Quantization of Poisson manifolds and the integrability of the modular function.

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    a r X i v : 1 3 0 6 . 4 1 7 5 v 1 [ m a t h . S G ] 1 8 J u n 2 0 1 3

    Quantization of Poisson manifolds from theintegrability of the modular function

    F. Bonechi , N. Ciccoli, J. Qiu , M. Tarlini June 18, 2013

    Abstract

    We discuss a framework for quantizing a Poisson manifold via the quan-tization of its symplectic groupoid, that combines the tools of geometricquantization with the results of Renaults theory of groupoid C -algebras.This setting allows very singular polarizations. In particular we considerthe case when the modular function is multiplicatively integrable , i.e. whenthe space of leaves of the polarization inherits a groupoid structure. If suit-able regularity conditions are satised, then one can dene the quantumalgebra as the convolution algebra of the subgroupoid of leaves satisfyingthe Bohr-Sommerfeld conditions.

    We apply this procedure to the case of a family of Poisson structureson C P n , seen as Poisson homogeneous spaces of the standard Poisson-Liegroup SU (n + 1). We show that a bihamiltoniam system on C P n denesa multiplicative integrable model on the symplectic groupoid; we com-pute the Bohr-Sommerfeld groupoid and show that it satises the neededproperties for applying Renault theory. We recover and extend Sheusdescription of quantum homogeneous spaces as groupoid C -algebras.

    INFN Sezione di Firenze, email: [email protected] di Matematica, Univ. di Perugia, email: [email protected], Universite du Luxembourg, email: [email protected]

    INFN Sezione di Firenze, email: [email protected]

    http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1http://arxiv.org/abs/1306.4175v1
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    1 Introduction

    The quantization of a Poisson manifold is well understood in terms of the de- formation quantization of the algebra of smooth functions but is much moreelusive if one looks for anoperatorial quantization . In fact Kontsevich formula[15], together with the globalization procedure [6], gives an explicit formula forthe deformation quantization of any smooth Poisson manifold. Nothing similarexists for the operatorial quantization, in whatever sense of the word.

    The concept of symplectic groupoid was introduced in the mid eighties byKarasev and Weinstein as a tool for quantization of Poisson manifolds. Indeed,when the Poisson manifold ( M, ) is integrable there exists a unique symplecticmanifold G(M, ), endowed with a compatible groupoid structure, that allowsto reconstruct the Poisson tensor on its space of units M . The compatibilitybetween the groupoid and symplectic structures is expressed by requiring thatthe graph of the multiplication is a lagrangian submanifold of G(M, )G(M, )G(M, ), where G(M, ) has the opposite symplectic structure. If His the spaceof states obtained by means of some quantization procedure of G(M, ), then, byapplying the semiclassical quantization dictionary, the graph of the multiplicationis quantized by a vector in H H H. This vector denes an algebra structureon H, that should be considered as the algebra of quantum observables ratherthan the space of states.

    The basic question that one has to address is what kind of quantization weare looking for. It seems a tall order to seek a map associating an operator to anyclassical observable, but rather one should look directly for the non commutativealgebra of quantum observables. In fact, from the point of view of geometricquantization, such a map needs the quantization of any observable, while we knowthat this is out of reach even in favourable cases, since the polarization procedureselects a very narrow class of quantizable functions. Very recently, E. Hawkinsin [14] proposed a general framework based on the geometric quantization of thesymplectic groupoid where the quantization output is a C -algebra. Basically, theidea of [14] is to look for amultiplicative polarization F T C G(M, ) that allowsfor the denition of a convolution product on the space of polarized sections.The best case is that of a real polarization dened by a Lie groupoid bration

    G(M, ) GF (M, ) over some Lie groupoid GF (M, ); the C -algebra in thiscase is the convolution algebra of the groupoid GF (M, ) of lagrangian leaves.The usual problem of geometric quantization is to nd such polarizations, inparticular real ones very often do not exist.

    Our starting observation is that thanks to the groupoid structure we canconsider much more general polarizations than those needed by geometric quan-tization. In fact, suppose that, instead of a smooth polarization, we just havea topological quotient of groupoids G(M, ) GF (M, ) with lagrangian bres.

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    Let us call GbsF (M, ) the subgroupoid of Bohr-Sommerfeld (BS) leaves, i.e. thoseleaves with trivial holonomy. In order to dene a convolution algebra it is enoughthat

    Gbs

    F (M, ) admits a Haar system; this can be fullled in cases of polariza-

    tions that are very singular and hard to use in geometric quantization. In thiscase, quantization then follows from Renault theory on groupoid C -algebras [22]and ts in a transparent way with the Poisson geometry.

    Indeed, given a Poisson manifold (M, ), there are two cohomology classesin H LP (M, ), the Lichnerowicz-Poisson cohomology, that are relevant for quan-tization. The Poisson tensor itself denes a class [ ] H 2LP (M, ); when aprequantization of the symplectic groupoid is chosen, there is dened a 2-cocycleof the symplectic groupoid with values in S 1 that we call the prequantization cocycle . This 2-cocycle can be 1 if and only if [] = 0. Moreover, for everyvolume form on M it is dened the modular vector eld that measures the non

    invariance of the volume form with respect to hamiltonian transformations. Itdenes a class in H 1LP (M, ), the modular class , independent on the choice of the volume form: it vanishes only if there exists an invariant volume form. Themodular vector eld is integrated to a 1-cocycle of the symplectic groupoid, thatwe call the modular function .

    It is then natural to require that for our generalized polarization, both themodular function and the prequantization cocycle should descend to GF (M, );the general theory of groupoid C -algebras tells us how to dene the convolu-tion algebra, twisted by the prequantization cocycle; the modular function isquantized to an automorphism of the algebra: the modular operator in noncom-mutative geometry. Moreover it determines a quasi invariant measure (equiva-lently the KMS state); nally we get an Hilbert algebra structure (compatiblescalar product and convolution) which corresponds to the algebra on the spacesof states that we are looking for.

    Integrable models on the symplectic groupoid are a possible source for suchsingular polarizations and appear very naturally when the Poisson structure isnot unimodular. We say that the modular function is multiplicatively integrable if there exists a maximal family of independent hamiltonians in involution withit such that the space of contour levels inherits the groupoid structure.

    A class of examples where such construction is possible is given by compactPoisson-Lie groups and their homogeneous spaces. They are the semiclassical

    limit of quantum groups and their quantum homogeneous spaces. A series of pa-pers by A.J. Sheu [25, 26] relates the quantum spaces to groupoid C -algebras. Incertain cases (for instance the standard C P n,q, 0 and the odd dimensional spheresS 2n +1q ) he showed that the C -algebra is a groupoid C -algebra; for SU q(n) orin the non standard structures C P n,q,t , for t (0, 1), his characterization of theC -algebra is just as a subalgebra of a groupoid C -algebra. His analysis is basedon representation theory of the quantum algebras.

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    We conjecture that we can interpret the groupoids appearing in Sheus pa-pers as BS-leaves of some polarization of the symplectic groupoid integratingthe semiclassical structure. Since they are in general non unimodular Poissonmanifolds, we conjecture that these BS groupoids appear from the multiplicativeintegrability of the modular function.

    In this paper we are going to show that this is actually true for the wholefamily of Poisson structures on C P n , thus recovering Sheus description for thestandard case and extending it in the non standard one. Moreover, as a byprod-uct of the construction we get the quantization of a class of Poisson submanifolds;in particular we recover the description of the odd dimensional spheres.

    The simplest example is the so called Podles two sphere and has been alreadystudied in [2] (the integrability issues were not explicit there, for this point of view see the discussion in [3]).

    In the general case, the relevant integrable model depends on a bihamiltonianstructure existing on all compact hermitian spaces. In fact it was shown in [ 16]that on compact hermitian symmetric spaces there exists a couple of compatiblePoisson structures ( 0, s ) dening a bihamiltonian system. The rst one 0, theBruhat or standard Poisson structure, is the quotient of the compact Poisson-Liegroup by a Poisson subgroup and s is the inverse of the Kirillov symplecticform, once that we realize the symmetric space as a coadjoint orbit. By taking alinear combination we get a Poisson pencil , i.e. a family of Poisson homogeneousspaces t = 0 + t s . For t = (0 , 1) t is called non standard . Since one of twoPoisson structures is symplectic, we can dene a maximal set of hamiltoniansthat are in involution with respect to every Poisson structure of the pencil (fora short account of this see [13]).

    We analyze here the case of complex projective spaces and we are inter-ested in the case where t [0, 1], when t is degenerate. The integrable modelcorresponds to the toric structure of C P n given by the action of the CartanT n SU (n + 1) and was studied in connection with the bihamiltonian structurein [13]. The action of the Cartan on C P n is hamiltonian in the symplectic casewith momentum map c : C P n t n , t n = Lie T n , and simply Poisson with respectto the Poisson structure t for t [0, 1]. This means that the action lifts to anhamiltonian action of T n on the symplectic groupoid G(C P n , t ) with momentummap h : G(C P n , t ) t n . The couple (c, h), where c is pulled back to G(C P n , t )with the source map, denes an integrable model. We show in Proposition 6.1that it is multiplicative. The modular vector eld with respect to the Fubini-Study volume form is just the fundamental vector eld of a particular elementof t n so that the modular function is in involution with ( c, h).

    Our main result is Theorem 6.3 where we give an explicit description of theBohr-Sommerfeld groupoid GbsF (C P n , , t ) and show that it has a unique Haarsystem. Moreover, in Proposition 5.4 we show that the maximal Poisson sub-

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    manifolds, including odd spheres, are quantized to subgroupoids of this BS-groupoid, showing the covariant nature of the correspondence. We show thatthe BS-groupoid for t = 0 and the Poisson submanifold ( S 2n 1,

    t) (C P

    N ,

    t)

    are the same appearing in Sheus description of C P n,q, 0 and S 2n 1q ; the result fort = (0 , 1) conjecturally improves his description for C P n,q,t .

    This is the plan of the paper. In Section 2 we give a very brief review of basicfacts on integration of Poisson homogeneous spaces and groupoid C -algebras.In Section 3 we introduce the denition of multiplicative integrability of themodular function and the general quantization scheme. In Section 4 we describethe geometry of ( C P n , t ) as Poisson homogeneous space of SU (n + 1) equippedwith the quasitriangular Poisson-Lie structure. In Section 5 we interpret thisfamily of Poisson structures as a Poisson pencil arising by the bihamiltoniansystem. In Section 6 we discuss the integrable model on the symplectic groupoid

    of (C P n , t ); we compute the Bohr-Sommerfeld leaves, discuss the Haar systemand the quasi invariant measure associated to the modular function; we discussthe quantization of maximal Poisson submanifolds. Finally in Section 7 we relateour results to Sheus description of quantum homogeneous spaces in terms of groupoid C -algebras.

