On Kostant-Kirillov Symplectic Structure and Quasi-Poisson Structures of the Euler-Arnold Systems

The concept of symplectic structure emerged between 1808 to 1810 through the works of Lagrange and Poisson on the trajectory of the planets of the solar system. In order to explain the variation of the orbital parameters, they introduced the symplectic structure associated to the manifold describing the states of the system and a fundamental operation on functions called Poisson’s bracket. But, the latter also comes from the Hamiltonian formalism which does not automatically lead to a Poisson structure. Although contrary to the Riemannian case, not every manifold necessarily admits a symplectic structure including even dimensional manifolds. The aim of this paper is to show the interaction between the Kostant-Kirillov symplectic structure and quasi-Poisson structures coming from the Euler-Arnold systems. The Lie algebra theoretical approach based on the Kostant-Kirillov coadjoint action will allow us to obtain a class of the quasi-Poisson structures resulting from the characterization of the Hamiltonian system and to prove some results on the Kostant-Kirillov symplectic structure in the quasi-Poisson context.


Introduction
Introduced by Alekseev and Yvette Kosmann-Schwarzbach (Alekseev & al., 2000), the the quasi-Poisson structures appeared as a finite-dimensional alternative to infinite-dimensional constructions of Poisson structures on moduli spaces. These constructions have been proposed particularly in (Goldman , 1986), (Jeffrey & al., 1992) and (Huebschmann, 1995). However, examples of quasi-Poisson structures appear in the study of the equations motion in mechanics (see (Euler, 1765), (kowalewski, 1889) and (Appel'rot, 1894)). According to (Arnold, 1966), the Euler equations for a perfect fluid is related to the geodesic equations of a Lie group with an invariant metric. This is referred as the generalised Euler equations known as the Euler-Arnold equations. The Euler-Arnold systems thus govern the Hamiltonian dynamics on Lie groups. The prototype being the equation of motion of a rotating solid formulated by Euler in 1765 (Euler, 1765). The generalisation of this formalism to infinite dimension (groups of diffeomorphisms) was introduced in 1966 by Arnold. He showed that the equation of motion of perfect fluids can be reformulated as a geodesic flow over the group of diffeomorphisms. The general theory of Lie groups requires some properties of differentiable manifold. In this paper, we will focus on a rotation group in which the elements are orthogonal matrices with determinant 1. In the case of three-dimensional space, the rotation group is known as the special orthogonal group often denoted SO(3). The latter is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space and its representations are important in physics, where they give rise to the elementary particles of integer spin ( i.e. an intrinsic form of angular momentum carried by elementary particles). A Lie group is a group that is also a differentiable manifold. Named after Norwegian mathematician Sophus Lie (1842-1899), who laid the foundations of the theory of continuous transformation groups, Lie groups play an enormous role in modern geometry, on several different levels. It is natural to associate any Lie group G to Lie algebra. There are two equivalent ways of introducing this Lie algebra. First is to introduce a space of vector fields on G, the other is to provide the tangent space at the neutral element with a Lie bracket, derived from the local expression of the internal operation of G. In the following, G denoted subgroup of SO(3) and the corresponding Lie algebra is g = T 1 G consists of skew-symmeric 3 × 3 matrices where 1 is the neutral element of G and T 1 G is the tangent space at 1. Example of basis of g is given by (1) We can explicitly describe the subgroup G. Namely, the exponential map exp permits to define the rotation around x-axis by the angle θ. We call it g θ = exp(θe x ) := Similarly, e y , e z generate rotations around y, z axes. Now, we introduce the commutator of two elements of g defined by Note that g has a canonical structure of a Lie algebra with this commutator and we have [e x , e y ] = e z , [e x , e z ] = −e y and [e y , e z ] = e x .
Let I = {x, y, z} be an index set and let X = i∈I j i e i be an element of g. Let's consider the following map : We call the Euler-Arnold system the differential equation : where [−, −] is the commutator defined by (2). From the characterization of the Hamiltonian field of (4), there exists an antisymmetric matrix J X and a differentiable function H (called Hamiltonian) such thaṫ According to (Weinstein, 1983), there exists a bivector field π J X associate to J X . However, π J X is not always a Poisson bivector field. The Jacobi identity is obviously not satisfied. We will call it "quasi-Poisson structures of the Euler-Arnold systems". The question arises is : can we construct a symplectic structure on G?
