On Extensions, Lie-Poisson Systems, and Dissipations
aa r X i v : . [ m a t h - ph ] J a n On Extensions, Lie-Poisson Systems, and Dissipations
January 12, 2021O˘gul Esen* , G¨okhan ¨Ozcan* and Serkan S¨utl¨u** *Department of Mathematics, Gebze Technical University,41400 Gebze-Kocaeli, Turkey** Department of Mathematics, I¸sık University,34980 S¸ile-˙Istanbul, Turkey Abstract
On the dual space of extended structure , equations governing the collective motion of two mutually interactingLie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providingthe most general realization of (de)coupling of Lie-Poisson systems. A double cross sum (matched pair) of2-cocycle extensions are constructed. The conditions are determined for this double cross sum to be a 2-cocycle extension by itself. On the dual spaces, Lie-Poisson equations are computed. We complement thediscussion by presenting a double cross sum of some symmetric brackets, such as double bracket, Cartan-Killingbracket, Casimir dissipation bracket, and Hamilton dissipation bracket. Accordingly, the collective motion of twomutually interacting irreversible dynamics, as well as mutually interacting metriplectic flows, are obtained. Thetheoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositionsof the BBGKY hierarchy are presented. As finite-dimensional examples, the coupling of two Heisenberg algebrasand coupling of two copies of 3 D dynamics are studied. MSC2020:
Key words:
Lie-Poisson Equation; Metriplectic System; Extended Structure.
Contents [email protected] [email protected] [email protected] Illustration: Decomposing 3 Particles BBGKY Hierarchy 22
It is a well-known fact that two mutually interacting dynamical/mechanical systems, when coupled, cannot preservetheir individual motions in the collective system. This is manifested in the equation of motion of the collectivesystem as the existence of additional terms, to those belonging to the individual subsystems. It is the Hamiltonianrealization of this collective motion, of two mutually interacting physical systems, that we study in the presentpaper. In order to be able to present the mutual interactions as Lie group / Lie algebra actions, we shall considerthe systems whose configuration spaces are Lie groups, [5, 7, 72, 80].If the configuration space of a physical system admits a Lie group structure, then the reduced Hamiltonian dynamicscan be achieved on the dual space of the Lie algebra [1, 4, 51, 57] since it carries a natural Poisson structure, calledthe Lie-Poisson bracket. Many physical systems fit into this geometry, such as the rigid body, fluid and plasmatheories [39,41]. Coupling two different characters of a physical system, such as the fluid motion under the EM fieldor the rigid body motion under the gravity [79], are particular instances of coupled systems where only one-sidedinteraction is allowed. Such systems have been studied, in literature, by the semidirect product theory, which hasbeen successfully established both in Lagrangian dynamics, see for example [14,54], and Hamiltonian dynamics, seefor example [59, 60], see also [87].
Double cross sum (matched pair) Lie groups.
Semidirect product Lie group constructions allow only one-sided action. As such, mutual actions are beyond the scope of the semidirect product theory. On the other hand,the double cross sum (matched pair) theory of [62–64] rises over two Lie groups with mutual actions, subject tocertain compatibility conditions. As a result, the Cartesian product of two such Lie groups becomes a Lie group byitself, through a suitable multiplication built over the mutual actions, called the “double cross sum” of these Liegroups [62–64]. A double cross product Lie group, then, contains the constitutive Lie groups as trivially intersecting2ie subgroups. Historically, generalizations of the semidirect product theory may be traced back to [83–85,90] underthe name of the Zappa-Sz´ep product, see also [9]. The idea was revived later in [53], as the “product of subgroups”(and as the “double Lie group” in [52]), from the representation theory point of view. However, the name “matchedpair” was first used in [81,82] for the Hopf algebra extensions. A pair of groups were referred to as “matched pairs”,for the first time, in [86] as yet another incarnation of the intimate relation between Hopf algebras and groups. Weadd that double cross sum (matched pair) Lie algebras are studied in [47] under the name of “twilled extension”.See [25] for more details and a list of applications of the theory both in mathematics and physics.
Dynamics on double cross sums (matched pairs).
Double cross sum (matched pair) of Lie groups (algebras)provides a promising geometrical framework for studying the collective motion of two mutually interacting physicalsystems. However, the Hamiltonian (Lie-Poisson) theory over double cross sum groups (and their double crosssum Lie algebras) is developed only recently in [23]. This Hamiltonian matched pair theory has immediately foundapplications in kinetic theory [21], and fluid theories [19]. The Lagrangian counterpart of the double cross sumshas been developed for the first order Euler-Poincar´e theories in [22], and for the higher order theories in [20]. Fordiscrete Lagrangian dynamics in the framework of Lie groupoids, the matched pair approach has found applicationsin [25].Evidently, one may use the matched pair strategy to decouple a physical system into two of its interacting subsystemsas well. So, one may study the constitutive subsystems in order to examine the whole system. In [24], it has beenshown that non-relativistic collisionless Vlasov’s plasma can be written as a non-trivial double cross sum (matchedpair) of its two subdynamics. One of the constitutive subdynamics is the compressible Euler fluid, while the other isthe kinetic moments of order > Extended structures.
Lie algebra actions are not the only sources to couple two Lie algebras. More generally,Lie algebra extensions may also be formed by the weak-actions and (twisted) 2-cocycles as well. An “extendedstructure”, introduced in [2, 3], provides a method to couple a Lie algebra and a vector space in a universal way.More precisely, a Lie algebra and a vector space are coupled using the action of the Lie algebra on the vector space,and the corresponding weak action (encoded by a twisted 2-cocycle) of the linear space on the Lie algebra. Fromthe decomposition point of view, this corresponds to the decomposition of a Lie algebra into a Lie subalgebra andits vector space complement. In particular, if the complementary space happens to be a Lie subalgebra, then theextended structure reduces to a double cross sum. Postponing the details into Section 3, we shall for now be content3ith the following diagram. Extended Structure twisted cocycle is zero sssssssssss y y ssssssssss one of the actions is zero ▲▲▲▲▲▲▲▲▲▲▲ % % ▲▲▲▲▲▲▲▲▲▲ Double Cross Sums (Matched Pairs)
The first goal of the present work.
For the semidirect product Hamiltonian theory, introductions of the centralextensions have already been studied, see for example, in [55]. But such an extension is missing for the matched pairHamiltonian theory. In accordance with this observation, the first goal of the present work is to fill this gap with atheory of Hamiltonian (Lie-Poisson) dynamics on the dual of extended structures . This will be achieved in Section4. The rich geometry of extended structures enables us to couple a Lie-Poisson bracket with some external variablesinteracting in all possible ways. From the decomposition point of view, this corresponds to the decompositionof Lie-Poisson dynamics to one of its Lie-Poisson subdynamics, and a complementary subsystem, which is notnecessarily Lie-Poisson. So, the Lie-Poisson bracket and the Lie-Poisson equations provided in Subsection 4.1evidently applicable to any Lie-Poisson model. In other words, they determine the most general form of thedecomposition. We remark that, in the matched pair Hamiltonian dynamics, the constitutive systems must beLie-Poisson by itself. In other words, the matched pair Hamiltonian dynamics turns out to be now a particularinstance of extended Hamiltonian dynamics in Subsection 4.1.Extended Hamiltonian framework in Subsection 4.1 fits very well with the cuts of order ≥ n > n = 3, which is missing in [56], weshall present two decompositions of the BBGKY dynamics; the matched pair (double cross sum) decomposition,and the extended structure decomposition. This way, we shall also be able to compare these two approaches. Thesedecompositions will precisely determine the relationship between the moments of the 3-particle plasma densityfunction. Without the extended Hamiltonian dynamics, on the other hand, the latter decomposition wouldn’t bepossible. The second goal of the present work.
Being Lie algebras, 2-cocycle extensions of Lie algebras may formmatched pairs along with the properly coupled 2-cocycle terms. We shall derive the conditions for the matchedpair of 2-cocycle extensions to be a 2-cocycle in Section 6 as the second goal of this note. We record the resultin Proposition 8. In addition, Lie-Poisson dynamics on the coupled system will be presented. The geometricframework, and the dynamical equations, will be illustrated in Section 8 via two copies of the Heisenberg algebra.4t is well-known that the symplectic two-form on the two-dimensional Euclidean space can be used to determine anextended Lie algebra structure on the three-dimensional Euclidean space. These results with the Heisenberg algebrain 3 D [37]. Being a nilpotent Lie algebra of class two, the Heisenberg algebra may be matched by itself, [64]. Thisprovides an application of the extended Hamiltonian theory as well as Proposition 8 and results with the physicallyinteresting equations on the dual spaces. The coadjoint orbits of the Heisenberg Lie algebras read the canonicalHamiltonian formalism. So that coupling of two Heisenberg algebras provides a mutual coupling of two canonicalHamiltonian dynamics under mutual interaction, which was also missing in the literature. Coupling of dissipative systems.
If a dynamical system is in the Hamiltonian form, then as a result of theskew-symmetry of the Poisson bracket, the Hamiltonian function is a conserved quantity [57]. This geometric factcorresponds to the conservation of energy when applied to some physical problems. The time-reversal characterof the Hamiltonian dynamics depends basically on this observation. Those systems violating the time-reversalproperty, therefore, could not be put into the Hamiltonian formulation. Nevertheless, one may achieve to add a(Rayleigh type) dissipative term to the Lie-Poisson dynamics by means of a linear operator from the dual space tothe Lie algebra [8]. This naive strategy works very well for many physical problems. More generally, and perhapsa more geometric approach is to add an additional feature to the manifold. There are methods, in the literature,to achieve this.
GENERIC - Metriplectic systems.
In the early ’80s, some extensions of the Poisson geometry are introducedindependently in order to add dissipative terms into Hamiltonian formulations (see Subsection 2.2). These geome-tries are known today under the name as metriplectic dynamics [44,45,69,70] or as GENERIC (General Equation forNon-Equilibrium Reversible Irreversible Coupling) [31, 34, 75]. In metriplectic systems, the geometry is determinedby two compatible brackets, namely a Poisson bracket and a (possibly semi-Riemannian) symmetric bracket. InGENERIC, which is more general [67], a dissipation potential is employed in order to arrive at the irreversible partof the dynamics. Accordingly, the Legendre transformation of dissipation potential determines the time irreversiblepart of the dynamics [32, 76]. If the potential is quadratic, then one arrives at a symmetry bracket as a particularcase.One of the problems in this coupling is to determine a proper symmetric bracket, or a dissipation potential com-patible with the Poisson geometry (see Subsection 2.3). In the present work, we shall refer to geometric ways toobtain symmetric brackets; such as the double bracket [10–12], Cartan-Killing bracket [71], and Casimir dissipationbracket [27]. We are interested in these geometries since, in the Lie-Poisson framework, they may be defined by theLie algebra bracket directly in an algorithmic way. In the present work, we shall study extensions/couplings of thesymmetric brackets as well as the dissipative systems. The latter will be the third goal here.
The third goal of the present work.
In order to present a complete picture, we shall study the couplings of thedissipative terms which are added to the Lie-Poisson dynamics as the third goal of this present work in Section 7.In other words, our third goal is to couple two metriplectic systems under mutual actions. First, we aim to providea way to couple two mutually interacting systems involving Rayleigh type dissipative terms. Later, we presentcouplings of the Double brackets, the Cartan-Killing brackets, and the Casimir dissipation brackets.As for the coupling problem, our emphasis shall be on 3 D systems. Accordingly, we present two illustrations.One, given in Section 9, will be on the rigid body dynamics. We shall provide couplings of both reversible andirreversible rigid body dynamics under mutual interactions. From the decomposition point of view, this correspondsto the Iwasava decomposition of SL (2 , C ). The other, in Section 8, will be to continue with the Heisenberg algebra,endowing the geometry with dissipations. 5 ontents. In the following section, we shall be exhibiting a brief summary of the preliminaries for the sake ofthe completeness of the work. These are including Hamiltonian dynamics and metriplectic dynamics. Section 3is reserved for the presentation of the extended structure, and its particular instances such as a matched pair ofLie algebras and 2-cocycle extensions. In Section 4, Lie-Poisson dynamics are studied in the dual space of theextended structure. In Section 5, we shall decompose BBGKY dynamics as an illustration of the theoretical resultsobtained in the previous sections. In Section 6, we shall determine the conditions for a matched pair 2-cocycle tobe a 2-cocycle of a matched pair. Couplings of the symmetric brackets are given in Section 7. In Sections 8 and 9,3 D examples are provided. Consider a Poisson manifold ( P, {• , •} ) [48,88]. On this geometry, Hamilton’s equation generated by a Hamiltonianfunction(al) H is defined to be ˙ z = { z , H} , (2)where z is in P . Define Hamiltonian vector field X H for a Hamiltonian function(al) H , as follows X H ( F ) = {F , H} . (3)A function(al) C is called a Casimir function(al) if it commutes with all other function(al)s that is, {F , C} = 0 forall F . If there does not exist any non-constant Casimir function(al) for a Poisson bracket, then we say that thePoisson bracket is non-degenerate. It should be noted that the Hamiltonian vector field generated by a Casimirfunction(al) C is identically zero. The characteristic distribution, that is the image space of all Hamiltonian vectorfields is integrable. This reads a foliation of P into a collection of symplectic leaves [89]. That is, on each leaf, thePoisson bracket turns out to be non-degenerate. If the bracket is already non-degenerate on P then there existsonly one leaf, and P turns out to be a symplectic manifold.Skew-symmetry of Poisson bracket verifies that Hamiltonian function(al) is preserved throughout the motion. SinceHamiltonian function(al) is taken as the total energy in classical systems, we may call this property as the conser-vation of energy. This manifests the reversible character of Hamiltonian dynamics.Referring to Poisson bracket, we define a bivector field Λ as followsΛ( d F , d H ) := {F , H} (4)for all F and H , [17]. Here, d F and d H denote the de-Rham exterior derivatives. So that, we may alternativelyintroduce a Poisson manifold by a tuple ( P, Λ) consisting of a manifold and a bivector field. There is Schouten-Nijenhuis algebra on bivector fields [6]. In this picture, the Jacobi identity turns out to be the commutation of Λwith itself under the Schouten-Nijenhuis bracket that is,[Λ , Λ] = 0 . (5)6 ie-Poisson systems. Consider a Lie algebra K equipped with a Lie bracket [ • , • ], [42]. Dual K ∗ admits a Poissonbracket, called Lie-Poisson bracket [39, 41, 49, 51, 57]. For two function(al)s F and H , the (plus/minus) Lie-Poissonbracket is defined to be {F , H} ( z ) = ± D z , (cid:20) δ F δ z , δ H δ z (cid:21) E (6)where δ F /δ z is the partial derivative (for infinite dimensional cases, the Fr´echet derivative) of the function(al) F .Here, the pairing on the right hand side is the duality between K ∗ and K whereas the bracket is the Lie algebrabracket on K . Note that, we assume the reflexivity condition on K , that is the double dual K ∗∗ = K . The dynamicsof an observable F , governed by a Hamiltonian function(al) H , is then computed to be˙ F = {F , H} ( z ) = ± D z , (cid:20) δ F δ z , δ H δ z (cid:21) E = ± D z , − ad δ H /δ z δ F δ z E = ± D ad ∗ δ H /δ z z , δ F δ z E . (7)Here, ad x x ′ := [ x , x ′ ] for all x and x ′ in K is the (left) adjoint action of the Lie algebra K on itself whereas ad ∗ isthe (left) coadjoint action of the Lie algebra K on the dual space K ∗ . Notice that ad ∗ x is defined to be minus of thelinear algebraic dual of ad x . Then, we obtain the equation of motion governed by a Hamiltonian function(al) H as˙ z ∓ ad ∗ δ H /δ z z = 0 . (8) Remark 1
There are a plus/minus notations in (6) and (7) . A plus sign appears if the reduction (Lie-Poissonreduction) is performed referring to a right symmetry whereas a minus sign appears if the reduction is performedreferring to a left symmetry. For the plasma dynamics (see Section 5), we shall refer plus Lie-Poisson bracket sinceVlasov’s plasma has a right (called relabelling) symmetry. For finite-dimensional rigid body motion (see Section 8),we shall employ minus Lie-Poisson bracket.
Coordinate realizations.
Assume a (local) coordinate chart ( z i ) (we prefer subscripts since we only focus on thedual spaces) around a point z in P . Then the Poisson bivector can be represented by a set of coefficient functionsΛ ij determining a Poisson bracket as {F , H} = Λ ij ∂ F ∂z i ∂ H ∂z i . (9)Then the equation of motion generated by a Hamiltonian function H becomes˙ z i = Λ ij ∂ H ∂z j . (10)Let us examine the Lie-Poisson structure which is defined on the dual of a finite dimensional Lie algebra. For this,assume that an K dimensional Lie algebra K admitting a basis { k i } = { k , . . . , k K } . The Lie algebra bracket on K determine a set of scalars C lij , called structure constants, satisfying[ k i , k j ] = C mij k m , (11)where the summation convention is assumed over the repeated indices. Note that, after fixing a basis, the structureconstants define a Lie bracket in a unique way. One has the dual basis { k i } = { k , . . . , k K } on the dual space K ∗ . We denote an element of K ∗ by z = z i k i where the coordinates { z , . . . , z N } are being real numbers. The(plus/minus) Lie-Poisson bracket (6) can be computed in this picture as {F , H} = ± C nij z n ∂ F δz i ∂ H δz j . (12)7he calculation in (12) reads that the coefficients Λ ij of the (plus/minus) Poisson bivector (9) are determinedthrough the linear relations Λ ij = ± C mij z m . (13)In this case, Lie-Poisson equations (8) are computed to be˙ z j ∓ C nij z n ∂ H δz i = 0 . (14) Rayleigh dissipation.
