Darboux coordinates on k-orbits of lie algebras

We prove that the existence of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for the linear transition to local canonical Darboux coordinates (p, q) on the coadjoint orbit.
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Darboux coordinates on k-orbits of lie algebrasDARBOUX COORDINATES ON K-ORBITSOF LIE ALGEBRASNguyen Viet HaiFaculty of Mathematics, Haiphong UniversityAbstract. We prove that the existence of the normal polarization associated with a linearfunctional on the Lie algebra is necessary and sufficient for the linear transition to localcanonical Darboux coordinates (p, q) on the coadjoint orbit.1IntroductionThe method of orbits discovered in the pioneering works of Kirillov (see [K]) is a universalbase for performing harmonic analysis on homogeneous spaces and for constructing newmethods of integrating linear differential equations. Here we describle co-adjoint Orbits O(the K-orbit) of a Lie algebra via linear algebraic methods. We deduce that in Darbouxcoordinates (p, q) every element F ∈ g = Lie G can be considered as a function F˜ on O,linear on pa ’s-coordinates, i.e.F˜ =nαia (q)pa + χi (q).(1)i=1Our main result is Theorem 3.2 in which we show that the existence of a normal polarization associated with a linear functional ξ is necessary and sufficient for the existence of localcanonical Darboux coordinates (p, q) on the K-orbit Oξ such that the transition to these coordinates is linear in the “momenta” as equation (1). For the good strata, namely families of withsome good enough parameter space, of coadjoint orbits, there exist always continuous fieldsof polarizations (in the sense of the representation theory), satisfying Pukanski conditions: foreach orbit Oξ and any point ξ in it, the affine subspace, orthogonal to some polarizations withrespect to the symplectic form is included in orbits themselves, i.e.ξ + H ⊥ ⊂ Oξ ,dim H = n −1dim Oξ .2In the next section, we construct K-orbits via linear algebraic methods. In Section 3 weconsider Darboux coordinates on K-orbits of Lie algebras and give the proof of Theorem 3.2.2The description of K-orbits via linear algebraic methodsLet G be a real connected n-dimensional Lie group and G be its Lie algebra. The actionof the adjoint representation Ad∗ of the Lie group defines a fibration of the dual space G ∗into even-dimensional orbits (the K-orbits). The maximum dimension of a K-orbit is n − r,where r is the index (ind G) of the Lie algebra defined as the dimension of the annihilator of1a general covector. We say that a linear functional (a covector) ξ has the degeneration degrees if it belongs to a K-orbit Oξ of the dimension dim Oξ = n − r − 2s, s = 0...., (n − r)/2.We decompose the space G ∗ into a sum of nonintersecting invariant algebraic surfacesMs consisting of K-orbits with the same dimension. This can be done as follows. We let fidenote the coordinates of the covector F in the dual basis, F = fi ei with ei , ej = δji , where{ej } is the basis of G. The vector fields on G ∗Yi (F ) ≡ Cij (F )∂,∂fjCij (F ) ≡ Cijk fkare generators of the transformation group G acting on the space G ∗ , and their linear spantherefore constitutes the space TF Oξ tangent to the orbit Oξ running through the point F .Thus, the dimension of the orbit Oξ is determined by the rank of the matrix Cij ,dim Oξ = rank Cij (ξ).It can be easily verified that the rank of Cij is constant over the orbit. Therefore, we obtainpolynomial equations that define a surface Ms ,M0 = {F ∈ G ∗ | ¬(λ1 (F ) = 0)};Ms = {F ∈ G ∗ | λs (F ) = 0, ¬(λs+1 (F ) = 0)}, s = 1, . . . ,M n−r = {F ∈ G ∗ | λ2n−r2n−r− 1;2(F ) = 0}.Here, we let λs (F ) denote the collection of all minors of Cij (F ) of the size n − r − 2s + 2,the condition λs (F ) = 0 indicates that all the minors of Cij (F ) of the size n − r − 2s + 2vanish at the point F , and ¬(λs (f ) = 0) means that the corresponding minors do not vanishsimultaneously at F .The space Ms can also be defined as the set of points F where all the polyvectors of degreen − r − 2s + 1 of the form Yi1 (F ) ∧ . . . ∧ Yin−r−2s+1 (F ) vanish, but not all the polyvectors ofdegree n − r − 2s − 1 vanish.We note that in the general case, the surface Ms consists of several nonintersecting invariant components, which we distinguish with subscripts as Ms = Msa ∪ Msb . . . . (To avoidstipulating each time that the space Ms is not connected, we assume the convention that s inparentheses, (s), denotes a specific type of the orbit with the degeneration degree s.) Each(s)component M(s) is defined by the corresponding set of homogeneous polynomials λα (F )satisfying the conditionsYi λ(s)(2)α (F )|λ(s) (F )=0 = 0.Although the invariant algebraic surfaces M(s) are not linear spaces, they are star sets, i.e.,F ∈ M(s) , implies tF ∈ M(s) for t ∈ R1 .The dual space G ∗ has a degenerate linear Poisson bracket{ϕ, ψ}(F ) ≡ F, [∇ϕ(F ), ∇ψ(F )];(s)ϕ, ψ ∈ C ∞(G ∗ ).(3)The functions Kµ (F ) that are nonconstant on M(s) are called the (s)-type Casimir functionsif they commute with any function on M(s) .2(s)The (s)-type Casimir functions Kµ (F ) can be found from the equations(s)∂Kµ (F ) Cij (F )= 0,∂fjF ∈M(s)i = 1, . . . , n.(4)It is obvious that the number r(s) of independent (s)-type Casimir functions is related to thedimension of the space M(s) as r(s) = dim M(s) + 2s + r − n. Because M(s) are star spaces, we(s)can assume without loss of generality that the Casimir functions Kµ (F ) are homogeneous,(s)(s)∂Kµ (F )(s)(s)mµfi = m(s)K(F)⇐⇒K(tF)=tKµ(s) (F );µµµ∂fiµ = 1, . . . , r(s) .In the general case, the Casimir functions are multivalued (for example, if the orbit spaceG ∗ /G is not semiseparable, the Casimir functions are infinitely valued), hi what follows, weuse the term ”Casimir function” to mean a certain fixed branch of the multivalued function(s)(s)Kµ . In the general case, the Casimir functions Kµ are only locally invariant under the(s)(s)coadjoint representation, i.e., the equality Kµ (Ad∗g F ) = Kµ (F ) holds for the elements gbelonging to some neighborho ...