    Notations . We will denote a groupoid with G= ( G, G0, lG, r G, mG, G, G); G0 isthe space of units, lG, r G : G G0 are the source and target maps, G2 G Gdenotes those elements ( 1, 2) G2 such that r G( 1) = lG( 2). Moreover, mG :G2 Gis the multiplication, G : G Gis the inversion and G : G0 Gis theembedding of units. We denote with xG= l

    1G (x) and Gx = r

    1G (x) for x G0.We say that Gis source simply connected (ssc) if xGis connected and simplyconnected for any x G0.We denote with Gk the space of strings of k-composable elements of G, withthe convention that G1 = G. The face maps are di : Gs Gs 1, i = 0 , . . . s ,dened for s > 1 as

    di( 1, . . . s ) =( 2, . . . s ) i = 0( 1, . . . i i+1 . . .) 0 < i < s( 1, . . . s 1) i = s

    (1)

    and for s = 1 as d0( ) = l

    G( ), d

    1( ) = r

    G( ). The simplicial coboundary

    operator : k(Gs ) k(Gs+1 ) is dened as

    () =s

    i=0

    ()i di () ,

    and 2 = 0. The cohomology of this complex for k = 0 is the real valuedgroupoid cohomology; s-cocycles are denoted as Z s (G, R ).

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    If is a line bundle onG, then denotes the line bundle d0 d1 d2over G2, where di : G2 Gdenote the face maps dened in ( 1).If S G0

    then we denote withGS

    = l 1G

    (S )

    r 1G

    (S ) the subgroupoid of

    Gobtained by restriction to S . If a group G acts on X , the action groupoid X G is dened on X G with structure maps as l(x, g) = x, r (x, g) = xg,m[(x, g)(xg,g)] = ( x,gg), (x) = ( x, e), (x, g) = ( xg,g 1), for x X andg, g G.

    A Poisson structure is denoted ( M, ), where is the two tensor dening thePoisson bracket on the smooth manifold M . We denote with also the bundlemap : T M T M . The Lichnerowicz-Poisson differential is denoted withdLP (X ) = [, X ]; its cohomology with H LP (M, ).

    2 Poisson manifolds, symplectic groupoids andHaar systems

    We recall in this Section the background material that we will need later. Inparticular we introduce basic denitions and properties of Poisson manifoldsand symplectic groupoids, of Poisson-Lie groups and their homogeneous spaces.Finally we discuss Haar systems on topological groupoids.

    2.1 Poisson manifolds and symplectic groupoids

    We say that a topological groupoid Gis a Lie groupoid if Gand G0 are smoothmanifolds, all maps are smooth and lG, r G are surjective submersions. A symplec-tic groupoid is a Lie groupoid, equipped with a symplectic form G, such thatthe graph of the multiplication is a lagrangian submanifold of GGG, where Gmeans Gwith the opposite symplectic structure. There exists a unique Poissonstructure on G0 such that lG and r G are Poisson and anti-Poisson morphisms re-spectively; the Poisson manifold G0, in such case, is said to be integrable . Given aPoisson manifold (M, ) is always possible to dene a (ssc) topological groupoidG(M, ) as the quotient of the space of cotangent paths with respect to cotangenthomotopies. A cotangent path is a bundle map (X, ) : T [0, 1]

    T M satisfying

    dX + () = 0; see [9, 5] for the denitions of cotangent homotopies and thegroupoid structure maps. The obstruction for the existence of the smooth struc-ture making G(M, ) to be the (ssc) symplectic groupoid integrating ( M, ) hasbeen studied in [10].

    Let (M, ) be a Poisson manifold. A vector eld Vect( M ) satisfyingdLP ( ) = 0 is called a Poisson vector eld . Two distinguished classes are relevantfor what follows. The rst one is the class [] H 2LP (M, ) dened by the Poisson

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    tensor itself. Let us assume that M is orientable, and let us choose a volume formV M on M . The modular vector eld V M = div V M is Poisson; its cohomologyclass

    V M H 1

    LP (M, ) does not depend on the choice of the volume form and

    is called the modular class .Let (M, ) be integrable so that G(M, ) is the ssc symplectic groupoid inte-grating it. Every Poisson vector eld Vect( M ) can be integrated to a unique

    real valued groupoid 1-cocycle f Z 1(G(M, ), R ) with the formulaf [X, ] = 10 (t), (X (t)) dt .

    We call f V M the modular function .

    2.2 Integration of Poisson homogeneous spaces

    A Poisson-Lie group is a Lie group G endowed with a Poisson structure G suchthat the group multiplication ( GG, G G ) (G, G ) is a Poisson map. Thismeans that for each g1, g2 G we have that

    G (g1g2) = lg1 G (g2) + r g2 G (g1) .

    As a consequence, there exists the simply connected dual Poisson-Lie group(G , G ), that acts innitesimally on G, the so called dressing action . We as-sume that this action is integrated to a group action (G is called complete ); asa consequence also G is complete. If g G and G we denote the left and

    right action of G on G as

    g and g

    , respectively; analogously the left and rightactions of G on G are denoted as g and g, respectively.Any PoissonLie group is integrable and its symplectic groupoid has been

    rst described in [18]. If G is complete then G(G, G ) = G G with l(g, ) = g,r (g, ) = g , m(g1, 1)(g 11 , 2) = ( g1, 1 2), (g) = ( g, e), (g, ) = ( g , 1). Thenon degenerate Poisson tensor reads

    G = G + G +a

    T a r T a , (2)

    where T a and r T a are left and right vector elds on G and on G with respectto a choice of dual basis of LieG and LieG .

    The (right) action of ( G, G ) on (M, ) is Poisson if the action seen as a map(M G, G ) (M, ) is Poisson. This means that for each m M andg G we have that

    (mg) = g (m) + m G (g) .

    If the action is transitive then ( M, ) is called a Poisson homogeneous space .If there exists m M where (m) = 0 then one can show that the stability

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    subgroup H m G is coisotropic and viceversa if a closed subgroup H Gis coisotropic then there exists a unique Poisson structure H \ G on H \G suchthat the projection map pr

    H is Poisson. We call these homogeneous spaces

    embeddable . We know that (LieH) (LieG) = LieG is a Lie subalgebra; letus denote with H G the connected subgroup integrating (LieH) , which weassume to be closed. Since H is coisotropic then there exists a canonical Poissonstructure G /H on G /H ; let prH denote the projection map. Moreover, thedressing action of H preserves H , and viceversa.

    Let us recall here how the integration of embeddable homogeneous spacesis done in [1]. The left G action on itself can be lifted to a symplectic ac-tion of G(G, G ); this action admits a Lu momentum map J : G(G, G ) G ,J (g, ) = g . If we denote J H = pr H J , then the symplectic groupoid integrat-ing (H \G, H \ G ) is the symplectic reduction H \J 1H (pr H (e)). The symplecticgroupoid structure descends from G(G, G ). We then get the following descrip-tion

    G(H \G, H \ G ) = {(pr H (g), ), g G, g H } .Let us denote with GH H H the bre bundle associated to the homo-geneous principal bundle with the dressing action of H on H . The map

    L : G(H \G, H \ G ) GH H H dened as L(pr H (g), ) = [g, g ] is a diffeomor-phism commuting l with the bundle projection; as a consequence l is a bration.By taking L we prove the same result for the target map r .

    2.3 Haar systems

    Let us recall basic facts from [22] (see also the very brief description given inSection 4 of [2]). Let T be a topological groupoid and let C c(T ) denote thespace of continuous functions with compact support. A left Haar system for T is a family of measures {x , x T 0}on T such that

    i) the support of x is xT = l 1T (x);ii) for any f C c(T ), and x T 0, (f )(x) = T fd x , denes (f ) C c(T 0);iii ) for any T and f C c(T ), T f ( )dr T ( )( ) = T f ( )dlT ( )( ).The composition of x with the inverse map will be denoted as x ; this family

    denes a right Haar system 1. Any measure on the space of units T 0 inducesmeasures , 1 on the whole T through

    T fd = (f )d , T fd 1 = 1(f )d .The measure is said to be quasi-invariant if and 1 are equivalent measures;in this case the Radon-Nikodym derivative D = d /d 1 is called the modular

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    function of . The function log D Z 1(T , R ) turns out to be a groupoid 1-cocycle with values in R and its cohomology class depends only on the equivalenceclass of .

    Let Z 2(T , S 1) be a continuous 2-cocycle. One can dene the convo-lution algebra C c(T , ) and correspondingly the groupoid C -algebra C (T , ).When = 1 we denote the convolution and C -algebras with C c(T ) and C (T ).Moreover, given a quasi invariant measure one can dene the structure of leftHilbert algebra on L2(T , 1 ) such that the modular operator is exactly givenby multiplication with D .

    We are interested in understanding when a given topological groupoid has a(possibly unique) Haar system. We will use the following result.

    Denition 2.1. A locally compact groupoid is etale if its unit space is open.

    As a consequence, xT , T x are discrete for all x T 0 and if there exists aleft Haar system then it is equivalent to the counting measure. We will use inparticular the following result (Prop. 2.8 in [22]):

    Proposition 2.2. A topological groupoid is etale and admits a left Haar system if and only if lT is a local homeomorphism.

    3 The quantization frameworkLet (M, ) be a Poisson manifold, G(M, ) be its symplectic groupoid, that isassumed to be smooth (this was referred in the previous Section as integrabilityof (M, ); it must not be confused with the notion of integrability of the modularfunction that we are going to discuss in this Section). Let us x a volume formV M and let f V M C

    (G(M, )) be the modular function.We are interested in studying cases where the hamiltonian dynamics of themodular function is integrable, i.e. f V M Poisson commutes with a set F =

    {f i}dim G/ 2i=1 of hamiltonians f i C (G(M, )) in involution. Such functionsshould be generically non degenerate in the usual sense, i.e. df 1 . . . df n = 0

    on an open and dense subset of G(M, ).Let us denote with GF (M, ) the topological space of connected contour levelsof F . We remark that if we denote by P F = X f i T G(M, ) the (generalized)real polarization dened by the hamiltonian vector elds X f i of the f is, we havethat GF (M, ) is not in general G(M, )/ P F . This is what happens for instancein [2]. We will not necessarily require that GF (M, ) be smooth, we will see laterwhat kind of regularity we need.

