Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras

Gh. Haghighatdoost; S. Abdolhadi-Zangakani; J. Abedi-Fardad

doi:10.2991/jnmp.k.201104.001

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Volume 28, Issue 2, June 2021, Pages 194 - 204

Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras

Authors

Gh. Haghighatdoost¹^{, *}, S. Abdolhadi-Zangakani², J. Abedi-Fardad¹

¹Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 53714-161, Iran

²Department of Mathematics, University of Bonab, Tabriz, Iran

^*Corresponding author. Email: gorbanali@azaruniv.ac.ir

Corresponding Author

Gh. Haghighatdoost

Received 29 June 2020, Accepted 31 October 2020, Available Online 17 December 2020.

DOI: 10.2991/jnmp.k.201104.001 How to use a DOI?
Keywords: Compatible Poisson structures; bi-Hamiltonian system; Lie groups
Abstract: In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of their Lie algebras and some vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then, using a theorem by Magri and Morosi, we obtain new integrable bi-Hamiltonian systems with two-, four- and six-dimensional symplectic real Lie groups as phase spaces.
Copyright: © 2020 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The pioneering work in the integrable bi-Hamiltonian systems is done by Magri [9] and then followed by the fundamental papers by Gelfand and Dorfman [6], Kosmann-Schwarzbach and Magri [8] and Magri and Morosi [11]. These works show that integrability of many systems in mathematical physics, mechanics, and geometry is closely related to their bi-Hamiltonian structures. It is shown that many classical systems have the bi-Hamiltonian structure, at the same time by using the bi-Hamiltonian methods many new nontrivial and interesting examples of integrable systems have been found (for more details see [4]). As we know, the study of bi-Hamiltonian structure is based on the very simple notion of compatible Poisson structures. It is proved that bi-Hamiltonian structure is very powerful in the theory of integrable Hamiltonian systems not only for finding new examples, but also for the integration of systems, constructing separation of variables and description of properties of solutions (see [17–21] for a review). In Abedi-Fardad et al. [2], by the adjoint representation of Lie algebra authors have calculated some compatible Poisson structure and bi-Hamiltonian systems on Lie groups as phase space. In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of its Lie algebras and some vector field X. Then we obtain new integrable bi-Hamiltonian systems by using Magri–Morosi theorem [11], for which the Lie group is the phase space.

The structure of this paper is as follows. In Section 2, we briefly review the definitions and notations of compatible Poisson structures and integrable bi-Hamiltonian systems. In Section 3, we give a method to obtain the compatible Poisson structures on low-dimensional Lie groups by means of structure constants of Lie algebras and some vector field X. In Section 4, we obtain these structures on two, four and nilpotent six-dimensional symplectic real Lie algebras. Finally, in Section 5, we obtain some compatible Poisson structures and integrable bi-Hamiltonian systems with two, four and nilpotent six-dimensional symplectic real Lie groups as phase spaces.

2. PRELIMINARIES

In this section, we recall some basic definitions and notations on compatible Poisson structures and integrable bi-Hamiltonian systems.

Definition: [11] A bi-Hamiltonian manifold M is a smooth manifold endowed with two compatible bi-vectors P and P′ such that

[P,P]=0,[P,P′]=0,[P′,P′]=0 , (1)

where [.,.] is the Schouten bracket.

The Poisson bracket corresponding to the Poisson bi-vector has the form

{f(x),g(x)}=<df,P dg>=Pij(x)∂f(x)∂xi∂g(x)∂xj , (2)

for all f, g ∈ F(M) and similar brackets {.,.}′ to P′. The brackets {.,.}′ and {.,.}′ satisfies the Jacobi identity: i.e. ∀ f, g, h ∈ C^∞(M),

{f,{g,h}}+{g,{h,f}}+{h,{f,g}}=[P,P]ijk∂f∂xi ∂g∂xj ∂h∂xk=0, (3)

if [P, P] = 0 and vice versa. A family of functions {H_i} on the manifold M that are in bi-involution with respect to these compatible Poisson brackets,

{Hi,Hj}={Hi,Hj}′=0, (4)

is called bi-integrable system or generalized bi-Hamiltonian system [10]. So to introduce the bi-Hamiltonian structure on the manifold M, we must determine a pair of compatible and independent Poisson bi-vectors P and P′.