Let us present a simply way to add dissipation to Lie-Poisson dynamics. Define a lineartransformation Υ from K ∗ to K . We equip a dissipative term to the right hand side of the Lie-Poisson system (8)by simply adding ∓ ad ∗ Υ( z ) z that is, ˙ z ∓ ad ∗ δ H /δ z z = ∓ ad ∗ Υ( z ) z (15)see [8]. We ask that Υ be a gradient relative to a certain metric at least on adjoint orbits. In the upcomingsubsection, we introduce a geometric framework for obtaining dissipation in the Lie-Poisson setting. In order to add dissipative terms to Hamiltonian dynamics, two geometric models are addressed in the literature,namely metriplectic systems [44, 45, 69, 70] and GENERIC (an acronym for General Equation for Non-EquilibriumReversible-Irreversible Coupling), [32, 34]. In metriplectic systems, Poisson geometry is equipped with a propersymmetric bracket (possibly semi-Riemannian). In GENERIC, which is more general [67], a dissipation potentialis employed. For convex potential functions, after the Legendre transformation, one arrives at the irreversible partof the dynamics [32, 76]. If the potential is quadratic, then one arrives at a symmetric bracket as a particular case.In this work, we are interested in dissipative dynamics defined through symmetric brackets [74]. Let us depict thisgeometry in detail.Consider a Poisson manifold ( P, {• , •} ) and assume, additionally, a symmetric bracket ( • , • ) on the space of smoothfunctions on P . The metriplectic bracket [ |• , •| ] on the manifold P is defined by the addition of the Poisson bracketand the symmetric bracket that is, for two function(al)s H and F ,[ |H , F| ] = {H , F} + a ( H , F ) , (16)where a being a scalar. Note that, a metriplectic bracket is an example of a Leibniz bracket [73]. There is no uniqueway to define a symmetric bracket. One way is to introduce a (maybe semi-)Riemannian metric G on M . Aftera bracket is established, the next task is to determine the generating function(al)s. In accordance with this, wedetermine two different kinds of the GENERIC (metriplectic) systems [35].In the first kind of GENERIC (metriplectic) systems, one refers a single function(al) F to generate the equationsof motion. So that, the dynamics is given as˙ z = [ | z , F| ] = { z , F} + a ( z , F ) (17)for z ∈ P . By, particularly, choosing the metric G positive definite, and by letting a be equal to −
1, one arrives atthe dissipation of the generating function F in time. In the second kind of GENERIC (metriplectic) systems, thereexist two function(al)s, namely a Hamiltonian function(al) H and an entropy-type function(al) S . In this time, the8ynamics is written as ˙ z = { z , H} + a ( z , S ) . (18)If the following identities {S , H} = 0 , ( H , S ) = 0 (19)hold, the metriplectic dynamics (18) can be generated by a single function (free energy function) F = H − S defined as the difference of the Hamiltonian and entropy-type function(al)s. For such kind of systems, Hamiltonianfunction(al) H is a conserved quantity whereas the dissipative behavior of the system is interpreted as the increaseof entropy along trajectories assuming that a is positive. This case is possible if the Poisson structure is degenerateand the symmetric tensor G is at most semi-definite. In the following subsection, we introduce some examples ofsymmetric brackets that can be attached to the Lie-Poisson bracket. In this subsection, we list some symmetric brackets available on the dual K ∗ of a Lie algebra K . After a symmetricbracket is determined, say ( • , • ) the irreversible dynamics governed by a generating function, say S , is computedto be ˙ z = a ( z , S ) , (20)where a is a real number. Assuming a basis { k i } , and the according local coordinates { z i } on K , the primary goalin this subsection is to define a symmetric tensor field G = G ij dz i ⊗ dz j (21)on K . In the dual space K ∗ , we employ the dual basis { k i } and the coordinates { z i } . There are two distinguishedfunctions on the Lie-Poisson picture. Here is a list: Double bracket.
Recall the structure of a Lie algebra K given in (11). In the Lie-Poisson setting, the coefficientsof the Poisson bivector are determined by the structure constants of the Lie algebra as (13) that is, Λ ij = C lij z l [71].For two functions F and H , we define a symmetric bracket, literarily called double bracket,( F , H ) ( D ) = X j Λ ij Λ lj ∂ F ∂z i ∂ H ∂z l = X j C rij C slj z r z s ∂ F ∂z i ∂ H ∂z l , (22)see [11]. So that we can write the coefficients of the symmetric bracket as in terms of the structure constants of theLie algebra as follows G ij = X l C ril C sjl z r z s . (23)Now we define a metriplectic bracket on K ∗ by adding Lie-Poisson bracket (12) and double bracket (22) that is˙ F = [ |F , H| ] ( D ) = {F , H} + a ( F , H ) ( D ) . (24)So that according to the definition (17) we compute the equation of motion as˙ z j ∓ C mij z m ∂ H ∂z i = a X i C rji C nmi z r z n ∂ H ∂z m (25)9here on the left hand side we have the reversible Hamiltonian dynamics whereas the dissipative term is located atthe right hand side. Cartan-Killing bracket.
Consider a Lie algebra given in coordinates as (11). Referring to skew-symmetry of thestructure constants the Cartan-Killing metric is defined as G ij = C nim C mjn . (26)It can be easily shown that the scalars G ij define a symmetric and bilinear covariant tensor [71]. We define asymmetric bracket for functions F and H in terms of the metric as follows( F , H ) ( CK ) = ∂ F ∂z i G ij ∂ H ∂z j = C nij C jmn ∂ F ∂z i ∂ H ∂z m . (27)To arrive at a metriplectic bracket on K ∗ , we add the Lie-Poisson bracket (12) and the symmetric bracket (27) thatis, ˙ F = [ |F , H| ] ( CK ) = {F , H} + a ( F , H ) ( CK ) . (28)Accordingly, the metriplectic dynamics is computed to be˙ z j ∓ C mij z m ∂ H ∂z i = aC nji C imn ∂ H ∂z m (29)where, on the left hand side, we have the reversible Hamiltonian dynamics whereas, on the right hand side, thedissipative term is exhibited. Casimir dissipation bracket.
Define a symmetric bilinear operator ψ on Lie algebra K . Referring to any Casimirfunction C of the Lie-Poisson bracket, define a symmetric bracket [27] of two functions F and H as( F , H ) ( CD ) = − ψ (cid:18)(cid:20) δ F δ z , δ H δ z (cid:21) , (cid:20) δCδ z , δ H δ z (cid:21)(cid:19) . (30)This bracket is suitable for the second type of metriplectic bracket. Note that the change of Hamiltonian functionover time is constant and the change of Casimir function can be given as˙ C = − ψ (cid:18)(cid:20) δ C δ z , δ H δ z (cid:21) , (cid:20) δ C δ z , δ H δ z (cid:21)(cid:19) = − (cid:13)(cid:13)(cid:13)(cid:13)(cid:20) δ C δ z , δ H δ z (cid:21)(cid:13)(cid:13)(cid:13)(cid:13) < . The dynamics of an arbitrary observable F governed by a Hamiltonian function H is deduced by this bracket as˙ F = − ψ (cid:18)(cid:20) δ F δ z , δ H δ z (cid:21) , (cid:20) δ C δ z , δ H δ z (cid:21)(cid:19) = *(cid:20) δ C δ z , δ H δ z (cid:21) ♭ , ad δH/δ z δ F δ z + = * − ad ∗ δ H /δ z (cid:20) δCδ z , δ H δ z (cid:21) ♭ , δ F δ z + . (31)Here, the musical mapping ♭ , from K to K ∗ , is defined through the symmetric operator ψ , satisfying the identity (cid:10) x ♭ , x ′ (cid:11) = ψ ( x , x ′ ) for two elements x and x ′ in K . In this case, the dissipative equation of motion can be writtenas follows ˙ z = − ad ∗ δ H δ z (cid:20) δCδ z , δ H δ z (cid:21) ♭ . (32)We collect the Lie-Poisson bracket (12) and the Casimir Dissipation bracket (30) together to arrive at the following10etriplectic bracket ˙ F = [ |F , H| ] ( CD ) = {F , H} + a ( F , H ) ( CD ) . (33)Then we compute the equation of motion as˙ z ∓ ad ∗ δ H δ z z = ( − a ) ad ∗ δHδ z (cid:20) δ C δ z , δ H δ z (cid:21) ♭ . (34) Hamilton dissipation bracket.
We start with assuming a symmetric (semi-positive definite) bilinear operator ψ defined on a Lie algebra K . We fix a Casimir function C of the Lie-Poisson bracket (6), and then introduce thefollowing symmetric bracket on the dual space K ∗ , for two function(al)s F and H , given by( F , H ) ( HD ) = − ψ (cid:16)h δ F δ z , δ C δ z i , [ δ H δ z , δ C δ z i(cid:17) (35)where the brackets on the right sides are Lie algebra brackets on K . An interesting feature of this symmetric bracketis to see that the Casimir function(al) C is a conserved quantity for the dynamics determined by the bracket (35)since ˙ C = 0 due to the skew-symmetry of the Lie-bracket. On the other hand the generating function H dissipatesthat is ˙ H = ( H , H ) ( HD ) = − ψ (cid:16)h δ H δ z , δ C δ z i , h δ C δ z , δ H δ z i(cid:17) = − (cid:13)(cid:13)(cid:13)(cid:13)h δ H δ z , δ C δ z i(cid:13)(cid:13)(cid:13)(cid:13) ≤ . More general, the gradient flow of an observable F generated by a function(al) H is computed to be˙ F = − ψ (cid:18)(cid:20) δ F δ z , δ H δ z (cid:21) , (cid:20) δ C δ z , δ H δ z (cid:21)(cid:19) = *(cid:20) δ C δ z , δ H δ z (cid:21) ♭ , ad δH/δ z δ F δ z + = * − ad ∗ δ H /δ z (cid:20) δCδ z , δ H δ z (cid:21) ♭ , δ F δ z + . (36)Accordingly, we can write the equation of motion generated by S is˙ z = − ad ∗ δC/δ z h δ C δ z , δ S δ z i ♭ . (37)Now we are ready to add the Lie-Poisson bracket (12) and the symmetric bracket exhibited in (35) in order todefine a metriplectic (Leibniz) bracket on K ∗ . By assuming that both the Lie-Poisson and the gradient dynamicsgenerated by a single function H we have that˙ F = [ |F , H| ] ( HD ) = {F , H} + a ( F , H ) ( HD ) . (38)Then we compute the equation of motion as˙ z ∓ ad ∗ δ H /δ z z = ( − a ) ad ∗ δC/δ z (cid:20) δ C δ z , δ H δ z (cid:21) ♭ . (39) In this section, we introduce a unifying construction, called extended structure , for factorization of Lie algebras.Then, we examine its particular instances such as matched pair Lie algebra and 2-cocycle extension.11 .1 Extended Structures
Let ( g , [ • , • ]) be a Lie algebra and, assume that, it acts on a vector space h from the right that is ⊳ : h ⊗ g → h , η ⊗ ξ η ⊳ ξ. (40)Our goal in this subsection is to construct the most general extension of g by h . To have this, we permit existenceof the following maps Φ : h ⊗ h −→ g , ( η, η ′ ) Φ( η, η ′ ) κ : h ⊗ h −→ h , ( η, η ′ ) κ ( η, η ′ ) (41)along with a linear map ⊲ : h ⊗ g → g , η ⊗ ξ η ⊲ ξ. (42)Note here that (42) is not an action since h is only a vector space. The need of the operations (41) and (42) willbe more evident in the sequel where we examine this in the point of view of decomposition. The following theoremdetermines the conditions to define a Lie algebra structure on the direct sum K = g ⊕ h , see also [2, 3]. Theorem 2
The direct sum K = g ⊕ h is a Lie algebra via [( ξ ⊕ η ) , ( ξ ′ ⊕ η ′ )] Φ ⊲⊳ = (cid:0) [ ξ, ξ ′ ] + η ⊲ ξ ′ − η ′ ⊲ ξ + Φ( η, η ′ ) (cid:1) ⊕ (cid:0) κ ( η, η ′ ) + η ⊳ ξ ′ − η ′ ⊳ ξ (cid:1) . (43) where the mappings are the ones in (40) , (41) and (42) , if and only if, for any η, η ′ , η ′′ ∈ h , and any ξ, ξ ′ ∈ g , κ ( η, η ) = 0 , Φ( η, η ) = 0 ,κ ( η, η ′ ) ⊳ ξ = κ ( η, η ′ ⊳ ξ ) − κ ( η ′ , η ⊳ ξ ) + η ⊳ ( η ′ ⊲ ξ ) − η ′ ⊳ ( η ⊲ ξ ) ,κ ( η, η ′ ) ⊲ ξ = [ ξ, Φ( η, η ′ )] + Φ( η, η ′ ⊳ ξ ) + Φ( η ⊳ ξ, η ′ ) + η ⊲ ( η ′ ⊲ ξ ) − η ′ ⊲ ( η ⊲ ξ ) ,η ⊲ [ ξ, ξ ′ ] = [ ξ, η ⊲ ξ ′ ] − [ ξ ′ , η ⊲ ξ ] + ( η ⊳ ξ ) ⊲ ξ ′ − ( η ⊳ ξ ′ ) ⊲ ξ,η ⊳ [ ξ, ξ ′ ] = ( η ⊳ ξ ) ⊳ ξ ′ − ( η ⊳ ξ ′ ) ⊳ ξ, (cid:8) Φ( η, κ ( η ′ , η ′′ ))+ (cid:8) η ⊲ Φ( η ′ , η ′′ ) = 0 , (cid:8) κ ( η, κ ( η ′ , η ′′ ))+ (cid:8) η ⊳ Φ( η ′ , η ′′ ) = 0 , (44) where (cid:8) refers to the cyclic sum over the indicated elements. Proof.
We first observe that [ η, η ] = (cid:0) Φ( η, η ) , κ ( η, η ) (cid:1) = 0 (45)if and only if Φ( η, η ) = 0 , κ ( η, η ) = 0 (46)for any η ∈ h . Next, we shall consider the mixed Jacobi identities. Let us begin with[ ξ, [ η, η ′ ]] + [ η, [ η ′ , ξ ]] + [ η ′ , [ ξ, η ]] = 0 , (47)12here [ ξ, [ η, η ′ ]] = [ ξ, (Φ( η, η ′ ) , κ ( η, η ′ ))] = (cid:0) − κ ( η, η ′ ) ⊲ ξ + [ ξ, Φ( η, η ′ )] g , − κ ( η, η ′ ) ⊳ ξ (cid:1) , [ η, [ η ′ , ξ ]] = [ η, ( − η ′ ⊲ ξ, − η ′ ⊳ ξ )] = (cid:0) − η ⊲ ( η ′ ⊲ ξ ) − Φ( η ′ ⊳ ξ, η ) , − κ ( η ′ ⊳ ξ, η ) − η ⊳ ( η ′ ⊲ ξ ) (cid:1) , [ η ′ , [ ξ, η ]] = [ η ′ , ( η ⊲ ξ, η ⊳ ξ )] = (cid:0) η ′ ⊲ ( η ⊲ ξ ) + Φ( η ⊳ ξ, η ′ ) , κ ( η ⊳ ξ, η ′ ) + η ′ ⊳ ( η ⊲ ξ )) (cid:1) . (48)Hence, (47) is satisfied if and only if κ ( η, η ′ ) ⊳ ξ = κ ( η ⊳ ξ, η ′ ) − κ ( η ′ ⊳ ξ, η ) + η ′ ⊳ ( η ⊲ ξ ) − η ⊳ ( η ′ ⊲ ξ )[ ξ, Φ( η, η ′ )] g = κ ( η, η ′ ) ⊲ ξ + η ⊲ ( η ′ ⊲ ξ ) − η ′ ⊲ ( η ⊲ ξ ) + Φ( η ′ ⊳ ξ, η ) − Φ( η ⊳ ξ, η ′ ) (49)for any η, η ′ ∈ h , and any ξ ∈ g .Next, we consider the Jacobi identity of an arbitrary η ∈ h , and any ξ, ξ ′ ∈ g , namely;[[ ξ, ξ ′ ] , η ] + [[ ξ ′ , η ] , ξ ] + [[ η, ξ ] , ξ ′ ] = 0 . (50)In this case, [[ ξ, ξ ′ ] , η ] = (cid:0) η ⊳ [ ξ, ξ ′ ] g , η ⊲ [ ξ, ξ ′ ] g ) (cid:1) , together with [[ ξ ′ , η ] , ξ ] = (cid:0) ( η ⊳ ξ ′ ) ⊳ ξ, − ( η ⊳ ξ ) ⊲ ξ ′ + [ ξ, η ⊲ ξ ′ ] g (cid:1) , [[ η, ξ ] , ξ ′ ] = (cid:0) − ( η ⊳ ξ ) ⊳ ξ ′ , ( η ⊳ ξ ′ ) ⊲ ξ − [ η ⊲ ξ, ξ ′ ] g (cid:1) . (51)As such, (50) is satisfied if and only if η ⊳ [ ξ, ξ ′ ] g = − ( η ⊳ ξ ) ⊳ ξ ′ + ( η ⊳ ξ ) ⊳ ξ ′ ,η ⊲ [ ξ, ξ ′ ] g = [ η ⊲ ξ, ξ ′ ] g + [ ξ, η ⊲ ξ ′ ] g + ( η ⊳ ξ ) ⊲ ξ ′ − ( η ⊳ ξ ′ ) ⊲ ξ (52)for any η ∈ h , and any ξ, ξ ′ ∈ g . Finally we consider the Jacobi identity[[ η, η ′ ] , η ′′ ] + [[ η ′ , η ′′ ] , η ] + [[ η ′′ , η ] , η ′ ] = 0 (53)for any η, η ′ , η ′′ ∈ h . We have,[[ η, η ′ ] , η ′′ ] = (cid:0) (Φ( η, η ′ ) , κ ( η, η ′ )) , η ′′ (cid:1) = (cid:0) η ′′ , κ ( κ ( η, η ′ )) + η ′′ ⊳ Φ( η, η ′ ) , η ′′ ⊲ (Φ( η, η ′ ) + Φ( η ′′ , κ ( η, η ′ )) (cid:1) , as well as [[ η ′ , η ′′ ] , η ] = [(Φ( η ′ , η ′′ ) , κ ( η ′ , η ′′ )) , η ]= (cid:0) η, κ ( κ ( η ′ , η ′′ )) + η ⊳ Φ( η ′ , η ′′ ) , η ′′ ⊲ Φ( η ′ , η ) + Φ( η, κ ( η ′ , η ′′ )) (cid:1) [[ η ′′ , η ] , η ′ ] = [(Φ( η, η ′′ ) , κ ( η, η ′′ )) , η ′ ]= (cid:0) η ′ , κ ( κ ( η, η ′′ )) + η ′ ⊳ Φ( η, η ′′ ) , η ′ ⊲ Φ( η, η ′′ ) + Φ( η ′ , κ ( η, η ′′ )) (cid:1) . (54)13ccordingly, (53) is satisfied if and only if X ( η,η ′ ,η ′′ ) κ ( η ′′ , κ ( η, η ′ )) + X ( η,η ′ ,η ′′ ) η ′′ ⊳ Φ( η, η ′ ) = 0 , X ( η,η ′ ,η ′′ ) η ′′ ⊲ Φ( η, η ′ ) + X ( η,η ′ ,η ′′ ) Φ( η ′′ , κ ( η, η ′ )) = 0 (55)for any η, η ′ , η ′′ ∈ h .We denote the direct product space K = g ⊕ h equipped with the Lie algebra bracket (43) by K = g Φ ⊲⊳ h . In [2, 3],the realization presented in Theorem 2 has already been introduced under the name of extended structure . We shallfollow this terminology as well. We remark that, the last two identities in (44) are called twisted cocycle identityfor Φ and twisted Jacobi identity for κ , respectively. In the following subsection, we shall exploit that extendedstructure realizes both matched (double cross sum) Lie algebra and 2-cocycle extension Lie algebra as particularinstances.Let us cite here some related studies presented in this section. See [13] for a coalgebra discussion related with theextension presented here. We refer [43] for the extensions of Hamiltonian vector fields. See [46] for extensions ofPoisson algebras. Decomposing a Lie algebra.
Instead of extending a Lie algebra with its representation space, one can decomposea Lie algebra into the internal direct sum of one of its Lie subalgebra and its complement. The latter is manifestingthe inverse of the statement in Theorem 2. So that, the (de)composition exhibited here is universal. Let us depictthis argument.We start with a Lie algebra K , and assume a subalgebra, say g , of it. It is always possible to define a complementarysubspace h ⊂ K so that K = g ⊕ h . In most general case, for η, η ′ in h , under the Lie algebra bracket of K , we havethat [ η, η ′ ] = Φ( η, η ′ ) ⊕ κ ( η, η ′ ) ∈ g ⊕ h . (56)Here, we define the mappings Φ : h × h → g , Φ( η, η ′ ) := proj g [ η, η ′ ] (57) κ : h × h → h , κ ( η, η ′ ) := proj h [ η, η ′ ] (58)where proj denotes the projection operator. Notice that, if Φ is identically zero then h becomes a Lie subalgebraof K . In this case, κ becomes the Lie algebra bracket on h . Lemma 3 (Universal Lemma)
Given a decomposition of Lie algebra K = g ⊕ h , where g is being a Lie subalgebra.The mappings Φ and κ in (41) recovered from (56) and the mutual actions computed through [0 ⊕ η, ξ ⊕
0] = η ⊲ ξ ⊕ η ⊳ ξ (59) satisfies the conditions in (44) . That is, K can be identified to the extended structure g Φ ⊲⊳ h hence, (43) reads adecomposition of the Lie bracket on K . Coordinate realizations.