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    Denition 3.1. The modular function f V M is said to be multiplicatively inte-grable if there exists on GF (M, ) a topological groupoid structure such that the quotient map

    G(M, )

    GF (M, ) is a groupoid morphism.

    Let us recall the prequantization of the symplectic groupoid: this is the or-dinary prequantization of the symplectic structure with additional requirementsthat make it compatible with the groupoid structure. The basic results comefrom [31], but the presentation is taken from [ 14]. If is a line bundle onGthenwith we denote the line bundle d0 d1 d2 on G2.Denition 3.2. A prequantization of the symplectic groupoid G(M, ) consists of the triple (, ; G ) where (, ) is a prequantization of G(M, ) as a symplectic manifold and G is a section of such that:

    i) G has norm one and is multiplicative, i.e. it satises the following cocycle condition: for ( 1, 2, 3) G3(M ) G ( 1, 2, 3) G ( 1, 2 3) G ( 2, 3) G ( 1, 2) 1 G ( 1 2, 3) 1= 1 . (3)

    ii) G is covariantly constant, i.e. if G is a (local) connection form for ,then G (locally) satises

    d G + ( G) G = 0 . (4)

    We call G the prequantization cocycle dened by G. From Theorem 3.2in [31] we get that if G(M, ) is prequantizable as a symplectic manifold andlG-locally trivial, then there exists a unique groupoid prequantization.The prequantization cocycle can be chosen to be 1 if G is multiplicatively

    exact, i.e. if there exists a primitive G such that G = 0. Theorem 4.2 in[8] shows that G is multiplicatively exact if and only if the class of the Poissonbivector [] H 2LP (M, ) is zero.

    Finally we say that a leaf GF (M, ) satises the Bohr-Sommerfeld con-ditions if the holonomy of along is trivial. We are able to prove that Bohr-Sommerfeld conditions select a subgroupoid of GF (M, ) only under the followingassumption.Assumption 1 : for each couple of composable leaves (1, 2) GF (M, )2 themap mG : 1 2 G(M, )2 12 induces a surjective map in homology.

    This assumption is veried by the integrable model we are going to considerin the following Sections.

    Lemma 3.3. If Assumption 1 is satised then the set of Bohr-Sommerfeld leaves

    GbsF (M, ) GF (M, ) inherits the structure of a topological groupoid.10

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    Proof . Let be a loop in the leaf = 12 and let (t) = 1(t) 2(t), for( 1(t), 2(t)) 1 2 G(M, )2. Then we have that

    G = 1

    0dt G, m ( 1 2) = ( 1 , 2 ) G + 1 + 2 .

    Since G satises (4) then we have that if 1, 2 GbsF (M, ) then 12GbsF (M, ).

    We refer to GbsF (M, ) as the groupoid of Bohr-Sommerfeld leaves .Assumption 2 : the groupoid of Bohr-Sommerfeld leaves GbsF (M, ) admits aHaar system.

    If Assumption 1 and 2 are satised then we get that the groupoid of Bohr-Sommerfeld leaves

    GbsF (M, ) admits a Haar system and inherits the modular

    cocycle f V M . Moreover, if the prequantization cocycle G can be chosen to beF -invariant it descends to GbsF too. These data give a quantization of the Poissonstructure. Indeed, by applying the results of Renaults theory summarized in theprevious Section, we can dene the convolution algebra C c(GbsF , G ). Moreover,we can determine the quasi invariant measure V M whose modular cocycle isf V M and then dene the left Hilbert algebra on L

    2(GbsF , 1V M ).In the rest of the paper, we will discuss a concrete example where such a

    quantization exists.

    4 A family of SU (n + 1)

    covariant Poisson struc-tures on C P n

    We introduce in this Section the basic example that we are going to quantize. Itis a family t , t [0, 1], of Poisson structures on the complex projective spaceC P n that are embeddable homogeneous spaces of the standard Poisson structureon SU (n + 1).

    We will rst recall the standard Poisson structure on SU (n + 1) (see [19])and then introduce a family of coisotropic subgroups U t (n) SU (n + 1).

    The standard multiplicative Poisson tensor of SU (n + 1) can be describedintrinsically integrating a 2 su valued Lie-algebra 2cocycle; here, in view of laterapplications, we describe it in terms of the dressing transformation of SB (n +1, C ) on SU (n + 1) . Let us start with the Iwasawa decomposition

    SL (n + 1 , C ) = SU (n + 1) SB (n + 1 , C ) , SB (n + 1 , C ) = An +1 N n +1 ,where An +1 = {diag(v1, . . . , vn +1 ), vi > 0, v1v2 . . . vn +1 = 1}and N n +1 is thegroup of upper diagonal complex matrices with 1 on the diagonal. It is easilyshown that pr An +1 : SB (n + 1 , C ) An +1 is a group morphism.

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    The Lie algebras su (n + 1) and sb (n + 1) are dual to each other through thepairing X, = ImTr( X ), for X su (n+1) , sb (n +1 , C ). We will thereforeoften identify implicitly sb with su . We will denote by p

    1,2the projections from

    sl to su and sb respectively. As said above, any d SL (n +1 , C ) can be Iwasawadecomposed into g with g SU (n + 1) and SB (n + 1 , C ). By taking theproduct g and rewriting it using the Iwasawa decomposition as g = g g,with g SU and g SB , one has dened the dressing transformation

    SB (n + 1 , C ) SU (n + 1)) SU (n + 1) ; ( , g) g .If sb then one has g = gAdg 1 , thus the innitesimal dressing transforma-tion generated by is

    lg( p1Adg 1 ), (5)

    where g SU (n + 1) and lg denotes left multiplication by g. Now regard asin su , one has Adg = p2Adg , which can also be seen by using the pairing.The Poisson tensor, when evaluated on two right invariant 1-forms ( r g 1 ) 1,2, isgiven by the paring of the 1-form (Adg) 2 with the vector eld ( 5) generated by 1, or in formulae:

    (r g 1 ) 1, (r g 1 ) 2 = p1Adg 1 1, p2Adg 1 2 . (6)From this formula one can show directly that is Poisson and it is furthermoremultiplicative (gh) = ( lg) (h) + ( r h ) (g).

    Dually, changing the roles of the two groups, one denes the dressing trans-formation of SU (n + 1) on SB (n + 1 , C ), and can write down a multiplicativePoisson tensor for the latter. In this way SB (n + 1 , C ) becomes the dual simplyconnected Poisson-Lie group of SU (n + 1).

    Let us denote for t [0, 1]

    t = 1 t 0 t0 idn 1 0 t 0 1 t

    SU (n + 1) . (7)

    Using the formula for the Poisson tensor ( 6), it can be shown that the sub-

    groupsU t (n) t S (U (1) U (n)) 1t SU (n + 1)

    are coisotropic for all t [0, 1] (this was rst proved in [27] at the innitesimallevel). Let us denote with u t(n) = Lie U t (n). Since U t (n) is coisotropic, u t (n)is a Lie subalgebra and we will denote with U t (n) SB (n + 1 , C ) the subgroupintegrating u t (n) sb (n + 1 , C ). An explicit expression for u t(n) can bededuced from [7], page 16. As a consequence of coisotropy there is a uniquely

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    dened Poisson structure t on C P n = U t (n)\SU (n + 1) such that the quotientmap pt : SU (n + 1) C P n is Poisson.It has to be remarked that Poisson structures corresponding to different valuesof t are not totally unrelated. Let X j , j = 1 . . . n + 1 denote the homogeneous

    coordinates of C P n .

    Lemma 4.1. The diffeomorphism : C P n C P n dened as [X 1, . . . , X n +1 ] = [X n +1 , . . . , X 1]

    sends t to 1 t .Proof . Let, in fact, J = ( i i,n +1 i) SU (n + 1) be the matrix with all antidi-

    agonal elements equal to i. We have clearly that is just the right multiplicationby J 1. Then a direct computation shows that pt (JgJ 1) = ( p1 t (g)) for allg SU (n + 1). On the other hand one has that A JAJ 1 is an antiPoissonmap, which therefore induces an anti-Poisson map on complex projective spaces.

    In general t and t for t = 1 t are not Poisson or anti Poisson diffeomor-phic. This is shown, for instance, for C P 1 in [4].The family of covariant Poisson structures t on C P n exhibits a sharply dif-

    ferent behaviour, depending on whether t is one of the limiting values t = 0 , 1,which will be called the standard or BruhatPoisson structure , or t (0, 1), thatwill be referred to as the non standard case. This is best described by analyzingthe corresponding symplectic foliation. It is an interesting fact that such folia-

    tion is essentially determined by the images under the projection of the Poissonsubgroups of SU (n + 1). For each k = 1 , . . . , n , let

    Gk = S (U (k) U (n + 1 k)) SU (n + 1) ; P k(t) = pt(Gk) C P n .Each Gk is a Poisson subgroup so that P k(t) is a Poisson submanifold of C P n .As can be easily deduced from [29], Proposition 2.1, this exhausts the list of maximal (with respect to inclusion) Poisson subgroups of SU (n + 1). We willexplicitly describe the Poisson submanifolds P k(t).

    When t = 0, i.e. in the case of the Bruhat-Poisson structure, Poisson sub-manifolds P k(0) C P k 1 are contained one inside the other giving rise to thefollowing chain of Poisson embeddings:

    {}= C P 0 C P 1 . . . C P n (8)where C P k corresponds to X j = 0 for j > k so that = [1, 0, . . . , 0]. Thesymplectic foliation, then, corresponds to the Bruhat decomposition

    C P n =n +1

    i=1S i , (9)

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    where each S i can be described as S i = {[X 1, . . . , X i , 0, . . . 0], X i = 0} C P i 1.Thus we have one contractible symplectic leaf in each even dimension, whichturns out to be symplectomorphic to standard C i . The maximal symplectic leaf (the maximal cell) S n +1 is open and dense in C P n . On such 2n-dimensional opencell S n +1 it is possible to dene the socalled Lus coordinates {yi}ni=1 as follows(see Appendix A for more details). Let z i = X i /X n +1 , 1 i n be the standardaffine coordinates on this cell and let us dene yn = z n and for i < n

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    such Poisson submanifolds intersect along lower dimensional symplectic leaves.Let us remark the following. First, if k < h , then P k(t) P h (t) =

    hi= k P i(t)

    and this is a simple consequence of inequalities (13). Furthermore, from theanalysis of the symplectic foliation of odd-dimensional Poisson spheres one canprove that the singular part of the symplectic foliation of each P k(t) is justP k(t) P k 1(t) P k(t) P k+1 (t); the limiting cases k = 1 , n of course behavedifferently in that only one of the two terms in the union appears. This can besummarized by saying that the foliation resembles more a stratication where thestrata of symplectic leaves of dimension 2m are given by the intersections of nmconsecutive Poisson submanifolds P k(t); furthermore each k+ n mi= k P i(t) is equalto the intersection of P k (t) with the linear subspace X k+1 = . . . = X k+ n m 1 = 0.Thus, for example, the subset of 0dimensional symplectic leaves equals

    n

    i=1

    P i(t) = [X 1, 0, . . . , 0, X n +1 ] t|X 1|2 (1 t)|X n +1 |2 = 0 S 1

    In each even dimension lower than 2n there appears continuous families of sym-plectic leaves, in sharp contrast with the standard case.