Definition: [13] In the coordinate basis, T_pM spanned by {e_μ} = {∂_μ} and Tp*M by {dx ^μ}, let us consider their linear combinations,

ei=eiμ ∂μ , Θi=eμi dxμ , eiμ∈GL(m,R), (5)

where m = dim(M), eiμ and eμi are non-singular (m × m)-matrices. In other words, {e_i} is the frame of basis vectors which is obtained by a GL(m, R)-rotation of the basis {e_μ}. In the above eμi is inverse of eiμ and we have

eμi eiν=δμν , eμi ejμ=δji, (6)

the bases {e_i} and {Θⁱ} are called the non-coordinate bases and coefficients eiμ are called the vielbeins. We have

[ei,ej]=fij kek, (7)

where fijk is a function of coordinates of the manifold M. When M is a Lie group G, we suppose that coefficients fijk be the structure constants of the Lie algebra g of the Lie group G and also e_i are (left or right) invariant vector fields on the group G (for more details see [13]). We have

fijk=eνk (eiμ ∂μ ejν-ejμ ∂μ eiν). (8)

Consider a symplectic manifold M endowed with a second compatible Poisson bracket. An important class of bi-Hamiltonian manifold occurs when one of the compatible Poisson structures is invertible, then one can define a linear map N: TM → TM acting on the tangent bundle by Magri et al. [10]

N=P′ P-1. (9)

Also by using Magri–Morosi’s theorem as follows, one can find the Hamiltonian and integrals of motions of bi-Hamiltonian systems.

Theorem (Magri–Morosi): [8,11] A remarkable consequence of the compatibility of P and P′ is that the torsion of Nijenhuis tensor N, i.e.

TN(X,Y)=[NX,NY]-N[NX,Y]-N[X,NY]+N2[X,Y], (10)

identically vanishes, where X and Y are arbitrary vector fields and the bracket [X, Y] denotes the Lie bracket (commutator) of vector fields. One of the main properties of N is that the normalized traces of the powers of N

Hk=12kTrNk, (11)

are in involution and satisfy Lenard–Magri recurrence relations [10]

P′ dHi=P dHi+1. (12)

3. COMPATIBLE POISSON STRUCTURES ON LOW-DIMENSIONAL LIE GROUPS

In this section, we give a method to obtain compatible Poisson structures on low-dimensional Lie groups by means of structure constants of related Lie algebras and some vector field X. For this purpose, we write the Poisson structure P (which is presented in Abedi-Fardad et al. [2]) in terms of the non-coordinate basis as

Pμν=eiμejνPij, (13)

where P^ij’s are constant antisymmetric matrix¹. One can rewrite the Schouten bracket [P, P] = 0 in the following matrix forms [2]:

P Xi Pij+P Yj P+Pij Xi t P=0, (14)

where (Xi)j k=-fij k and (Yk)ij=-fij k. In this way, having the structure constants of the Lie algebra g, one can solve (14) and obtain constant antisymmetric matrix P. Then, by substituting P in Eq. (13) we can obtain Poisson structure P of the Lie group G. The list of some Poisson structure P obtained by Eq. (14) are brought in Abedi-Fardad et al. [2].

Now, according to Tsiganov [18,19] and Vershilov and Tsiganov [21], let us suppose that the desired second Poisson bi-vector P′ is the Lie derivative of P along some unknown vector field X

P′=LX(P) , (15)

which must satisfy the equation

[P′,P′]=[LX(P),LX(P)]=0 ⇔[LX2(P),P]=0 , (16)

with respect to the Schouten bracket [.,.]. By (15) bi-vector P′ is compatible with a given bi-vector P, i.e. [P, P′] = 0. Obviously, Eq. (16) is too difficult to be solved, because it has infinitely many solutions labeled by different separated coordinates (see [18] and [17]). To solve Eq. (16), we will use the vector field [18]

