Poisson tensor compatible with canonical ones is  P6 = 

0

A1 xM

xM A1

(A1 J)M + (A2 x)M

 .

(2.16)

The corresponding first coefficients of the polynomial Casimir functions depend on the variable xi only: 1 H1α =
A1 x, x and H1β = −
A2 x, x. 2  =
c, x, where c is an arbitrary numerical vector. They are in involution with the linear function H 2835

2.3. Poisson pencils with constant terms On the Poisson manifold V  e∗ (3), we can find only one constant Poisson tensor,  Pconst

 =

0 dM

dM eM

,

(2.17)

that is compatible with the canonical tensor P (see (2.2)) and is parameterized by two numerical vectors d = (d1 , d2, d3 ) and e = (e1 , e2 , e3 ). In this section, we consider the following Poisson pencils: P (λ, µ) = P (λ) + µP  = P + λP  + µP  ,

(2.18)

where P is the canonical tensor given by (2.2), P  is a linear Lie–Poisson tensor from the previous sections, and P  is a constant tensor that satisfies the additional compatibility conditions: [P, P  ] = [P  , P  ] = 0.

(2.19)

 Of course, a solution P  of system (2.19) is a particular case of the tensor Pconst given by (2.17). In the generic case where all the eigenvalues of the symmetric matrix A in formulas (2.7) and (2.10) are different, solution of Eqs. (2.19) are presented in the table.

1 2 3 4 5 6

Tensor P  Vector d P1 (2.7) d=0 P2 (2.10) d = 0 P3 (2.12) d = (d1 , d2, d3 ) P4 (2.14) d = (0, 0, d3) P5 (2.15) d = (d1 , d2, d3 ) P6 (2.16) d = 0

Vector e e=0 e = (e1 , e2 , e3 ) e = −a−1 d e = (0, 0, e3) e = (e1 , e2 , e3 ) e = (e1 , e2 , e3 )

If two eigenvalues of the symmetric matrix A in formulas (2.7) and (2.10) coincide, ai = aj , then dk = 0 and ek =  0 if i = j = k. The definition of the polynomial Casimir functions H(λ) (see (1.6)) implies that if we add a constant tensor P  to the Poisson pencil P (λ) (2.18), then we have to add linear terms: H1α → H1α + 2
d, x,

H1β → H1β +
d, J +
e, x

to the coefficients H1α and H1β from the previous sections. Thus, in the Clebsch case we cannot add any linear terms to integrals of motion (d = e = 0). In the Steklov– Lyapunov case, we can add linear terms which depend on three parameters. The corresponding integrable system is the so-called Rubanovski system (see [10]). The remaining three integrable systems admit addition of linear terms to quadratic integrals of motion as well. Let us consider some examples related with the first Lie–Poisson pencil. If the matrix A equals zero in P1 (formula (2.7)), then the corresponding second Poisson tensor, 

P1 =



aJM 0

0  ∗ 0 ∗ = 0 ∗  ∗ ∗

aJ3 0 ∗ ∗ ∗ ∗

−aJ2 aJ1 0 ∗ ∗ ∗

0 0 0 0 ∗ ∗

0 0 0 0 0 ∗

corresponds to the Euler top with the following Hamiltonian: H1α = −a
J, J = −a(J12 + J22 + J32 ). 2836

 0 0  0 , 0  0 0

(2.20)

Adding a constant term, one gets the second Poisson tensor for the  0 aJ3 −aJ2 0 aJ1   ∗ aJ d ∗ ∗ 0  M M P 1 = = dM 0 ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗

usual Lagrange top [8]:  0 0 0 0 0 d1   0 −d1 0  , 0 0 0   ∗ 0 0 ∗ ∗ 0

(2.21)

with the following integrals of motion:

α = −a(J 2 + J 2 + J 2 ) + 2d1 x1 H 1 1 2 3

β = d1 J1 . and H 1

If we set A = −diag(a−1 b2 , 0, 0) and w = (a−1 b, 0, 0) in (2.9) and add a constant term with d = (d1 , 0, 0), then we obtain the Poisson tensor for the generalized Lagrange top:   0 aJ3 − bx2 −aJ2 − bx3 0 bJ2 bJ3 0 aJ1 0 −bJ1 d1  ∗   ∗ 0 0 −d1 −bJ1  ∗   (2.22) P1 =  , ∗ ∗ 0 0 0  ∗   ∗ ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ ∗ 0 with the Hamilton function  α = −a(J 2 + J 2 + J 2 ) + 2b(x2 J3 − x3 J2 ) + 2d1 x1 . H 1 1 2 3 2.4. Rational Poisson pencils Let us consider Poisson tensors P  that are compatible with the canonical tensor and are polynomials of higher order in the variables (x, J). In this case, the polynomial Casimir functions H α (λ) and H β (λ) are higher-order polynomials in (x, J) as well. For instance, the quadratic Poisson tensor,   0 0 0 0 a3 x23 −a2 x22 2 2 0 a1 x1   ∗ 0 0 −a3 x3   −a1 x21 0  ∗ ∗ 0 a2 x22   Psec =  , 0 b3 x23 + 2a3 x3 J3 −b2 x22 − 2a2 x2 J2  ∗ ∗ ∗   ∗ ∗ ∗ ∗ 0 b1 x21 + 2a1 x1 J1 ∗ ∗ ∗ ∗ ∗ 0 gives rise to cubic integrals of motion: H1α = −