Choose a basis { e α } on an N -dimensional Lie algebra g and a basis { f a } on M -dimensional vector space h . Notice that we reserve the Greek scripts while denoting the basis of the Lie algebra g h . We denote[ e α , e β ] = C θαβ e θ , [ f a , f b ] = Φ αab e α + κ dab f d , (60)where the set C θαβ determines the structure constants of the Lie subalgebra g whereas the sets of constants Φ αab and κ dab are coordinate realizations of the mappings Φ and κ determined in (56). We identify the mappings (40) and(42) in terms of the basis e α , f a as f a ⊳ e α = R baα f b , f a ⊲ e α = L βaα e β . (61)It is needless to say that, the scalars L βaα and R baα determine the mappings in a unique way.In the present finite case, the extended structure g Φ ⊲⊳ h is an N + M dimensional vector space. Referring to thebasis of the constitutive subspaces, one can define a basis { ¯ e , . . . , ¯ e N + M } on g Φ ⊲⊳ h as { ¯ e α , ¯ e a } ⊂ g Φ ⊲⊳ h , ¯ e α = e α ⊕ , ¯ e a = 0 ⊕ f a . (62)In the light of these local choices (60) and (61), one can calculate the structure constants of the extended Lie bracket(43) via [¯ e β , ¯ e α ] Φ ⊲⊳ = ¯ C γβα ¯ e γ + ¯ C aβα ¯ e a = [ e β ⊕ , e α ⊕ Φ ⊲⊳ = C γβα e γ ⊕ , [¯ e β , ¯ e a ] Φ ⊲⊳ = ¯ C γβa ¯ e γ + ¯ C dβa ¯ e d = [ e β ⊕ , ⊕ f a ] Φ ⊲⊳ = − L γaβ e γ ⊕ − R daβ f d , [¯ e b , ¯ e a ] Φ ⊲⊳ = ¯ C γba ¯ e γ + ¯ C dba ¯ e d = [0 ⊕ f b , ⊕ f a ] Φ ⊲⊳ = Φ γba e γ ⊕ κ dba f d . (63)As a result, structure constants of g Φ ⊲⊳ h can be written as¯ C γβα = C γβα , ¯ C aβα = 0 , ¯ C γβa = − L γaβ , ¯ C dβa = − R daβ , ¯ C γba = Φ γba , ¯ C dba = κ dba . (64) We shall recall the matched pair construction from [62, 63]. Let ( g , [ • , • ] g ) and ( h , [ • , • ] h ) be two Lie algebrasadmitting mutual actions ⊲ : h ⊗ g → g , ⊳ : h ⊗ g → h . (65)That is, we have that the following identities hold, for any ξ, ξ ′ ∈ g , and any η, η ′ ∈ h ,[ η, η ′ ] ⊲ ξ = η ⊲ ( η ′ ⊲ ξ ) − η ′ ⊲ ( η ⊲ ξ ) ,η ⊳ [ ξ, ξ ′ ] = ( η ⊳ ξ ) ⊳ ξ ′ − ( η ⊳ ξ ′ ) ⊳ ξ. (66)The direct sum K = g ⊕ h can be endowed with a Lie algebra structure if some compatibility conditions are satisfied. Theorem 4
The direct sum of two Lie algebras g and h under mutual actions (65) is a Lie algebra if it is equippedwith the bilinear mapping [( ξ ⊕ η ) , ( ξ ′ ⊕ η ′ )] ⊲⊳ = (cid:0) [ ξ, ξ ′ ] + η ⊲ ξ ′ − η ′ ⊲ ξ (cid:1) ⊕ (cid:0) [ η, η ′ ] + η ⊳ ξ ′ − η ′ ⊳ ξ (cid:1) . (67)15 atisfying the following compatibility conditions [ η, η ′ ] ⊳ ξ = [ η, η ′ ⊳ ξ ] − [ η ′ , η ⊳ ξ ] + η ⊳ ( η ′ ⊲ ξ ) − η ′ ⊳ ( η ⊲ ξ ) ,η ⊲ [ ξ, ξ ′ ] = [ ξ, η ⊲ ξ ′ ] − [ ξ ′ , η ⊲ ξ ] + ( η ⊳ ξ ) ⊲ ξ ′ − ( η ⊳ ξ ′ ) ⊲ ξ (68) for any η, η ′ ∈ h , and any ξ, ξ ′ ∈ g . If a Lie algebra K is constructed in the realm of Proposition 4, then we call K a matched pair (double cross sum)Lie algebra, and denote it as K = g ⊲⊳ h , see also [91]. Notice that, we denote the matched pair Lie bracket (67) by[ • , • ] ⊲⊳ . If one of the actions in (65) is trivial then we arrive at a semidirect product Lie algebra. So that matchedpair Lie algebras are generalizations of the semidirect Lie algebras [14–16, 40]. From the extended structure to matched pairs.
Recall the extended structure in Subsection 3.1. In thatrealization, a more relaxed construction is given where h is not necessarily assumed to be a Lie algebra. Considerparticular case of that structure where the mapping Φ in (41) is taken to be identically zero. In the realm ofTheorem 2, for the case of Φ ≡
0, the last condition in the list (44) gives that κ mapping satisfies the Jacobiidentity that is (cid:8) κ ( η, κ ( η ′ , η ′′ )) = 0 . (69)This reads that the vector space h turns out to be a Lie algebra:[ η, η ′ ] := κ ( η, η ′ ) . (70)Further, for Φ ≡
0, the third line and the fifth line in the compatibility list (44) reduce to the action conditions (66),and the second and fourth lines in (44) become the matched pair compatibility conditions in (68). This observationsays that Theorem 4 is a particular case of Theorem 2. That is, every matched pair Lie algebra is an extendedstructure. For the matched pair theory, Universal Lemma (3) takes the following particular form
Lemma 5 (Universal Lemma II)
Let K be a Lie algebra with two Lie subalgebras g and h such that K is iso-morphic to the direct sum of g and h as vector spaces through the vector addition in K . Then K is isomorphic tothe matched pair g ⊲⊳ h as Lie algebras, and the mutual actions (65) are derived from [ η, ξ ] = η ⊲ ξ ⊕ η ⊳ ξ. (71) Here, the inclusions of the subalgebras are defined to be g −→ K : ξ → ( ξ ⊕ , h −→ K : η → (0 ⊕ η ) . (72) Coordinate realizations.
Assume that the coordinates are chosen as in Subsection 3.1. To have the localcharacterization for the matched pair Lie algebra, we first analyse the structure constants given in (63). See thatthe first and the second lines remain the same but since Φ αab are all zero and, the constants κ dba turn out to bestructure constants of the Lie algebra h . In order to highlight this, we denote the structure constants by D dba . Letus record this [ e α , e β ] = C γαβ e γ , [ f a , f b ] = Φ αab e α + κ dab f d , (73)Therefore, we have [¯ e b , ¯ e a ] = ¯ C γba ¯ e γ + ¯ C dba ¯ e d = [0 ⊕ f b , ⊕ f a ] ⊲⊳ = 0 ⊕ D dba f d . (74)16o that the structure constants of the matched Lie algebra are definitely equal to (64) except that, in the presentcase, ¯ C γba = 0, and ¯ C dba = D dba : ¯ C γβα = C γβα , ¯ C aβα = 0 , ¯ C γβa = − L γaβ , ¯ C dβa = − R daβ , ¯ C γba = 0 , ¯ C dba = D dba . (75) In this subsection, we discuss another particular instance of extended structures exhibited in Subsection 3.1. Inthis case, we assume that the right action (40) of g on h and the Lie bracket on g are trivial while all the othergeometric ingredients of Theorem 2 are remaining the same. In other words, in this subsection, η ⊲ ξ = 0 , [ ξ, ξ ′ ] = 0 (76)for all ξ and ξ ′ in g , and for all η in h . As a result of this selection, one observes that the extended Lie bracket(43) and the list of conditions (44) will take particular forms. Let us examine them from the bottom to the top.Since the right action is trivial, the last condition in (44) turns out out to be the Jacobi identity (69) for κ . Thismanifests that the two-tuple ( h , κ ) becomes a Lie algebra. Accordingly, in this section, we denote κ by a bracketnotation [ • , • ] as in (70). On the other hand, the penultimate condition, namely the twisted 2-cocycle condition, in(44) takes the particular form (cid:8) Φ( η, [ η ′ , η ′′ ])+ (cid:8) η ⊲ Φ( η ′ , η ′′ ) = 0 . (77)This determines Φ as a g -valued 2-cocycle on h [26]. The second, the fourth and the fifth conditions in (44) areidentically satisfied whereas the third line[ η, η ′ ] ⊲ ξ = η ⊲ ( η ′ ⊲ ξ ) − η ′ ⊲ ( η ⊲ ξ ) (78)is exploiting that ⊲ is a left Lie algebra action of h on g . Eventually, we arrive at the following reduced form of theLie algebra bracket (43) [ ξ ⊕ η, ξ ′ ⊕ η ′ ] Φ ⋊ = (cid:0) η ⊲ ξ ′ − η ′ ⊲ ξ + Φ( η, η ′ ) (cid:1) ⊕ [ η, η ′ ] . (79)over the direct product space. In this case, we denote the total space by g Φ ⋊ h equipped with the Lie bracket (79)and call it 2-cocycle extension of h by a vector field g .Once again, as a manifestation of Universal Lemma 3 we can discuss the decomposition point of view as follows.Assume a Lie algebra K and one of its nontrivial centers, say g . Consider the decomposition g ⊕ h inducingnontrivial Φ and κ mappings as in (57) and (58) and a left action (42) then Universal Lemma 3 reads that, K canbe decomposed into a 2-cocycle extension of h by g that is K = g Φ ⋊ h . Coordinate realizations.
Assume once more that the coordinates are chosen as in Subsection 3.1. In this case,the Lie bracket on g and, the right action of g on h are trivial. So that, the structure constants C γβα of g given in (60)will be all zero while the constants Φ αab and D dab determining Φ and κ mappings remain the same. Since the rightaction is zero, R daβ in (61) will be zero whereas the scalars L βaα are giving the left action. If these aforementionedchanges are applied to the system of equations (63), one can reach the structure constants of 2-cocycles.17 Lie-Poisson Dynamics on Extensions
Dual of a Lie algebra admits Lie-Poisson bracket according to the definition in (6). In the present section, followingthe order of the extensions and couplings in Section 3, we compute associated Lie-Poisson brackets.
Assume the Lie algebraic framework in Subsection 3.1, and let that all the conditions in Theorem 2 hold. Now westart with the left action ⊲ in (42) then, freeze an element η in h in this operation to have a linear mapping η ⊲ onthe subalgebra g . This linear mapping and the dual action ∗ ⊳ η are η ⊲ : g −→ g , ξ η ⊲ ξ ∗ ⊳ : g ∗ ⊗ h −→ g ∗ , h µ ∗ ⊳ η, ξ i = h µ, η ⊲ ξ i . (80)This dual mapping is a right representation of h on the dual space g ∗ . Later, by freezing ξ ∈ g in the left action ⊲ in (42), we define a linear mapping b ξ : h g . We record here this linear mapping b ξ and the dual mapping b ∗ ξ as b ξ : h −→ g , b ξ ( η ) = η ⊲ ξ, (81) b ∗ ξ : g ∗ −→ h ∗ , h b ∗ ξ µ, η i = h µ, b ξ η i = h µ, η ⊲ ξ i . (82)Consider the right action ⊳ in (40). In this operation, we freeze ξ in g to have an automorphism on h , denoted by ⊳ ξ . We write ⊳ ξ and its dual ξ ∗ ⊲ in the following display ⊳ ξ : h −→ h , η η ⊳ ξ ∗ ⊲ : g × h ∗ −→ h ∗ , h ξ ∗ ⊲ ν, η i = h ν, η ⊳ ξ i . (83)See that, ∗ ⊲ is a left representation of g on the dual h ∗ . Further, we freeze an element, say η in h , in the right action(40). This enables us to define a linear mapping a η from g to h . Here are the mapping a η and its dual a ∗ η in arespective order a η : g h , a η ( ξ ) = η ⊳ ξ (84) a ∗ η : h ∗ g ∗ , h a ∗ η ν, ξ i = h ν, a η ξ i = h ν, η ⊳ ξ i . (85)Let us recall mappings Φ and κ displayed in (57) and (58), respectively. Define two functions κ η and Φ η as κ η : h → h κ η ( η ′ ) := κ ( η, η ′ ) (86)Φ η : h → g Φ η ( η ′ ) := Φ( η, η ′ ) (87)where η, η ′ ∈ h , ν ∈ h ∗ and µ ∈ g ∗ . According to these definitions, the dual mappings are calculated as κ ∗ η : h ∗ → h ∗ h κ ∗ η ν, η ′ i = h ν, − κ η ( η ′ ) i = −h ν, κ ( η, η ′ ) i , (88)Φ ∗ η : g ∗ → h ∗ h Φ ∗ η µ, η ′ i = h µ, − Φ η ( η ′ ) i = −h µ, Φ( η, η ′ ) i , (89)18espectively. Proposition 6
The adjoint action on g Φ ⊲⊳ h is being the extended Lie bracket in (43) , the coadjoint action ad ∗ of an element ξ ⊕ η in g Φ ⊲⊳ h onto an element µ ⊕ ν in the dual space g ∗ ⊕ h ∗ is computed to be ad ∗ ( ξ ⊕ η ) ( µ ⊕ ν ) = (cid:0) ad ∗ ξ µ − µ ∗ ⊳ η − a ∗ η ν (cid:1)| {z } ∈ g ∗ ⊕ (cid:0) κ ∗ η ν + Φ ∗ η µ + ξ ∗ ⊲ ν + b ∗ ξ µ (cid:1)| {z } ∈ h ∗ . (90) Here, ad ∗ (in italic font) is for the infinitesimal coadjoint actions of subalgebras to their duals. By using the equations (88) and (89), (plus/minus) extended Lie-Poisson bracket is computed as {H , F} Φ ⊲⊳ ( µ ⊕ ν ) = ± * µ ⊕ ν, (cid:20)(cid:16) δ H δµ ⊕ δ H δν (cid:17) , (cid:16) δ F δµ ⊕ δ F δν (cid:17)(cid:21) Φ ⊲⊳ + = ± (cid:28) µ, (cid:20) δ H δµ , δ F δµ (cid:21)(cid:29) ± (cid:28) ν, κ (cid:18) δ H δν , δ F δν (cid:19)(cid:29) ± (cid:28) µ, Φ (cid:18) δ H δν , δ F δν (cid:19)(cid:29)| {z } A: from twisted cocycle ± (cid:28) µ, δ H δν ⊲ δ F δµ (cid:29) ∓ (cid:28) µ, δ F δν ⊲ δ H δµ (cid:29)| {z } B: action of h on g from the left ± (cid:28) ν, δ H δν ⊳ δ F δµ (cid:29) ∓ (cid:28) ν, δ F δν ⊳ δ H δµ (cid:29)| {z } C: action of g on h from the right (91)for two functions H , F . We assume the reflexivity condition which reads that δ H /δµ and δ F /δµ are elements of g whereas δ H /δν and δ F /δν are elements of h . The Lie bracket on the first line in (91) is the extended Lie bracket[ • , • ] Φ ⊲⊳ in (43). In the Poisson bracket, the term labelled by A is a manifestation of the existence of twisted cocycleΦ. The terms labelled by B are due to the left action of h on g whereas the terms labelled by C are due to the rightaction of g on h .Recall the (plus/minus) Lie-Poisson equation in (8) determined as a coadjoint flow. In the light of the Lie-Poissonbracket (91), governed by a Hamiltonian function H = H ( µ, ν )), for the present picture, the (plus/minus) Lie-Poisson equation is computed as ˙ µ = ± ad ∗ δ H δµ ( µ ) | {z } Lie-Poisson Eq. on g ∗ ∓ µ ∗ ⊳ δ H δν | {z } action of h ∓ a ∗ δ H δν ν, | {z } action of g ˙ ν = ± κ ∗ δ H δν ( ν ) ± Φ ∗ δ H δν ( µ ) | {z } twisted cocycle ± δ H δµ ∗ ⊲ ν | {z } action of g ± b ∗ δ H δµ µ. | {z } action of h (92)Notice that we have determined and labelled the terms in the Lie-Poisson in order to identify them. As depicted in(92), one can easily follow how the Lie-Poisson dynamics on g ∗ is extended by addition of terms coming from themutual actions of h and g on each other as well as from the twisted 2-cocycle term. Coordinate realizations.