X=Xμ∂μ. (17)

In this way the relation (15) has the following form:

P′μν=Xλ(∂λPμν)+(∂λXμ)Pνλ+∂λXνPλμ. (18)

We suppose that

Xμ=Xiei μ, (19)

where Xⁱ is a linear function of group coordinates x_i of the Lie group G. Note that one can obtain another set of vector field and second Poisson bi-vector by setting X_i as a quadratic function of the Lie group coordinates and in this way obtain new bi-Hamiltonian systems. Now using (8), (13) and (19) one can rewrite the relation (18) as follows²:

P′mn=Xifij mPjn+Xifik nPmk-(∂λXm)ek λPkn-(∂λXn)ej λPmj. (20)

Also, we can rewrite Eq. (16) in the following matrix forms [2]:

P′ Xi P′iγ+P′ Yγ P′+P′iγ Xi t P′+(etP′)kγ∂kP′+A+B=0, (21)

where e^t is a transpose of the vielbein eα μ and A and B have the following forms:

A = ((etP′)k1∂kP′1γ(etP′)k2∂kP′1γ(etP′)k1∂kP′2γ(etP′)k1∂kP′mγ........(etP′)km∂kP′1γ(etP′)km∂kP′2γ(etP′)km∂kP′mγ), (22)

B= ((etP′)k1∂kP′γ1(etP′)k1∂kP′γ2(etP′)k2∂kP′γ1(etP′)km∂kP′γ1........(etP′)k1∂kP′γm(etP′)k2∂kP′γm(etP′)km∂kP′γm). (23)

In this way, having the structure constants of the Lie algebra g and using the relation (20), we can solve the matrix equation (21) in order to obtain P′ and then by inserting P′ in Eq. (13) and using the related vielbeins, the second Poisson bi-vector P′ on Lie groups is obtained.

Note that in Abedi-Fardad et al. [2], by the adjoint representation of the Lie algebra, the authors have calculated some compatible Poisson structure and bi-Hamiltonian systems on Lie groups as phase space. Indeed, the Schouten bracket (1) has been rewritten in the matrix forms (2.17), (2.21) and (2.22) in Abedi-Fardad et al. [2] and they have obtained the set of compatible Poisson bracket by setting P′ as linear functions of the Lie group coordinates and new bi-Hamiltonian systems. In this work according to Tsiganov [18,19] and Vershilov and Tsiganov [21] we consider

P′=LX(P) , X=Xμ∂μ ,

and by using adjoint representation, we rewrite (15) in the matrix form (18) and obtain completely different new compatible Poisson bracket and bi-Hamiltonian systems on Lie groups as phase space.

The method which is used in Abedi-Fardad et al. [2] is completely different from the method of current work. By the new method we can find new integrable bi-Hamiltonian systems, for example, for two-dimensional symplectic real Lie groups, but by using the method which applied in Abedi-Fardad et al. [2] one cannot obtain them for the two-dimensional cases.

Our new results are not isomorphic to the systems that have been found in Abedi-Fardad et al. [2]. In this work we suppose vector field X to be linear. Study of non-linear vector field can be a new complicated question, but maybe gives us some newer systems.

4. SOME COMPATIBLE POISSON STRUCTURE ON LOW-DIMENSIONAL LIE ALGEBRAS

In this section, we will consider all of the two, four and nilpotent six-dimensional symplectic real Lie algebras and solve matrix equation (21) in order to obtain the vector field X and P′. For this purpose, we use the classification of two-, four- and six-dimensional real Lie algebras (A₂, A₄ and A₆) which have been presented in Patera et al. [15]. Let us consider an example; for Lie algebra A₂ ⊕ A₂ we have the following non-zero commutators:

[e1,e2]=e2,[e3,e4]=e4.

Also, according to Mojaveri and Rezaei-Aghdam [12] for Lie algebra A₂ ⊕ A₂ the matrix (eij) and P are as follows:

(eij)=(1x2000100001x40001), P=(0p1200*00p24**0p34***0).