2
ai x3i 3

and H1β = −

1
3
bi x i − ai x2i Ji . 3

Thus, if we want to consider quadratic Hamilton functions, we have to use very specific polynomial or even rational second Poisson tensors. In this section, we consider one remarkable property of compatible Poisson tensors P and P  that have a common Casimir function C. Namely, in this case we can easily construct some special solutions of the Yang– Baxter equation and a special family of rational Poisson tensors,  Pr = P  + C −1 Ppol .

(2.23)

 In (2.23), P  is one of the Poisson tensors obtained in Sec. 2.1, and the tensor Ppol is a polynomial tensor that  is a solution of Eqs. (1.1). For such rational tensors Pr , we can easily find functionally independent polynomial integrals of motion among rational solutions Hi of the complementary equations (1.5). For instance, let us substitute tensor P1 given by (2.20) and associated with the Euler top into tensor (2.23)  and resolve the compatibility conditions (1.1) in the class of cubic tensors Ppol . In this case, C =
x, J, and the general solution of Eqs. (1.1),  JM 0 Pr = + ∆P1 (c), 0 0

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is parameterized by an arbitrary vector c = (c1 , c2 , c3 ):  ∆P1 (c) =
c, J 

−cM 0

   2 J ∧ (c ∧ x)
c, J M +  C 0 (J ∧ c) ⊗ J 0



J ⊗ (c ∧ J)

 ,

(2.24)

0

where (u ⊗ v)ij = ui vj . The corresponding polynomial integrals of motion,   I1 = H1α = − 1 +
c, c
J, J +
c ∧ J, c ∧ J

and I2 = −2 C H1β =
J, J
c, J2 ,

are second-order and fourth-order polynomials in momenta J. If we substitute the tensor P 1 (see (2.21)) associated with the Lagrange top into tensor (2.23) and resolve the compatibility conditions (1.1), we get the following particular solution:  JM dM P r = + ∆P1 (c) + ∆P 1 (c, d). dM 0 Here c and d are two orthogonal vectors, ∆P1 (c) is given by formula (2.24), and the third term is equal to      0 (c ∧ x) ⊗ J ∧ (c ∧ d) −
c, J (c ∧ x) ⊗ d − (d ∧ x) ⊗ c 1 . ∆P 1 (c, d) =  C ∗
d, JJM −
c ∧ d, x(c ∧ d)M Using group transformations (2.5), we can always put c = (0, 0, 1) and d = (d1 , 0, 0). In this case, the first

α and H

β coincide with the Hamilton function for the Kowalevski top, coefficients of the Casimir functions H 1 1

1α = J12 + J22 + 2J32 + 2d1 x1 , H and the second integral of motion,

β = (J 2 + J 2 + J 2 )J 2 + 2d1 (x1 (J 2 + J 2 ) + x2 J1 J2 ) − d2 x2 , −2 C H 1 2 3 3 1 3 1 2 1 which are fourth-order polynomials in momenta. The Poisson tensor P r has been implicitly constructed in [5] by using the classical r-matrix on the auxiliary space so∗ (3, 2). If we substitute the tensor P1 (see (2.21)) associated with the generalized Lagrange top into tensor (2.23) and resolve the compatibility conditions (1.1), we can get the second Poisson tensor for the Kowalevski top with the Hamilton function  α = J 2 + J 2 + 2J 2 + 2b(x2 J3 − x3 J2 ) + 2d1 x1 . H 1 1 2 3 According to [3], this integrable system is isomorphic to the Kowalevski top on the manifold so∗ (4). The systems on so(4) will be considered in detail in forthcoming publications. 3. Conclusion In the present paper, we construct all of linear Poisson tensors on e∗ (3) that are compatible with the canonical tensor. The corresponding dynamical systems are described. We have to underline that all the considered Poisson tensors are degenerate. According to the generic biHamiltonian theory, in order to integrate the equation of motion for the corresponding integrable system, we have to propose a reduction of the Poisson pencil that preserves all the bi-Hamiltonian properties of the systems. At present, we do not know how to construct such a reduction in the generic case. For the Lagrange top, such a reduction was consructed by using the known Lax representation and the corresponding r-matrix [7]. The author is grateful to I. V. Komarov for useful discussions. The research was partially supported by the RFBR (grants 06-01-00140 and NSc-5403.2006.1). Translated by A. V. Tsiganov. 2838

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