We follow the notation in Subsection 3.1. Recall, ( N + M )-dimensional extendedstructure K = g Φ ⊲⊳ h . Denote the dual basis of g ∗ and h ∗ by { e α } and { f a } , respectively. Then, define the dualbasis { ¯ e α , ¯ e a } ⊂ g ∗ ⊕ h ∗ , ¯ e α = e α ⊕ , ¯ e a = 0 ⊕ f a (93)19n the dual space g ∗ ⊕ h ∗ . Using this basis, we can write an element of g ∗ ⊕ h ∗ as follows( µ, ν ) = µ α ¯ e α + ν a ¯ e a . (94)In this picture, the mappings (80) and (83) turn out to be( µ α e α ) ∗ ⊳ ( η a f a ) = µ α η a L αaβ e β , ( ξ α e α ) ∗ ⊲ ( ν a f a ) = ν a ξ α R abα f b , (95)where L αaβ and R abα are scalars in (61) determining the actions . Later, we compute the dual mappings in (82), (85)and also (88), (89) in terms of local coordinates as follows: b ∗ ( ξ α e α ) ( µ α e α ) = µ α ξ β L αaβ f a , a ∗ ( η a f a ) ( ν a e a ) = ν a η b R abα e α , (96) κ ∗ η ν = − κ abd ν a η b f d , Φ ∗ η µ = − Φ αbk µ α η b f k . (97)Therefore, the (plus/minus) Lie-Poisson bracket (91) is written in coordinates as {H , F} Φ ⊲⊳ ( µ ⊕ ν ) = ± µ α C αβγ ∂ H ∂µ β ∂ F ∂µ γ ± ν a κ abd ∂ H ∂ν b ∂ F ∂ν d ± µ α Φ αbk ∂ H ∂ν b ∂ F ∂ν k ± µ α L αaβ (cid:0) ∂ H ∂ν a ∂ F ∂µ β − ∂ F ∂ν a ∂ H ∂µ β (cid:1) ± ν a R abβ (cid:0) ∂ H ∂ν b ∂ F ∂µ β − ∂ F ∂ν b ∂ H ∂µ β (cid:1) . (98)whereas the (plus/minus) Lie-Poisson dynamics (92) as˙ µ β = ± µ ρ C ρβα ∂ H ∂µ α ∓ µ α L αaβ ∂ H ∂ν a ∓ ν a R abβ ∂ H ∂ν b , ˙ ν d = ± µ α Φ αdb ∂ H ∂ν b ± ν a κ adb ∂ H ∂ν b ± ν a R adα ∂ H ∂µ α ± µ α L αdβ ∂ H ∂µ β . (99) In Subsection 3.2, matched pair Lie algebra g ⊲⊳ h is realized as a particular instance of extended structure bychoosing the twisted 2-cocycle Φ as trivial. So that, it is argued that, for a matched pair Lie algebra, both ofthe constitutive spaces, g and h are Lie subalgebras. Therefore, in this case, the duals of each of these subspaces,namely g ∗ and h ∗ , admit Lie-Poisson flows. This lets us to claim that the Lie-Poisson dynamics on the dual g ∗ ⊕ h ∗ of matched pair can be considered as the collective motion of two Lie-Poisson subdynamics [22]. Algebraically, thiscorresponds to matching of two Lie coalgebras. We refer [66] for the Lie coalgebra structure of the dual space.On the dual of a matched pair, both of the dual actions ∗ ⊳ and ∗ ⊲ , exhibited in (80) and (83) respectively, are equallyvalid. Notice that, for the present discussion ⊲ is a true left action of h on g so that ∗ ⊳ is a true right dual action of h on g ∗ . This is not true for extended structure since h is not assumed to be a Lie subalgebra. It is immediate toobserve that the dual mappings b ∗ ξ and a ∗ η , in (82) and (85) respectively, are remaining the same. The difference ofmatched pair construction form the extended structure is that the κ mapping, in (58), is a Lie bracket and that Φmapping, in (57), is zero. As previously stated, we prefer to denote κ by a bracket, so we write the mapping (88) as κ ∗ η ν = ad ∗ η ν. (100)These observations lead us to the following proposition as a particular case of Proposition 6, see also [21, 22].20 roposition 7 The adjoint action on g ⊲⊳ h is being the matched pair Lie bracket in (67) , the coadjoint action ad ∗ of an element ξ ⊕ η in g ⊕ h onto an element µ ⊕ ν in the dual space g ∗ ⊕ h ∗ is computed to be ad ∗ ( ξ ⊕ η ) ( µ ⊕ ν ) = (cid:0) ad ∗ ξ µ − µ ∗ ⊳ η − a ∗ η ν (cid:1)| {z } ∈ g ∗ ⊕ (cid:0) ad ∗ η ν + ξ ∗ ⊲ ν + b ∗ ξ µ (cid:1)| {z } ∈ h ∗ . (101)Later, on the dual space g ∗ ⊕ h ∗ , the (plus/minus) Lie-Poisson bracket of double cross sum is computed to be {H , F} ⊲⊳ ( µ ⊕ ν ) = ± (cid:28) µ, (cid:20) δ H δµ , δ F δµ (cid:21)(cid:29) ± (cid:28) ν, (cid:20) δ H δν , δ F δν (cid:21)(cid:29)| {z } A: direct product ∓ (cid:28) µ, δ H δν ⊲ δ F δµ (cid:29) ± (cid:28) µ, δ F δν ⊲ δ H δµ (cid:29)| {z } B: via the left action of h on g ∓ (cid:28) ν, δ H δν ⊳ δ F δµ (cid:29) ± (cid:28) ν, δ F δν ⊳ δ H δµ (cid:29)| {z } C: via the right action of g on h . (102)Notice that, the terms labelled by A are just the sum of individual Poisson brackets on the dual spaces g ∗ and h ∗ of the constitutive Lie subalgebras g and h , respectively. The terms labeled by B is a result of the left actionof h on g whereas the terms labelled by C is due to the right action of g on h . For the case of one-sided actions,that is semidirect product theories, B or C drops. If there is no action then, both B and C drop. In the lightof the (plus/minus) matched pair Lie-Poisson bracket (102), matched pair Lie-Poisson equations generated by aHamiltonian function H = H ( µ, ν ) on g ∗ ⊕ h ∗ is computed to be˙ µ = ± ad ∗ δ H δµ ( µ ) | {z } Lie-Poisson Eq. on g ∗ ∓ µ ∗ ⊳ δ H δν | {z } action of h ± a ∗ δ H δν ν | {z } action of g , ˙ ν = ± ad ∗ δ H δν ( ν ) | {z } Lie-Poisson Eq. on h ∗ ∓ δ H δµ ∗ ⊲ ν | {z } action of g ± b ∗ δ H δµ µ | {z } action of h . (103)The first terms on the right hand sides are the individual equations of motions. The other terms are the dual andcross actions appearing as manifestations of the mutual actions. Coordinate realizations.
Recall the Lie-Poisson bracket in (98) and the Lie-Poisson equations (99) computedfor the case of extended structures. In the matched pair case, we take the constants determining twisted 2-cocycleas zero that is Φ γba = 0, and the structure constants for the Lie algebra h as κ dba = D dba . So that, the (plus/minus)matched Lie-Poisson bracket (102) takes the following form in the coordinates {H , F} Φ ⊲⊳ ( µ ⊕ ν ) = ± µ α C αβγ ∂ H ∂µ β ∂ F ∂µ γ ± ν a D abd ∂ H ∂ν b ∂ F ∂ν d ± µ α L αaβ (cid:0) ∂ H ∂ν a ∂ F ∂µ β − ∂ F ∂ν a ∂ H ∂µ β (cid:1) ± ν a R abβ (cid:0) ∂ H ∂ν b ∂ F ∂µ β − ∂ F ∂ν b ∂ H ∂µ β (cid:1) . (104)The matched (plus/minus) Lie-Poisson dynamics in (103) is computed to be˙ µ β = ± µ ρ C ρβα ∂ H ∂µ α ∓ µ α L αaβ ∂ H ∂ν a ∓ ν a R abβ ∂ H ∂ν b , ˙ ν d = ± ν a D adb ∂ H ∂ν b ± ν a R adα ∂ H ∂µ α ± µ α L αdβ ∂ H ∂µ β . (105)21 .3 Lie Poisson Dynamics on Duals of 2-cocycles In Subsection 3.3, it is shown that 2-cocycle extension g Φ ⋊ h of a Lie algebra h over its representation space g ,as a particular case of extended structure g Φ ⊲⊳ h . Thus, the Lie-Poisson dynamics on the dual space of 2-cocycleextensions would be derived through the Lie-Poisson dynamics on the dual space of extended structures which isgiven in Subsection 4.1. So that, we follow Subsection 3.3 and make particular choice that the Lie bracket on g is trivial. This results with several consequences. One is that the left action ⊲ and the right dual action ∗ ⊳ , in(80), are trivial. Also, observe that the coadjoint action on g ∗ becomes identically zero. In addition, in this case,Φ turns out to be a true 2-cocycle and, κ becomes a Lie bracket on the vector space h . Thus, the dual of κ suitsthe coadjiont action as in (100). We employ all these modifications to the Lie-Poisson bracket (91) on the dual ofextended structure to arrive at the (plus/minus) Lie-Poisson bracket on the dual of 2-cocycle extension g Φ ⋊ h as {H , F} Φ ⋊ ( µ ⊕ ν ) = ± (cid:28) ν, [ δ H δν , δ F δν ] (cid:29) ± (cid:28) µ, Φ (cid:18) δ H δν , δ F δν (cid:19)(cid:29)| {z } A: 2-cocycle ± (cid:28) µ, δ H δν ⊲ δ F δµ (cid:29) ∓ (cid:28) µ, δ F δν ⊲ δ H δµ (cid:29)| {z } B: left action of h on g . (106)Here, the first term on the right hand side is the Lie-Poisson bracket on h ∗ . The term labelled as A is due to2-cocycle Φ whereas the terms labelled as B are due to the left action of h on g . For the Lie-Poisson bracket (106),the Lie-Poisson equations governed by a Hamiltonian function H = H ( µ, ν )is computed to be˙ µ = ∓ µ ∗ ⊳ δ H δν | {z } action of h , ˙ ν = ± ad ∗ δ H δν ( ν ) | {z } Lie-Poisson Eq. on h ∗ ± Φ ∗ δ H δν ( µ ) | {z } ± b ∗ δ H δµ µ | {z } action of h . (107)A direct observation gives that the Lie-Poisson equation (107) is a particular case of the Lie Poisson equation (92)where ∗ ⊲ , a ∗ are zero. Coordinate realizations.
In Subsection 3.3, 2-cocycle extension is written in terms of coordinates for finitedimensional cases. Referring to Subsection 4.1, we write the Lie-Poisson bracket (106) in coordinates as {H , F} Φ ⊲⊳ ( µ ⊕ ν ) = ± ν a D abd ∂ H ∂ν b ∂ F ∂ν d ± µ α Φ αbk ∂ H ∂ν b ∂ F ∂ν k ± µ α L αaβ (cid:0) ∂ H ∂ν a ∂ F ∂µ β − ∂ F ∂ν a ∂ H ∂µ β (cid:1) . (108)Notice that, we write the structure constants on h as D abd . Further, we can write the Lie Poisson equations in (107)as ˙ µ β = ∓ µ α L αaβ ∂ H ∂ν a , ˙ ν d = ± µ α Φ αdb ∂ H ∂ν b ± ν a D adb ∂ H ∂ν b ± µ α L αdβ ∂ H ∂µ β . (109) In this section, we provide a concrete example to Lie algebraic constructions and the Lie-Poisson structures in-troduced up to now. For this, we focus on BBGKY hierarchy in plasma dynamics [38]. In [56], it is proved thatBBGKY hierarchy can be recasted as a Lie-Poisson equation. The formulations presented in that study is for n > n = 3 which is missing in [56]. Accordingly, we first determine the dynamics ofBBGKY hierarchy for n = 3. Then, we shall investigate its Lie-Poisson form. Two decomposition of the BBGKYdynamics will be presented (1) as a matched pair and, (2) as an extended structure.22 .1 BBGKY Dynamics for 3 Particles Assume that a plasma rests in a finite 3 D manifold Q in R . Being a cotangent bundle, P = T ∗ Q is a symplecticand a Poisson manifold [51, 57]. Define the product symplectic space P = P × P × P endowed with the productsymplectic and product Poisson structures.We denote the 3 particle density function by f = f ( z , z , z ) on P . The dynamics of 3-particle plasma densityfunction is governed by the Vlasov equation ∂f ∂t = { H ( z , z , z ) , f ( z , z , z ) } (110)where H is the total energy of the plasma particles [58,61,68]. Here, {• , •} denotes the canonical Poisson bracket on P with respect to the variables z , z , z . We refer [18, 19, 28, 36] for some recent studies on Vlasov motion relatedwith the geometry here. In the present work, we assume that, the particle energy function is in form H = X i H ( z i ) + X i Now, we determine the moments of the plasma density function f = f ( z , z , z ) on P as f ( z ) := 3 Z f ( z , z , z ) dz dz f ( z , z ) := 6 Z f ( z , z , z ) dz . (112)To find the dynamics of the moments f and f , we simply take the partial derivatives of (112) then, directlysubstitute the Vlasov equation (110) into these expressions.In order to arrive at the equation of governing the moment functions, we record the following identity, on thePoisson space P , Z { h ( z ) , f ( z ) } z dz = 0 , (113)which is valid for any two functions. The equation (113) is the result of that we have omitted the boundary terms.In (113), {• , •} z stands for the Poisson bracket on P . We shall be referring to this identity in the sequel. We23ompute the dynamics of f as ∂f ∂t = 3 Z { H ( z , z , z ) , f ( z , z , z ) } dz dz = 3 Z n X i H ( z i ) + X i Assume that the adjoint action of the A on itself is the Lie bracket [ • , • ] A in (119). Thecoadjoint action of the space A on its dual A ∗ is D ad ∗ ( L ,L ,L ) ( f , f , f ) , ( K , K , K ) E = − D ( f , f , f ) , ad ( L ,L ,L ) ( K , K , K ) E = D ( f , f , f ) , (cid:2) ( K , K , K ) , ( L , L , L ) (cid:3) A E = D f , { K , L } + { K , L (3)2 } + { K , L (3)1 } + { K (3)2 , L } + { K (3)1 , L } + { K (3)2 , L (3)2 } E + (cid:10) f , { K , L (2)1 } z ,z + { K (2)1 , L } z ,z (cid:11) + (cid:10) f , { K , L } z (cid:11) . (121)In the second line, we have substituted the Lie algebra bracket (119). In the last equality, the first pairing is theone available between A ∗ and A with the symplectic volume dz dz dz whereas the second one is between A ∗ and A with the symplectic volume dz dz , and finally the last one is between A ∗ and A with the symplectic volume dz . It is evident that, in order to arrive at the explicit expression of the coadjoint action from (121), we need tosingle out the functions K , K and K . For this, we first recall the following association property of the smooth25unctions Z h ( z ) { k ( z ) , l ( z ) } z dz = Z k ( z ) { l ( z ) , h ( z ) } z dz. (122)To see this identity, we simply consider the Leibniz identity { h ( z ) k ( z ) , l ( z ) } z = h ( z ) { k ( z ) , l ( z ) } z + k ( z ) { h ( z ) , l ( z ) } z , (123)then, take the integral of this expression. In this case, the left hand side turns out to be zero due to (113). Theintegrals on the right hand side of (123) is exactly (122) after a reordering.Let us apply the identity (122) to the first pairing on the last equality in (121). We have that Z f ( z , z , z ) (cid:16) { K , L } + { K , L (3)2 } + { K , L (3)1 } + { K (3)2 , L } + { K (3)1 , L } + { K (3)2 , L (3)2 } (cid:17) ( z , z , z ) dz dz dz = Z (cid:16) K { L , f } ( z , z , z ) + K { L (3)2 , f } ( z , z , z ) + K { L (3)1 , f } ( z , z , z ) (cid:17) dz dz dz + Z K ( z , z ) (cid:16) Z { L , f } z ,z dz (cid:17) dz dz + Z K ( z ) (cid:16) Z { L , f } z dz dz (cid:17) dz + 2 Z K ( z , z ) { L , f } z ,z dz dz + Z K ( z , z ) (cid:16) Z { L ( z , z ) + L ( z , z ) , f } z ,z dz (cid:17) dz dz , (124)where we have employed the identities (113) and (122) and the definitions of moments in (112). In a similar way,we compute the pairings on the last line of (121) as follows Z f ( z , z ) (cid:16) { K , L (2)1 } z ,z + { K (2)1 , L } z ,z (cid:1) ( z , z ) dz dz + Z f ( z ) { K , L } z ( z ) dz = Z K ( z , z ) { L (2)1 , f } z ,z ( z , z ) dz dz + Z K ( z ) (cid:16) Z { L ( z , z ) , f ( z , z ) } z dz (cid:17) dz + Z K ( z ) { L , f } z ( z ) dz . (125)In (124) and (125), we collect the terms involving K , K and K in an order and then, arrive at the coadjoint flow ad ∗ ( L ,L ,L ) ( f , f , f ) = ( ˜ f , ˜ f , ˜ f ) (126)where˜ f ( z , z , z ) = { L + L (3)2 + L (3)1 , f } ( z , z , z ) , ˜ f ( z , z ) = { L (2)1 + 2 L , f } z ,z ( z , z ) + 12 Z { L ( z , z ) + L ( z , z ) , f ( z , z , z ) } z ,z dz + 6 Z { L , f } z ,z ( z , z , z ) dz ˜ f ( z ) = { L , f } z ( z ) + 2 Z { L ( z , z ) , f ( z , z ) } z dz + 3 Z { L ( z , z , z ) , f ( z , z , z ) } z dz dz (127)26 ie-Poisson equation. We assume now the following functional H ( f , f , f ) = Z H ( z , z , z ) f ( z , z , z ) dz dz