    In low dimensions we can say that non standard C P 1 contains a singular S 1of 0dimensional leaves of equation t|X 1|2 (1 t)|X 2|2 = 0, separating two 2dimensional symplectic cells. As for non standard C P 2, it contains two copies of S 3 separating three affine 4dimensional cells: the two embedded Poisson spheresS 3 are made of two distinct S 1families of 2dimensional symplectic leaves andare intersecting along a common circle S 1 of 0dimensional leaves. The singularpart of the symplectic foliation for C P 3 is summarized in the following diagram,which is, of course, also the starting diagram of leaves of dimension up to 4 ineach higher dimensional C P n (and can be easily extended in any dimension):

    S 5 S3 S 3

    S 1S 5

    S 3 S 3 S 1

    (14)

    We summarize the description of the symplectic foliation of (C

    P n , t ) in thefollowing Proposition.

    Proposition 4.1. For t (0, 1), (C P n , t ) decomposes in the following regular Poisson submanifolds

    C P n =n

    r,s 0, r + s=0

    C P (r,s )n

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    where for r + s = n, C P (r,s )n is an open connected component of C P n \ P sing , and for r + s < nC P (r,s )n = n r s j =1 P r + j (t) \ P r (t) P n +1 s (t) . (15)

    For r + s = n, (C P (r,s )n , t ) is non degenerate. For r + s < n , (C P (r,s )n , t ) is

    Poisson diffeomorphic to S 2(r + s)r,s S 1, where S

    2(r + s)r,s is a symplectic manifold of

    dimension 2(r + s).

    We will need the following fact.

    Proposition 4.2. All symplectic leaves of (C P n , t ), for t = [0, 1], are con-tractible.

    Proof . For t = 0 , 1 the symplectic foliation corresponds to the Bruhat decom-position so that the result is straightforward. Let us consider t (0, 1). Let usdiscuss rst the open symplectic leaves C P (r,n r )n of dimension 2n. For r = 0 , nthey are dened by 0 < F 1,t and F n,t < 0 respectively; it is easy to check thatthey are open disks. For 0 < r < n , C P (r,n r )n is characterized by the inequalityF r,t < 0 < F r +1 ,t , that implies F r +1 ,t F r,t = |X r +1 |2 = 0. So let us dene for [0, 1]

    (, [X 1, . . . X n +1 ]) = [X 1, . . . , X r , X r +1 , X r +2 , . . . , X n +1 ] .

    It is straightforward to verify that preserves the inequalities, i.e. : [0, 1] C P (r,n r )n C P (r,n r )n , (1, p) = p and (0, p) = [0, . . . , 0, 1, 0, . . . , 0] for all p

    C P (r,n r )n .Let us consider now the singular part. C P (r,s )n is identied by

    F r,t < 0 = F r +1 ,t = F r +2 ,t = . . . = F n s,t < F n s+1 ,t . (16)

    By using the denition of the F s it easy to conclude that [ X 1, . . . , X n +1 ] C P (r,s )n

    iff X r +2 = X r +3 = . . . = X n s = 0, X r +1 , X n s+1 = 0 and

    t

    r +1

    j =1 |X j |2

    = (1 t)n +1

    j = n s+1 |X j |2

    .

    Note if r + s = n 1 no homogeneous coordinate X is zero, but it is easyto adjust the following construction also to this case. We can then describeC P (r,s )n = S 1 S

    2(r + s )r,s () where

    S 2(r + s )r,s () = {(z, z ) = [z 1, . . . , z r , z r +1 = , 0, . . . , 0, 1, z n s+2 , . . . , z n +1 ],

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    > 0, tr

    j =1|z j |2 + t2 = (1 t)(1 +

    n +1

    j = n s+2|z j |2)}

    is a symplectic leaf parametrized by S 1. From the properties of the actionof the Cartan subgroup, we conclude that z r +1 z r +1 ei is a Poisson diffeomor-phism from S

    2(r + s )r,s () to S

    2(r + s)r,s (ei ). This proves that C P (r,s )n = S

    2(r + s)r,s (1)S 1as a Poisson manifold.

    The following map : [0, 1] S 2(r + s)r,s () S

    2(r + s)r,s ()

    (, (z, z )) = ( 0(z, )z, z ) 0(z, ) = 2 + (1 t)(1 2)t ri |z i|2 + t2contracts S

    2(r + s)r,s () to the open disk {(z 1, . . . , z r ),

    ri=1 |z i|

    2

    < (1 t)/t }.Finally let us describe the (ssc) symplectic groupoid integrating ( C P n , t ).

    It can be constructed as a symplectic reduction of the Lu-Weinstein symplecticgroupoid structure on SL (n + 1 , C ) integrating the Poisson Lie group ( SU (n +1), ), along the lines described in Subsection 2.2. So that we have

    G(C P n , t ) = {[g ], g SU (n + 1) , SB (n + 1 , C ), g U t(n) } , (17)where we denoted with [g ] the class of g SL (n+1 , C ) under the quotient mapSL (n + 1 , C ) U t (n)\SL (n + 1 , C ). As a smooth manifold, it can be describedas the bre bundle over

    CP n with bre U t (n) associated to the homogeneousprincipal bundle with the dressing action of U t (n) on U t (n) . We will denote the

    symplectic form as G. Since the l (or r ) bers are diffeomorphic to U t (n) thatis contractible, the restriction of the symplectic form G to the l-bres is exact;by applying Corollary 5.3 of [8] we conclude that G is exact.

    The homogenous action of SU (n + 1) on C P n is Poisson; thus it can be liftedto an action on the symplectic groupoid given by right SU (n + 1) multiplication.This action is symplectic and admits Lu momentum map. We are interested inits restriction to the action of the Cartan subgroup T n , which is hamiltonianin the usual sense. Under the identication t n = LieAn +1 , the momentum maph :

    G(C P n , t)

    t n is dened for each [g ]

    G(C P n , t ) as

    h[g ] = log prAn +1 ( ) . (18)

    For each H t n we denote the component of h along H as hH [g ] = H, h [g ] =ImTr( H logprAn +1 ). It is easy to check that hH is the R -valued groupoid cocyclethat lifts the Poisson fundamental vector eld realizing the t n action on C P n .

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    5 The bihamiltonian system on C P nWe introduce here the integrable model relevant for the quantization of the co-variant Poisson structures introduced in the previous section. It is, basically, thetoric structure on C P n dened by the Cartan subgroup action; we feel howevernecessary to emphasize its bihamiltonian structure.

    Let be the inverse of the Fubini-Study symplectic form on C P n denedin (35). We know from the results of [16], valid on compact Hermitian symmetricspaces, that the Bruhat-Poisson structure 0 and are compatible, i.e.

    [0, ] = 0 ,

    so that a Poisson pencil 0 + t t R , of SU (n +1)-covariant Poisson structuresis dened (see also [13] for an alternative proof). We are going to show that thefamily of Poisson structures t that we discussed in Section 4 coincide with thisPoisson pencil for t [0, 1].

    Lemma 5.1. For each t [0, 1], t = 0 + t .

    Proof . The proof is quite general and depends on the fact that there existsonly one Poisson pencil of SU (n + 1)covariant Poisson strcutures on C P n . Infact given the two covariant Poisson structures t and 0 then their difference is aSU (n +1)invariant 2tensor. In principle there is no guarantee that such tensoris Poisson; however, as proven in [27], up to scalar multiples, the only SU (n + 1)invariant 2tensor on C P n is the inverse of the Fubini-Study symplectic form.

    In Appendix B we give a self contained alternative proof of the above Lemmabased on a direct computation; this will x the scalar multiplying appearingin the above proof.

    From now on, we will use the notation t = 0 + t for t R , keeping in mindthat t is obtained as a quotient by a coisotropic subgroup only for t [0, 1].

    From the general theory of bihamiltonian systems (see [20] for an introduc-tion) we get that the recursion operator J = 0 has vanishing Nijenhuis torsion , i.e. for each couple (v1, v2) of vector elds

    T (J )(v1, v2) = [J v1, J v2]

    J ([J v1, v2) + [v1, J v2]

    J [v1, v2]) = 0 .

    Let us dene I k = 1k Tr J k , k = 1 . . . n . It is shown in Section 4 of [20] that

    these global functions satisfy the so called Lenard recursion relations

    dI k+1 = J dI k ;

    moreover, they are in involution with respect to both Poisson structures 0 and . Since the modular vector eld is the divergence of the Poisson tensor, we can

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    conclude that the modular vector eld with respect to the Fubini-Study volumeform V F S = 1n !

    n is independent on t. Moreover, it is a consequence of Theorem

    3.5 of [11] that this modular vector eld is the

    hamiltonian vector eld of I 1,

    i.e. F S divV F S t = (dTr I 1) .

    We are going to show that the fundamental vector elds of the Cartan actionsolve the spectral problem for the recursion operator.

    The homogeneous action of the Cartan subgroup T n SU (n + 1) is hamil-tonian with respect to with hamiltonian c : C P n t n . The component of calong H t n reads

    cH [g] c[g], H = Tr( g 1gH ) ,where is dened in (36). Let us introduce the following basis H k , k = 1 , . . . nfor t n = Lie T n , where for each k we have

    H k =i

    n + 1(n + 1 k) idk 00 k idn +1 k

    . (19)

    In terms of homogeneous and Lus coordinates we have

    ck cH k +k

    n + 1= 1 j k |X j |

    2

    j |X j |2= 1 ki=1

    11 + |yi|2

    , (20)

    respectively. It is clear that

    0 c1 c2 . . . cn 1 (21)so that the image of the momentum map is the standard simplex n R n .

    The action of the Cartan is hamiltonian with respect to but only Poissonwith respect to t for all t [0, 1], i.e. the fundamental vector elds of the actiondene non trivial classes in the Lichnerowicz-Poisson cohomology. It is possibleto dene local hamiltonians, as we are going to show in the following Proposition.