Substituting fijk, P and eij in Eq. (20) and solving (21) one can obtain the vector field X and Poisson structure P′ for this Lie algebra. One of the solutions has the following form:

X1=a1x1-d2p12x2p24,X2=b1x1+b2x2+b3x3+c4+p24x4p34,X3=c3x3+c4x4,X4=d1x1+d2x2+d3x3+d4x4,

P′=(0-a1p12-b2p12+a1p12x100*00d1p12-b2p24-d4p24-b3p34+a1p24x1+c3p24x30*0-c3p34-d4p34+c3p34x3***0).

In this way, we have obtained vector field X and compatible Poisson structure on two, four and nilpotent six-dimensional symplectic real Lie algebras. The results are summarized in Table 1 ³. Note that in the Table 1 we present some Lie algebras in which we can construct integrable bi-Hamiltonian systems over their related Lie groups. Also, all parameters a_i, b_i, c_i, d_i, e_i, f_i and p_ij are arbitrary real constants.

g	Vector field X	Non-zero Poisson structure relations P′
A₂	X¹ = a₁x₁ + a₂x₂	{x₁, x₂} = −a₁p₁₂ − b₂p₁₂ + a₁p₁₂x₁
A₂	X² = b₁x₁ + b₂x₂	{x1,x2}=-a2p12-d3x3p12+a2x4p122
A_4,1	X¹ = a₂x₂ + a₃x₃ + a₄x₄	{x1,x3}=a2p142-a4p14-b4p14-c4p14+b4x4p14+c4x4p14
	X² = b₁x₁ + b₂x₂ − c₃x₃ + b₄x₄	{x1,x3}=a2p142-a4p14-b4p14-c4p14+b4x4p14+c4x4p14
	X³ = −b₁x₁ − b₂x₂ + c₃x₃ + c₄x₄	{x₂,x₃}=−a₂p₂₃−a₃p₂₃−d₃p₂₃−₂d₃x₃p₂₃
	X⁴ = d₃x₃ − (a₂x₄)/2	{x₂,x₃}=−a₂p₂₃−a₃p₂₃−d₃p₂₃−₂d₃x₃p₂₃
A_4,3	X¹ = a₁x₁ + a₂x₂ + a₃x₃ + a₄x₄	{x1,x2}=-b2p12-b1e4xp12-d3p12x3+b2p12x4
	X² = b₁x₁ + b₂x₂ − b₃x₃ + b₄x₄	{x1,x4}=(-a4+b2-b4-c4)p14-b1e4xp14+a3p14x3-d3p14x3+(-a3+d3)p14x3+(a4p14+b2p14+c4p14)x4
	X³ = −a₁x₁ − a₂x₂ + (d₃ − a₃) + c₄x₄
	X⁴ = d₃x₃ − b₂x₄	{x₂,x₃}=(-a₃-b₂-b₃-d₃+a₃-d₃)p₂₃-d₃p₂₃x₃
II ⊕ R	X¹ = d₁x₁ + a₂x₂ + a₃x₃ + a₄x₄	{x₁, x₂} = − a₂p₁₂ − c₂p₁₂ − d₂p₁₂ − c₂p₁₂x₂
	X² = b₄x₄	{x₁, x₄} = − a₃p₁₃ − d₃p₁₃ + b₄p₁₃x₄
	X³ = c₂x₂	{x₃, x₄} = − a₃p₃₄ − a₄p₃₄ − b₄p₃₄ − d₃p₃₄ − d₄p₃₄ + b₄p₃₄x₄
