Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras
- DOI
- 10.2991/jnmp.k.201104.001How 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
The Poisson bracket corresponding to the Poisson bi-vector has the form
Definition: [13] In the coordinate basis, T_{p}M spanned by {e_{μ}} = {∂_{μ}} and
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]
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.
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
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
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
In this way the relation (15) has the following form:
We suppose that
Also, we can rewrite Eq. (16) in the following matrix forms [2]:
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
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 (A_{2}, A_{4} and A_{6}) which have been presented in Patera et al. [15]. Let us consider an example; for Lie algebra A_{2} ⊕ A_{2} we have the following non-zero commutators:
Also, according to Mojaveri and Rezaei-Aghdam [12] for Lie algebra A_{2} ⊕ A_{2} the matrix
Substituting
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′ |
---|---|---|
A_{2} | X^{1} = a_{1}x_{1} + a_{2}x_{2} | {x_{1}, x_{2}} = −a_{1}p_{12} − b_{2}p_{12} + a_{1}p_{12}x_{1} |
X^{2} = b_{1}x_{1} + b_{2}x_{2} | ||
A_{4,1} | X^{1} = a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} | |
X^{2} = b_{1}x_{1} + b_{2}x_{2} − c_{3}x_{3} + b_{4}x_{4} | ||
X^{3} = −b_{1}x_{1} − b_{2}x_{2} + c_{3}x_{3} + c_{4}x_{4} | {x_{2},x_{3}}=−a_{2}p_{23}−a_{3}p_{23}−d_{3}p_{23}−_{2}d_{3}x_{3}p_{23} | |
X^{4} = d_{3}x_{3} − (a_{2}x_{4})/2 | ||
A_{4,3} | X^{1} = a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} | |
X^{2} = b_{1}x_{1} + b_{2}x_{2} − b_{3}x_{3} + b_{4}x_{4} | ||
X^{3} = −a_{1}x_{1} − a_{2}x_{2} + (d_{3} − a_{3}) + c_{4}x_{4} | ||
X^{4} = d_{3}x_{3} − b_{2}x_{4} | {x_{2},x_{3}}=(-a_{3}-b_{2}-b_{3}-d_{3}+a_{3}-d_{3})p_{23}-d_{3}p_{23}x_{3} | |
II ⊕ R | X^{1} = d_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} | {x_{1}, x_{2}} = − a_{2}p_{12} − c_{2}p_{12} − d_{2}p_{12} − c_{2}p_{12}x_{2} |
X^{2} = b_{4}x_{4} | {x_{1}, x_{4}} = − a_{3}p_{13} − d_{3}p_{13} + b_{4}p_{13}x_{4} | |
X^{3} = c_{2}x_{2} | {x_{3}, x_{4}} = − a_{3}p_{34} − a_{4}p_{34} − b_{4}p_{34} − d_{3}p_{34} − d_{4}p_{34} + b_{4}p_{34}x_{4} | |
X^{4} = d_{1}x_{1} + d_{2}x_{2} + d_{3}x_{3} + d_{4}x_{4} | ||
III ⊕ R | X^{1} = −a_{3}x_{2} + a_{3}x_{3} + a_{4}x_{4} | {x_{1}, x_{2}} = − p_{13}(b_{1} + c_{1} + d_{1} − 2b_{1}x_{1} − 2c_{1}x_{1} + 2d_{3}x_{3} + a_{3}( − 1 + 8x_{3})) |
X^{2} = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + a_{4}x_{4} | {x_{1}, x_{3}} = − p_{13}(b_{1} + c_{1} + d_{1} − 2b_{1}x_{1} − 2c_{1}x_{1} + 2d_{3}x_{3} + a_{3}( − 1 + 8x_{3})) | |
X^{3} = c_{1}x_{1} − b_{2}x_{2} − (2a_{3} + b_{3} + d_{3})x_{3} | {x_{2}, x_{4}} = p_{24}( − d_{4} + a_{3}(1 + 2x_{2} − 2x_{3}) − 2a_{4}(1 + x_{4})) | |
X^{4} = d_{1}x_{1} + d_{3}x_{3} + d_{4}x_{4} | ||
VI_{0} ⊕ R | X^{1} = −b_{1}x_{1} − b_{2}x_{2} + a_{3}x_{3} + a_{4}x_{4} | {x_{1}, x_{2}} = − 2p_{12}(c_{1}x_{1} + c_{2}x_{2}) |
X^{2} = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + b_{4}x_{4} | {x_{3}, x_{4}} = p_{34}( − b_{3} − b_{4} − d_{3} − d_{4} + a_{3}( − 1 + x_{3}) + b_{3}x_{3} + a_{4}( − 1 + x_{4}) + b_{4}x_{4}) | |