    Proposition 5.1. On each open subset of C P n where ck = 1

    t, the function

    bk = log |ck 1 + t|is a local hamiltonian for the fundamental vector eld of H k , i.e. H k = {bk , } t .Moreover,

    J (H k ) = ( ck 1)H k .

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    Proof . Let f C (C P n ); then we compute

    {bk , f } t =bkck {ck , f } t =

    bkck ({ck , f }0 + tH k (f )) .

    If we assume also that X n +1 = 0, we can use Lu coordinates {yi}, dened in(10), where the Poisson bivector 0 takes the form ( 11). This assumption doesnot spoil the global validity of the result and the calculation is done in this waybecause the computation in these coordinates is particularly transparent, whilea coordinate free computation is lengthy and unilluminating.

    Let us rst remark that in these coordinates

    H k = ik

    j =1

    y j yj y j y j .

    We then compute

    {bk , f } t =1

    ck 1 + t 1

    k j =1 (1 + |y j |2)+ t H k (f ) = H k (f ) ,

    where in the last passage we used ( 20). Finally, if we denote b0k = bk|t=0 =log |ck 1| then we computeJ (H k ) = 0dck =

    ck

    b0k

    0db0k =ck

    b0k

    H k = ( ck

    1)H k ,

    where in the rst equality we used the denition of the recursion operator J =0 .

    We then conclude the following result on the modular vector eld and classof (C P n , t ).

    Lemma 5.2. The modular vector eld of (C P n , t), t R , with respect to the Fubini-Study volume form is

    F S =n

    i=1

    H i ,

    where H i is the fundamental vector eld of H i t n . The modular class is non zero for each t [0, 1].

    We note that ni=1 H i =12 > 0 , the half sum of positive roots.

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    6 The quantization of (C P n , t)We introduce here the multiplicative integrable model on the symplectic groupoid

    G(C P n , t ), where t = 0 + t for t [0, 1]; we compute its groupoid of Bohr-Sommerfeld leaves and prove that it gives the quantization of ( C P n , t ).

    6.1 The multiplicative integrable model

    Let us introduce the integrable system. The action of the Cartan T n SU (n +1)on (C P n , ) is hamiltonian with momentum map c dened in (20); similarly theright action on G(C P n , t ) is hamiltonian with momentum map h given in (18).With {ci}and {hi}we denote the components of the momentum maps alongthe basis H i dened in (19). We remark that since pr An +1 : SB (n + 1 , C ) =An +1 N n +1 An +1 is a group homomorphism the components of h are groupoidcocycles. In particular, as a consequence of Lemma 5.2 the modular functionwith respect to the Fubini-Study volume form is

    f F S =n

    i=1

    hi . (22)

    Lemma 6.1. The 2n hamiltonians F = {l (ci), h i}ni=1 , , are in involution.Proof . Indeed, the ci s are in involution with respect to t for all t due to

    the bihamiltonian nature of the integrable model and l is a Poisson morphism;moreover for each [g ] G(C P n , t) we compute that

    {l (ci ), h j}([g ]) =ddt

    l (ci)(getH j )|t=0 =ddt

    ci(getH j )|t=0 = 0 ,where we used the T n -invariance of c and the fact that T n normalizes SB (n +1).

    It is important to compute the following formula.

    Lemma 6.2. On [g ] G(C P n , t ), we have that hk[g ] = det k , where det kdenotes the determinant of the rst k k minor.Proof . We have that

    hk[g ] = ImTr( H k logprAn +1 ) =k

    i=1

    log ii k

    n + 1

    n +1

    i=1

    log ii = logdet k ,

    where we used the fact that det = 1.

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    Let us rst show that the integrable system is multiplicative . For each (c, h)R n R n , let Lch = {[g ] G(C P n , t ) | c(g) = c, h( ) = h}denote the contourlevel set. The proof that L

    chis connected is postponed to Proposition 6.2 ii ).

    It is easy to check that the map that sends ( c, h) R n R n tor (c, h)i = 1 t + e h i (ci + t 1) , (23)

    denes a R n -action. Let R n R n denote the action groupoid and let R n R n | ndenote the action groupoid restricted to the standard simplex n R n .Proposition 6.1. The space of contour level sets GF (C P n , t ) inherits the struc-ture of topological groupoid. For t (0, 1), it is equivalent to the following wide subgroupoid of R n R n | n ,

    GF (C P

    n,

    t) =

    {(c, h) R n R n

    | n |c

    i= c

    i+1= 1

    t = h

    i= h

    i+1 }.

    If t {0, 1}then it is equivalent to the wide subgroupoid GF (C P n , t ) = {(c, h) R n R n | n | ci = 1 t = hi = 0}.

    As a consequence, the modular function with respect to the Fubini-Study volume form is multiplicatively integrable.

    Proof . Let the space of units ( GF )0 be the contour level sets of c. Let [g ]Lch ; we dene l(Lch ) = Lc and r (Lch ) = Lc(r [g ]). We want to show that thismap is well dened and coincides with (23). If ci = 1 t, then also ci(r ([g ])) =1

    t, since xing ci = 1

    t denes the Poisson submanifold P i(t), see (12) and

    (20). If ck(r (g )) = 1 t then the corresponding generator of the Cartan ishamiltonian with respect to t on every connected component of ck = 1 t withhamiltonian bk , where bk is the action variable dened in Proposition 5.1. As aconsequence, when restricted to this component hk has the following local exactform in groupoid cohomology, i.e. hk = l (bk) r (bk), that is, even though bkis not globally dened, the difference l (bk) r (bk) is. We then get

    hk[g ] = logck + t 1

    r (c, h)k + t 1,

    i.e. formula (23). Since h is a groupoid cocycle, the map that sends Lch to (c, h)R n R n | n respects the groupoid multiplication. We have to characterize itsimage. If ck = 1 t and [g ] Lch then we can choose g S (U (k)U (n +1 k))so that = g 1g for some U t (n) , see (17). Since S (U (k) U (n + 1 k))is a Poisson subgroup, and so it is invariant with respect to the dressing action,we can write in blocks

    g = g11 00 g22, = 11 120 22

    , g = g 11 00 g 22

    .

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    We can then compute

    exp hk[g ] = det k( ) = det( g 111 ) det( 11) det( g 11) = det( 11) = det k() ,

    since the det is invariant under the dressing action in U (k). Let us recall thatu t (n) = Ad t u0(n) (see (7)), so that we can write = expAd t for u0(n) , i.e. ij = 0 only if i = 1 , j > 1. A direct computation shows that

    prAn +1 (Adt ) = t(1 t) 1,n +1 diag(1, 0, . . . , 1) ,so that we see from [18] that hk[g ] = t(1 t) 1,n +1 and the result easilyfollows.Remark 6.3. The orbits of

    GF (C P n , t ) correspond to the decomposition of

    C P n in regular Poisson strata given in ( 9) for t = 0 , 1 and in Proposition 4.1 fort (0, 1). Moreover, the restrictions on h reect the fact that when t (0, 1) oneach non maximal stratum the Cartan vector elds which are transversal to thesymplectic leaves collapse to a single S 1 generator, while for t = 0 , 1 they are allzero.

    6.2 The Bohr-Sommerfeld groupoid GbsF (C P n , t )In order to compute the Bohr-Sommerfeld leaves we need to show that Lch is

    connected and describe H 1(Lch , Z ). It is obvious that for each c n Lc is aT n -orbit.It is easy to produce 1-cycles in Lch . The hamiltonian ux of h j , j = 1 . . . n

    and of l (b j ) for c j = 1 t dene a set of 1-cycles in Lch . Indeed, x [g ] Lch ; j (t) = [getH j ], for j = 1 , . . . n , is the ux of h j . Let us compute the ux of l (b j ), for c j = 1 t, and show that it is periodic.Lemma 6.4. The hamiltonian ux of l (b j ) passing through [g ] Lch is given by

    j (t) = [geB H j [g]t ] ,

    where BH j [g] = H j +

    a

    T a (b j )([g])T a sl (n + 1 , C ) ,

    for some basis {T a}of su (n + 1) and the dual basis {T a}of sb (n + 1) . Moreover, j (t) = j (t + 2 ) for each t.

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    Proof . Let j (t) = [g j (t) j (t)] be the hamiltonian ux of l (b j ). Sincel( j (t) ) = [ getH j ] then [g j (t) j (t) ] = [getH j j (t)]. By using the denition of the Poisson tensor ( 2), especially the third term thereof that entangles G andG , we get that

    j (t) = (a

    T a (b j )[getH j ]T a ) j (t) = e tH j (a

    T a (b j )[g]T a)etH j j (t) , (24)

    note that in the second equality we used the equivariance of the momentum map,which can be seen explicitly from

    T a (c j )([g]) = Tr[ g 1g[T a , H j ]], (25)

    where is the weight of the fundamental representation of SU (n + 1).

    The solution to the equation ( 24) is

    j (t) = e tH j etB H j [g] .

    We just have to show that j is periodic. It is a direct computation to obtaina block matrix form for BH j [g]: choose the generators of the Lie algebra of N n +1 SB (n + 1 , C ) as (E ij )kl = ik jl and iE ij with i < j , and the dualgenerators i(E ij + E ji ) and (E ij E ji ). For k , l < j or k , l > j the lattercommute with H j , and hence from (25) the corresponding blocks of the matrixT a (c j )([g]) vanishes, the blocks that remain are

    BH j = in + 1 (n + 1 k) idk A0 k idn +1 k. (26)

    It can be diagonalized to H j with the similarity transformation by

    1 A/ (n + 1)0 1 .

    The periodicity of j then follows from the observation that exp 2 H j Z nU t (n).

    We are going to show that { j , k}generate H 1(Lch , Z ). We will use argu-ments based on the use of the homotopy long exact sequence associated to abration. We need to introduce some denitions and preliminary results.

    Let x t y denote that x, y C P n live in the same symplectic leaf of t . LetLch = {(x, y) Lc Lr (c,h ) | x t y} Lc Lr (c,h ); we have clearly that

    (l r ) : Lch Lch . (27)Lemma 6.5. Let x Lc and let T x = {s T n |xs t x}, Then we see that

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    i) For each x, y Lc, T x = T y T c.ii ) For each x Lc, T x (xT c) = H i (x) , ci = 1

    t .

    iii ) The hamiltonian ux of l (b j ), for c j = 1 t, described in Lemma 6.4denes an action of T c on Lch that commutes with the T n -action.iv) T c T n acts transitively on Lch as (x, y)(s, k ) = ( xsk,yk ), where (x, y)Lch , (s, k ) T c T n ; the map ( 27 ) intertwines the actions.