	X⁴ = d₁x₁ + d₂x₂ + d₃x₃ + d₄x₄
III ⊕ R	X¹ = −a₃x₂ + a₃x₃ + a₄x₄	{x₁, x₂} = − p₁₃(b₁ + c₁ + d₁ − 2b₁x₁ − 2c₁x₁ + 2d₃x₃ + a₃( − 1 + 8x₃))
	X² = b₁x₁ + b₂x₂ + b₃x₃ + a₄x₄	{x₁, x₃} = − p₁₃(b₁ + c₁ + d₁ − 2b₁x₁ − 2c₁x₁ + 2d₃x₃ + a₃( − 1 + 8x₃))
	X³ = c₁x₁ − b₂x₂ − (2a₃ + b₃ + d₃)x₃	{x₂, x₄} = p₂₄( − d₄ + a₃(1 + 2x₂ − 2x₃) − 2a₄(1 + x₄))
	X⁴ = d₁x₁ + d₃x₃ + d₄x₄
VI₀ ⊕ R	X¹ = −b₁x₁ − b₂x₂ + a₃x₃ + a₄x₄	{x₁, x₂} = − 2p₁₂(c₁x₁ + c₂x₂)
	X² = b₁x₁ + b₂x₂ + b₃x₃ + b₄x₄	{x₃, x₄} = p₃₄( − b₃ − b₄ − d₃ − d₄ + a₃( − 1 + x₃) + b₃x₃ + a₄( − 1 + x₄) + b₄x₄)
	X³ = c₁x₁ − c₂x₂
	X⁴ = (− c₁ − c₂ − d₂)x₁ + d₂x₂ + d₃x₃ + d₄x₄
A₂ ⊕ A₂	X¹ = a₁x₁ − (d₂p₁₂x₂)/p₂₄	{x₁, x₂} = − a₁p₁₂ − b₂p₁₂ + a₁p₁₂x₁
	X² = b₁x₁ + b₂x₂ + b₃x₃ + (c₄p₂₄x₄)/p₃₄	{x₂, x₄} = d₁p₁₂ − b₂p₂₄ − d₄p₂₄ − b₃p₃₄ + a₁p₂₄x₁ + c₃p₂₄x₃
	X³ = c₃x₃ + c₄x₄	{x₃, x₄} = c₃p₃₄ − d₄p₃₄ + c₃p₃₄x₃
	X⁴ = d₁x₁ + d₂x₂ + d₃x₃ + d₄x₄
A_6,7	X¹ = a₁x₁ + a₃x₃,	{x₁,x₅}=a₁p₁₅−d₁p₁₅−f₁p₁₅−h₁p₁₅−d₁p₁₅x₁−c₃p₁₅x₃−(−c₃−h₄)p₁₅x₃−h₄p₁₅x₃
	X² = − 2a₁x₁ + b₃x₃,	{x₂,x₆}=(−f₂−(−c₃−f₆)−f₆−h₂−c₂x₂−c₃x₃−(−c₃−f₆)x₃−f₆x₃−c₆x₆)p₂₆
	X³ = c₂x₂ + c₃x₃ + c₆x₆	{x₃,x₄}=(−a₃−b₃−c₃−f₃−h₃−(−c₃−h₄)−h₄+₂a₃x₃+b₃x₃)p₃₄
	X⁴ = d₁x₁ + c₂x₂ + (−c₃ − h₄)x₃ − c₆x₆	x₄,x₅}=(−h₄+a₁x₁+a₃x₃)p₄₅
	X⁵ = f₁x₁ + f₂x₂ + f₃x₃ + f₅x₅ + f₆x₆,
	X⁶ = h₁x₁ + h₂x₂ + h₃x₃ + h₄x₄ − f₅x₅ + (−c₃ − f₆)x₆
A_6,25	X1=a3x3+12(-c6-f5)x6,X2=b1x1,X3=c1x1-2a3x3+c4x4+c5x5+c6x6,X4=d5x5,X5=f2x2+a3x3+f4x4+f5x5+12(-c6-f5)x6,X6=h1x1+h2x2+h4x4+h5x5	{x1,x3}=(-b1-c1-h1-b1x1)p13{x2,x4}=(-c4-f2-f4-h2-h4-f2x2-f4x4-c6x5-(-c6-f5)x5-f5x5)p24{x3,x6}=-c6p36-(-c6-f5)p36{x4,x5}=(-h4+a1x1+a3x3)p45{x5,x6}=(-c5-c6-d5-(-c6-f5)-f5-h5+d5x5)p56

Table 1

Some vector fields and compatible Poisson structures on two, four and nilpotent six-dimensional symplectic real Lie algebras

