A Global Quantum Duality Principle

for Subgroups and Homogeneous Spaces

Nicola Ciccoli, Fabio Gavarini

Received: May 5, 2013 Revised: February 10, 2014

Communicated by Joachim Cuntz

Abstract. For a complex or real algebraic group G, with 𝔤 := Lie(G) , quantizations of global type are suitable Hopf algebras F_{q}[G] or U_{q}(𝔤) over ℂ[q, q^{−1}]. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on 𝔤: correspondingly, one has dual Poisson groups G^{∗} and a dual Lie bialgebra 𝔤^{∗}. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called flat in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever.

The global quantum duality principle (GQDP), as developed in [F. Gavarini, The global quantum duality principle, Journ. für die Reine Angew. Math. 612 (2007), 17–33.], associates with any global quantization of G, or of 𝔤, a global quantization of 𝔤^{∗}, or of G^{∗}. In this paper we present a similar GQDP for quantum subgroups or quantum homogeneous spaces. Roughly speaking, this associates with every quantum subgroup, resp. quantum homogeneous space, of G, a quantum homogeneous space, resp. a quantum subgroup, of G^{∗}. The construction is tailored after four parallel paths — according to the different ways one has to algebraically describe a subgroup or a homogeneous space — and is “functorial”, in a natural sense.

Remarkably enough, the output of the constructions are always quantizations of proper type. More precisely, the output is related to the input as follows: the former is the coisotropic dual of the coisotropic interior of the latter — a fact that extends the occurrence of Poisson duality in the original GQDP for quantum groups. Finally, when the input is a strict quantization then the output is strict as well — so the special rôle of strict quantizations is respected.

We end the paper with some explicit examples of application of our recipes.

2010 Mathematics Subject Classification: Primary 17B37, 20G42, 58B32; Secondary 81R50 Keywords and Phrases: Quantum Groups, Poisson Homogeneous Spaces, Coisotropic Subgroups