    Proof . i) Since the action of the Cartan is by Poisson diffeomorphisms, itstangent action commutes with the sharp map, so that if x t y then x t y for any T n . Let x Lc and s T x ; let y = x , for some T n . We havethat x t xs implies y t ys so that s T y .

    ii ) Let x Lc and let S x be the symplectic leaf through x. If ci = 1 t thenH i is locally hamiltonian and clearly preserves the leaf. Let ci = 1 t and letH i (x) = 0 so that di(x) is dened and non zero. Clearly H i (x), di(x) = 1.We can choose an open U x containing x such that the points y such that di(y)is dened and ci(y) = 1 t are dense. For such y, t (di(y)) = / bi = ( ci 1 +t)/ ci that extends to 0 in x. So we have that di(x) N x S x and H i (x) T xS x .iii ) This is just a rephrasing of Lemma 6.4.

    iv) Let us rst show transitivity. Let ( x, y), (x0, y0) Lch ; then we know that(x, y) = ( x0k1, y0k2) = ( x0k1k 12 , y0)(1, k2) for k1, k2 T n , so that x0k1k 12y0 x0 so that k1k 12 T c and (x, y) = ( x0, y0)(k1k

    12 , k2). It is a trivial fact

    that l

    r intertwines the T n -action; the same property with respect to T c follows

    from the properties of the hamiltonian vector elds of l (b j ).For each [g] C P n let us denote N n +1 [g] = {n N n +1 | gn U t (n) , [gn ] =[g]}= g 1U t (n) U t (n)g N n +1 ; it is clearly a subgroup of N n +1 . Next we shallshow that this subgroup is of exponential type (and hence contractible) and that

    two such subgroups are isomorphic if the base points live in the same T n -orbit.

    Lemma 6.6. i) N n +1 [g] N n +1 is connected and then of exponential type.

    ii) For s T n , the map n s 1 n is a group isomorphism from N n +1 [gs] N n +1 [g].

    Proof . i) We can easily see that N n +1 [g] = [g]G(C P n , t )[g] N n +1 . FromLemma 1 in [10], we know that0([g]G(C P n , t)[g]) = 1(S [g]) ,

    where S [g] is the symplectic leaf through [g]. Since in our case the symplecticleaves are all contractible (see Lemma 4.2), we conclude that N n +1 [g] is the inter-section of two connected subgroups of SB (n +1 , C ); since connected subgroups of

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    an exponential group are of exponential type and are closed under the intersec-tion (see [24] pg.9) we get the result. Point ii ) is obtained by direct computation.

    Let N c denote the group bundle over Lc with bre N n +1 [g] over [g] Lc andlet p : N c Lc denote the projection; it acts on Lch with anchor l : Lch Lcand action a : N c pl Lch Lch given by

    a(n, [g ]) = [gn ] . (28)

    This action preserves the bres of ( 27).

    Proposition 6.2. i) The map ( 27 ) denes a bre bundle and the groupoid action of N c on Lch Lch

    N c p

    Lch

    l ~ ~

    l r

    Lc Lch

    denes a principal groupoid bundle.

    ii) Lch is connected and H 1(Lch , Z ) is generated by j , k .

    Proof. i) Let us show rst that we have a groupoid principal bundle accordingto denition in section 5.7 of [21]. The map l r : Lch Lch is a surjectivesubmersion as a consequence of Lemma 6.5 (iv). Next we have to show that themap

    (a, pr2) : N c Lc L c Lch Lch L ch L ch Lch , (n, [g ]) ([gn ], [g ])is a diffeomorphism. Let ([g ], [g ]) Lch L chL ch Lch , i.e. (lr )[g ] = (lr )[g ].Then we see that [ g] = [g] and g = hg for h U t (n). Dene n = 1 N n +1 .Since we also have that g, g U t (n) and recalling that the dressing action of U t (n) preserves U t (n) (and viceversa), one shows that

    gn = g g 1 = g h (g

    1) = g h (g ) 1 U t (n) ,

    gn

    = g 1

    = ( hg

    ) 1

    = h(g ) 1

    g = h

    g ,for h, h U t (n). We conclude that n N n +1 [g] and the map ([ g ][g ]) (n, [g ]) denes an inverse of (a, pr2). The map N c : Lch L ch L ch Lch N c,([g ][g ]) 1, is called the division map.Due to Lemma 6.5 iv), Lch is homeomorphic to T c T n /T for T being thestability subgroup. Let us x [ g0 0] Lch and let ( x0, y0) = ( l r )[g0 0] andV 0 an open containing ( x0, y0) and admitting a section ( s, k ) : V 0 T c T n

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    such that ( x, y) = ( x0, y0)(s(x, y), k(x, y)) for each (x, y) V 0. Let [g ] Lch |V 0and (x, y) = ( l r )[g ]; then [g ](s(x, y), k(x, y)) 1 (l r ) 1(x0, y0) and[g ]

    N c

    ([g0,

    0], [g ](s(x, y), k(x, y)) 1) N

    [g0 ]gives the local trivialization.

    ii ) We apply the long exact homotopy sequence to the bre bundle ( 27).Since the bre is contractible we get that k(Lch ) = k(Lch ), in particular Lch isconnected. The same homotopy sequence applied to T T c T n Lch , sinceT is connected, shows that the map 1(T c T n ) 1(Lch ) = 1(Lch ), sendingthe generators of T c T n to { j , k}, is surjective. The statement on homologyeasily follows.

    We are now ready for the computation of the groupoid of Bohr-Sommerfeldleaves. From the discussion at the end of Section 3, we know that the symplecticform G is exact. In the following Lemma we show that there exists a primitiveG, i.e. a one form G satisfying dG = G, adapted to the momentum maph of the action of the Cartan T n . Remark that the fact that h is an R n -valuedcocycle plays a role.

    Lemma 6.7. There exists a primitive G of the symplectic form G such that for each H t n

    H G = hH ,where H is the fundamental vector eld corresponding to H .

    Proof. Since T n is compact, by averaging, we can always nd a T n -invariantprimitive G, i.e. LH G = 0. By using the denition of the momentum map

    we easily see that H G + hH = aH R .

    Since the submanifolds of units G0 = C P n is lagrangian then G|G0 is closed andsince H 1(C P n ) = 0 then G|G0 is exact. The T n -action on G(C P n , t ) preservesG0; moreover for any orbit H C P n of H we have that hH | = 0 because hH ,being an R -valued cocycle, is zero on the units. We then conclude that

    H G = aH = 0 .From now on, G will denote a xed primitive satisfying the condition of Lemma 6.7.

    Theorem 6.3. For R , the groupoid of Bohr-Sommerfeld leaves is the fol-lowing subgroupoid of GF (C P n , t ) described in Proposition 6.1

    GbsF (C P n , t ) = {(c, h) GF (C P n , t ) | log |ck 1 + t| Z , hk Z } .

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    It is an etale groupoid and admits a unique left Haar system for al l t [0, 1].The generators of the Cartan subgroup are quantized to the groupoid cocycles h

    i(c, h) = h

    i; in particular the modular cocycle reads

    f F S (c, h) =n

    i=1

    h i Z .

    Proof . Let Lch GF (C P n , t ) and let ck = 1 t. Let G be a primitive of Gas in Lemma 6.7; we have to pair it with the generators j , k of H 1(Lch , Z ).We then compute the Bohr-Sommerfeld conditions as

    k G = k G|G[g ] = k l S [g ] = l(k ) S [g ] = bk 2 = 2 log |ck1+ t| 2 Z ,where k : D G(C P n , t)[g] extends k to the disk and in the second equalitywe use the fact that |G[g ] = l S [g ] . Since H G = hH , for each H t n , weeasily compute

    j G = 20 H j G dt = 2h j 2 Z .The fact that the set of Bohr-Sommerfeld leaves GbsF (C P n , t ) is a subgroupoid isa straightforward computation.

    Let (c, h) GbsF (C P n , t) and let I (c, h) = f 1F S (I (f F S (c, h))) the open neigh-borhood of f F S (c, h) of radius > 0. It is clear that for < 1, I (c, h) = {(c , h)GbsF (C P n , t )}so that l is a local homeomorphism. By applying Proposition 2.2we conclude that GbsF (C P n , t ) is etale and admits a unique left Haar system, thecounting measure on the l-bres.

    The space of units of GbsF (C P n , t) is the following subset of the standardsimplex Zn (t) = {c n | ck = 1 t e

    n k , nk Z }.For > 0, the quasi invariant measure on Zn (t) associated to f F S is computed

    as

    F S

    (c) = det( J (c) + t) =

    k|c

    k 1 + t

    |= exp

    n

    k=1

    nk

    .

    The components of GbsF (C P n , t ) are labeled by the couples ( r, s ) of non neg-ative integers such that r + s n. Indeed the component GbsF (C P n , t )(r,s ) is therestriction of GbsF (C P n , t ) to ck = 1 t for r < k n s. Let us describe thiscomponent more explicitly. Let Z n act by translation on Z n , where Z = Z {}

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    0 1 t 10 1 t 1

    1 t

    1

    Figure 1: Z1 (t) and Z2 (t)

    be the one point compactication. Let Zn

    Z n be the action groupoid. Let Zr,s (t) Z

    nbe dened as

    {(m, . . . , n ) Zr

    Zn r s

    Zs

    | 1

    log(1 t) m i m i+1 ,n i n i+1

    1

    log t} . (29)Note that Zr,s (t) corresponds to the stratum C P (r,s )n of dimension 2(r + s) of

    (C P n , t ) described in Proposition 4.1 for t (0, 1). Indeed one can check that(16) can be restated as

    cr < 1 t = cr +1 = cr +2 = . . . = cn s < c n s+1 ,whose BS leaves are exactly those in Zr,s (t). Analogously r, 0(0) correspondsto the 2r -dimensional Bruhat cell for t = 0.

    For t (0, 1), the component GbsF (C P n , t )(r,s ) is isomorphic to the followingsubgroupoid of the action groupoid Z n Z n | Zr,s

    {(q, p) Z

    n

    Z n

    |Z

    r,s |q

    i= q

    i+1=

    = p

    i= p

    i+1 }.

    If t = 0 there are only components like GbsF (C P n , t )(r, 0) = Gbs,(r, 0) that is isomor-phic to the following subgroupoid of the action groupoid Z n Z n | Zr, 0{(q, p) Z

    n

    Z n | Zr, 0 | q i = = pi = 0}.The description for t = 1 is analogous.

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    6.3 Quantization of Poisson morphisms

    We discuss in this Section the covariance of the correspondence that sends(C P n , t ) to the etale groupoid GbsF (C P n , t ), established in Theorem 6.3, withrespect to some of the Poisson isomorphisms that we discussed in previous Sec-tions.

