A method for the deformation quantization of coadjoint orbits of semisimple
Lie groups is proposed. It is based on the algebraic structure of the orbit.
Its relation to geometric quantization and differentiable deformations is
explored.Comment: Talk presented at the meeting "Noncommutative geometry and Hopf
  algebras in Field Theory and Particle Physics", Torino, 199

Cahen M.

M. A. LIEDÓ

Varadarajan V. S.

English

arXiv

A method for the deformation quantization of coadjoint orbits of semisimple Lie groups is proposed. It is based on the algebraic structure of the orbit. Its relation to geometric quantization and differentiable deformations is explored

Lledó Barrena, María Antonia

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:math/0003142v1  [math.QA]  23 Mar 2000Deformation Quantization of Coadjoint OrbitsM. A. Lledo´Dipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi 24, I-10129 Torino, Italy, andINFN, Sezione di Torino, Italy.e-mail: lledo@athena.polito.itAbstractA method for the deformation quantization of coadjoint orbits ofsemisimple Lie groups is proposed. It is based on the algebraic struc-ture of the orbit. Its relation to geometric quantization and differen-tiable deformations is explored.Let G be a complex Lie group of dimension n and GR a real form of G.Let G and GR be their respective Lie algebras with Lie bracket [ , ]. As it iswell known, G∗R has a Poisson structure,{f1, f2}(λ) =< [(df1)λ, (df2)λ], λ >, f1, f2 ∈ C∞(G∗R), λ ∈ G∗R. (1)Choosing a basis {X1, . . .Xn} of GR and its dual, {ξ1, . . . ξn}, the Poissonbracket can be written as{f1, f2}(x) = ckijλk∂f1∂xi∂f2∂xj, x =n∑i=1xiξi ∈ G∗.Notice that this Poisson bracket is never symplectic, in particular it is 0 atthe origin. Under the action of g ∈ GR, it satisfies g∗{f1, f2} = {g∗f1, g∗f2},so GR is a group of automorphisms of the Poisson algebra C∞(G∗R). Theaction of GR on G∗R is not transitive, so G∗R is foliated in orbits. This foliationcoincides with the foliation given by the Hamiltonian vector fields of (1). Sothe orbits of the coadjoint action of a Lie group are symplectic manifolds.We want to describe formal deformations of the Poisson algebra C∞(Θ)(Θ is a coadjoint orbit) or of some subalgebra of it. It is convenient to workwith the complexification of the Poisson algebra.An associative algebra Ah over C[[h]] is a formal deformation of a Poissonalgebra (A, { , }) over C if there exists an isomorphism of C[[h]]-modulesψ : A[[h]] −→ Ah satisfying the following properties:1a. ψ−1(F1F2) = f1f2 mod(h) where Fi ∈ Ah are such that ψ−1(Fi) = fimod(h), fi ∈ A. (By mod(h) we mean that the projections p : A[[h]] −→A[[h]]/hA[[h]] of both quantities coincide).b. ψ−1(F1F2 − F2F1) = h{f1, f2} mod(h2).An example of interest for our purposes is the polynomial algebra on G∗.A formal deformation of Pol(G∗) [1] is given by the algebra Uh = TC[[h]](G)/Lh,where TC[[h]](G) is the tensor algebra over C[[h]] and Lh is the proper two sidedidealLh =∑X,Y ∈GTC[[h]](G)⊗ (X ⊗ Y − Y ⊗X − h[X, Y ])⊗ TC[[h]](G) ⊂ TC[[h]](G).The isomorphism ψ : Pol(G∗) −→ Uh is not canonical. A possible choice isin terms of a Poincare´-Birkhoff-Witt basis,ψ(xi1xi2 · · ·xik) = Xi1 ·Xi2 · · ·Xik , 1 ≤ i1 ≤ · · · ≤ ik ≤ n. (2)Another choice is the symmetrizer map,Sym(xi1xi2 · · ·xik) =1k!∑σ∈SkXσ(i1) ·Xσ(i2) · · ·Xσ(ik), (3)where Sk is the group of permutations of order k.Given a choice for ψ one can define an associative product (star product)on A[[h]] bya ⋆ψ b = ψ−1(ψ(a) · ψ(b)).Then, for any choice of ψ, (Ah, ⋆ψ) is an algebra isomorphic to Ah. Withthe star product we recover the semiclassical interpretation of the elementsof the algebra as functions on the phase space. The star product can alwaysbe written as a formal seriesa ⋆ψ b = ab+∑n>0hnCnψ(a, b),where Cnψ are some bilinear operators. Let ⋆ and ⋆′ be two isomorphic starproducts,a ⋆ b = T−1(T (a) ⋆ T (b)), T : A[[h]] −→ A[[h]].2It is clear that T can be written asT (a) =∑n≥0hnT n(a),and because of property a, T 0 must be an automorphism of the commutativealgebra A[[h]]. If T0 is the identity we say that ⋆ and ⋆′ are equivalent (orgauge equivalent) star products. (2) and (3) are two equivalent star products.For A being the full algebra of C∞ functions on the Poisson manifold,if the operators Cnψ are bidifferential operators we say that the star productis differentiable. Gauge equivalence can be restricted to the class of dif-ferentiable star products by considering only differentiable T n. Notice thatthe differentiability is a property of the particular star product and not ofthe formal