1 Introduction

A Global QDP for Subgroups and Homogeneous Spaces 335

In this paper we work with quantizations of (algebraic) complex and real groups, their subgroups and homogeneous spaces, and a special symmetry among such quantum objects which we refer to as the “Global Quantum Duality Principle”. This is just a last step in a process, which is worth recalling in short. In any possible sense, quantum groups are suitable deformations of some algebraic objects attached with algebraic groups, or Lie groups. Once and for all, we adopt the point of view of algebraic groups: nevertheless, all our analysis and results can be easily converted in the language of Lie groups. The first step to deal with is describing an algebraic group G via suitable algebraic object(s). This can be done following two main approaches, a global one or a local one. In the global geometry approach, one considers U(𝔤) — the universal enveloping algebra of the tangent Lie algebra 𝔤 := Lie(G) — and F [G] — the algebra of regular functions on G. Both these are Hopf algebras, and there exists a nondegenerate pairing among them so that they are dual to each other. Clearly, U(𝔤) only accounts for the local data of G encoded in 𝔤, whereas F [G] instead totally describes G: thus F [G] yields a global description of G, which is why we speak of “global geometry” approach. In this context, one describes (globally) a subgroup K of G — always assumed to be Zariski closed — via the ideal in F [G] of functions vanishing on it; alternatively, an infinitesimal description is given taking in U(𝔤) the subalgebra U(𝔨) , where 𝔨 := Lie(K) . For a homogeneous G–space, say M, one describes it in the form M ≅ G/K — which amounts to fixing some point in M and its stabilizer subgroup K in G. After this, a local description of M ≅ G/K is given by representing its leftinvariant differential operators as U(𝔤)/U(𝔤) 𝔨: therefore, we can select U(𝔤) 𝔨 — a left ideal, left coideal in U(𝔤) — as algebraic object to encode M ≅ G/K, at least infinitesimally. For a global description instead, obstructions might occur. Indeed, we would like to describe M ≅ G/K via some algebra F [M] ≅ F [G/K] strictly related with F [G] . This varies after the nature of M ≅ G/K — hence of K — and in general might be problematic. Indeed, there exists a most natural candidate for this job, namely the set F [G]^{K} of K–invariants of F [G] , which is a subalgebra and left coideal. The problem is that F [G]^{K} permits to recover exactly G/K if and only if M ≅ G/K is a quasi-affine variety (which is not always the case). This yields a genuine obstruction, in the sense that this way of (globally) encoding the space M ≅ G/K only works with quasi-affine G–spaces; for the other cases, we just drop this approach — however, for a complete treatment of the case of projective G–spaces see [6]. In contrast, the approach of formal geometry is a looser one: one replaces F [G] with a topological algebra F [[G]] = F [[G_{f}]] — the algebra of “regular functions on the formal group G_{f}” associated with G — which can be realized either as the suitable completion of the local ring of G at its identity or as the (full) linear dual of U(𝔤) . In any case, both algebraic objects taken into account now only encode the local information of G. In this formal geometry context, the description of (formal) subgroups and (formal) homogeneous spaces goes essentially the same. However, in this case no problem occurs with (formal) homogeneous space, as any one of them can be described via a suitably defined subalgebra of invariants F [[G_{f}]]^{K_{f}}: in a sense, “all formal homogeneous spaces are quasi-affine”. As a consequence, the overall description one eventually achieves is entirely symmetric. When dealing with quantizations, Poisson structures arise (as semiclassical limits) on groups and Lie algebras, so that we have to do with Poisson groups and Lie bialgebras. In turn, there exist distinguished subgroups and homogeneous spaces — and their infinitesimal counterparts — which are “well-behaving” with respect to these extra structures: these are coisotropic subgroups and Poisson quotients. Moreover, the well-known Poisson duality — among Poisson groups G and G^{∗} and among Lie bialgebras 𝔤 and 𝔤^{∗} — extends to similar dualities among coisotropic subgroups (of G and G^{∗}) and among Poisson quotients (of G and G^{∗} again). It is also useful to notice that each subgroup contains a maximal coisotropic subgroup (its “coisotropic interior”), and accordingly each homogeneous space has a naturally associated Poisson quotient. As to the algebraic description, all properties concerning Poisson (or Lie bialgebra) structures on groups, Lie algebras, subgroups and homogeneous spaces have unique characterizations in terms of the algebraic codification one adopts for these geometrical objects. Details change a bit according to whether one deals with global or formal geometry, but everything goes in parallel in either context. By (complex) “quantum group” of formal type we mean any topological Hopf algebra H_{ℏ} over the ring ℂ[[ℏ]] whose semiclassical limit at ℏ = 0 — i.e., H_{ℏ}/ℏ H_{ℏ} — is of the form F [[G_{f}]] or U(𝔤) for some formal group G_{f} or Lie algebra 𝔤. Accordingly, one writes H_{ℏ} := F_{ℏ}[[G_{f}]] or H_{ℏ} := U_{ℏ}(𝔤) , calling the former a QFSHA and the latter a QUEA. If such a quantization (of either type) exists, the formal group G_{f} is Poisson and 𝔤 is a Lie bialgebra; accordingly, a dual formal Poisson group G_{f}^{∗} and a dual Lie bialgebra 𝔤^{∗} exist too.