    Let us rst dene the map : Zn

    Zn

    as (q )i = q n +1 i . The followingLemma, proved by a direct check, shows that the Poisson isomorphism between(C P n , t ) and (C P n , 1 t ) in Lemma 4.1 is quantized by a groupoid isomor-phism.Lemma 6.8. The map from GbsF (C P n , t )(r,s ) to GbsF (C P n , 1 t )(s,r ) sending (q, p) to ((q ), ( p)) extends to a groupoid isomorphism : GbsF (C P n , t )

    GbsF (C P n ,

    1 t ) for each t [0, 1].

    We are then going to show that the Poisson embeddings

    ik : P k(t) pt (S (U (k) U (n + 1 k)) (C P n , t )described in Section 4 are quantized to subgroupoids of GbsF (C P n , t ). As a resultwe get at the same time the quantization of these Poisson structures and showthe covariance of the quantization with respect to these Poisson morphisms.

    We know from general results that G(C P n , t )|P k (t) = {[g ] | ck[g] = 1 t} is coisotropic and the symplectic groupoid integrating P k(t) is obtained ascoisotropic reduction. The coisotropic foliation is generated by the hamiltonianux of ck

    G(P k(t), t ) = G(C P n , t )// {ck = 1 t} .The set of hamiltonians F = {ci , i = k, h j}clearly descends to a set dening amultiplicative integrable model. All the analysis of the previous Section descendsto the quotient, so that we get the following Proposition.

    Proposition 6.4. For t [0, 1], the Poisson submanifold P k(t) is quantized by the subgroupoid

    GbsF (P k(t), t ) = (c, h) GbsF (C P n , t ) | ck = 1 t .By applying the contravariant functor assigning to a groupoid its C -algebra,

    we get a family of C algebra short exact sequences0 I k C (GbsF (C P n , t )) C (GbsF (P k(t), t )) 0 .

    It can be remarked that putting cn 1 = 1 t in GF (P n (t), t ), and then cn 2and so on, we obtain a chain of subgroupoids, that corresponds to the chain of Poisson embeddings of spheres appearing as the right side of ( 14).

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    7 Quantum homogeneous spaces as groupoidC -algebras

    In this Section we will discuss how the quantization of the Poisson pencil andPoisson submanifolds that we described in Theorem 6.3 and Proposition 6.4 isrelated to quantum spaces appearing in quantum group theory.

    The family of quantum homogeneous spaces C P n,q,t was rst introduced in[12], generalizing previous work by Podles on C P 1. In such paper a parametrizedfamily of right coideal subalgebras of the Hopf algebra A(SU q(n + 1)) of rep-resentative functions was introduced. The algebra A(C P n,q,t ) is then dened asthe subalgebra of coinvariants and is characterized in terms of generators andrelations: it is a noncommutative deformation of the algebra of polynomial func-tions on C P n whose semiclassical limit gives the Poisson structure ( C P n , t ). In asimilar manner the odd-dimensional quantum spheres A(S 2n 1q ) had already beenpreviously dened as right coideal subalgebras of A(SU q(n + 1)) in [30] as oneof the rst examples of quantum homogeneous spaces. It has to be remarked herethat quantum odd spheres and standard complex projective spaces are quotientsby quantum subgroups, while in the non standard case the quotient procedurerequires the use of coisotropic quantum subgroups, in perfect agreement withwhat happens at the semiclassical level.

    There is a canonical way to complete both these polynomial algebras tothe C algebra C (C P n,q,t ) and C(S 2n 1q ). The C norm is dened as a =sup (a) of the operator norms among bounded representations of thepolynomial algebra.In [25] Sheu, relying on the representation theory of A(C P n,q,t ), proved thatsuch algebra is in fact, the groupoid C algebra of a topological groupoid whent = 0 , 1, and also, for arbitrary t in the case n = 1. In the more general case inwhich n > 1 and 0 < t < 1 his results were less conclusive though, basing on thesame construction, he was able to show that C (C P n,q,t ) is contained in a groupoidC algebra. On the other hand he was able to prove the existence of a shortexact sequence of C algebras 0 I C (C P n,q,t ) C (S 2n 1q ) 0 relating thenon standard quantum complex projective spaces to odddimensional quantumspheres.

    We are going to show how our quantization of ( C P n , t), for t [0, 1], isrelated to these results. In the cases covered by Sheus description like ( C P n , 0)and (P k(t) = S 2n 1, t ), for k = 1 , n, we will then establish that our quantizationof the symplectic groupoid produces the same C -algebra of the quantum spacesobtained via quantum group techniques. We conjecture that this is also true for(C P n , t ) and (P k(t), t ), for t = (0 , 1) and 1 < k < n .

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    7.1 The standard C P n,q, 0In [25], it was shown that the C -algebra of standard quantum projective spaces isthe C -algebra of a groupoid T n dened as follows. Let us consider the translationaction of Z n on Z

    n(trivial on the points at innity) and restrict the action

    groupoid to Nn:

    T n = Z n Zn

    |N n = {( j ; k) Z n Zn ki , j i + ki 0}.

    Consider then the subgroupoid T n T n given by:T n = {( j ; k) T n | ki = =

    i

    k=1

    jk = 0 , j l = 0 , l i + 1} .

    Finally let T n be the quotient T n / where ( j ; k) ( j ; k1, . . . , k i 1, , . . . , ) if ki = . We denote with ( j ; k) the equivalence class. It is shown in Theorem1 of [25] that C (C P n,q, 0) C (T n ). The choice of this particular descriptionof the groupoid C algebra depends on a representation-theoretic constructionof C (C P n,q, 0) which could be seen as an analogue of the symplectic foliationdescription in terms of Lus coordinates (see Appendix A).

    Proposition 7.1. The map : GbsF (C P n , 0) T n , dened as (c, h) = ( p1, p2 p1, . . . ; q 1, q 2 q 1, . . .) , ci = 1 e

    qi , hi = pi ,

    is an isomorphism of topological groupoids.Proof . The map is easily seen to be well dened once one recalls the range of

    (c, h) described in Proposition 6.1 and the characterization of Bohr-Sommerfeldleaves given in Theorem 6.3. Remark that in the denition of the map thereis no need of specifying the denition of q i q i 1 = for i r + 1, whencr 1 < c r = 1. Indeed, since q r q r 1 is well dened and equal to , theseentries are not relevant in the equivalence class ( c, h).

    It is an easy check to verify that is a continuous groupoid morphism. Theinverse is given by

    1( j ; k) = ( c, h), ci = 1 e il =1 k l , hi = i

    l=1

    j l,

    that is well dened and continuous.Let us also remark that the functorial quantization of Poisson submanifolds

    P 1(0), . . . , P n (0), in this case, exactly gives rise to the composition sequencedescribed in Corollary 3 of [25], as a simple checking shows.

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    7.2 The non standard C P 1,q,t , for t (0, 1)Let us now move to the non standard case. In such case Sheu produced anexplicit groupoid description only for n = 1. In the n > 1 case, the non standardquantum complex projective space is described just as a sub C algebra of agroupoid C algebra.

    For n = 1 in [25] it was proved that C (C P 1,q,t ) C (G ), t (0, 1), where Gis the subgroupoid of the action groupoid Z 2 Z

    2

    |N 2G = {( j, j, k 1, k2) Z 2 Z

    2

    |N 2 |k1 or k2 = }.Let us recall that GbsF (C P 1, t) has three components

    GbsF (C P 1, t )(1,0) = {(q, p) Z Z |q q t , q + p q t }G

    bs

    F (C P

    1,

    t)(0,1) =

    {(q, p) Z

    Z

    |q

    q

    t, q + p

    q

    t }GbsF (C P 1, t )(0,0) = {(, p) Z Z }where q t = 1 ln(1 t) and q t = 1 ln t. A direct computation shows that:Proposition 7.2. The map : GbsF (C P 1, t) G dened on each component by:

    (q, p) =( p, p,, q q t ) if (q, p) GbsF (C P 1, t )(1,0)( p, p, q q t , ) if (q, p) GbsF (C P 1, t )(0,1)( p, p,, ) if (q, p) GbsF (C P 1, t )(0,0)

    is an isomorphism of topological groupoids for all t (0, 1).

    The existence of this isomorphism also proves that the groupoids of BS-leavesand so the groupoid C algebras are not depending on t, as long as 0 < t < 1.

    7.3 Odd dimensional quantum spheres S 2n1q From the discussion in Section 4 we know that the Poisson submanifold P n (t)(C P n , t ) is isomorphic to the odd Poisson sphere S 2n 1. We want to compare thegroupoid GbsF (P n (t), t ) appearing in Proposition 6.4 with the characterization of the quantum sphere given in [ 26], where it is shown that C (S 2n 1q ) C (F n 1),where F n 1 is a groupoid that we are going to describe.

    Let

    F n 1 be the following subgroupoid of the action groupoid (with the rst

    Z acting trivially)

    {(z ; x; w) Z (Z n 1 Zn 1)|N n 1 |wi = z =

    j

    x j , xk = 0 , k > i }Dene on it the equivalence relation ( z ; x; w) (z ; x; w1, . . . , w i, ) if wi = and let F n 1 = F n 1/ . Let us denote with ( z ; x; w) F n 1 the equivalenceclass. Let, as before, q t = 1 log(1 t).

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    system exists for all compact hermitian symmetric spaces; most of the resultsextend to the general case. In particular, the proof of the multiplicativity of themodular function is general; given the eigenvalues of the Nijenhuis operator, thestructure maps of the groupoid of leaves are expressed by the same formulas.