X^{3} = c_{1}x_{1} − c_{2}x_{2} | ||
X^{4} = (− c_{1} − c_{2} − d_{2})x_{1} + d_{2}x_{2} + d_{3}x_{3} + d_{4}x_{4} | ||
A_{2} ⊕ A_{2} | X^{1} = a_{1}x_{1} − (d_{2}p_{12}x_{2})/p_{24} | {x_{1}, x_{2}} = − a_{1}p_{12} − b_{2}p_{12} + a_{1}p_{12}x_{1} |
X^{2} = b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{3} + (c_{4}p_{24}x_{4})/p_{34} | {x_{2}, x_{4}} = d_{1}p_{12} − b_{2}p_{24} − d_{4}p_{24} − b_{3}p_{34} + a_{1}p_{24}x_{1} + c_{3}p_{24}x_{3} | |
X^{3} = c_{3}x_{3} + c_{4}x_{4} | {x_{3}, x_{4}} = c_{3}p_{34} − d_{4}p_{34} + c_{3}p_{34}x_{3} | |
X^{4} = d_{1}x_{1} + d_{2}x_{2} + d_{3}x_{3} + d_{4}x_{4} | ||
A_{6,7} | X^{1} = a_{1}x_{1} + a_{3}x_{3}, | {x_{1},x_{5}}=a_{1}p_{15}−d_{1}p_{15}−f_{1}p_{15}−h_{1}p_{15}−d_{1}p_{15}x_{1}−c_{3}p_{15}x_{3}−(−c_{3}−h_{4})p_{15}x_{3}−h_{4}p_{15}x_{3} |
X^{2} = − 2a_{1}x_{1} + b_{3}x_{3}, | {x_{2},x_{6}}=(−f_{2}−(−c_{3}−f_{6})−f_{6}−h_{2}−c_{2}x_{2}−c_{3}x_{3}−(−c_{3}−f_{6})x_{3}−f_{6}x_{3}−c_{6}x_{6})p_{26} | |
X^{3} = c_{2}x_{2} + c_{3}x_{3} + c_{6}x_{6} | {x_{3},x_{4}}=(−a_{3}−b_{3}−c_{3}−f_{3}−h_{3}−(−c_{3}−h_{4})−h_{4}+_{2}a_{3}x_{3}+b_{3}x_{3})p_{34} | |
X^{4} = d_{1}x_{1} + c_{2}x_{2} + (−c_{3} − h_{4})x_{3} − c_{6}x_{6} | x_{4},x_{5}}=(−h_{4}+a_{1}x_{1}+a_{3}x_{3})p_{45} | |
X^{5} = f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3} + f_{5}x_{5} + f_{6}x_{6}, | ||
X^{6} = h_{1}x_{1} + h_{2}x_{2} + h_{3}x_{3} + h_{4}x_{4} − f_{5}x_{5} + (−c_{3} − f_{6})x_{6} | ||
A_{6,25} |
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 A_{2}:
Substituting X^{i} in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:
By means of Eqs. (9) and (11), the integral of motion can be found for this Lie group as follows:
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:
For this Lie group, by means of (9) and (11), the integrals of motion can be found as follows:
Lie group A_{4,3}:
Again, substituting X^{i} 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:
Also the integrals of motion can be found as follows:
Lie group A_{2} ⊕ A_{2}:
Substituting X^{i} 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:
By means of (9) and (11), the integral of motion can be found for this Lie group as follows:
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:
By means of (9) and (11), the integral of motion can be found for this Lie group as follows:
Lie group II ⊕ R:
Through substituting X^{i} 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:
By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:
Lie group VI_{0} ⊕ R:
Again, substituting X^{i} in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:
By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:
Lie group A_{6,7}:
Substituting X^{i} in Eq. (19) and P′ in Eq. (13) one can obtain the vector field X and compatible Poisson structures P and P′ as follows:
By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:
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:
By means of (9) and (11), the integrals of motion can be found for this Lie group as follows:
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
The study of non-constant P^{ij} can be a new problem.
Note that here P^{ij}’s are constant.
REFERENCES
Cite this article
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 -