dz + Z H ( z , z ) f ( z , z ) dz dz + Z H ( z ) f ( z ) dz . (128)Then we have that δ H δ ( f , f , f ) = (cid:16) δ H δf , δ H δf , δ H δf (cid:17) = (cid:0) H ( z , z , z ) , H ( z , z ) , H ( z ) (cid:1) ∈ A (129)which are the energy function in (111) under the isomorphism α in (118). If the three tuple H = ( H , (1 / H , H )is substituted in to the coadjoint action then we arrive at ∂∂t ( f , f , f ) = ad ∗ δ H /δ ( f ,f ,f ) ( f , f , f ) = ad ∗ ( H , (1 / H ,H ) ( f , f , f ) . (130)A direct calculation proves that the coadjoint flow (130) is precisely the system of equations (114), (115) and (110)governing the dynamics of the moments. Recall the direct sum A = A ⊕ A ⊕ A in (117). One evident decomposition of A is given by A = g ⊕ h , g = A ⊕ A and h = A . (131)In the present subsection, we examine this realization in the matched pair decomposition point of view. Decomposition of the Lie algebra. It is a direct calculation two show that the Lie bracket (119) is closedif it is restricted to the constitutive subspaces g and h . This reads that both g and h are Lie subalgebras.So, as a manifestation of Universal Lemma 5, this decomposition can be written as a matched pair decompositionintroduced in Subsection 3.2. We first determine the Lie algebra brackets on g and h by restricting the bracket(119) to the subspaces g and h [ • , • ] : g ⊗ g −→ g , [( K , K ) , ( L , L )] = (cid:16) { K + K (3)2 , L } + { K + K (3)2 , L (3)2 } , (cid:17) , [ • , • ] : h ⊗ h −→ h , [ K , L ] = { K , L } z , (132)respectively. In order to compute mutual actions, we recall the identity (59). In the present case, we compute (cid:0) K ⊲ ( L , L ) (cid:1) ⊕ (cid:0) K ⊳ ( L , L ) (cid:1) := [(0 , , K ) , ( L , L , A = (cid:0) { K (3)1 , L } , { K (2)1 , L } z ,z (cid:1) ⊕ h on g , and the right action of g on h as ⊲ : h ⊗ g −→ g , K ⊲ ( L , L ) = (cid:0) { K (3)1 , L } , { K (2)1 , L } z ,z (cid:1) , ⊳ : h ⊗ g −→ h , K ⊳ ( L , L ) = 0 , (134)respectively. Notice that, the right action ⊳ is trivial so that the Lie algebra A = g ⋊ h is a semidirect productLie algebra. The Lie bracket [ • , • ] A in (119) admits the following decomposition[( K , K ) ⊕ K , ( L , L ) ⊕ L ] = (cid:0) [( K , K ) , ( L , L )] + K ⊲ ( L , L ) − L ⊲ ( K , K ) (cid:1) ⊕ [ K , L ] . (135)27ere, the subalgebras [ • , • ] and [ • , • ] are the ones in (132), and the left action is in (134). This realization isprecisely in the matched pair Lie bracket form (67) where the left action ⊳ is trivial. Let us apply this to theLie-Poisson formulation of BBGKY Dynamics (102). Decomposition of BBGKY dynamics. The dual spaces of the constitutive Lie subalgebras g and h are g ∗ = A ∗ ⊕ A ∗ and h ∗ = A ∗ , respectively. So that, we can write A ∗ = g ∗ ⊕ h ∗ . The coadjoint action of g on g ∗ ,and the coadjoint action of h on h ∗ are ad ( L ,L ) ( f , f ) = (cid:16) { L , f } + 6 Z { L ( z , z ) , f ( z , z , z ) } z ,z dz , { L ( z , z ) , f ( z , z ) } z ,z + 6 Z { L ( z , z , z ) , f ( z , z , z ) } z ,z dz + 12 Z { L ( z , z ) + L ( z , z ) , f } z ,z dz (cid:17) ,ad K f = { K ( z ) , f ( z ) } z , (136)respectively. Recall the mutual actions of g and h on each other given in (134). The dual of these actions arecomputed to be ∗ ⊳ : g ∗ ⊗ h −→ g ∗ , ( f , f ) ∗ ⊳ K = (cid:0) { f ( z , z , z ) , K ( z ) } z , { f ( z , z ) , K ( z ) } z (cid:1) ∗ ⊲ : g ⊗ h ∗ −→ h ∗ , ( L , L ) ∗ ⊲ f = 0 . (137)The mapping b in (81) and its dual (82) are computed to be b ( L ,L ) : h −→ g , b ( L ,L ) ( K ) := K ⊲ ( L , L ) b ∗ ( L ,L ) : g ∗ −→ h ∗ , b ∗ ( L ,L ) ( f , f ) = 3 Z { L , f } z dz dz + 2 Z { L , f } z dz . (138)Since the left action is trivial both the mapping a in (84) and its dual (85) are trivial. It is now straight forward tomodify the matched Lie-Poisson equation (103) to the present version and determine the coadjoint flow as d ( f , f ) dt = ad ∗ ( L ,L ) ( f , f ) − ( f , f ) ∗ ⊳ K df dt = ad ∗ K f + b ∗ ( L ,L ) ( f , f ) . (139)This equations is precisely the coadjoint flow (130) realization of BBGKY dynamics. Evidently takes the classicalform if ( L , L , K ) = ( H , (1 / H , H ). Recall once more the direct sum A = A ⊕ A ⊕ A in (117). We have examined a matched pair decomposition(131) of this sum. An alternative decomposition of A can be given by A = g ⊕ h , g := A , and h = A ⊕ A . (140) Decomposition of the Lie algebra. It is straightforward to see that, g is a subalgebra of A with inducedbracket [ • , • ] : g ⊗ g −→ g , [ K , L ] = { K , L } , (141)28here {• , •} is the Poisson bracket on P . On the other hand, the subspace h fails to be so. Indeed, the Liebracket (119) of two generic elements (0 , K , K ) and (0 , L , L ) in h is[(0 , K , K ) , (0 , L , L )] A = { K (3)2 , L (3)2 } ⊕ (cid:0) { K , L (2)1 } z ,z + { K (2)1 , L } z ,z , { K , L } z (cid:1) ∈ g ⊕ h , (142)where the first term on the right hand side falls into g whereas the the second and the third terms are in h .So that, this decomposition should be analysed in the light of extended structures presented in Subsection 3.1.Accordingly, by referring to (56), we defineΦ : h ⊗ h −→ g , (( K , K ) , ( L , L )) 7→ { K (3)2 , L (3)2 } κ : h ⊗ h −→ h , (( K , K ) , ( L , L )) (cid:0) { K , L (2)1 } z ,z + { K (2)1 , L } z ,z , { K , L } z (cid:1) . (143)Now, we are ready to compute mutual actions defined in (42) and (40) between the constitutive spaces h and g .To obtain the fomulas, we employ the identity (59) that is( K , K ) ⊲ L ⊕ ( K , K ) ⊳ L = [(0 , K , K ) , ( L , , A = (cid:0) { K (3)2 , L } + { K (3)1 , L } (cid:1) ⊕ (0 , ∈ g ⊕ h (144)which gives that ⊲ : h ⊗ g −→ g , ( K , K ) ⊲ L = { K (3)2 , L } + { K (3)1 , L } ⊳ : h ⊗ g −→ h , ( K , K ) ⊳ L = (0 , . (145)Due to Universal Lemma 3, the decomposition g ⊕ h of the Lie algebra A reads the decomposition of the Liebracket {• , •} A given in (119) into the form of extended Lie bracket 3.1 where the right action ⊳ is trivial[ K ⊕ ( K , K ) , L ⊕ ( L , L )] Φ ⊲⊳ = (cid:0) { K , L } + ( K , K ) ⊲ L − ( L , L ) ⊲ K + Φ(( K , K ) , ( L , L )) (cid:1) ⊕ κ (( K , K ) , ( L , L )) , (146)where the left action is the one given in (145) whereas Φ and κ mapping are those available in (143). An alternative decomposition of the Lie algebra. The hierarchy of the moment functions suggests analternative formulation of Φ and κ mappings in (143). This is due to the fact that the term [ K (3)2 , L (3)2 ] can bewritten as a sum of some terms in h and some terms in g . Indeed,[ K (3)2 , L (3)2 ] = 2( { K ( z , z ) , L ( z , z ) } z ,z ) (3) ( z , z , z )+ 4 { K ( z , z ) , L ( z , z ) + L ( z , z ) } z ,z + 4 { K ( z , z ) , L ( z , z ) + L ( z , z ) } z ,z + 4 { K ( z , z ) , L ( z , z ) + L ( z , z ) } z ,z . (147)Here, as depicted in the display, the first term on the right hand side can be written as the image of the symmetricfunction { K ( z , z ) , L ( z , z ) } under the embedding A A given in (116). Accordingly, instead of Φ and κ h ⊗ h −→ g , (( K , K ) , ( L , L )) { K ( z , z ) , L ( z , z ) + L ( z , z ) } + 4 { K ( z , z ) , L ( z , z ) + L ( z , z ) } + 4 { K ( z , z ) , L ( z , z ) + L ( z , z ) } ˜ κ : h ⊗ h −→ h , (( K , K ) , ( L , L )) (cid:16) { K ( z , z ) , L ( z ) + L ( z ) } z ,z + { K ( z ) + K ( z ) , L ( z , z ) } z ,z + 2 { K ( z , z ) , L ( z , z ) } z ,z , { K ( z ) , L ( z ) } z (cid:17) . (148)Evidently, this observation reads an alternative Lie bracket operation on A as well. We denote this by a tildenotation [ • , • ] ˜Φ ⊲⊳ and record as follows[ K ⊕ ( K , K ) , L ⊕ ( L , L )] ˜Φ ⊲⊳ = (cid:0) { K , L } + ( K , K ) ⊲ L − ( L , L ) ⊲ K + ˜Φ(( K , K ) , ( L , L )) (cid:1) ⊕ ˜ κ (( K , K ) , ( L , L )) , (149)where { K , L } is the Poisson bracket on P , and ⊲ is the left action in (145). Decomposition of the dynamics: A ∗ = g ∗ ⊕ h ∗ . We start with the dualization of the mutual actions in (145).Notice that, the left action is trivial, so that it induces a trivial dual action. For the right action, we compute thedual action as g ∗ ∗ ⊳ h −→ g ∗ , f ∗ ⊳ ( K , K ) = { f , K (3)2 } + { f , K (3)1 } . (150)Using the right action in (145), and in the lights of the definitions in (81) and(82), we compute the followingmapping along with its dual b L : h −→ g , b L ( K , K ) = ( K , K ) ⊲ L = [ K (3)2 , L ] + [ K (3)1 , L ] b ∗ L : g ∗ −→ h ∗ , b ∗ L f = (cid:0) Z { L , f } z ,z dz , Z { L , f } z dz dz (cid:1) . (151)Further, according to (86) and (87), by freezing the first entries of ˜Φ and ˜ κ in (148), we arrive at linear mappings.One od these mappings is ˜Φ ( K ,K ) from h to g , and other is ˜ κ ( K ,K ) from h to h . Dualizations of thesemappings result with˜Φ ∗ ( K ,K ) : g ∗ −→ h ∗ , ˜Φ ∗ ( K ,K ) f = (cid:16) Z { K ( z , z ) , f ( z , z , z ) } z dz , (cid:17) , ˜ κ ∗ ( K ,K ) : h ∗ −→ h ∗ , ˜ κ ∗ ( K ,K ) ( f , f ) = (cid:16) { K ( z ) , f ( z , z ) } z + 2 { K ( z , z ) , f ( z , z ) } z ,z , Z { K ( z , z ) , f ( z , z ) } z dz + { K ( z ) , f ( z ) } z (cid:17) . (152)Now, recall the decomposed Lie-Poisson equation (92). Since, in the present case, the right action is trivial, we takethe terms involving ∗ ⊳ and a ∗ as zero. After substituting all these into the Lie-Poisson equation, we have that df dt = ad ∗ K ( f ) − f ∗ ⊳ ( K , K ) ,d ( f , f ) dt = ˜ κ ∗ ( K ,K ) ( f , f ) + ˜Φ ∗ ( K ,K ) f + b ∗ L f . (153)Here, ad ∗ K ( f ) = { K , f } is the coadjoint action of g on its dual space g ∗ . If we take K = H , K = (1 / H K = H then, the system is exactly the dynamics of the moments in (114), (115) and (110) by decomposingthe coadjoint flow (130). In Subsection 3.3, 2-cocycle extensions are exhibited as particular instances of extended structures. In this section,we couple two 2-cocycle extensions under mutual actions. This will be achieved by matched pair theory availablein Subsection 3.2. Our goal is to explore conditions for a matched pair of 2-cocycles to be a 2-cocycle of a matchedpair. We shall, further, study dynamics on the coupled system to have the Lie-Poisson equations for the collectivemotion. We start with two Lie algebras, say l and k , and two vector spaces V and W . Assume a V -valued 2-cocycle ϕ on l ,and a W -valued 2-cocycle φ on k given by ϕ : l × l → V, φ : k × k → W. (154)Further, we consider a left action of the Lie algebra l on the vector spaces V , and a left action of the Lie algebra k on the vector spaces W that is ⇃ : l ⊗ V → V, l ⊗ v l ⇃ v, ⇂ : k ⊗ W → W, k ⊗ w k ⇂ w. (155)If these actions are compatible with the 2-cocycle maps as in (77) then, one arrives at the following 2-cocycleextensions g := V ϕ ⋊ l , h := W φ ⋊ k . (156)In the light of the discussions done in Subsection 3.3, after some proper modifications of (79), we have the followingLie algebra brackets on the extended Lie algebras g and h [ v ⊕ l, v ′ ⊕ l ′ ] ϕ ⋊ = (cid:0) l ⇃ v ′ − l ′ ⇃ v + ϕ ( l, l ′ ) (cid:1) ⊕ [ l, l ′ ] , [ w ⊕ k, w ′ ⊕ k ′ ] φ ⋊ = (cid:0) k ⇂ w ′ − k ′ ⇂ w + φ ( k, k ′ ) (cid:1) ⊕ [ k, k ′ ] , (157)respectively. Here, the bracket [ l, l ′ ] is the Lie algebra bracket on l whereas the bracket [ k, k ′ ] is the Lie bracket on k . Matching of 2-cocycles. We now examine matched pair of 2-cocycle extensions g = V ϕ ⋊ l and h = W φ ⋊ k . Tothis end, we consider the followings 3 sets of mappings: (1) We first consider mutual Lie algebras actions of l and k on each other ◮ : k ⊗ l → l , k ⊗ l k ◮ l, ◭ : k ⊗ l → k , k ⊗ l k ◭ l. (158)31e assume that these actions satisfy the compatibility condition (68) hence, determine a matched pair Lie algebradenoted by l ⊲⊳ k . (2) In order to extend the mutual actions given in (158) to the product spaces g and h in (156), we introduce aright action of l on W and, a left action k on V given by y : k ⊗ V → V, k ⊗ v k y v, x : W ⊗ l → W, w ⊗ l w x l, (159)respectively. (3) In addition, it is always possible to have the following cross representations ǫ : k ⊗ l → V, ǫ ( k, l ) ∈ V,ι : k ⊗ l → W, ι ( k, l ) ∈ W. (160)Referring to the mappings (158), (159) and (160), we define mutual actions of 2-cocycle extensions g = V ϕ ⋊ l and h = W φ ⋊ k as follows ⊲ : ( W φ ⋊ k ) × ( V ϕ ⋊ l ) −→ V ϕ ⋊ l , (( w ⊕ k ) , ( v ⊕ l )) ( k y v + ǫ ( k, l )) ⊕ ( k ◮ l ) , ⊳ : ( W φ ⋊ k ) × ( V ϕ ⋊ l ) −→ W φ ⋊ k , (( w ⊕ k ) , ( v ⊕ l )) ( w x l + ι ( k, l )) ⊕ ( k ◭ l ) . (161)It is possible to see that ⊲ is a left action whereas ⊳ is a right action. In order to construct a matched pair of h = W φ ⋊ k and g = V ϕ ⋊ l , one needs to justify the compatibility conditions in (68). A direct observation givesthat, for the actions (161), the compatibility conditions (68) consist of 4 equations. Two of them, those for thesecond terms in the decompositions, involve only the left ◮ and the right ◭ actions in (158). These two equationsare precisely the matched pair compatibility conditions for l ⊲⊳ k . Since, we assume that l ⊲⊳ k is a matched pair,these two compatibility conditions are automatically satisfied. So, we left with other two compatibility conditions.For any k, k ′ in k , l, l ′ in l , v, v ′ in V , and w, w ′ in W , these equations are computed to be k y ( l ⇃ v ′ − l ′ ⇃ v + ϕ ( l, l ′ )) + ǫ ( k, [ l, l ′ ]) = l ⇃ ( k y v ′ + ǫ ( k, l ′ )) − ( k ◮ l ′ ) ⇃ v + ϕ ( l, k ◮ l ′ ) − l ′ ⇃ ( k y v + ǫ ( k, l )) + ( k ◮ l ) ⇃ v ′ − ϕ ( l ′ , k ◮ l ) + ( k ◭ l ) y v ′ − ( k ◭ l ′ ) y v + ǫ ( k ◭ l, l ′ ) − ι ( k ◭ l ′ , l ) − ( w x l ′ + ι ( k, l ′ )) x l ( k ⇂ w ′ − k ′ ⇂ w + φ ( k, k ′ )) x l + ι ([ k, k ′ ] , l ) = k ⇂ ( w ′ x l + ι ( k ′ , l )) − ( k ′ ◭ l ) ⇂ w + φ ( k, k ′ ◭ l ) − k ′ ⇂ ( w x l + ι ( k, l )) + ( k ◭ l ) ⇂ w ′ − φ ( k ′ , k ◭ l ) − w x ( k ′ ◮ l )+ w ′ x ( k ◮ l ) + ι ( k ′ , k ◮ l ) − ǫ ( k, k ′ ◮ l ) + k y ( k ′ y v + ǫ ( k ′ , l )) , (162)where ⇃ and ⇂ are the left actions in (155), ◮ and ◭ are actions in (158), y and x are the actions in (159), ǫ and ι are the mappings in (160). By assuming that these conditions are satisfied, we define the following matched pairLie algebra g ⊲⊳ h = ( V ϕ ⋊ l ) ⊲⊳ ( W φ ⋊ k ) . (163)To compute the matched Lie algebra bracket on this total space, we recall the general formula of