    While the existence of the bihamiltonian system is known since long time, atthe best of our knowledge, C P n is the only case that has been investigated. Thekey characterization that we need is the diagonalization of the Nijenhuis operator:for C P n is a result easy to obtain (Proposition 5.1). In the general case we dontknow if global eigenvalues of the Nijenhuis operator exist, how to describe themand nally if the whole quantization procedure works. This requires furtherinvestigation, that makes a reasonable plan for our future research.Prequantization cocycle . The prequantization cocycle did not appear in our nalquantization setting; this means that we produce our quantum algebras by tak-ing it to be trivial. On one side, in those cases where we can compare our resultswith quantum group constructions, via Sheus results, there is no need of a nontrivial 2-cocycle. On the other side we know that, at least in some examplesbut we expect it in general, its cohomology class is not trivial. It is possiblethat it makes a difference what kind of cohomology we consider, differentiableor just continuous, and that for the C -algebras smooth properties are not rele-vant. An indication in this direction is given by recent results of [ 28] where it isshown that certain Poisson-homogeneous structures which are cohomologicallynon trivial may become trivial if one admits vector elds which are smooth alongthe symplectic leaves and globally only continuous. In any case, the role of this

    prequantization cocycle is still unclear.Morita equivalences . The Poisson structures ( C P n , t) are not in general Poissondiffeomorphic for different t. In [4, 23] it was proven that non standard C P 1 Pois-son structures are non diffeomorphic for different values of t. In fact the Poissoninvariant allowing to distinguish them is the so called regolarized Liouville vol-ume V () = ln 1 t1+ t . However, since the modular periods are the same, apartfrom the singular t = 0 , 1 case, non standard Poisson C P 1 are all Poisson-Moritaequivalent. The fact that the corresponding groupoid of Bohr-Sommerfeld leavesdoes not depend, as topological groupoid, on t can be seen as a quantum ana-logue of this fact. It would be interesting to analyze these properties for highern. We remark that the example developed in [ 3] shows that even non Poisson-Morita equivalent structures can produce the same topological groupoid of BSleaves. We expect that, like in the n = 1 case, all (C P n , t) are at least Moritaequivalent. We plan to investigate this equivalence and how it is quantized tothe Bohr-Sommerfeld groupoids.

    Acknowledgments The research of J.Q. is supported by the Luxembourg FNRgrant PDR 2011-2, and by the UL Grant GeoAlgPhys 2011-2013

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    A Review of Lus Coordinates

    In this section, we recall the derivation of Lus parametrization of Bruhat cells[17]. Do notice that, in contrast to the main text, the homogeneous spaces such as the ag manifolds are presented as right cosets, e.g. C P n = SU (n + 1) /U (n).This is to avoid certain awkwardness in the presentation, no conclusions in themain text will be changed.

    We focus on the case K = SU (n), K = SB (n) and G = K C = SL (n, C ),for such groups, the concepts such as the Bruhat Poisson structure, Bruhat cells,dressing actions, etc can all be understood rather concretely.

    Our focus is the ag manifold ag(n1, , nk), where the string of increasingintegers 0 = n0 < n 1 < n 2 < < n k < n k+1 = n denotes the nested subspacesof C n of dimension n1, n2, , nk . A point in the ag can be represented by amatrix g SL (n), suppose g = [v1, vn ], where vi are column vectors, then theag in question is span(v1, vn 1 ) span( v1, vn 2 ) span( v1, vn k ) C n .There are certain redundancies of such a parametrization, for example, a rightmultiplication by a matrix in SB does not change the ag, since these matricesare upper-triangular. In general, unless n i n i 1 = 1 for all 1 i k, i.e. thefull-ag manifold, the stability group is bigger than SB .

    In contrast, the left multiplication by SB on g = [v1, vn ] SU induces a non-trivial action on the ag, so one can decompose the ag manifoldinto orbits of this action. The action is related to the dressing action reviewedin the text, write g = g g, with g SU and g SB . One can discard gsince g

    g and g give the same ag. If the stability group were exactly SB ,

    then the action can be identied with the dressing transformation.To cut down the stability group one can do the following. The identity

    matrix corresponds to [ e1, , en ], where ei are the standard basis vectors, whilean element of the Weyl group W , when represented as a matrix, corresponds to[e(1) , , e(n ) ], where is a permutation (recall that the Weyl group for SU arenone other than permutations). One can certainly reach any point in the ag byleft multiplying SB to a suitable w W , thus one can label the orbits of the leftSB action by elements of W . Further introduce a subset of permutations calledshuffles, denoted as n (n1, n2, , n k), consisting of { S n |(i) < ( j ) if nk

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    Once this condition is enforced, the stability group is cut down to being the rightmultiplication by SB . To summarize, the ag has the following decomposition

    ag(n1 , nk) = w SB w SB/SB,where the summation is over n (n1, n2, , nk)-shuffles.Next, we shall identify a subgroup N w SB whose left action on a particularshuffle w is free. Let sb , then the action of on w is nonzero if p1Adw 1 = 0,where p1 : sl su is the projection. Recall that SB is a subgroup generated bythe positive roots of su and it , where t is the Cartan. One sees that for a givenw, the subset of roots w = { > 0|Adw 1 < 0}will generate a nonzero actionsince p1Adw 1 = 0, and thus N w is generated by w .

    One can always decompose an element w of the Weyl group into a string of Weyl reections w = w k w 1 , where each i is a simple root and w i is theWeyl reection of i . If we denote by C w the image of the left SB action on w,then Lu showed that there is a diffeomorphism

    C k C 1 C w, (30)where we have abbreviated C w i as C i . The map is given by taking the productof the lhs, furthermore, this map is a Poisson diffeomorphism, where the lhsis given the product Poisson structure. This gives us a very practical way of parametrizing the orbit C w by that of C . Recall that a simple root gives anembedding of SU (2) into SU (n), so we may focus on the SU (2) case.

    For SU (2), the non-trivial Weyl group element can be chosen as w = i2,where i are Pauli matrices. There is only one simple root + = ( 1 + i2)/ 2,and w 1+ w = , so N w is generated by + , and hence can be parametrizedas

    (y) = ey+ = 1 y0 1 , yC

    it is then easy to work out the orbit of the dressing action

    w = 1 y 11 y

    00 1

    1 2y0 1 , = (1 + |y|2)1/ 2,where the last two factors belong to SB . Thus the orbit of the dressing actionis parametrized as

    g(y) =1 y 1

    1 y, = (1 + |y|2)1/ 2. (31)

    This orbit actually covers all points of the ag C P 1 except the point z = , i.e.the unique dimension 2 Schubert cell of C P 1.37

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    To work out the general formula Eq .30 for C P n , we x the simple roots forSU (n + 1) to be: i = [0, 1, 1

    i,i +1

    , 0, 0], then the Weyl reection w i is anidentity matrix except that the ( i, i + 1) diagonal block is occupied by i2. Nextwe work out as an example the parametrization of the largest cell of C P n . TheWeyl group element corresponding to this cell is the shuffle n +1 (1), and it canbe decomposed into

    n +1 (1) = w n w 1 . (32)The orbit of each C i is parametrized as a rank ( n + 1) block diagonal matrix,

    g i(yi) =1 i 1 0 0

    0 g(yi) 0

    0 0 1 n i, (33)

    where g(y) is dened in Eq.31. It is then not hard to compute the product Eq. 30explicitly, in fact, we will only need the rst column of C w , which is given by

    [g n (yn ) g 1(y1)] ,1=

    11 n

    (1)n y1 2 n , yn 2 n 1 n , yn 1 n , yn , 1T ,

    where i = (1 + |yi|)1/ 2. This rst column of C w should be identied with thehomogeneous coordinates [X 1, X n +1 ] of C P n , and C w corresponds to the cellwhere X n +1 = 0. For convenience of the identication, we will make a redenitionof variables yn 1 yn 1, yn 3 yn 3, , then we have

    z n =X n

    X n +1= yn , z n 1 =

    X n 1X n +1

    = yn 1 n , .And we have the inverse map

    yn = z n ,yn 1 = z n 1(1 + |z n |2) 1/ 2,yn 2 = z n 2(1 + |z n 1|2 + |z n |2) 1/ 2,

    y1 = z 1(1 + |z 2|2 + + |z n |2) 1/ 2. (34)These are the Lus coordinates quoted in ( 10). They can be thought as coordi-nates on leaves of different copies of SU (2) projected onto the complex projectivespace. Let us remark that in principle all that is contained in here applies also tothe nonstandard case, the only difference being in the projection map pt , whichwill pick a t-depending linear combination of the rst and last column of C w.

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    A.1 The Kirillov-Kostant symplectic form

    In the text, we shall also need the symplectic Kirillov-Kostant symplectic formfor C P n , we review here the construction of this form and some concrete formulaeand show that it takes a very simple form in Lus coordinates.

    For a Lie group K , the coadjoint orbit associated with g is O ={(Ad 1g ) |g K }, and it is diffeomorphic to K/H where H is the stabilitygroup of . Over O , one can write down a form

    = idTr[g 1dg] = iTr[( g 1dg)2], (35)It is non-degenerate and closed (but not exact).

    In the case of K = SU (n + 1), the coadjoint orbit passing through

    =1

    n + 1diag[n,

    1,

    ,

    1

    n] (36)

    is nothing but C P n , since H in this case is clearly S (U (1) U (n)).For SU (2) the computation is simple, letting = [1/ 2, 1/ 2] and parametrizea group element as g(y)ei 3 (where g(y) is dened in Eq.31), one gets =

    i2

    dTr[g 1dg3] = idy dy(1 + |y|2)2, (37)

    the Fubini-Study Kahler form on C P 1.To obtain the Kirillov-Kostant for C P n , we take as in Eq.36, i.e. the highest

    weight of the fundamental representation of SU (n +1). Then we shall use the Luparametrization for SU (n + 1) /S (U (1)

    U (n)). The largest cell in this quotient

    is labeled by the shuffle w = n +1 (1), and it can be decomposed as in Eq. 32 Thecell C w is now a product

    C w = C n C 2 C 1 ,We parametrize this product C n C 2 C 1 as g n (yn ) g 1(y1), where g i (yi) isdened in Eq.33.

    One may now proceed to compute the Liouville form Tr[ g 1dg] leading to

    Tr[g 1dg] = iA(z 1) +i21A(y2) +

    i21

    22A(y3) +

    i21 2n 1

    A(yn ), (38)

    A(y) =i2

    ydy

    ydy

    (1 + |y|2) .Let us plug in the change of variables Eq. 34 and get the neat result

    Tr[g 1dg] = 12

    ni=1 z i dz i c.c

    1 + ni=1 |z i|2.

    The differential of the above 1-form is recognized as the Fubini-Study Kahlerform for C P n .

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    A.2 Concrete formulae for the Bruhat Poisson structure

    In this section we collect some formulae for the Poisson structure of SU (2) andits dual group SB (2), as well as the symplectic form on the double SL (2). Thecomputation leading to these results are quite lengthy and is omitted.

    Parametrize an element of SU (2) as g(y)ei 3 , and plug it into Eq. 6, a straight-forward calculation shows

    SU (2) = i(1 + |y|2)

    y y

    ,

    which vanishes when y . As mentioned above the map Eq. 30 is a Poissonmap, so the Poisson tensor for a general cell is just the sum of the results above.The group SB (2) is dual group of SU (2) , and has a simple parametrization

    = e 3

    eu +

    , and the dual Poisson structure in this coordinate system is

    2SB (2) = 2 i (1 + |u|2