5. COMPATIBLE POISSON STRUCTURES AND INTEGRABLE BI-HAMILTONIAN SYSTEMS ON TWO, FOUR AND NILPOTENT SIX-DIMENSIONAL SYMPLECTIC REAL LIE GROUPS

In this section, we construct the compatible Poisson structures and integrable bi-Hamiltonian systems with real Lie groups separately as follows. Substituting P′ in Eq. (13) and using the related vielbeins, the compatible Poisson structure P is obtained. Using relations (15) and (16), the compatible Poisson structure P′ and vector field X on Lie groups are obtained. In this way we find new bi-Hamiltonian systems over two, four and nilpotent six-dimensional symplectic real Lie groups as phase spaces.

Lie group A₂:

Substituting Xⁱ in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:

X1=a1x1+a2x2+a1x1x2+a2x22,X2=b1x1+b2x2,

P=(0p12*0),P′=(0-a1p12-b2p12+a1p12x1*0).

By means of Eqs. (9) and (11), the integral of motion can be found for this Lie group as follows:

H=-a1-b2+a1x1.

Lie group A_4,1:

Similar to previous case, from (13) and (19) we calculate the vector field X and compatible Poisson structures P and P′ as follows:

X1=a2x2+a3x3+a4x4,X2=b1x1+b2x2-c3x3+b4x4+b1x1x4+b2x2x4-c3x3x4+b4x42,X3=-b1x1-b2x2+c3x3+c4x4-b1x1x4-b2x2x4+c3x3x4+c4x42-12b1x1x42-12b2x2x42+12c3x3x42+(c4x43)/2,X4=d3x3-(a2x4)/2,

P=(0p23x422+p12p23x4p14*0p230**00***0),P′=(0p12p13p14*0p230**00***0),

where

p12=-12a2p23x42-12a3p23x42-12d3p23x42-d3p23x3x42+a2p12x42-a2p12-d3p12x3,p13=-p23(a2+a3+d3+2d3x3)x4,p14=12p14(a2-2(a4-(b4+c4)(-1+x4))),p23=-a2p23-a3p23-d3p23-2d3x3p23.

For this Lie group, by means of (9) and (11), the integrals of motion can be found as follows:

H1=-a22-a3-a4-b4-c4-d3-2d3x3+b4x4+c4x4,H2=12((a2+a3+d3+2d3x3)2+(a22-a4+(b4+c4)(-1+x4))2).

Lie group A_4,3:

Again, substituting Xⁱ in (19) and P′ in (13) one can obtain the vector field X and compatible Poisson structures P and P′ for this Lie group as follows:

X1=a1ex4x1+a2ex4x2+a3ex4x3+a4ex4x4,X2=b1x1+b2x2+b3x3+b4x4,X3=-a1x1-a2x2(d3-a3)x3+c4x4-a1x1x4-a2x2x4-a3x3x4+c4x42,X4=d3x3-b2x4,

P=(0p12ex40p14ex4*0p230**00***0),P′=(0p120p14*0p230**00***0),

where

p12=-b2ex4p12-b1e2x4p12-d3ex4x3p12+b2ex4x4p12,p14=-a4ex4p14+b2ex4p14-b4ex4p14-c4ex4p14-b1e2x4p14+a4ex4x4p14+b2ex4x4p14+c4ex4x4p14,p23=-b2p23-b3p23-2d3p23-d3x3p23.

Also the integrals of motion can be found as follows:

H1=-a4-b3-b4-c4-2d3-b1ex4-d3x3+a4x4+b2x4+c4x4,H2=12((b2+b3+d3(2+x3))2+(a4+b4+c4+b1ex4-a4x4-c4x4-b2(1+x4))2).