In this context, as formal quantizations of subgroups or homogeneous spaces one typically considers suitable subobjects of either F_{ℏ}[[G_{f}]] or U_{ℏ}(𝔤) such that: (1) with respect to the containing formal Hopf algebra, they have the same relation as a in the “classical” setting — such as being a one-sided ideal, a subcoalgebra, etc.; (2) taking their specialization at ℏ = 0 is the same as restricting to them the specialization of the containing algebra (this is typically mentioned as a “flatness” property). This second requirement has a key consequence, i.e. the semiclassical limit object is necessarily “good” w.r. to the Poisson structure: namely, if we are quantizing a subgroup, then the latter is necessarily coisotropic, while if we are quantizing a homogeneous space then it is indeed a Poisson quotient. In the spirit of global geometry, by (complex) “quantum group” of global type we mean any Hopf algebra H_{q} over the ring ℂ[q, q^{−1}] whose semiclassical limit at q = 1 — i.e., H_{q}/( q − 1) H_{q} — is of the form F [G] or U(𝔤) for some algebraic group G or Lie algebra 𝔤. Then one writes H_{q} := F_{q}[G] or H_{q} := U_{ℏ}(𝔤) , calling the former a QFA and the latter a QUEA. Again, if such a quantization (of either type) exists the group G is Poisson and 𝔤 is a Lie bialgebra, so that dual formal Poisson groups G^{∗} and a dual Lie bialgebra 𝔤^{∗} exist too. As to subgroups and homogeneous spaces, global quantizations can be defined via a sheer reformulation of the same notions in the formal context: we refer to such quantizations as strict. In this paper, we introduce two more versions of quantizations, namely proper and weak ones, ordered by increasing generality, namely {strict} ⊊ {proper} ⊊ {weak}. This is achieved by suitably weakening the condition (2) above which characterizes a quantum subgroup or quantum homogeneous space. Remarkably enough, one finds that now the existence of a proper quantization is already enough to force a subgroup to be coisotropic, or a homogeneous space to be a Poisson quotient. The Quantum Duality Principle (=QDP) was first developed by Drinfeld (cf. [7], §7) for formal quantum groups (see [10] for details). It provides two functorial recipes, inverse to each other, acting as follows: one takes as input a QFSHA for G_{f} and yields as output a QUEA for 𝔤^{∗}; the other one as input a QUEA for 𝔤 and yields as output a QFSHA for G_{f}^{∗}. The Global Quantum Duality Principle (=GQDP) is a version of the QDP tailored for global quantum groups (see [11, 12]): now one functorial recipe takes as input a QFA for G and yields a QUEA for 𝔤^{∗}, while the other takes a QUEA for 𝔤 and provides a QFA for G^{∗}. An appropriate version of the QDP for formal subgroups and formal homogeneous spaces was devised in [5]. Quite in short, the outcome there was an explicit recipe which taking as input a formal quantum subgroup, or a formal quantum homogeneous space, respectively, of G_{f} provides as output a quantum formal homogeneous space, or a formal quantum subgroup, respectively, of G_{f}^{∗}. In short, these recipes come out as direct “restriction” (to formal quantum subgroups or formal quantum homogeneous spaces) of those in the QDP for formal quantum groups. This four-fold construction is fully symmetric, in particular all duality or orthogonality relations possibly holding among different quantum objects are preserved. Finally, Poisson duality is still involved, in that the semiclassical limit of the output quantum object is always the coisotropic dual of the semiclassical limit of the input quantum object. The main purpose of the present work is to provide a suitable version of the GQDP for global quantum subgroups and global quantum homogeneous spaces — extending the GQDP for global quantum groups — as much general as possible. The inspiring idea, again, is to “adapt” (by restriction, in a sense) to these more general quantum objects the functorial recipes available from the GQDP for global quantum groups. Remarkably enough, this approach is fully successful: indeed, it does work properly not only with strict quantizations (which should sound natural) but also for proper and for weak ones. Even more, the output objects always are global quantizations (of subgroups or homogeneous spaces) of proper type — which gives an independent motivation to introduce the notion of proper quantization. Also in this setup, Poisson duality, in a generalized sense, shows up again as the link between the input and the output of the GQDP recipes: namely, the semiclassical limit of the output quantum object is always the coisotropic dual of the coisotropic interior of the semiclassical limit of the input quantum object. Besides the wider generality this GQDP applies to (in particular, involving also non-coisotropic subgroups, or homogeneous spaces which are not Poisson quotients), we pay a drawback in some lack of symmetry for the final result — compared to what one has in the formal quantization context. Nevertheless, such a symmetry is almost entirely recovered if one restricts to dealing with strict quantizations, or to dealing with “double quantizations” — involving simultaneously a QFA and a QUEA in perfect (i.e. non-degenerate) pairing. At the end of the paper (Section 6) we present some applications of our GQDP: this is to show how it effectively works, and in particular that it does provide explicit examples of global quantum subgroups and global quantum homogeneous spaces. Among these, we also provide an example of a quantization which is proper but is not strict — which shows that the former notion is a non-trivial generalization of the latter.