Note that a symplectic structure or symplectic form on G is defined to be a differential 2-form ω on G that is closed and is non-degenerate. According to (Kirillov, 1976), the dual space g * of the corresponding Lie algebra g plays an important role in the the Kirillov-Kostant bracket which is always degenerate at the origin in g * . In this paper, we will establish the symplectic structure coincide with orbits of the coadjoint action of g * , by extending the results contained in (Lesfari, 2009). We show that the Kostant-Kirillov symplectic structure is given by with j, k ∈ C 3 where τ 1 = f ∧ j, τ 2 = f ∧ k and ∧ is the usual vector product. Some properties on the Kostant-Kirillov symplectic structure of G and quasi-Poisson structures of the Euler-Arnold systems are are described in section 2. the interaction between the Kostant-Kirillov symplectic structure and quasi-Poisson structures coming from the Euler-Arnold systems is detailed in the section 3 of this article.

Some Properties on the Kostant-Kirillov Symplectic Structure of the Lie Group G and Quasi-Poisson Structures of the Euler-Arnold Systems
In this section, we describe the Lie algebra theoretical approach based on the Kostant-Kirillov coadjoint action and we present a useful result on the Kostant-Kirillov symplectic structure in the quasi-Poisson context.
be an element of G, X belongs to g and t be a real number. We have : g θ exp(tX)g −1 θ = exp(tg θ Xg −1 θ ). Therefore, g θ Xg −1 θ in an element of g and considering the following automorphism is a Lie algebra, we have the following proprietes : Let X ∈ g, We call Aut(g) the group of automorphisms of g and adX the endomorphism of g defined by The Jacobi identity shows that ad(X) is a derivation and the space Der(g) of derivations of g is a Lie algebra for the commutator defined by (2). The application ad : g −→ Der(g) is a homomorphism of Lie algebras, The application ad : g −→ Der(g) ⊂ End(g) is a representation of the Lie algebra g called the adjoint representation.
It's a symmetric bilinear form on g which is associative, i.e. the ad(X) transformation is skew-symmetric with ·, · .
Definition 2.1. The bilinear form ·, · associated to the adjoint representation ad is called the Killing form.
To illustrate, let us take a few examples.
-In the set M(n, R) of n × n matrices with elements in R, the Killing form is defined as -In so(n), for n ≥ 2, X, Y = (n − 2)T r(XY). We can deduce that Let's consider the Killing form of g. From the following identity i.e. Ad(g) belongs to the orthogonal group of the Killing form. As a result | det(Ad(g))| = 1.
Let g * the dual space of the Lie algebra g. The coadjoint representation of a Lie group is the dual of the adjoint representation. The corresponding action of G on g * is called the coadjoint action. The orbits of that action are called coadjoint orbits, which are especially important in the orbit method of representation theory or, more generally, geometric quantization. An important class of symplectic stuctures consists of the coadjoints orbits by the coadjoint action (Kirillov, 1976).
In the Kirillov method of orbits, representations of G are constructed geometrically starting from the coadjoint orbits. Specifically, Definition 2.2. The coadjoint representation Ad * of G is the dual representation of the adjoint representation Ad, and is given by Ad * (g) := Ad(g −1 ) T where g ∈ G and (g −1 ) T denotes the transpose of g −1 . The representation space of Ad * is g * , the dual of the Lie algebra g.
In terms of Ad, we consider the following description of Ad * , In this paper, the description of the coadjoint representation simplifies to According to (Lu & al., 1990), the Killing form , is non-degenerate and we can identify the dual space g * with g via the map f −→ X f . Now, we will use the notation Lemma 2.1. (Kirillov,1976) Let g f be the Lie algebra of the stabilizing group Theorem 2.1. Let f ∈ g * and let O f be the coadjoint orbit at f . Consider the application Then ω f is a skew-symmetric bilinear form.