deformation. The star products (2) and (3) can be extended toC∞(Θ) as differentiable star products, but we will see later an example ofa star product corresponding to the same formal deformation which is notdifferentiable [2].We will consider only semisimple Lie groups. The semisimple coadjointorbits of G on G are complex algebraic varieties defined over R. They aregiven by the invariant polynomials. If l is the rank of G, we can choosel homogeneous polynomials pi(x), i = 1, . . . , l generating the subalgebra ofinvariant polynomials, C[p1, . . . pn]. Then the semisimple coadjoint orbits aregiven by the algebraic equationspi(x) = ci. (4)(see for example [3]). The intersection of the complex orbit with GR is a realalgebraic variety consisting on a finite number of connected components,which are orbits of the real form of the group. For the compact real formthere is only one connected component.It is easy to check that the star products 2 and 3 do not restrict well tothe orbit, that is, in generala ⋆ pi|Θ 6= 0.We want to know if there is some choice of ψ that gives a star product whichrestricts to Θ.In the approach of geometric quantization, the algebra of quantum ob-servables is given by the quotient of Uh by a certain ideal. This ideal, Ih,3is prime and AdG-invariant, so there is a well defined action of G on Uh/Ih.We summarize here the results of [4], where the quantization of the coadjointorbits are obtained in terms of the quotient of the enveloping algebra by aprime, AdG-invariant ideal. We consider the polynomial algebra over the realalgebraic manifold (union of orbits) defined by (4),Pol(Θ) = Pol(G∗)/I0, I0 = {p ∈ Pol(G∗)/p|Θ = 0}.This is the Poisson algebra that we want to deform. We quote first a resultfrom Varadarajan [5] that we need.Lemma (1). Let x ∈ G∗ be a regular element of G∗ (or equivalently, apoint in which the centralizer has dimension equal to the rank of G∗). Then(dp1)x, ..., (dpl)x are linearly independent.From now on we will restrict to regular orbits only. It is clear that in thiscase I0 is generated by p1 − c1, . . . pl − cl. We consider now the elements inUh that are the image of pi by the symmetrizer, Pi = Sym(pi), called Casimiroperators. Pi are central elements in Uh and they are also AdG-invariant (thesymmetrizer commutes with the action of G).Let Ih be the ideal generated by Pi − Ci(h), where Ci(0) = ci. ThenUh/Ih is a formal deformation of Pol(Θ). The technical assumption of reg-ularity is needed to prove the existence of a C[[h]]-module isomorphismψ : Pol(Θ)[[h]] −→ Uh/Ih, which is not obvious. The ideal Ih itself isAdG-invariant, so G has a natural action by automorphisms on the algebraUh/Ih. For special values of Ci(h), Ih is in the kernel of an irreducible unitaryrepresentation of GR.This deformation of polynomials can be specialized for any value of h.For SU(2),[H,X] = ~2X, [H, Y ] = −~2Y, [X, Y ] = ~H,the Casimir operator isP =12(XY + Y X +12H2).It was shown in [4] that with the choiceC = l(l + ~)), l = ~m/2,the algebra obtained is the same than the one obtained in geometric quanti-zation.4According to our definition, a star product in Pol(Θ) is given by a C[[h]]-module isomorphism ψ˜ : Pol(Θ)[[h]] −→ Uh/Ih. In particular, to obtaina star product in Pol(G∗) which restricts to the orbit one should look foran isomorphism ψ : Pol(G∗)[[h]] −→ Uh such that the following diagramcommutesPol(G∗)[[h]]ψ−−−→ UhypiypˆiPol(Θ)[[h]]ψ˜−−−→ Uh/Ih.(5)π and πˆ are the natural projections. If ψ(I0) = Ih, then ψ˜ is defined uniquelyby the diagram (5). An example of such star product is given in [2], whereit is also shown that it is not differentiable. Since ψˆ is not unique one couldask if there is some choice that renders it differentiable. This issue will beaddressed in [6].References[1] J. Hoppe, Quantum theory of a massless relativistic surface and a twodimensional bound state problem. MIT PhD thesis, (1982).[2] M. Cahen and S. Gutt, C.R. Acad. Sc. Paris 296 (1983) 821-823; inGravitation and Cosmology, Monographs and Textbooks in PhysicalSciences, Bibliopolis. Eds W. Rundler and A. Trautman, (1987); Bull.Soc. Math. Belg. 36 B, 207-221, (1987).[3] V. S. Varadarajan, Harmonic Analysis on Real Reductive Groups. Lec-ture Notes in Mathematics, no. 576. Springer-Verlag, (1977).[4] R. Fioresi and M. A. Lledo´, math.QA/9906104. To appear in the PacificJournal of Mathematics.[5] V. S. Varadarajan, Amer. J. Math, 90, (1968).[6] R. Fioresi, A. Levrero and M. A. Lledo´. In preparation.5

Deformation Quantization of Coadjoint Orbits

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