2 General Theory

The main purpose of the present section is to collect some classical material about Poisson geometry for groups and homogeneous spaces. Everything is standard, we just need to fix the main notions and notations we shall deal with.

2.1 Subgroups and homogeneous spaces

Let G be a complex affine algebraic group and let 𝔤 be its tangent Lie algebra. Let us denote by F [G] its algebra of regular functions and by U(𝔤) its universal enveloping algebra. Both such algebras are Hopf algebras, and there exists a natural pairing of Hopf algebras between them, given by evaluation of differential operators onto functions. This pairing is perfect if and only if G is connected, which we will always assume in what follows. A real form of either G or 𝔤 is given once a Hopf ∗–algebra structure is fixed on either F [G] or U(𝔤) — and in case one take such a structure on both sides, the two of them must be dual to each other. Thus by real algebraic group we will always mean a complex algebraic group endowed with a suitable ∗–structure. A subgroup K of G will always be considered as Zariski–closed and algebraic. For any such subgroup, the quotient G/K is an algebraic left homogeneous G–space, which is quasi-projective as an algebraic variety. Given an algebraic left homogeneous G–space M and choosing m ∈ M, the stabilizer subgroup K_{m} will be a closed algebraic subgroup of G such that G/K_{m} ≃ M ; changing point will change the stabilizer within a single conjugacy class. We shall describe the subgroup K, or the homogeneous space G/K, through either an algebraic subset of F [G] — to which we will refer as a global coding — or an algebraic subset of U(𝔤) — to which we will refer as a local coding. The complete picture is the following:

— subgroup K :

(local) letting 𝔨 = Lie(K) we can consider its enveloping algebra U(𝔨) which is a Hopf subalgebra of U(𝔤) ; we then set ℭ ≡ ℭ(K) := U(𝔨) ;

(global) functions which are 0 on K form a Hopf ideal I ≡ I(K) inside F [G] , such that F [K] ≃ F [G]/I .

— homogeneous space G/K :

(local) let ℑ ≡ ℑ(K) = U(𝔤) · 𝔨 : this is a left ideal and two-sided coideal in U(𝔤) , and U(𝔤)/ℑ is the set of left–invariant differential operators on G/K.

(global) regular functions on the homogeneous space G/K may be identified with K–invariant regular functions on G. We will let C = C(K) = F [G]^{K} ; this is a subalgebra and left coideal in F [G] .

\underline{Warning} : this needs clarification! The point is: can one recover the homogeneous space G/K from C(K) = F [G]^{K} ? The answer depends on geometric properties of G/K itself — or (equivalently) of K — which we explain later on.

For any Hopf algebra H we introduce the following notations: ≤^{1} will stand for “unital subalgebra”, ⊴ for “two-sided ideal”, ⊴_{l} for “left ideal” and similarly ˙≤ will stand for “subcoalgebra”, ˙⊴ for “two-sided coideal” and ˙⊴_{ℓ} for “left coideal”. When the same symbols will be decorated by a subindex referring to a specific algebraic structure their meaning should be modified accordingly, e.g. ⊴_{H} will stand for “Hopf ideal” and ≤_{H} for “Hopf subalgebra”.

