Darboux coordinates on k-Orbits of lie algebras - Nguyen Viet Hai

DARBOUX COORDINATES ON K-ORBITS OF LIE ALGEBRAS Nguyen Viet Hai Faculty of Mathematics, Haiphong University Abstract. 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. 1 Introduction The method of orbits discovered in the pioneering works of Kirillov (see [K]) is a universal base for performing harmonic analysis on homogeneous spaces and for constructing new methods 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 Darboux coordinates (p, q) every element F ∈ g = LieG can be considered as a function F˜ on O, linear on pa’s-coordinates, i.e. F˜ = n∑ i=1 αai (q)pa + χi(q). (1) Our main result is Theorem 3.2 in which we show that the existence of a normal polariza- tion associated with a linear functional ξ is necessary and sufficient for the existence of local canonical Darboux coordinates (p, q) on the K-orbit Oξ such that the transition to these coordi- nates is linear in the “momenta” as equation (1). For the good strata, namely families of with some good enough parameter space, of coadjoint orbits, there exist always continuous fields of polarizations (in the sense of the representation theory), satisfying Pukanski conditions: for each orbit Oξ and any point ξ in it, the affine subspace, orthogonal to some polarizations with respect to the symplectic form is included in orbits themselves, i.e. ξ +H⊥ ⊂ Oξ, dimH = n− 1 2 dimOξ. In the next section, we construct K-orbits via linear algebraic methods. In Section 3 we consider Darboux coordinates on K-orbits of Lie algebras and give the proof of Theorem 3.2. 2 The description of K-orbits via linear algebraic methods Let G be a real connected n-dimensional Lie group and G be its Lie algebra. The action of 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 (indG) of the Lie algebra defined as the dimension of the annihilator of 1 a general covector. We say that a linear functional (a covector) ξ has the degeneration degree s if it belongs to a K-orbit Oξ of the dimension dimOξ = n− r − 2s, s = 0...., (n− r)/2. We decompose the space G∗ into a sum of nonintersecting invariant algebraic surfaces Ms consisting of K-orbits with the same dimension. This can be done as follows. We let fi denote the coordinates of the covector F in the dual basis, F = fiei with 〈ei, ej〉 = δij , where {ej} is the basis of G. The vector fields on G∗ Yi(F ) ≡ Cij(F ) ∂ ∂fj , Cij(F ) ≡ C k ijfk are generators of the transformation group G acting on the space G∗, and their linear span therefore constitutes the space TFOξ 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, dimOξ = rankCij(ξ). It can be easily verified that the rank of Cij is constant over the orbit. Therefore, we obtain polynomial equations that define a surface Ms, M0 = {F ∈ G ∗ | ¬(λ1(F ) = 0)}; Ms = {F ∈ G ∗ | λs(F ) = 0, ¬(λs+1(F ) = 0)}, s = 1, . . . , n− r 2 − 1; Mn−r 2 = {F ∈ G∗ | λ n−r 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 + 2 vanish at the point F , and ¬(λs(f) = 0) means that the corresponding minors do not vanish simultaneously at F . The spaceMs can also be defined as the set of points F where all the polyvectors of degree n− r− 2s+ 1 of the form Yi1(F ) ∧ . . .