Proof. According to the Lemma 2.1, we can consider elements of T f (O f ) as elements on the form X + Z for some Z ∈ g f .
We prove that ω f is well-defined and it is obviously skew-symmetric and bilinear given that the Lie bracket is also.
The form ω f defined by (6) is called Kostant-Kirillov structure. Now, let's show that the coadjoint orbits are symplectic structures in G. Recall that, giving a symplectic structure (or symplectic form) on G is to define a closed non-degenerate differential 2-form.
Proof. According to the Theorem 2.1, ω f is a well-defined 2-form. The Lie algebra g has trivial center and thus ω f is non-degenerate. It remains to be shown that ω f is closed. Let d(Ad * ) be the differential of Ad * defined by d(Ad * ) := ad * . This means that, the representation of g on g * corresponding to Before calculating the exterior derivative of ω f , we need to recall the expression for the exterior derivative of an n-form ω which can be written explicitly as (Gengoux & al.,2013) : for all X 0 , . . . , X n ∈ g, n ∈ N. In particular, for X, Y, Z in ∈ g, we have the formula for the exterior derivative of a 2-form : Applying the formula (7), we have : From (8), we obtain the equality in (9) is equivalent to : Consequently, dω f (X, Y, Z) = 0. Thus, ω f is closed. This concludes that ω f is a symplectic structure on T f (O f ).
Note that, the symplectic Kostant-Kirillov structure ω f is written (Berndt, 2007) as Another natural way to define a symplectic structure is to consider the cotangent bundle T * (O f ). Let's consider the following map : Lemma 2.2. The map defined by (12) is an isomorphism generated by the symplectic structure ω f .
Since, ω f is a symplectic form, it follows that ω f is non-degenerate. The non-degeneracy condition means that It suffices to conclude that, is an isomorphism.
According to (Lesfari, 2009), any symplectic structure induces a Hamiltonian vector field associated to a differentiable function H (called Hamiltonian) expressed in terms of a differential system. Let's consider the local coordinate system (x, y, z), the differential system can be expressed as follows : We have : Let a i j (i, j = 1, 2, 3) be the components of the matrix such that we have (dx) = a 11 Since is skew-symmetric, we have : which can be written in more compact formẊ = J X ∂H ∂X . This is a complete characterization of hamiltonian vector field. The associated matrix J X belongs to g and determine a symplectic structure. Note that J X ∈ g is not necessarily a Poisson bivector. The vector field corresponding to J X expressed in a coordinate system (x, y, z) is defined by The bracket of two functions F and G being given by (Weinstein,1983) {F, G} = π J X (dF, dG) In this paper, π 12 = −2a 12 , π 13 = 2a 13 and π 23 = 2a 23 i.e.
Let's consider the partial differential equation where u, v, w are the unknown functions. There exists at least one solution. Indeed, (x, y, z) satisfies the equation (17). π J X is called Poisson bivector field if and only if (a 23 , a 13 , a 12 ) is a solution of the partial differential equation (17). Otherwise, it will be called Poisson quasi-bivector field. As a result, π J X is a Poisson quasi-bivector field for a 23 ∈ C[y, z], a 13 ∈ C[x, z] and a 12 ∈ C[x, y].
Unless otherwise stated, y n is not identically equal to zero. From this, π 0 is a Poisson quasi-bivector field. Let (A 0 , ., {., .} 0 ) be the quasi-Poisson algebra defined by A 0 = C[x, y, z] with quasi-Poisson structure {., .} 0 associated to π 0 defined by It follows It is the matrix associated to π 0 on A 0 = C[x, y, z].
The following section is devoted to the explicit calculation of a Kostant-Kirillov structure induced by π 0 in J n 0 .