With such notations, with any subgroup K of G there is associated one of the following algebraic objects:

(a) I ⊴_{H} F [G] , (b) C ≤^{1} ˙⊴_{ℓ} F [G] , (c) ℑ ⊴_{l} ˙⊴ U(𝔤) , (d) ℭ ≤_{H} U(𝔤) (2.1)

In the real case, one has to consider, together with (2.1), additional requirements involving the ∗ structure and the antipode S, namely

(a) I^{∗} = I , (b) S(C)^{∗} = C , (c) S(ℑ)^{∗} = ℑ , (d) ℭ^{∗} = ℭ (2.2)

In the connected case algebraic objects of type I, ℑ and ℭ in (2.1) are enough to reconstruct either K or G/K:

K = Spec(F [G]/I ) = exp(Prim(ℭ)) = exp(Prim(ℑ))

where Prim(X) denotes the set of primitive elements of a bialgebra X. In contrast, C(K) = F [G]^{K} might be not enough to reconstruct K, due to lack of enough global algebraic functions; this happens, for example, when G/K is projective and therefore C(K) = ℂ. Any group K which can be reconstructed from its associated C is called observable: we shall now make this notion more precise. Let us call τ the map that to any subgroup K associates the algebra of invariant functions F [G]^{K} and let us call σ the map that to any subalgebra A of F [G] associates its stabilizer σ(A) = { g ∈ G | g · f = f ∀ f ∈ A }. These two maps are obviously inclusion–reversing. Furthermore they establish what is also known as a simple Galois correspondence: namely, for any subgroup K and any subalgebra A one has

(σ ∘ τ)(K) ⊇ K , (τ ∘ σ)(A) ⊇ A

so that (τ ∘ σ ∘ τ) (K) = τ(K), (σ ∘ τ ∘ σ) (A) = σ(A). A subgroup K of G such that (σ ∘ τ) (K) = K is said to be observable: this means exactly that such a subgroup can be fully recovered from its algebra of invariant functions τ(K). If K is any subgroup, then K̂ := (σ ∘ τ) (K) is the smallest observable subgroup containing K; we will call it the observable hull of K. Remark then that C(K) = C(K̂). The following fact (together with many properties of observable subgroups), which gives a characterization of observable subgroups in purely geometrical terms, may be found in [13]: Fact: a subgroup K of G is observable if and only if G/K is quasi–affine. Let us now clarify how to pass from algebraic objects directly associated with subgroups to those corresponding to homogeneous spaces. Let H be a Hopf algebra, with counit ε and coproduct Δ . For any submodule M ⊆ H define

M^{+} := M ∩ Ker(ε) , H^{coM} := { y ∈ H | (Δ(y) − y ⊗ 1) ∈ H ⊗ M } (2.3)

Let C be a (unital) subalgebra and left coideal of H and define Ψ(C) = H ·C^{+}. Then Ψ(C) is a left ideal and two-sided coideal in H. Conversely, let I be a left ideal and two-sided coideal in H and define Φ(I) := H^{coI} . Then Φ(I) is a unital subalgebra and left coideal in H. Also, this pair of maps (Φ, Ψ) defines a simple Galois correspondence, that is to say (a) Ψ and Φ are inclusion-preserving;

(b) (Φ ∘ Ψ) (C) ⊇ C , (Ψ ∘ Φ) (I) ⊆ I ;

(c) Φ ∘ Ψ ∘ Φ = Φ , Ψ ∘ Φ ∘ Ψ = Ψ .

(where the third property follows from the previous ones; see [19, 21, 22] for further details). Let now K be a subgroup of G and let I, C, ℑ, ℭ the corresponding algebraic objects as described in (2.1). We can thus establish the following relations among them:

subgroup vs. homogeneous space: objects directly related to the subgroup (namely, I and ℭ) and objects directly related to the homogeneous space (namely, C and ℑ) are linked by Ψ and Φ as follows:

ℑ = Ψ(ℭ) ,  ℭ = Φ(ℑ) ,  I ⊇ Ψ(C) ,  C = Φ(I) (2.4)

In particular, K is observable if and only if I = Ψ(C) ; on the other hand, we have in general Ψ(C(K)) = I(K̂) .

orthogonality with respect to the natural pairing between F [G] and U(𝔤) : this is expressed by the relations

I = ℭ^{⊥} ,  ℭ = I^{⊥} ,  C = ℑ^{⊥} ,  ℑ ⊆ C^{⊥}  (2.5)

In particular, K is observable if and only if ℑ = C^{⊥}; on the other hand, we have in general C(K)^{⊥} = ℑ(K̂) .