∧ Yin−r−2s+1(F ) vanish, but not all the polyvectors of degree n− r − 2s− 1 vanish. We note that in the general case, the surface Ms consists of several nonintersecting in- variant components, which we distinguish with subscripts as Ms = Msa ∪Msb . . . . (To avoid stipulating each time that the space Ms is not connected, we assume the convention that s in parentheses, (s), denotes a specific type of the orbit with the degeneration degree s.) Each component M(s) is defined by the corresponding set of homogeneous polynomials λ (s) α (F ) satisfying the conditions Yiλ (s) α (F )|λ(s)(F )=0 = 0. (2) 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 ∈ R 1. The dual space G∗ has a degenerate linear Poisson bracket {ϕ, ψ}(F ) ≡ 〈F, [∇ϕ(F ),∇ψ(F )]〉; ϕ, ψ ∈ C∞(G∗). (3) The functions K(s)µ (F ) that are nonconstant on M(s) are called the (s)-type Casimir functions if they commute with any function on M(s). 2 The (s)-type Casimir functions K(s)µ (F ) can be found from the equations Cij(F ) ∂K (s) µ (F ) ∂fj ∣∣∣ F∈M(s) = 0, i = 1, . . . , n. (4) It is obvious that the number r(s) of independent (s)-type Casimir functions is related to the dimension of the spaceM(s) as r(s) = dimM(s)+2s+r−n. BecauseM(s) are star spaces, we can assume without loss of generality that the Casimir functions K(s)µ (F ) are homogeneous, ∂K (s) µ (F ) ∂fi fi = m (s) µ K (s) µ (F )⇐⇒ K (s) µ (tF ) = t m (s) µ K(s)µ (F ); µ = 1, . . . , r(s). In the general case, the Casimir functions are multivalued (for example, if the orbit space G∗/G is not semiseparable, the Casimir functions are infinitely valued), hi what follows, we use the term ”Casimir function” to mean a certain fixed branch of the multivalued function K (s) µ . In the general case, the Casimir functions K (s) µ are only locally invariant under the coadjoint representation, i.e., the equality K(s)µ (Ad∗gF ) = K (s) µ (F ) holds for the elements g belonging to some neighborhood of unity in the group G. Remark. Without going into detail, we note that the spaces M(s) are critical surfaces for some polynomial (s − 1)-type Casimir functions, which gives a simple and efficient way to construct the functions λ(s). We now let Ω(s) ⊂ Rr(s) denote the set of values of the mapping K(s) : M(s) → R r(s) and introduce a locally invariant subset O(s)ω as the level surface, O(s)ω = {F ∈M(s) | K (s) µ (F ) = ω (s) µ , µ = 1, . . . , r(s); ω (s) ∈ Ω(s)}. The dimension of O(s)ω is the same as the dimension of the orbit Oξ ∈ M(s), where ω (s) = K(s)(F ). If the Casimir functions are single valued, the orbit space in separable, and the set O (s) ω then consists of a denumerable (typically, finite) number or orbits; accordingly, we call this level surface the Strata of orbits. The space G∗ thus consists of the union of connected invariant nonintersecting algebraic surfaces M(s); these, in turn, are union of the Strata of orbits O (s) ω : G∗ = ⋃ (s) M(s) = ⋃ (s) ⋃ ω(s)∈Ω(s) O(s)ω . (5) Example 1. (see [H3]): The so called real diamond Lie algebra is the 4-dimensional solvable Lie algebra g with basis e1, e2, e3, e4 satisfying the following commutation relations: [e1, e2] = e3, [e4, e1] = −e1, [e4, e2] = e2, [e3, e1] = [e3, e2] = [e4, e3] = 0. The real diamond Lie algebra is isomorphic to R4 as vector spaces. We identify its dual vector space g∗ with R4 with the help of the dual basis e1, e2, e3, e4 and with the local coordinates as (α, β, γ, δ). Denote K-orbit of G in g, passing through F by O = {K(g)F |g ∈ G}, with 〈K(g)F, U〉 = 〈F,Ad(g−1)U〉, ∀F ∈ g∗, g ∈ G and U ∈ g. By a direct computation one obtains (see [H3]): 3 i. Each point of the line α = β = γ = 0 is a 0-dimensional co-adjoint orbit Oδ = (0, 0, 0, δ). ii. The set α = 0, β = γ = 0 is union of 