the matched Liebracket in (67) then, by employing the actions in (161), compute (cid:2)(cid:0) ( v ⊕ l ) ⊕ ( w ⊕ k ) (cid:1) , (cid:0) ( v ′ ⊕ l ′ ) ⊕ ( w ′ ⊕ k ′ ) (cid:1)(cid:3) ⊲⊳ = (¯ v ⊕ ¯ l ) ⊕ ( ¯ w ⊕ ¯ k ) (164)32here ¯ v = l ⇃ v ′ − l ′ ⇃ v + k y v ′ − k ′ y v + ǫ ( k, l ′ ) − ǫ ( k ′ , l ) + ϕ ( l, l ′ ) , ¯ l = [ l, l ′ ] + k ◮ l ′ − k ′ ◮ l, ¯ w = k ⇂ w ′ − k ′ ⇂ w + w x l ′ − w ′ x l + ι ( k, l ′ ) − ι ( k ′ , l ) + φ ( k, k ′ ) , ¯ k = [ k, k ′ ] + k ◭ l ′ − k ′ ◭ l. (165) Matched pair as a 2-cocycle. Now, we investigate that under which conditions the matched pair g ⊲⊳ h in (163)turns out to be a 2-cocycle extension by itself. To show this, we need to determine a left action a 2-cocycle. Let usdetermine these one by one. (Left action). Recall the mutual actions in (158) and the matched pair algebra l ⊲⊳ k . It is evident that l ⊲⊳ k is aLie subalgebra of g ⊲⊳ h . Define a left action of l ⊲⊳ k on the product space V ⊕ W as follows: · ⊲ : ( l ⊲⊳ k ) × ( V ⊕ W ) −→ ( V ⊕ W ) , ( l ⊕ k ) · ⊲ ( v ⊕ w ) = (cid:0) l ⇃ v + k y v (cid:1) ⊕ (cid:0) − w x l + k ⇂ w (cid:1) , (166)where we have employed the left actions ⇃ and ⇂ in (155), the actions y and x in (159). To be a left action, (166)needs to satisfy the first condition in (66). We compute this as( k ◮ l ′ ) ⇃ v − ( k ′ ◮ l ) ⇃ v + ( k ◭ l ′ ) y v − ( k ′ ◭ l ) y v = l ⇃ ( k ′ y v ) − l ′ ⇃ ( k y v ) k y ( l ′ ⇃ v ) − k ′ y ( l ⇃ v ) (167)for the first entry in (166). Notice that, this is an equation defined on the vector space V . For the second entry, wehave − w x ( k ◮ l ′ ) + w x ( k ′ ◮ l ) + ( k ◭ l ′ ) ⇂ w − ( k ′ ◭ l ) ⇂ w = − ( k ′ ⇂ w ) x l + ( k ⇂ w ) x l ′ − k ⇂ ( w x l ′ ) + k ′ ⇂ ( w x l ) (168)where ◮ and ◭ are the mutual actions exhibited in (158). (2-cocycle). Later, we introduce a ( V ⊕ W )-valued 2-cocycle on l ⊲⊳ k , in terms of the 2-cocycles ϕ and φ given in(154), as followsΘ : ( l ⊲⊳ k ) × ( l ⊲⊳ k ) −→ V ⊕ W, Θ(( l ⊕ k ) , ( l ′ ⊕ k ′ )) = (cid:0) ϕ ( l, l ′ )+ ǫ ( k, l ′ ) − ǫ ( k ′ , l ) (cid:1) ⊕ (cid:0) φ ( k, k ′ )+ ι ( k, l ′ ) − ι ( k ′ , l ) (cid:1) , (169)33ne needs to ask the compatibility conditions, in (77),0 = ϕ ( l, k ′′ ◮ l ′ − k ′ ◮ l ′′ ) + ǫ ( k, [ l ′′ , l ′ ]) + ǫ ( k, k ′′ ◮ l ′ − k ′ ◮ l ′′ ) − ǫ ([ k ′′ , k ′ ] , l ) − ǫ ( k ′′ ◭ l ′ − k ′ ◭ l ′′ , l ) − l ⇃ ( ϕ ( l ′ , l ′′ ) + ǫ ( k ′ , l ′′ ) − ǫ ( k ′′ , l ′ )) − k y ( ϕ ( l ′ , l ′′ ) + ǫ ( k ′ , l ′′ ) − ǫ ( k ′′ , l ′ )) ϕ ( l ′ , k ◮ l ′′ − k ′′ ◮ l ) + ǫ ( k ′ , [ l, l ′′ ]) + ǫ ( k ′ , k ◮ l ′′ − k ′′ ◮ l ) − ǫ ([ k, k ′′ ] , l ′ ) − ǫ ( k ◭ l ′′ − k ′′ ◭ l, l ′ ) − l ′ ⇃ ( ϕ ( l ′′ , l ) + ǫ ( k ′′ , l ) − ǫ ( k, l ′′ )) − k y ( ϕ ( l ′′ , l ) + ǫ ( k ′′ , l ) − ǫ ( k, l ′′ )) ϕ ( l ′′ , k ′ ◮ l − k ◮ l ′ ) + ǫ ( k ′′ , [ l ′ , l ]) + ǫ ( k ′′ , k ′ ◮ l − k ◮ l ′ ) − ǫ ([ k ′ , k ] , l ′′ ) − ǫ ( k ′ ◭ l − k ◭ l ′ , l ′′ ) − l ′′ ⇃ ( ϕ ( l, l ′ ) + ǫ ( k, l ′ ) − ǫ ( k ′ , l )) − k y ( ϕ ( l, l ′ ) + ǫ ( k, l ′ ) − ǫ ( k ′ , l ))0 = φ ( l, k ′′ ◮ l ′ − k ′ ◮ l ′′ ) + ι ( k, [ l ′′ , l ′ ]) + ι ( k, k ′′ ◮ l ′ − k ′ ◮ l ′′ ) − ι ([ k ′′ , k ′ ] , l ) − ι ( k ′′ ◭ l ′ − k ′ ◭ l ′′ , l ) − l ⇃ ( φ ( l ′ , l ′′ ) + ι ( k ′ , l ′′ ) − ι ( k ′′ , l ′ )) − k y ( φ ( l ′ , l ′′ ) + ι ( k ′ , l ′′ ) − ι ( k ′′ , l ′ )) φ ( l ′ , k ◮ l ′′ − k ′′ ◮ l ) + ι ( k ′ , [ l, l ′′ ]) + ι ( k ′ , k ◮ l ′′ − k ′′ ◮ l ) − ι ([ k, k ′′ ] , l ′ ) − ι ( k ◭ l ′′ − k ′′ ◭ l, l ′ ) − l ′ ⇃ ( φ ( l ′′ , l ) + ι ( k ′′ , l ) − ι ( k, l ′′ )) − k y ( φ ( l ′′ , l ) + ι ( k ′′ , l ) − ι ( k, l ′′ )) ϕ ( l ′′ , k ′ ◮ l − k ◮ l ′ ) + ι ( k ′′ , [ l ′ , l ]) + ι ( k ′′ , k ′ ◮ l − k ◮ l ′ ) − ι ([ k ′ , k ] , l ′′ ) − ι ( k ′ ◭ l − k ◭ l ′ , l ′′ ) − l ′′ ⇃ ( φ ( l, l ′ ) + ι ( k, l ′ ) − ι ( k ′ , l )) − k y ( φ ( l, l ′ ) + ι ( k, l ′ ) − ι ( k ′ , l )) . (170)We are ready now to define a 2-cocycle extension of the Lie alegbra l ⊲⊳ k via V ⊕ W . We record this by( V ⊕ W ) Θ ⋊ ( l ⊲⊳ k ) . (171)To arrive at the Lie bracket on this space, one only needs to employ the explicit definitions of the left action · ⊲ in2-cocycle Θ into the generic formula (79) of Lie bracket for 2-cocycle extensions. This results with[( v ⊕ w ) ⊕ ( l ⊕ k ) , ( v ′ ⊕ w ′ ) ⊕ ( l ′ ⊕ k ′ )] Θ ⋊ = (cid:0) ( l ⊕ k ) · ⊲ ( v ′ ⊕ w ′ ) − ( l ′ ⊕ k ′ ) · ⊲ ( v ⊕ w )+ Θ(( l ⊕ k ) , ( l ′ ⊕ k ′ )) (cid:1) ⊕ [( l ⊕ k ) , ( l ′ ⊕ k ′ )] , (172)where · ⊲ is the left action in (166), and the bracket [( l ⊕ k ) , ( l ′ ⊕ k ′ )] is the matched pair Lie bracket on l ⊲⊳ k . It isimmediate to see that extended Lie algebra bracket on (172) is precisely equal to the matched Lie bracket given in(164) and (165), up to some reordering. Eventually, we are ready now to collect all the discussions done so far inthe following proposition. Proposition 8 The matched pair Lie algebra g ⊲⊳ h , given in (163) , of 2-cocycle extensions g = V ϕ ⋊ l and h = W φ ⋊ k is a 2-cocycle extension admitting a left action · ⊲ in (166) and a 2-cocycle Θ in (169) that is ( V ϕ ⋊ l ) ⊲⊳ ( W φ ⋊ k ) ∼ = ( V ⊕ W ) Θ ⋊ ( l ⊲⊳ k ) . (173)Even though, we have derived all the mappings and conditions up to now explicitly. There is a short but implicitway to arrive at that proposition by employing Universal Lemma 5. For this, we first embed Lie subalgebras to thespace ( V ⊕ W ) Θ ⋊ ( l ⊲⊳ k ) as follows g −→ ( V ⊕ W ) Θ ⋊ ( l ⊲⊳ k ) , ( v ⊕ l ) ( v ⊕ ⊕ ( l ⊕ h −→ ( V ⊕ W ) Θ ⋊ ( l ⊲⊳ k ) , ( w ⊕ k ) (0 ⊕ w ) ⊕ (0 ⊕ k ) . (174)Then Universal Lemma 5 reads that the total space admits a matched pair decomposition.34 Particular case. For future reference, we now examine a particular case of Proposition 8. First, we choose theleft actions ⇃ and ⇂ in (155) as trivial. So that, the Lie brackets on the 2-cocycle extensions in (157) turn out to be[ v ⊕ l, v ′ ⊕ l ′ ] ϕ ⋊ = ϕ ( l, l ′ ) ⊕ [ l, l ′ ] , [ w ⊕ k, w ′ ⊕ k ′ ] φ ⋊ = φ ( k, k ′ ) ⊕ [ k, k ′ ] . (175)In addition, consider that the mutual actions of k and l in (158) and the cross representations in (159) are all zero.Hence, we have that, the mutual actions of h and g in (161) are ⊲ : (( w ⊕ k ) , ( v ⊕ l )) ( ǫ ( k, l ) ⊕ , ⊳ : (( w ⊕ k ) , ( v ⊕ l )) ( ι ( k, l ) ⊕ . (176)A direct calculation gives us that, in the present setting, ⊲ is a left action if and only if ǫ ( k, [ l, l ′ ]) = 0, and ⊳ is a right action if and only if ι ([ k, k ′ ] , l ) = 0. Eventually, we claim that, the assumptions in this case reduce thematched Lie algebra bracket in (164) and (165) as (cid:2)(cid:0) ( v ⊕ l ) ⊕ ( w ⊕ k ) (cid:1) , (cid:0) ( v ′ ⊕ l ′ ) ⊕ ( w ′ ⊕ k ′ ) (cid:1)(cid:3) ⊲⊳ = (cid:16)(cid:0) ǫ ( k, l ′ ) − ǫ ( k ′ , l ) + ϕ ( l, l ′ ) (cid:1) ⊕ (cid:17) ⊕ (cid:16)(cid:0) ι ( k, l ′ ) − ι ( k ′ , l ) + φ ( k, k ′ ) (cid:1) ⊕ (cid:17) (177)where ǫ and ι are as in (160). In order to implement Proposition 8, we exploit a proper left action and a 2-cocycleoperator. Here, according to (166), we take the left action as trivial whereas the 2-cocycle term Θ is precisely equalto the one in (169). We have exhibited matched pair of 2-cocycle extensions in the previous subsection. To arrive at Hamiltoniandynamics on the dual picture, we make use the Lie-Poisson formalism on 2-cocycle extensions, that is the theoryin Subsection 4.3. In that subsection, there exist both the Lie-Poisson bracket (106) and the Lie-Poisson equations(107) for the case of 2-cocycles. Lie-Poisson brackets. Let us consider the following notation on the dual spaces µ = µ V ∗ ⊕ µ W ∗ ∈ V ∗ ⊕ W ∗ , ν = ν l ∗ ⊕ ν k ∗ ∈ l ∗ ⊕ k ∗ . (178)Substitute the 2-cocycle Θ in (169) and the left action · ⊲ in (166) in the Lie-Poisson bracket (106). Therefore, forthis situation, (plus/minus) Lie-Poisson bracket is {H , F} Θ ⋊ ( µ ⊕ ν ) = ± D ν l ∗ ⊕ ν k ∗ , (cid:2)(cid:0) δ H δν l ∗ ⊕ δ H δν k ∗ (cid:1) , (cid:0) δ F δν l ∗ ⊕ δ F δν k ∗ (cid:1)(cid:3)E ± D µ V ∗ ⊕ µ W ∗ , Θ (cid:16) δ H δν l ∗ ⊕ δ H δν k ∗ , δ F δν l ∗ ⊕ δ F δν k ∗ (cid:17)E ± D µ V ∗ ⊕ µ W ∗ , (cid:0) δ H δν l ∗ ⊕ δ H δν k ∗ (cid:1) · ⊲ (cid:0) δ F δµ V ∗ ⊕ δ F δµ W ∗ (cid:1)E ∓ D µ V ∗ ⊕ µ W ∗ , (cid:0) δ F δν l ∗ ⊕ δ F δν k ∗ (cid:1) · ⊲ (cid:0) δ H δµ V ∗ ⊕ δ H δµ W ∗ (cid:1)E , (179)where the bracket on the first line is the matched pair Lie bracket on l ⊲⊳ k . Here, the first pairing is the one between35 ∗ × k ∗ and l ⊲⊳ k , the others are the pairing between V ∗ ⊕ W ∗ and V ⊕ W . It is maybe needless to say that wehave assumed that all the vector spaces studied in here are reflexive. If the explicit expressions for the Lie bracketon l ⊲⊳ k , the left action · ⊲ in (166), and the 2-cocycle Θ in (169) are substituted into the bracket (179), one arrivesat the bracket as {H , F} Θ ⋊ ( µ ⊕ ν ) = ± (cid:10) ν l ∗ , (cid:2) δ H δν l ∗ , δ F δν l ∗ (cid:3) + δ H δν k ∗ ◮ δ F δν l ∗ − δ F δν k ∗ ◮ δ H δν l ∗ (cid:11) ± (cid:10) ν k ∗ , (cid:2) δ H δν k ∗ , δ F δν k ∗ (cid:3) + δ H δν k ∗ ◭ δ F δν l ∗ − δ F δν k ∗ ◭ δ H δν l ∗ (cid:11) ± (cid:10) µ V ∗ , δ H δν l ∗ ⇃ δ F δµ V ∗ + δ H δν k ∗ y δ F δµ V ∗ + δ F δν l ∗ ⇃ δ H δµ V ∗ + δ F δν k ∗ y δ H δµ V ∗ (cid:11) ± (cid:10) µ V ∗ , ǫ ( δ H δν k ∗ , δ F δν l ∗ ) − ǫ ( δ F δν k ∗ , δ H δν l ∗ ) + ϕ ( δ H δν l ∗ , δ F δν l ∗ ) ∓ (cid:10) µ W ∗ , − δ F δµ W ∗ x δ H δν l ∗ + δ H δν k ∗ ⇂ δ F δµ W ∗ − δ H δµ W ∗ x δ F δν l ∗ + δ F δν k ∗ ⇂ δ H δµ W ∗ (cid:11) . ± (cid:10) µ W ∗ , ι ( δ H δν k ∗ , δ F δν l ∗ ) − ι ( δ F δν k ∗ , δ H δν l ∗ ) + φ ( δ H δν k ∗ , δ F δν k ∗ ) (cid:11) (180)Here, ◮ and ◭ are the actions in (158), x and y are in (159) whereas ⇃ and ⇃ are those in (155).In the light of Proposition 8, one can write the bracket (179) as a matched pair Lie-Poisson bracket. To this end,by reordering the elements in (178), define˜ µ = µ V ∗ ⊕ ν l ∗ ∈ g = V ϕ ⋊ l , ˜ ν = µ W ∗ ⊕ ν k ∗ ∈ h = W φ ⋊ k . (181)Then the matched pair Lie-Poisson bracket in (102) takes the form {H , F} ⊲⊳ (˜ µ ⊕ ˜ ν ) = ± {H , F} ϕ ⋊ (˜ µ ) ± {H , F} φ ⋊ (˜ ν ) ∓ D µ V ∗ ⊕ ν l ∗ , (cid:0) δ H δµ W ∗ ⊕ δ H δν k ∗ (cid:1) ⊲ (cid:0) δ F δµ V ∗ ⊕ δ F δν l ∗ (cid:1) − (cid:0) δ F δµ W ∗ ⊕ δ F δν k ∗ (cid:1) ⊲ (cid:0) δ H δµ V ∗ ⊕ δ H δν l ∗ (cid:1)E ∓ D µ W ∗ ⊕ ν k ∗ , (cid:0) δ H δµ W ∗ ⊕ δ H δν k ∗ (cid:1) ⊳ (cid:0) δ F δµ V ∗ ⊕ δ F δν l ∗ (cid:1) − (cid:0) δ F δµ W ∗ ⊕ δ F δν k ∗ (cid:1) ⊳ (cid:0) δ H δµ V ∗ ⊕ δ H δν l ∗ (cid:1)E , (182)where the actions ⊳ and ⊲ are in (161). Notice that, the Lie-Poisson brackets on the right hand side of the firstline are the individual Lie-Poisson brackets on the dual spaces g ∗ and h ∗ , respectively. Those terms available in thesecond line of (182) are manifestations of the left action of h on g whereas the third line is due to the right action of g on h . A direct computation gives that the Lie-Poisson bracket (182) is equal to the Lie-Poisson bracket in (180). Dual actions: First, define the dual actions of y and x in (159) as ∗ x : V ∗ → V ∗ , h µ V ∗ ∗ x k, v i = h µ V ∗ , k y v i , ∗ y : W ∗ → W ∗ , h l ∗ y µ W ∗ , w i = h µ W ∗ , w x l i , (183)respectively. See that, ∗ x is a right action whereas ∗ y is a left action. Secondly, introduce the dual (right) actionsof ⇃ and ⇂ in (155) as ∗ ⇃ k : W ∗ → W ∗ , h µ W ∗ ∗ ⇃ k, w i = h µ W ∗ , k ⇂ w i , ∗ ⇂ l : V ∗ → V ∗ , h µ V ∗ ⇂ l, v i = h µ V ∗ , l ⇃ v i , (184)36espectively. Then, determine the dual actions of ◭ and ◮ in (158) ∗ ◭ : k ⊗ l ∗ −→ l ∗ , h k ∗ ◭ ν l ∗ , l i = h ν l ∗ , k ◮ l i , ∗ ◮ : l ⊗ k ∗ −→ k ∗ , h l ∗ ◮ ν k ∗ , k i = h ν k ∗ , k ◭ l i , (185)respectively. We define the dual mappings of the 2-cocycles ϕ and φ as ϕ ∗ l : V ∗ −→ l ∗ , h ϕ ∗ l µ V ∗ , l ′ i = −h µ V ∗ , ϕ l l ′ i ,φ ∗ k : W ∗ −→ k ∗ , h φ ∗ k µ W ∗ , k ′ i = −h µ W ∗ , φ k k ′ i . (186)Lastly, exhibit the dual mappings of ǫ and ι in (160) as ǫ ∗ k : V ∗ −→ l ∗ , h ǫ ∗ k µ V ∗ , l i = −h µ V ∗ , ǫ k l i ,ι ∗ k : W ∗ −→ l ∗ , h ι ∗ k µ W ∗ , l i = −h µ W ∗ , ι k l i . (187) Lie-Poisson equations. According to the equations (107), it suffice to define dual mappings of the action · ⊲ (166)and Θ (169) to write Lie-Poisson dynamics. For · ⊲ , by the definiton, we compute h ( µ V ∗ ⊕ µ W ∗ ) ∗ · ⊳ ( l ⊕ k ) , ( v ⊕ w ) i = h µ V ∗ , l ⇃ v + k y v i + h µ W ∗ , − w x l + k ⇂ w i , = h µ V ∗ , l ⇃ v i + h µ V ∗ , k y v i + h µ W ∗ , − w x l i + h µ W ∗ , k ⇂ w i . (188)Then, we have ( µ V ∗ ⊕ µ W ∗ ) ∗ · ⊳ ( l ⊕ k ) = (cid:0) µ V ∗ ∗ x k + µ V ∗ ∗ ⇂ l (cid:1) ⊕ (cid:0) l ∗ y µ W ∗ + µ W ∗ ∗ ⇃ k (cid:1) . (189)For Θ, we computeΘ ∗ ( l ⊕ k ) ( µ V ∗ ⊕ µ W ∗ ) = ( ϕ ∗ l µ V ∗ + ǫ ∗ k µ V ∗ − ι ∗ k µ W ∗ ) ⊕ ( φ ∗ k µ W ∗ + ι ∗ k µ W ∗ − ǫ ∗ k µ V ∗ ) , (190)where ϕ ∗ l , φ ∗ k defined as (186) and ǫ ∗ k , ι ∗ k defined as (187). There are two more dual mappings we need to get for theleft side of the equation (107). These are b ∗ ( l ⊕ k ) ( µ V ∗ ⊕ µ W ∗ ) and the coadjoint action of ( δ H δν l ∗ ⊕ δ H δν k ∗ ) on the dualelement ( ν l ∗ ⊕ ν k ∗ ). Being a matched pair, we can employ the equation (101) for the case of l ⊲⊳ k . Accordingly, wearrive at ad ∗ ( δ H δν l ∗ ⊕ δ H δν k ∗ ) ( ν l ∗ ⊕ ν k ∗ ) = (cid:0) ad ∗ δ H δν l ∗ ν l ∗ + ν l ∗ ∗ ◭ δ H δν k ∗ + ν k ∗ ∗ ◮ δ H δν k ∗ (cid:1) ⊕ (cid:0) ad ∗ δ H δν k ∗ ν k ∗ − ν l ∗ ∗ ◭ δ H δν l ∗ − ν k ∗ ∗ ◮ δ H δν l ∗ (cid:1) , (191)where ∗ ◭ and ∗ ◮ denote dual actions in the equation (185). Finally, using equation (82), we arrive b ∗ ( l,k ) µ V ∗ = ( µ V ∗ ∗ ⇂ l + µ V ∗ ∗ x k ) ⊕ ( µ W ∗ ∗ ⇃ k − µ W ∗ ∗ y l ) . (192)Therefore, according to the equations (189), (190), (191),(192) the (plus/minus) Lie-Poisson equations (governed37y Hamiltonian function H = H (( µ V ∗ , µ W ∗ ) , ( ν l ∗ , ν k ∗ )) is computed as˙ µ V ∗ = µ V ∗ ∗ ⇂ δ H δν l ∗ + µ V ∗ ∗ x δ H δν k ∗ , ˙ µ W ∗ = − δ H δν l ∗ ∗ y µ W ∗ + µ W ∗ ∗ ⇃ δ H δν k ∗ , ˙ ν l ∗ = ϕ ∗ l µ V ∗ + ǫ ∗ k µ V ∗ − ι ∗ k µ W ∗ + ad ∗ δ H δν l ∗ ν l ∗ + ν l ∗ ∗ ◭ δ H δν k ∗ + ν k ∗ ∗ ◮ δ H δν k ∗ + µ V ∗ ∗ ⇂ l + µ V ∗ ∗ x k, ˙ ν k ∗ = φ ∗ k µ W ∗ + ι ∗ k µ W ∗ − ǫ ∗ k µ V ∗ + ad ∗ δ H δν k ∗ ν k ∗ − ν l ∗ ∗ ◭ δ H δν l ∗ − ν k ∗ ∗ ◮ δ H δν l ∗ + µ W ∗ ∗ ⇃ k − µ W ∗ ∗ y l. (193) Particular case: Now, we examine how Lie-Poisson equations look like for the particular case we gave in (6.1). Itis possible to briefly remind: we took the left actions ⇃ and ⇂ in (155), the mutual actions of k and l in (158) and thecross representations in (159) as trivial. Since the calculations of these choices are made in the previous section, itis possible to see the effects directly for Lie-Poisson equations (193). See that, both ˙ µ V ∗ and ˙ µ W ∗ are zero and˙ ν l ∗ = ϕ ∗ l µ V ∗ + ǫ ∗ k µ V ∗ − ι ∗ k µ W ∗ + ad ∗ δ H δν l ∗ ν l ∗ ˙ ν k ∗ = φ ∗ k µ W ∗ + ι ∗ k µ W ∗ − ǫ ∗ k µ V ∗ + ad ∗ δ H δν k ∗ ν k ∗ . (194) In the present section, we consider two Lie algebras g and h under mutual actions (65) satisfying the compatibilityconditions in (65). So that, we have a well-define matched pair Lie algebra structure g ⊲⊳ h equipped with thematched pair Lie bracket (67). As explained in Subsection 4.2, on the dual space g ∗ ⊕ h ∗ , there exists matchedLie-Poisson bracket {• , •} ⊲⊳ displayed in (102). This Poisson structure lets us to arrive at matched pair Lie-Poisson equations (103) governing the collective motion of the individual Lie-Poisson dynamics on g ∗ and h ∗ . Inall the subsection, we follow this construction. After presenting discussions on coupling of Rayleigh dissipation inthe following subsection, we shall follow the order given in Subsection 2.3, and examine the coupling problem ofsymmetric brackets. In (15), Rayleigh type dissipation is introduced to the Lie-Poisson system by means the coadjoint action and alinear operator, from the Lie algebra the dual space to the algebra. In this subsection, we provide a way to coupletwo Lie-Poisson dynamics admitting Rayleigh type dissipative terms. For this, we first determine the dynamics ofconstitutive systems. Assume that, on the dual space g ∗ , for the Lie-Poisson dynamics, Rayleigh type dissipationis provided by a linear operator Υ g : g ∗ g that is˙ µ ∓ ad ∗ δ H δµ µ = ∓ ad ∗ Υ g ( µ ) µ. (195)Assume also that on h ∗ , for the Lie-Poisson dynamics, Rayleigh type dissipation is provided by a linear operatorΥ h : h ∗ h . so that, ˙ ν ∓ ad ∗ δ H δν ν = ∓ ad ∗ Υ h ( ν ) ν. (196)38o couple the dynamics in (195) and (196), we introduce a linear operator from the dual space g ∗ ⊕ h ∗ to thematched pair Lie algebra g ⊲⊳ h given by g ∗ ⊕ h ∗ −→ g ⊕ h , ( µ ⊕ ν ) (Υ g ( µ ) ⊕ Υ h ( ν )) , (197)where Υ g and Υ h are the linear mappings, in (195) and (196), generating the dissipation for the individual systems.The dissipative term generated by the mapping Λ is computed to be ∓ ad ∗ Υ g ( µ ) ⊕ Υ h ( ν ) ( µ ⊕ ν ) = (cid:0) ∓ ad ∗ Υ g ( µ ) µ ± µ ∗ ⊳ Υ h ( ν ) ∓ a ∗ Υ h ( ν ) ν (cid:1) ⊕ (cid:0) ∓ ad ∗ Υ h ( ν ) ν ± Υ g ( µ ) ∗ ⊲ ν ∓ b ∗ Υ g ( µ ) µ (cid:1) , (198)where the dual actions ∗ ⊳ and ∗ ⊲ are those given in (80) and (83), respectively. Notice that, the cross actions a ∗ and b ∗ are the ones in (85) and (82), respectively. dissipation term will be obtained as above. Observe that, whilecoupling the dissipative terms in (198), we respect the mutual actions. So that, the collective dissipative term ismanifesting the mutual actions. It reduces to the direct sum of the dissipative terms of the individual motions ifthe actions are trivial.Obeying the general construction in (15), we merge the dissipative terms in (198) with the matched pair Lie-Poissonequations (103). This reads the coupled system˙ µ ∓ ad ∗ δ H /δµ µ ± δ H δµ ∗ ⊳ ν ∓ a ∗ δ H /δν ν = ∓ ad ∗ Υ g ( µ ) µ ± µ ∗ ⊳ Υ h ( ν ) ∓ a ∗ Υ h ( ν ) ν ˙ ν ∓ ad ∗ δ H /δν ν ± δ H δν ∗ ⊲ ∓ b ∗ δ H /δµ µ = ∓ ad ∗ Υ h ( ν ) ν ± Υ g ( µ ) ∗ ⊲ ν ∓ b ∗ Υ g ( µ ) µ. (199)It is evident that, this formulation respects the mutual actions. By taking one these actions, one arrives at thesemidirect product theory for the Lie-Poisson system with Rayleigh type dissipation. If both of the actions aretrivial, it is immediate to see that the system (199) turns out to be simple collection of the individual motions in(195) and (196). Recall that, in (22), we have presented double bracket in terms of the structure constants of Lie algebra. So that,to have a symmetric bracket on the matched Lie-Poisson geometry, we first recall the structure constants of thematched pair Lie algebra given in (75). Then referring to the coordinate realization of matched pair Lie-Poissonbracket, in (98), we compute associated Poisson bivector Λ asΛ αβ = ± C γβα µ γ , Λ αb = ∓ R dbα ν d ∓ L γbα µ γ , Λ aβ = ± R dβa ν d ± L γβa µ γ , Λ ab = ± D dab ν d , (200)where C γβα ’s are structure constants on g , D dab ’s are structure constants on h . Here, R dbα ’s and L γbα ’s are constantsdefining the right and the left actions according to the exhibitions in (61), respectively. In accordance with thiscoordinate realizations and in the light of the definition (22), matched Double bracket dissipation ( F , S ) ( mD ) fortwo functions F and S defined on g ∗ ⊕ h ∗ is( F , S ) ( mD ) ( µ, ν ) =( X b Λ αb Λ βb + X γ Λ αγ Λ βγ ) ∂ F ∂µ α ∂ S ∂µ β + ( X b Λ ab Λ αb + X γ Λ aγ Λ αγ ) ∂ F ∂ν a ∂ S ∂µ α + ( X a Λ ba Λ ca + X β Λ bβ Λ cβ ) ∂ F ∂ν b ∂ S ∂ν c + ( X β Λ αβ Λ bβ + X a Λ αa Λ ba ) ∂ F ∂µ α ∂ S ∂ν b . (201)39eferring to this bracket, the dissipative dynamics (20) for a = 1, generated by a functional S , is computed to be˙ µ β = ( X b Λ βb Λ αb + X γ Λ βγ Λ αγ ) ∂ S ∂µ α + ( X α Λ βα Λ aα + X b Λ βb Λ ab ) ∂ S ∂ν a ˙ ν d = ( X b Λ db Λ αb + X γ Λ dγ Λ αγ ) ∂ S ∂µ α + ( X a Λ da Λ na + X α Λ dα Λ nα ) ∂ S ∂ν n . (202)In order to arrive at the explicit expression of the symmetric bracket (201) and the disspative dynamics in (202)in terms of the local characterizations of left and the right actions and the structure constants of the constitutivesubalgebras, one needs to substitute the calculations (200) into (201) and (202).Now, we add the matched Lie-Poisson bracket {• , •} ⊲⊳ , given in (104), and the matched double bracket ( • , • ) ( mD ) in(201). This reads the matched metriplectic bracket. The matched metriplectic dynamics generated by a Hamiltonianfunction H and an entropy type function S is computed to be˙ z = [ | z , H| ] ⊲⊳,D = { z , H} ⊲⊳ + a ( z, S ) ( mD ) (203)In order to arrive at the explicit expression of the equations of motion in (203), it is enough to add the reversiblematched pair dynamics in (105) and the irreversible matched pair dynamics in (202). By taking one of the actionstrivial, one arrives semidirect product metriplectic bracket and semidirect product dynamical equation. If, both ofthe actions are trivial, then the coupling turns out to be a simple addition. Once more, we recall the structure constants (75) of the matched pair Lie algebra. Referring to (26), we firstcompute the matched pair Cartan metric as ¯ G aβ and ¯ G αb ¯ G αb = − R daα D abd + L βaα R abβ + C γαβ L βbγ , ¯ G ab = L βaα L αbβ + D kad D dbk ¯ G aβ = L γaǫ C ǫβγ − D dab R bdβ + R daγ L γdβ , ¯ G αβ = R baα R abβ + C ǫαγ C γβǫ (204)on the matched pair Lie algebra g ⊕ h . We write an element of g ∗ ⊕ h ∗ as ( µ, ν ) = µ α ¯ e α + ν a ¯ e a . We compute thematched pair Cartan-Killing bracket, defined in (27), as( F , H ) ( mCK ) = ∂ F ∂µ α ¯ G αβ ∂ H ∂µ β + ∂ F ∂µ α ¯ G αb ∂ H ∂ν b + ∂ F ∂ν a ¯ G aβ ∂ H ∂µ β + ∂ F ∂ν a ¯ G ab ∂ H ∂ν b =( R bαa R aβb + C ǫαγ C γβǫ ) ∂ F ∂µ α ∂ H ∂µ β + ( − R daα D abd + L βaα R abβ + C γαβ L βγb ) ∂ F ∂µ α ∂ H ∂ν b + ( R dγa L γdβ + L γαa C αβγ − D dab R bdβ ) ∂ F ∂ν a ∂ H ∂µ β + ( L βαa L αβb + D kad D dbk ) ∂ F ∂ν a ∂ H ∂ν b . (205)According to this formulation (205), for a functional S the equation of motion will be˙ µ β = ¯ G βa ∂S∂ν a + ¯ G βα ∂S∂µ α , ˙ ν d = ¯ G dβ ∂S∂µ β + ¯ G da ∂S∂ν a . (206)40e substitute the explicit represetations of the metric (204) into the system (206) and arrive at that˙ µ β = ( − R daβ D abd + L αaβ R abα + C γβα L αγb ) ∂S∂ν b + ( R bαa R aβb + C ǫαγ C γβǫ ) ∂S∂µ β , ˙ ν d = ( R aγd L γaβ + L γαd C αβγ − D adb R baβ ) ∂S∂µ β + ( L βαa L αβd + D kab D bdk ) ∂S∂ν a . (207) Recall the Casimir dissipation bracket given in (30). In order to carry this discussion to coupled systems on g ∗ × h ∗ ,as it can deduced from that equation, we first need to determine a real valued bilinear operator on the matchedpair Lie algebra g ⊲⊳ h and then employ the pairing equipped with a Casimir function(al). Let us first determinedissipations individually on g ∗ and h ∗ then couple them.Consider symmetric bilinear operators on g ve h , denoted by ψ and ϑ , respectively. Assume that C is a Casimirfunction g ∗ and D is a Casimir function on h ∗ . Then, Casimir dissipation brackets are( F , H ) ( CD ) g ∗ ( µ ) = − ψ (cid:20) δ F δµ , δ H δµ (cid:21) g , (cid:20) δ C δµ , δ H δµ (cid:21) g ! ( F , H ) ( CD ) h ∗ ( ν ) = − ϑ (cid:20) δ F δν , δ H δν (cid:21) h , (cid:20) δ D δν , δ H δν (cid:21) h ! . (208)Define a real valued symmetric bilinear operator on g ⊕ h , using ψ and ϑ , as follows( ψ, ϑ ) : ( g ⊲⊳ h ) × ( g ⊲⊳ h ) −→ R , ( ξ ⊕ η, ξ ′ ⊕ η ′ ) ψ ( ξ, ξ ′ ) + ϑ ( η, η ′ ) . (209)In terms of the Casimir functions C and D on g ∗ and h ∗ , respectively, we define a Casimir function ( C , D ) on g ∗ × h ∗ ,for example, as follows ( C , D )( µ, ν ) = C ( µ ) + D ( ν ) . (210)So that, matched Casimir dissipation bracket is defined to be( F , H ) ( mCD ) ( µ ⊕ ν ) = − ( ψ, ϑ ) (cid:18)(cid:20)(cid:18) δ F δµ ⊕ δ F δν (cid:19) , (cid:18) δ H δµ ⊕ δ H δν (cid:19)(cid:21) ⊲⊳ , (cid:20)(cid:18) δ C δµ ⊕ δ D δν (cid:19) , (cid:18) δ H δµ ⊕ δ H δν (cid:19)(cid:21) ⊲⊳ (cid:19) (211)where [ • , • ] ⊲⊳ is the matched Lie algebra bracket in (67). Referring to this bracket, the dissipative dynamics (20)for a = 1, generated by a functional H , is a system of equations. The dynamics on g ∗ is˙ µ = − ad ∗ δ H δµ (cid:2) δ C δµ , δ H δµ (cid:3) ♭ − ad ∗ δ H δµ (cid:0) δ D δν ⊲ δ H δµ (cid:1) ♭ + ad ∗ δ H δµ (cid:0) δ H δν ⊲ δ C δµ (cid:1) ♭ − (cid:2) δ C δµ , δ H δµ (cid:3) ♭ ∗ ⊳ δ H δν − (cid:2) δ D δν ⊲ δ H δµ (cid:3) ♭ ∗ ⊳ δ H δν + (cid:2) δ H δν ⊲ δ C δµ (cid:3) ♭ ∗ ⊳ δ H δν − a ∗ δ H δν (cid:2) δ D δν , δ H δν (cid:3) ♭ − a ∗ δ H δν (cid:0) δ D δν ⊳ δ H δµ (cid:1) ♭ − a ∗ δ H δν (cid:0) δ H δν ⊳ δ C δµ (cid:1) ♭ (212)whereas the dynamics on h ∗ is˙ ν = − ad ∗ δ H δν (cid:2) δ D δν , δ H δν (cid:3) ♭ − ad ∗ δ H δν (cid:0) δ D δν ⊳ δ H δµ (cid:1) ♭ + ad ∗ δ H δν (cid:0) δ H δν ⊳ δ C δµ (cid:1) ♭ − δ H δµ ∗ ⊲ [ δ D δν , δ H δν ] ♭ − δ H δµ ∗ ⊲ (cid:2) δ D δν ⊳ δ H δµ (cid:3) ♭ + δ H δµ ∗ ⊲ (cid:2) δ H δν ⊳ δ C δµ (cid:3) ♭ − b ∗ δ H δµ (cid:2) δ C δµ , δ H δµ (cid:3) ♭ − b ∗ δ H δµ (cid:0) δ D δν ⊲ δ H δµ (cid:1) ♭ + b ∗ δ H δµ (cid:0) δ H δν ⊲ δ C δµ (cid:1) ♭ . (213)41ere, the right action ⊲ and the left action ⊳ are those in (65) whereas ∗ ⊲ and ∗ ⊳ are the dual actions in (80)and (83), respectively. Notice that dual operators a ∗ is in (85) , and b ∗ is in (82). Here, superscript ♭ denotes thedualization obtained through for the symmetric operators ψ and ϑ given by h ξ ♭ , ξ ′ i = ψ ( ξ, ξ ′ ) , h η ♭ , η ′ i = ϑ ( η, η ′ ) . (214)In order to prevent to notation inflation, we denote these two mappings by the same notation as we did whiledenoting the Lie algebra brackets on g and h .We can couple the matched irreversible motion, that is matched Casimir dissipation motion, in (212) and (213)with matched reversible motion, that is matched Lie-Poisson dynamics, in (103). This results with matched pair ofmetriplectic system involving Casimir dissipation terms, and simple achieved by adding the right hand sides of thesystems obeying the order. this collective motion, can be determined by a single matched metriplectic bracket[ |F , H| ] ⊲⊳,CD = {F , H} ⊲⊳ + a ( F , H ) ( mCD ) , where {• , •} ⊲⊳ is the matched Lie-Poisson bracket in (102) and ( • , • ) ( mCD ) is the matched Casimir dissipationbracket in (211). In this case, the dynamics governed by a Hamiltonian function(al) H , is implicitly written by( ˙ µ ⊕ ˙ ν ) = [ | ( µ ⊕ ν ) , H| ] ⊲⊳,CD . At first recall the Hamilton dissipation bracket given in (35) and the pure irreversible motion in (37). In thissubsection, we couple (match) two Hamilton dissipation bracket in form (35) and two pure irreversible motion in(37). Accordingly, obeying the notation presenting in the presvious subsection we introduce the following Hamiltondissipations brackets on the constitutive spaces g ∗ and h ∗ , for two bilinear operators ψ and ϑ , as follows( F , H ) HD g ( µ ) = − ψ (cid:20) δ F δµ , δ C δµ (cid:21) g , (cid:20) δ H δµ , δ C δµ (cid:21) g ! ( F , H ) HD h ( ν ) = − ϑ (cid:20) δ F δν , δ D δν (cid:21) h , (cid:20) δ H δν , δ D δν (cid:21) h ! . (215)where C and D are Casimir functions on g ∗ and h ∗ , respectively. In order to match these symmetric brackets, recallthe real valued bilinear map (209) defined on matched pair Lie algebra g ⊲⊳ h . Then, introduce matched Hamiltondissipation bracket( F , H ) ( mHD ) ( µ ⊕ ν ) = − ( ψ, ϑ ) (cid:18)(cid:20)(cid:18) δ F δµ ⊕ δ F δν (cid:19) , (cid:18) δ C δµ ⊕ δ D δν (cid:19)(cid:21) , (cid:20)(cid:18) δ H δµ ⊕ δ H δν (cid:19) , (cid:18) δ C δµ ⊕ δ D δν (cid:19)(cid:21)(cid:19) , (216)42here the brackets inside the paring are the matched Lie bracket in (67). Irreversible the dynamics on g ∗ × h ∗ canbe obtained as˙ µ = − ad ∗ δ C δµ (cid:20) δ H δµ , δ C δµ (cid:21) ♭ − ad ∗ δ C δµ (cid:18) δ H δν ⊲ δ C δµ (cid:19) ♭ + ad ∗ δ C δµ (cid:18) δ D δν ⊲ δ H δµ (cid:19) ♭ − [ δ H δµ , δ C δµ ] ♭ ∗ ⊳ δ D δν − [ δ H δν ⊲ δ C δµ ] ♭ ∗ ⊳ δ D δν + [ δ D δν ⊲ δ H δµ ] ♭ ∗ ⊳ δ D δν − a ∗ δ D δν [ δ H δν , δ D δν ] ♭ − a ∗ δ D δν (cid:18) δ H δν ⊳ δ C δµ (cid:19) ♭ − a ∗ δ D δν (cid:18) δ D δν ⊳ δ H δµ (cid:19) ♭ ˙ ν = − ad ∗ δ D δν [ δ H δν , δ D δν ] ♭ − ad ∗ δ D δν (cid:18) δ H δν ⊳ δ C δµ (cid:19) ♭ + ad ∗ δ D δν (cid:18) δ D δν ⊳ δ H δµ (cid:19) ♭ − δ C δµ ∗ ⊲ [ δ H δν , δ D δν ] ♭ − δ C δµ ∗ ⊲ [ δ H δν ⊳ δ C δµ ] ♭ + δ C δµ ∗ ⊲ [ δ D δν ⊳ δ H δµ ] ♭ − b ∗ δ C δµ [ δ H δµ , δ C δµ ] ♭ − b ∗ δ C δµ (cid:18) δ H δν ⊲ δ C δµ (cid:19) ♭ + b ∗ δ C δµ (cid:18) δ D δν ⊲ δ H δµ (cid:19) ♭ . (217)Here, the right action ⊲ and the left action ⊳ are those in (65) whereas ∗ ⊲ and ∗ ⊳ are the dual actions in (80) and(83), respectively and a , a ∗ defined as in (84), (85) and b , b ∗ defined as in (81), (82). We start with a 3 dimensional Heisenberg algebra which we denote