Lie group A₂ ⊕ A₂:

Substituting Xⁱ in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X, compatible Poisson structures P, P′ for this Lie group are obtained as follows:

X1=a1x1-d2p12x2p24+a1x1x2-d2p12x22p24,X2=b1x1+b2x2+b3x3+c4p24x4p34,X3=c3x3+c4x4+c3x3x4+c4x42,X4=d1x1+d2x2+d3x3+d4x4,

P=(0p1200*00p24**0p34***0),P′=(0(-a1-b1-2c1+a1x1-b1x1)p1200*00a1p24x1+c1p24x1+b3p24x3+c3p24x3**0-2b3p34-c3p34-d3p34+c3x3p34-d3x3p34***0).

By means of (9) and (11), the integral of motion can be found for this Lie group as follows:

H1=-a1-b1-2b3-2c1-c3-d3+a1x1-b1x1+c3x3-d3x3 ,H2=12((a1+b1+2c1-a1x1+b1x1)2+(2b3+c3+d3-c3x3+d3x3)2).

Lie group III ⊕ R:

Also for this Lie group, from (13) and (19) one can obtain the vector field X, compatible Poisson structures P, P′ as follows:

X1=-a3x2+2a3x22+a3x3-2a3x32+a4x4-2a4x2x4-2a4x3x4,X2=b1x1+b2x2+b3x3+a4x4,X3=c1x1-b2x2-(2a3+b3+d3)x3,X4=d1x1+d3x3+d4x4,

P=(0p13p130*00p24**00***0),P′=(0p12p130*00p24**00***0),

where

p12=a3p13-b1p13-c1p13-d1p13+2b1x1p13+2c1x1p13-8a3x3p13-2d3x3p13,p13=a3p13-b1p13-c1p13-d1p13+2b1x1p13+2c1x1p13-8a3x3p13-2d3x3p13,p24=a3p24-2a4p24-d4p24+2a3x2p24-2a3x3p24-2a4x4p24.

By means of (9) and (11), the integral of motion can be found for this Lie group as follows:

H1=2a3-2a4-b1-c1-d1-d4+2b1x1+2c1x1+2a3x2-10a3x3-2d3x3-2a4x4,H2=12((b1+c1+d1-2b1x1-2c1x1+2d3x3+a3(-1+8x3))2+(d4+a3(-1-2x2+2x3)+2a4(1+x4))2).

Lie group II ⊕ R:

Through substituting Xⁱ in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ for this Lie group as follows:

X1=-d1x1+a2x2+a3x3+a4x4,X2=b4x4+b4x3x4,X3=c2x2,X4=d1x1+d2x2+d3x3+d4x4,

P=(0p12p130*000**0p34***0),P′=(0p12p130*000**0p34***0),

where

p12=-a2p12-c2p12-d2p12-c2x2p12,p13=-a3p13-d3p13+b4x4p13,p34=-a3p34-a4p34-b4p34-d3p34-d4p34+b4x4p34.

By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:

H1=-a2-a3-a4-b4-c2-d2-d3-d4-c2x2+b4x4,H2=12((a2+c2+d2+c2x2)2+(a3+a4+b4+d3+d4-b4x4)2).

Lie group VI₀ ⊕ R:

Again, substituting Xⁱ in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:

X1=(-b1x1-b2x2+a3x3+a4x4)(Cosh[x3]+Sinh[x3]),X2=(b1x1+b2x2+b3x3+b4x4)(Cosh[x3]+Sinh[x3]),X3=c1x1+c2x2,X4=-(c1+c2+d2)x1+d2x2+d3x3+d4x4,

P=(0p12cosh2(x3)-p12sinh2(x3)00*000**0p34***0),P′=(0p1200*000**0p34***0),

where

p12=-2c1p12x1cosh2(x3)-2c2p12x2cosh2(x3)+2c1p12x1sinh2(x3)+2c2p12x2sinh2(x3),p34=-a3p34-a4p34-b3p34-b4p34-d3p34-d4p34+a3x3p34+b3x3p34+a4x4p34+b4x4p34.

By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:

H1=-a3-a4-b3-b4-d3-d4-2c1x1-2c2x2+a3x3+b3x3+a4x4+b4x4,H2=14(8(c1x1+c2x2)2+2(a3+a4+b3+b4+d3+d4-a3x3-b3x3-a4x4-b4x4)2).