2-dimensional co-adjoint orbits, which are just half-planes Oα = {(f1, 0, 0, f4) | f1, f4 ∈ R, αf1 > 0} iii. The set α = γ = 0, β = 0 is union of 2-dimensional co-adjoint orbits, which are just half-planes Oβ = {(0, f2, 0, f4) | f2, f4 ∈ R, βf2 > 0}. iv. The set αβ = 0, γ = 0 is decomposed into a family of 2-dimensional co-adjoint orbits, which are just hyperbolic-cylinders Oαβ = {(f1, f2, 0, f4) |f1, f2, f4 ∈ R & αf1 > 0, βf2 > 0, f1f2 = αβ}. v. The open set γ = 0 is decomposed into a family of 2-dimensional co-adjoint orbits , which are just hyperbolic- paraboloids Oγ = {(f1, f2, γ, f4) |f1, f2, f4 ∈ R & f1f2 − γf4 = αβ − γf4)}. In this example, G∗ = ⋃ (s)M(s) = Oδ ⋃ Oα ⋃ Oβ ⋃ Oαβ ⋃ Oγ, each Strata consists of several K-orbits. We now consider the quotient space B(s) = M(s)/G, whose points are the orbitsOξ ∈M(s). It is obvious that dimB(s) = r(s). We introduce local coordinates j on B(s). For this, we parameterize an (s)-covector ξ ∈M(s) by real-valued parameters j = (j1, . . . , jr(s)), assuming that ξ depends linearly on j (this can be done because M(s) is a star surface): ξ = ξ(j) with λ(s)α (ξ(j)) ≡ 0, K (s) µ (ξ(j)) = ω (s) µ (j), det ∂ω (s) µ (j) ∂jν = 0. Elementary examples show that global parameterization does not exist on the whole of M(s) in general, i.e., the manifold B(s) is not covered by one chart. In this case, we define an atlas of charts on B(s) and parameterize the corresponding connected invariant nonintersecting subsets MA(s),M B (s), . . . with a nonvanishing measure in M(s) as follows: M(s) = M A (s) ∪M B (s) ∪ . . . . (6) The corresponding domains of values JA, JB, . . . of the j parameters then satisfy the relation Ω(s) = ω(s)(JA) ∪ ω(s)(JB) ∪ . . .. We illustrate decomposition (6) with a simple example. Example 2. (The group SO(2, 1)). In the case of the algebra so(2, 1), [e1, e2] = e2, [e2, e3] = 2e1, and [e3, e1] = e3. Decompo- sition (5) becomes O0ω = {f 2 1 + f2f3 = ω, ¬(F = 0)}, O 1 = {F = 0}. 4 For ω > 0, the class O0ω consists of two orbits. For nondegenerate orbits, Ω = R 1. There is no single parameterization in this case. Indeed, the most general form of the parameterization ξ(j) = (a1j, a2j, a3j) (where ai, are some numbers) leads to ω(j) = aj2, where a = a21+a2a3, and therefore (depending on the sign of a) ω(j) is always greater than zero, less than zero, or equal to zero, i.e., ω(R1) = Ω. We introduce two spectral types, typeA : ξ(j) = (0, j, j); JA = [0,∞); O0Aω(j) = {f 2 1 + f2f3 = j 2, F = 0}; typeB : ξ(j) = (0, j,−j); JB = (0,∞); O0Bω(j) = {f 2 1 + f2f3 = −j 2, F = 0}. 3 Darboux coordinates We let ωξ denote the Kirillov form on the orbit Oξ. It defines a symplectic structure and acts on the vectors a and b tangent to the orbit as ωξ(a, b) = 〈ξ, [α, β]〉, where a = ad∗αξ and b = ad ∗ βξ. The restriction of Poisson brackets (3) to the orbit coincides with the Poisson bracket generated by the symplectic form ωξ. According to the well-known Darboux theorem, there exist local canonical coordinates (Darboux coordinates) on the orbit Oξ such that the form ωξ becomes ωξ = dpa ∧ dq a; a = 1, . . . , 1 2 dimOξ = n− r 2 − s, where s is the degeneration degree of the orbit. Let ξ be an (s)-type covector and F = (f1, f2, ...) ∈ Oξ. It can be easily seen that the transition to canonical Darboux coordinates (fi) → (pa, q a) amounts to constructing analytic functions fi = fi(q, p, ξ) of the variables (p, q) satisfying the following conditions: fi(0, 0, ξ) = ξi; (7) ∂fi(q, p, ξ) ∂pa ∂fj(q, p, ξ) ∂qa − ∂fj(q, p, ξ) ∂pa ∂fi(q, p, ξ) ∂qa = Ckijfk(q, p, ξ); (8) λ(s)α (F (q, p, ξ)) = 0, K (s) µ (F (q, p, ξ)) = K (s) µ (ξ). (9) We require that the transition to the canonical coordinates be linear in pa, fi(q, p, ξ) = α a i (q)pa + χi(q, ξ); rankα a i (q) = 1 2 dimOξ. (10) Obviously, a transition of form (10) does not exist in the general case; however, assuming that αai (q) and χi(q; ξ) are holomorphic functions of the complex variables q, we considerably broaden the class of Lie algebras and K-orbits for which this transition does exist. Let Xi(x) = Xai (x)∂xa be transformation group generators that generate an n-dimensional Lie algebra G of vector fields on a homogeneous space M = G/H : [Xi, Xj] = C k ijXk (here and in what follows, xa (a = 1, . . . ,m = dimM), are local coordinates of a point x ∈M ); H 5 is the isotropy group of a base point x0 and H is its Lie algebra. Inhomogeneous first-order operators X˜i = Xi+χi(x) are called the continuations of the generators Xi if they still satisfy the commutation relations of the algebra G (χi(x) and are smooth functions on M). By definition, the n-component function χ(x) is to be found from the system of equations Xai (x) ∂χj(x) ∂xa −Xaj (x) ∂χi(x) ∂xa = Ckijχk(x). (11) Solutions of this system span a linear space, in which we can single out the module of trivial solutions of the form χ0 = { χ0i (x) = X a i (x) ∂S(x) ∂xa } , where S(x) is an arbitrary smooth function. Constructing trivial continuations X˜i = e−SXieS is equivalent to performing the ”gauge” transformation ∂xa → ∂xa+∂xaS(x). In what follows, we are interested in only nontrivial continuations that generate the quotient space of all solutions of system (11) modulo trivial solutions. Proposition 3.1. The space of nontrivial continuations is finite dimensional and is isomorphic to the quotient space H∗/[H,H]∗. We do not give the complete proof here; however, we give the explicit form of all nontrivial solutions of system (11), which implies the validity of Proposition 3.1. We relabel and change the basis in G, Xa(x0) = ∂ ∂xa ∣∣∣ x0 , a = 1, . . . ,m; Xα(x0) = 0, α = m+ 1, . . . , n. We consider the right action of G on M . An arbitrary element g ∈ G is represented as g = hs(x), where h ∈ H and s(x) is a smooth Borel mappingM → G assigning the right coset class Hs(x) ⊂ G to each point x ∈ M . In the coordinates g = (hα, xa) (assuming that e = (0, x0)), the left-invariant vector fields ξi(g) have the form ξi(g) = Xai (x)∂xa + ξ α i (h, x)∂hα . Direct calculation verifies that the continuations given by the operators X˜i = Xi + ξ α i (0, x)ξα; ξ ∈ H ∗, 〈ξ, [H,H]〉 = 0, are nontrivial. It can also be shown that there are no other nontrivial continuations. Now we can give main result of this article. Theorem 3.2. The linear transition to canonica coordinates on the orbit Oξ exists if and only if there exists a normal polarization (in general, complex) associated with the linear functional ξ, i.e., a subalgebra H ⊂ Gc such that dimH = n− 1 2 dimOξ, 〈ξ, [H,H]〉 = 0, ξ +H ⊥ ⊂ Oξ. Chøng minh. We consider following steps: Step 1. Assume fi(q, p, ξ) = αai (q)pa+χi(q, ξ); rankα a i (q) = 1 2 dimOξ. We write equation (8) in more detail as αai (q)∂qaα b j(q)− α a j (q)∂qaα b i(q) = C k ijα b k(q); (12) 6 αai (q)∂qaχj(q; ξ)− α a j (q)∂qaχi(q; ξ) = C k ijχk(q; ξ). (13) Equation (12) is equivalent to [ai, aj] = Ckijak, where ai = α a i (q)∂qa , and the operators ai, are therefore generators of the