An Explicit Calculation of a Kostant-Kirillov Symplectic Structure on the Quasi-Poisson Tensor π 0
In this section, we calculate explicitly the coadjoint orbits of the Lie group G and their Kostant-Kirillow symplectic structures coadjoint orbits of the Lie group G. Let g θ be the element of G defined above. Like any Lie group, there are two known operations on G called the left translation L g θ : G −→ G, g θ −→ g θ g θ , and the right translation These applications are diffeomorphisms of G and we have We can define the automorphism of G as Let 1 be the unit of G. Since g = T 1 G is the Lie algebra of G, its adjoint representation Ad g θ is the derivative of R g −1 θ L g θ in 1 defined by : . By a simple computation, we show that Ad g θ is an algebra homomorphism defined by (20) Now consider the notation with d dt g θ (t) | t=0 = ξ and g θ (0) = 1, where End(g) is the space of endomorphisms of g. We have ad ξ (η) = [ξ, η] f or all ξ ∈ g and η ∈ End(g).
Definition 3.1. Let f ∈ g * and let O * f = {Ad * g θ ( f ) : g θ ∈ G} ⊂ g * , the dual of coadjoint orbit on the Lie group G. A vector tangent τ to the orbit O * f of f is expressed as an element A ∈ g by τ = {A, f }, A ∈ g.
We have to use the following lemma : Since g * = g, we can obviously remark that the symplectic Kostant-Kirillov structure ω f defined on (6) is written as Recall that G is a subgroup of SO(3) and the corresponding Lie algebra is g = T 1 G consists of skew-symmetric 3 × 3 matrices where 1 is the neutral element of G and T 1 G is the tangent space at 1. Consider the basis (e x , e y , e y ) defined . From the Euler-Arnold system the differential systems defined on (4), we haveẊ = (J X )X where is the matrix associated to the Poisson quasi-bivector field defined on (18).
Proposition 3.2. Let n ≥ 1 be an integer and let θ be a given constant parameter. The adjoint orbit of the group G is O(J X ) = 1 2n ye x + (− 1 2 (sin θ)y n + 1 2n (cos θ)x)e y + ( 1 2 (cos θ)y n + 1 2n (sin θ)x)e z Proof. Consider the invertible matrix g θ = We have which completes the proof of O(J X ).
Corollary 3.1. For n 2 and θ π + 2kπ (k be an integer), the adjoint orbit of the group G induces a Poisson quasibivector field From the above, we have the following classification : Proposition 3.3.
Let's consider A = k∈I j k e k ∈ g where I = {x, y, z} be an index set. Then The spectrum of A is From ( where k∈I j k e k ∈ g and I = {x, y, z} be an index set. Let's consider V = (z 1 , z 2 , z 3 ) ∈ C 3 : 1 4n 2 z 2 1 + 1 4n 2 z 2 2 + 1 4 z 2n 2 = k∈I j 2 k .
Therefore, the orbit O * (J X ) is isomorphic to V defined on (33).
We obtain a similar result as before about a classification of the Kostant-Kirillov orbit : For n = 1, θ = 2kπ, k ∈ Z For n = 2, θ = π + 2kπ, k ∈ Z For n ≥ 3, θ = 2kπ, k ∈ Z O * (J X ) 1 2 ye x + 1 2 xe y + 1 2 ye z 1 4 ye x − 1 4 xe y − 1 2 y 2 e z J X Note that, we have a Poisson structure through the action of g π+2kπ with n = 2 and k an integer.
To determine the symplectic structure on O * (J X ), we will use the following result.
Theorem 3.1. Let f ∈ g * . The Kostant-Kirillov symplectic structure is given by with j, k ∈ C 3 where τ 1 = f ∧ j, τ 2 = f ∧ k and ∧ is the usual vector product.
Let j = ( j x , j y , j z ) and k = (k x , k y , k z ) be two elements of C 3 . The Killing form can be written as : where ∧ is the usual vector product. Let's consider the isomorphism where j, k ∈ C 3 and X, Y ∈ g. From (25), we also have : ω f (τ 1 , τ 2 ) = f, j ∧ k , τ 1 = f ∧ j and τ 2 = f ∧ k.
It follows that, the tangent vectors to O * (J X ) at a given point are also the tangent vectors to V at this point. In the following, ( f 1 , f 2 , f 3 ) be the local coordinate system and f = ( f 1 , f 2 , f 3 ) be the given point. The main result is :