Lie group A_6,7:

Substituting Xⁱ in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:

X1=a1x1+a3x3+a1x1x3+a3x32+a1x1x4+a3x3x4,X2=-2a1x1+b3x3-2a1x1x3+b3x32,X3=c2x2+c3x3+c6x6,X4=d1x1-c2x2-(c3+h4)x3+d6x6,X5=f1x1+f2x2+f3x3+f5x5+f6x6,X6=h1x1+h2x2+h3x3+h4x4-f5x5-(c3+f6)x6,

P=(0000p150*0000p16**0p3400***0p45+p15x30****00*****0),P′=(0000p120*0000p26**0p3400***0p450****00*****0),

where

p15=(a1-d1-f1-h1-d1x1)p15,p26=(c3-f2-h2-c2x2-c6x6)p26,p34=(-a3-b3-f3-h3+2a3x3+b3x3)p34,p45=-h4p45+a1p45x1+a1p15x3-d1p15x3-f1p15x3-h1p15x3+a3p45x3-d1p15x1x3.

By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:

H1=a1-a3-b3+c3-d1-f1-f2-f3-h1-h2-h3-d1x1-c2x2+2a3x3+b3x3-c6x6,H2=12((-a1+d1+f1+h1+d1x1)2+(a3+b3+f3+h3-2a3x3-b3x3)2+(-c3+f2+h2+c2x2+c6x6)2),H3=13(-(-a1+d1+f1+h1+d1x1)3-(a3+b3+f3+h3-2a3x3-b3x3)3-(-c3+f2+h2+c2x2+c6x6)3).

Lie group A_6,25:

Finally, the vector field X, compatible Poisson structures P, P′ and the integrals of motion of this Lie group are obtained as follows:

X1=a3x3+a3x2x3-1/2(c6+f5)x6-1/2(c6+f5)x2x6,X2=b1x1,X3=c1x1-2a3x3+c4x4+c5x5+c6x6,X4=d5x5+d5x52,X5=f2x2+a3x3+f4x4+f5x5-1/2(c6+f5)x6,X6=h1x1+h2x2+h4x4+h5x5,

P=(00p13000*00p240p24x5**000p36***000****0p56*****0),

where

P′=(00p13000*00p240p26**000p36***000****0p56*****0),

p13=(-b1-c1-h1-b1x1)p13,p24=(-c4-f2-f4-h2-h4-f2x2-f4x4)p24,p26=(-c4x5-f2x5-f4x5-h2x5-h4x5-f2x2x5-f4x4x5)p24,p36=f5p36,p56=-c5p56-d5p56-h5p56+d5p56x5.

By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:

H1=-b1-c1-c4-c5-d5-f2-f4-h1-h2-h4-h5-b1x1-f2x2-f4x4+d5x5,H2=12((b1+c1+h1+b1x1)2+(c4+f2+f4+h2+h4+f2x2+f4x4)2+c5+d5+h5-d5x5)2),H3=13(-(b1+c1+h1+b1x1)3-(c4+f2+f4+h2+h4+f2x2+f4x4)3-(c5+d5+h5-d5x5)3).

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT

The corresponding author would like to thank Professor Norbert Euler for kindly and worthy comments. The authors would like to thank the referees for carefully reading the paper and their valuable comments.

Footnotes

1

The study of non-constant P^ij can be a new problem.

2

Note that here P^ij’s are constant.

3

Note that in Abedi-Fardad et al. [2] the Poisson structure P on Lie algebras have been given.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 2
Pages: 194 - 204
Publication Date: 2020/12/17
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.2991/jnmp.k.201104.001 How to use a DOI?
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TY  - JOUR
AU  - Gh. Haghighatdoost
AU  - S. Abdolhadi-Zangakani
AU  - J. Abedi-Fardad
PY  - 2020
DA  - 2020/12/17
TI  - Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras
JO  - Journal of Nonlinear Mathematical Physics
SP  - 194
EP  - 204
VL  - 28
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.201104.001
DO  - 10.2991/jnmp.k.201104.001
ID  - Haghighatdoost2020
ER  -

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