by g , see [64]. We assume a basis { e , e , e } for g , and define the Lie algebra operation as[ e , e ] = 0 , [ e , e ] = e , [ e , e ] = 0 . (218)These read that the structure constants as C = − C = 1 while the rest is zero. Heisenberg algebra can be writtenas a 2-cocycle extension Lie algebra. So that we can examine it in the realms of the discussions done in Subsection3.3. To see this, referring to the basis of the algebra g , we define two linear spaces V = h e i and l = h e , e i .Accordingly, we introduce a V -valued skew-symmetric bilinear mapping on l as ϕ : l × l −→ V, ϕ ( e , e ) = 0 , ϕ ( e , e ) = 0 , ϕ ( e , e ) = e . (219)It is straightforward to verify that ϕ is a 2-cocycle. Taking the left action of l on V (see the first action in the list(155)) is zero, we arrive easily that the bracket (218) is indeed in a 2-cocycle extension form (79). Coadjoint flow. The dual space is denoted by g ∗ with the dual basis { e , e , e } . Assume the following coordinates ξ = ( ξ , ξ , ξ ) in g , and µ = ( µ , µ , µ ) in g ∗ . Then, the coadjoint action of a Lie algebra element ξ in g to a dualelement µ in g ∗ is computed to be ad ∗ : g × g ∗ −→ g ∗ , ad ∗ ξ µ = ( µ ξ , − µ ξ , . (220)Referring to this calculation, we write the Lie-Poisson dynamics (14) generated by a Hamiltonian function H asfollows ˙ µ = µ ∂ H ∂µ , ˙ µ = − µ ∂ H ∂µ ˙ µ = 0 . (221)Here, the latter gives that µ is a constant. In equation (221), if we choose µ = q , µ = p , and µ = 1, then we43rrive at the Hamilton’s equations in its very classical form˙ q = ∂ H ∂p , ˙ p = − ∂ H ∂q . (222)Therefore we claim that, in the present geometry, the Hamiltonian dynamics can be realized as a coadjoint flow. Double Bracket dissipation. For the present case, the symmetric double bracket in (22) is computed to be( F , S ) ( D ) ( µ ) = µ (cid:0) ∂ S ∂µ ∂ F ∂µ + ∂ S ∂µ ∂ F ∂µ (cid:1) . (223)Therefore for a function S , the irreversible dynamics due to the symmetric bracket is computed to be˙ µ = µ ∂ S ∂µ , ˙ µ = µ ∂ S ∂µ , ˙ µ = 0 . (224)Then the metriplectic equations of motion (25) are computed to be˙ µ = µ ∂ H ∂µ + µ ∂ S ∂µ , ˙ µ = − µ ∂ H ∂µ + µ ∂ S ∂µ , ˙ µ = 0 . (225)If we choose µ = q , µ = p , and µ = 1, then the metriplectic dynamics (225) turns out to be˙ q = ∂ H ∂p + ∂ S δq , ˙ p = − ∂ H ∂q + ∂ S ∂p . (226)Two interesting oarticular instances of the present dynamics: (1) Let us take the Hamiltonian function H = p + V ( q ) to be the total energy of the system and S = S ( q ) thenthe system (226) reduces to ¨ q − S qq ˙ q − V q = 0 . (227)We cite [67] for a more elegant geometrization of the second order ODE (227) in terms of the GENERIC framework. (2) As another naive application of the dissipative system (226), we consider a general Hamiltonian function H butchoose S = ap / a , then a fairly straight-forward calculation gives (226) that˙ q = ∂ H ∂p , ˙ p = − ap + ∂ H ∂q . (228)This is the conformal Hamiltonian dynamics as described in [65]. To see the geometry behind this dynamics, atfirst consider the vector field X = ( ∂ H /∂p ) ∂ q + ( − ap + ∂ H /∂q ) ∂ p generating (228) and then define the symplectictwo-form Ω = dq ∧ dp . A Hamiltonin vector field preserves the symplectic two-forms but the vector field X satisfies L X Ω = d (cid:0) d H − apdq (cid:1) = adq ∧ dp = a Ω (229)where L denotes the Lie derivative. This reads that X preserved the symplectic two-form up to some conformalfactor a . 44 .2 Coupling of Two Heisenberg Algebras and Matched Lie-Poisson Dynamics Consider two 3 D Heisenberg algebras g and h . We choose a basis { e , e , e } ∈ g , { f , f , f } ∈ h . (230)To fix the notation, we record here the coordinate realizations of two arbitrary elements ξ and ξ ′ in g , and twoarbitrary elements η and η ′ in h as follows ξ = ξ e + ξ e + ξ e , ξ ′ = ξ ′ e + ξ ′ e + ξ ′ e ,η = η f + η f + η f , η ′ = η ′ f + η ′ f + η ′ f . (231)In accordance with these choices, the Lie algebra brackets on g and h can be exhibited in the form[ ξ, ξ ′ ] = ( ξ ξ ′ − ξ ′ ξ ) e , [ η, η ′ ] = ( η η ′ − η ′ η ) f (232)respectively. Obeying the notation in (42), and referring to the coordinate realizations (231), we introduce a rightaction of g on h , and a left action h on g as [64] ⊲ : h ⊗ g → g , η ⊲ ξ = − η ξ e , ⊳ : h ⊗ g → g , η ⊳ ξ = − ξ η f . (233)This reads that the action constants R = − L = − g ⊲⊳ h is computed to be[( ξ, η ) , ( ξ ′ , η ′ )] ⊲⊳ = ( ξ ξ ′ − ξ ξ ′ − η ξ ′ + η ′ ξ ) e ⊕ ( η η ′ − η η ′ − η ξ ′ + η ′ ξ ) f . (234) As coupling of two cocycle extensions. Referring to the basis (230), Heisenberg algebras g and h can berealized as 2-cocycle extensions V ϕ ⋊ l := h e i ϕ ⋊ h e , e i , W φ ⋊ k := h f i φ ⋊ h f , f i , (235)in the light of the following 2-cocycles ϕ : l × l −→ V, ϕ ( e , e ) = 0 , ϕ ( e , e ) = 0 , ϕ ( e , e ) = e ,φ : k × k −→ W, φ ( f , f ) = 0 , φ ( f , f ) = 0 , φ ( f , f ) = f (236)while the left actions in (155) are zero. We now examine Proposition 8 for the matched pair of two Heisenbergalgebras. That is, we show that g ⊲⊳ h is a 2-cocycle extension by itself. For this, following the notation in Subsection6.1, we take that both (1) mutual actions ◭ and ◮ of l and k on each other exhibited in (158), (2) cross actions x and y in (159) are all zero mappings whereas (3) mappings ǫ and ι given in (160) are computed to be ǫ ( f , e ) = − e ι ( f , e ) = − f , (237)while the rest are zero. Referring to these choices, it is now straight forward to observe that both the left action ⊲ and the right action ⊳ of the Heisenberg algebras given in the display (233) can be recasted in the form of (161).45his reads that the matched Lie algebra bracket (234) on Heisenberg algebras is indeed fitting (164).On the other, now we show that the matched pair g ⊲⊳ h is a 2-cocycle extension of l ⊲⊳ k = h e , e i ⊲⊳ h f , f i over its representation space V ⊕ W := h e i ⊕ h f i according to the decomposition in (235). We remark here that,the mutual actions in l ⊲⊳ k are trivial since we take that ◭ and ◮ of l and k as zero mappings. Nevertheless, wedetermine the left action · ⊲ of the Lie algebra l ⊲⊳ k onto V ⊕ W as trivial. We introduce a cocycleΘ : ( l ⊲⊳ k ) × ( l ⊲⊳ k ) −→ V ⊕ W : (cid:0) h e , e i ⊲⊳ h f , f i (cid:1) × (cid:0) h e , e i ⊲⊳ h f , f i (cid:1) −→ h e i ⊕ h f i , (238)which, referring to the formula in (169), can be exploited asΘ( ξ e + ξ e + η f + η f , ξ ′ e + ξ ′ e + η ′ f + η ′ f ) = ( ξ ξ ′ − ξ ′ ξ − η ξ ′ + η ′ ξ ) e ⊕ ( η η ′ − η η ′ − η ξ ′ + η ′ ξ ) f . (239)So, the trivial action · ⊲ of l ⊲⊳ k on V ⊕ W with the cocycle (239), it is immediate to see the matched Lie algebrabracket (234) is also a 2-cocycle extension bracket by obeying (79). To sum up, we can argue that coupling of twoHeisenberg algebras is a nontrivial example of Proposition 8. Matched Lie-Poisson equations. Let us first fix the notation for the dual elements as µ = µ e + µ e + µ e ∈ g ∗ , ν = ν f + ν f + ν f ∈ h ∗ . (240)The dual actions ∗ ⊳ in (83) and ∗ ⊲ in ( 80), the cross actions a ∗ in (85) and b ∗ in (82) are computed to be µ ∗ ⊳ η = − µ η e , ξ ∗ ⊲ ν = − ν ξ f , b ∗ ξ µ = − µ ξ f , a ∗ η ν = − ν η e . (241)Then the matched Lie-Poisson equations (103) generated by a Hamiltonian function H = H ( µ, ν ) on g ∗ ⊕ h ∗ iscomputed to be ˙ µ = − ν ∂ H ∂ν + µ ∂ H ∂µ , ˙ µ = µ ∂ H ∂ν − µ ∂ H ∂µ , ˙ µ = 0˙ ν = − µ ∂ H ∂µ + ν ∂ H ∂ν , ˙ ν = − ν ∂ H ∂ν + ν ∂ H ∂µ , ˙ ν = 0 . (242)Let us study some particular instances on this system of equations. In equation (242), if we choose µ = q , µ = p , ν = u , ν = w , and µ = ν = 1 then we arrive at the Hamilton’s equations in a coupled form˙ q = ∂ H ∂p − ∂ H ∂w , ˙ p = ∂ H ∂u − ∂ H ∂q , ˙ u = − ∂ H ∂p + ∂ H ∂w , ˙ w = − ∂ H ∂u + ∂ H ∂q . (243) Rayleigh type dissipation. For the present case, in order to add Rayleigh type dissipation term to the Lie-Poissondynamics on g ∗ ⊕ h ∗ as g ∗ ⊕ h ∗ −→ g ⊲⊳ h , µ ⊕ ν (Υ g e + Υ g e + Υ g e , Υ h f + Υ h f + Υ h f ) . (244)Referring to the matched Lie-Poisson equation of motions (199) for the dual spaces g ∗ ⊕ h ∗ can be obtained as46ollows˙ µ − µ ∂ H ∂µ + ν ∂ H ∂ν = ν Υ h − µ Υ g , ˙ µ − µ ∂ H ∂ν + µ ∂ H ∂µ = µ Υ g − µ Υ h , ˙ µ = 0 , ˙ ν + µ ∂ H ∂µ − ν ∂ H ∂ν = µ Υ g − ν Υ h , ˙ ν + ν ∂ H ∂ν − ν ∂ H ∂µ = ν Υ h − ν Υ g , ˙ ν = 0 . (245) Double Bracket dissipation. For the present discussion, matched double bracket (201) takes the particular form( F , H ) ( mD ) ( µ ⊕ ν ) = µ ∂ F ∂µ ∂ H ∂µ + (2 µ + 2 ν µ + ν ) ∂ F ∂µ ∂ H ∂µ − ( µ + ν ) ∂ F ∂ν ∂ H ∂µ + (2 ν + 2 µ ν + µ ) ∂ F ∂ν ∂ H ∂ν − ( µ + ν ) ∂ F ∂µ ∂ H ∂ν + ν ∂ F ∂ν ∂ H ∂ν . (246)Then the irreversible dynamics (202) is computed to be˙ µ = ∂ S ∂µ µ , ˙ µ = ∂ S ∂µ µ , ˙ µ = 0˙ ν = ∂ S ∂ν ν , ˙ ν = ∂ S ∂ν ν , ˙ ν = 0 . (247)Let us collect the reversible Lie-Poisson dynamics in (242) and the irreversible dynamics generated by S under therealm of the symmetric (double) bracket in (246). Then the metriplectic equations of motion are computed to be˙ µ = µ ∂ H ∂µ − ν ∂ H ∂ν + ∂ S ∂µ µ ˙ µ = µ ∂ H ∂ν − µ ∂ H ∂µ + ∂ S ∂µ µ , ˙ ν = − µ ∂ H ∂µ − ν ∂ H ∂ν + ∂ S ∂ν ν , ˙ ν = ν ∂ H ∂ν + ν ∂ H ∂µ + ∂ S ∂ν ν . (248)After the choice µ = q , µ = p , ν = u , ν = w , and µ = ν = 1 read the following system˙ q = ∂ H ∂p − ∂ H ∂w + ∂ S ∂q , ˙ p = ∂ H ∂u − ∂ H ∂q + ∂ S ∂p ˙ u = − ∂ H ∂p − ∂ H ∂w + ∂ S ∂u , ˙ w = ∂ H ∂u + ∂ H ∂q + ∂ S ∂w . (249)Let us more particularly take the Hamiltonian function H = 1 / p + w ) + V ( q, u ) to be the total energy of thesystem then from (249) we obtain ¨ q − S qq ˙ q + 2 V q + S w − S p = 0¨ u − S uu ˙ u + 2 V u + S w + S p = 0 (250) We consider two identical three dimensional Euclidean spaces denoted by g = R and h = R k . These are Liealgebras equipped with the following Lie algebra structures[ ξ, ξ ′ ] R = ξ × ξ ′ , [ η, η ′ ] R k = k × ( η × η ′ ) , (251)respectively, [63]. Here, × denotes cross product on R , and k is the unit vector (0 , , 1) available in the standardbasis on R . Notice that, the subscript R k is just to remind that the bracket is not classical cross product on R g on it dual g ∗ ≃ R , and the coadjoint action of h on it dual h ∗ ≃ R are computed to be ad ∗ ξ µ = µ × ξ, ad ∗ η ν = ( ν · η ) k − ν ( k · η ) . (252)In order to employ the notation in (73), we introduce the following basis ( e , e , e ) on R and ( f , f , f ) on R k e = f = (1 , , , e = f = (0 , , , e = f = k = (0 , , . (253)So that, the strcuture constants in (73) turn out to be C = C = C = 1 , D = D = 1 , (254)where the rest is zero. Mutual actions. Left action of R k on R , and right action R on R k are defined through ⊳ : R k × R → R k , η ⊳ ξ := η × ξ ⊲ : R k × R → R , η ⊲ ξ := η × ( ξ × k ) . (255)It is straight forward to prove that these actions are indeed satisfying the matched pair Lie algebra conditions in(68). Referring to the notations fixed in (61), the constants determining the actions (255), can be computed to be L = L = − , L = L = 1 , R = R = R = − , R = R = R = 1 , (256)with the rest is zero. According to Theorem 4, one can define the matched pair Lie algebra R ⊲⊳ R k equipped withthe matched Lie bracket (67) calculated as[ ξ ⊕ η, ξ ′ ⊕ η ′ ] ⊲⊳ = (cid:0) ξ × ξ ′ + η × ( ξ ′ × k ) − η ′ × ( ξ × k ) (cid:1) ⊕ (cid:0) k × ( η × η ′ ) + η × ξ ′ − η ′ × ξ (cid:1) . (257) Matched Lie-Poisson dynamics. The Lie-Poisson dynamics on the dual space g ∗ × h ∗ ≃ R × R can be obtainedby employing Proposition 7. To have this, first we compute the duals of the actions (255), in a respective order, as µ ∗ ⊳ η = µ ( η · k ) − ( µ · k ) η, ξ ∗ ⊲ ν = ξ × ν (258)whereas the mappings (85) and (82) are a ∗ η ν = ν × η, b ∗ ξ µ = ( µ · ξ ) k − ( µ · k ) ξ. (259)Notice that, the Lie-Poisson formulation on the dual of g ∗ corresponds to the the rigid body dynamics in 3 D , [39,57].So we can consider the matched pair dynamics in this setting as the coupling of two bodies in 3 D . Since the dynamicsfor the rigid bodies are given by minus Lie-Poisson equation, we refer minus Lie-Poisson bracket and minus Lie-Poisson equations. Assume the following coordinates µ = ( µ , µ , µ ) and ν = ( ν , ν , ν ). Recall that, in (200),the explicit realization of the matched Lie-Poisson bivector, defined for the matched pair Lie-Poisson bracket (108),has already been given. In the following table, we exhibit the coefficients of the matched Lie-Poisson bivector forthe present case. 48 Λ αβ Λ αb Λ aβ Λ ab Λ µ − µ - µ - ν ν µ ν − µ − ν + µ ν Λ µ ν − ν µ − µ µ - ν − µ ν + µ ν Λ - µ - ν ν − ν Λ - µ ν − ν − ν Λ g ∗ whereas the last one is the Lie-Poissonbivector on h ∗ whereas the second and third columns are manifesting the mutual actions (255). Collecting all theseresults, we compute the matched Lie-Poisson equations (103) generated by a Hamiltonian function H on the dualspace as ˙ µ = ∂ H ∂µ × µ + ( ∂ H ∂ν · k ) µ − ( µ · k ) ∂ H ∂ν − ν × ∂ H ∂ν , ˙ ν = ( k · ∂ H ∂ν ) ν − ( ν · ∂ H ∂ν ) k + ∂ H ∂µ × ν + ( µ · k ) ∂ H ∂µ − ( µ · ∂ H ∂µ ) k . (260)In Section 7, couplings of various ways of dissipations are listed. We examine now these couplings for the concreteexample given in previous subsection. Rayleigh dissipation. In this 3 D framework, and for a linear operator( R ) ∗ ⊕ ( R k ) ∗ −→ R ⊲⊳ R k , µ ⊕ ν (Υ g ( µ ) ⊕ Υ h ( ν )) , (261)the matched Lie-Poisson systems with Rayleigh type dissipations, that is the system (199), takes the particularform˙ µ − ∂ H ∂µ × µ − µ ( ∂ H ∂ν · k ) + ( µ · k ) ∂ H ∂ν + ν × ∂ H ∂ν = µ × Υ g ( µ ) − µ (Υ h ( ν ) · k ) + ( µ · k )Υ h ( ν ) + ν × Υ h ( ν )˙ ν − ν ( k · ∂ H ∂ν ) + ( ν · ∂ H ∂ν ) k − ∂ H ∂µ × ν − ( µ · k ) ∂ H ∂µ + ( µ · ∂ H ∂µ ) k = ( ν · Υ h ( ν )) k − ν ( k · Υ h ( ν )) − Υ h ( ν ) × ν + ( µ · Υ g ( µ )) k − ( µ · k )Υ g ( µ ) . (262) Cartan-Killing dissipation. Determine the matched Cartan-Killing metric (204) as[ G ij ] = − − − − − − − . (263)49hen, irreversible dynamics generated by a function S , presented in (206), is computed to be˙ µ = − ∂ S ∂µ + 5 ∂ S ∂µ , ˙ µ = − ∂ S ∂µ − ∂ S ∂µ , ˙ µ = − ∂ S ∂µ , ˙ ν = − ∂ S ∂ν − ∂ S ∂ν , ˙ ν = 5 ∂ S ∂ν − ∂ S ∂ν , ˙ ν = ∂ S ∂ν . 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