transformation group acting in the domain Q (Q = G/H for a real polarization and Q = Gc/H for a complex polarization, where H is the Lie group corresponding to H). Because rankαai (q) = dimOξ/2 = dimQ, we conclude that whenever solutions of system (12) exist, there exists an isotropy algebra H of the point q = 0 of the dimension n − dimOξ/2. It is obvious that the converse is also true: the existence of an (n − dimQ)-dimensional subalgebra H is sufficient for the existence of solutions to system (12). Step 2. As follows from Proposition 3.1, the existence of solutions to system (13) with initial conditions (7) (or the existence of nontrivial continuations of the operators ai) is equivalent to the condition that the algebra H be adapted to the linear functional ξ. Therefore, the existence of the polarization H for a covector ξ is necessary and sufficient for relations (7) and (8) to be satisfied. Step 3. We now show that the normality of the polarization, i.e., that the polarization satisfies the Pukanski conditions ξ + H⊥ ⊂ Oξ, is necessary and sufficient for relations (9) to be satisfied. We let {eA} denote a basis of the isotropy algebra H and {ea} denote the comple- mentary basis vectors in Gc: ei = {eA, ea}. By definition of the isotropy algebra, αaA(0) = 0, whence detαba(0) = 0. Making a linear change of coordinates q and of the basis in G c, we can assume without loss of generality that αba(0) = δ b a. In our notation, H ⊥ = {(0, pa)}, where pa are arbitrary numbers, and the Pukanski condition becomes (ξA, pa + ξa) ∈ Oξ. Let relations (9) be satisfied:λ(s)α (F (q, p, ξ)) = 0, K (s) µ (F (q, p, ξ)) = K (s) µ (ξ). Setting q = 0 we then have Φ(ξA, ξa) = (λ (s) α (ξA, ξa), K (s) µ (ξA, ξa) = Φ(fA(q, p; ξ), fa(q, p; ξ))|q=0 = Φ(ξA, pa + ξa). (We recall the notation Φ = (λ(s), K(s))). This implies that for any value of pa, the point (ξA, pa + ξa) belongs to the same class of orbits as the point (ξA, ξa); because the Strata of orbits consists of adenumerable number of K-orbits, we then conclude that these two points belong to the same orbit. Conversely, let the Pukanski condition be satisfied, i.e. F (0, p; ξ) = (ξA, pa + ξa) ∈ Oξ and Φ(ξA, pa + ξa) = Φ(ξA, ξa). We then show that equation (9) is satisfied. Because the functions Φ(F ) satisfy equations (2) and (4), we have( αab (q) ∂Φ(F (q, p; ξ)) ∂qa − ∂fb(q, p; ξ) ∂qa ∂Φ(F (q, p; ξ)) ∂pa )∣∣∣ q=0 = ∂Φ(F (q, p; ξ)) ∂qb ∣∣∣ q=0 = 0. Therefore, Φ(F (q, p; ξ)) = Φ(F (0, p; ξ)) = Φ(ξ). This implies λ(s)α (F (q, p, ξ)) = λ (s) α (ξ) = 0,K (s) µ (F (q, p, ξ)) = K (s) µ (ξ), and the theorem is proved. 7 Tµi liÖu [H1] Do Ngoc Diep and Nguyen Viet Hai, Quantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Straight Line, Contributions to Algebra and Geometry, 42(2) (2001), 418-427. [H2] Nguyen Viet Hai, Quantum co-adjoint orbits of MD4-groups, Vietnam J. Math., 29 (2001), 131-158. [H3] Nguyen Viet Hai, Quantum Co-adjoint Orbits of the Real Diamond Lie Group, Journal of Science, TXXI, N03, Vietnam National University, Hanoi, 2005. [K] A. A. Kirillov, Elements of the Theory of Representation, Springer Verlag, Berlin - New York - Heidelberg, 1976. To¹ ®é §ac-bu trªn c¸c K-quÜ ®¹o cña c¸c ®¹i sè Lie NguyÔn ViÖt H¶i Khoa To¸n, tr−êng §¹i häc H¶i-Phßng Tãm t¾t. Chóng t«i chøng minh sù tån t¹i cña ph©n cùc chuÈn kÕt hîp víi mét hµm tuyÕn tÝnh trªn ®¹i sè Lie lµ ®iÒu kiÖn cÇn vµ ®ñ ®èi víi biÕn ®æi tuyÕn tÝnh vÒ to¹ ®é Darboux chÝnh t¾c (p, q) trªn quÜ ®¹o ®èi phô hîp. 8