Let us also remark that orthogonality intertwines the local and global description.

The “formal” vs. “global” geometry approach. In the present approach we are dealing with geometrical objects — groups, subgroups and homogeneous spaces — which we describe via suitably chosen algebraic objects. When doing that, universal enveloping algebras or subsets of them only provide a local description — around a distinguished point: the unit element in a (sub)group, or its image in a coset (homogeneous) space. Instead, function algebras yield a global description, i.e. they do carry information on the whole geometrical object; for this reason, we refer to the present approach as the “global” one. The “formal geometry” approach instead only aims to describe a group by a topological Hopf algebra, which can be realized as an algebra of formal power series; in short, this is summarized by saying that we are dealing with a “formal group”. Subgroups and homogeneous spaces then are described by suitable subsets in such a formal series algebra (or in the universal enveloping algebra, as above): this again yields only a local description — in a formal neighborhood of a distinguished point — rather than a global one. Now, the analysis above shows that an asymmetry occurs when we adopt the global approach. Indeed, we might have problems when describing a homogeneous space by means of (a suitably chosen subalgebra of invariant) functions: technically speaking, this shows up as the occurrence of inclusions — rather than identities! — in formulas 2.4 and 2.5. This is a specific, unavoidable feature of the problem, due to the fact that homogeneous spaces (for a given group) do not necessarily share the same geometrical nature — beyond being all quasi-projective — in particular they are not necessarily quasi-affine. The case of those homogeneous spaces which are projective is treated in [6], where their quantizations are studied; in particular, there a suitable method to solve the problematic “C–side” of the QDP in that case is worked out, still in terms of “global geometry” but with a different tool (semi-invariant functions, rather than invariant ones). In contrast, in the formal geometry approach such a lack of symmetry does not occur: in other words, it happens that every formal (closed) subgroup is observable, or every formal homogeneous space is quasi-affine. This means that there is no need of worrying about observability, and the full picture — for describing a subgroup or homogeneous space, in four different ways — is entirely symmetric. This was the point of view adopted in [5], where this complete symmetry of the formal approach is exploited to its full extent.

2.2 Poisson subgroups and Poisson quotients

Let us now assume that G is endowed with a complex Poisson group structure corresponding to a Lie bialgebra structure on 𝔤, whose Lie cobracket is denoted δ : 𝔤 ⟶ 𝔤 ∧ 𝔤. At the Hopf algebra level this means that F [G] is a Poisson– Hopf algebra and U(𝔤) a co-Poisson Hopf algebra, in such a way that the duality pairing is compatible with these additional structures (see [4] for basic definitions). Let us recall that the linear dual 𝔤^{∗} inherits a Lie algebra structure; on the other hand, it has a natural Lie coalgebra structure, whose cobracket δ : 𝔤^{∗} ⟶ 𝔤^{∗} ∧ 𝔤^{∗} is the dual map to the Lie bracket of 𝔤. Altogether, this makes 𝔤^{∗} into a Lie bialgebra, which said to be dual to 𝔤. Therefore, there exist Poisson groups whose tangent Lie bialgebra is 𝔤^{∗}; we will assume one such connected group is fixed, we will denote it with G^{∗} and call it the dual Poisson group of G. In the real case the involution in F [G] is a Poisson algebra antimorphism and the one in U(𝔤) is a co-Poisson algebra antimorphism. A closed subgroup K of G is called coisotropic if its defining ideal I(K) is a Poisson subalgebra, while it is called a Poisson subgroup if I(K) is a Poisson ideal, the latter condition being equivalent to K ↪ G being a Poisson map. Connected coisotropic subgroups can be characterized, at an infinitesimal level, by one of the following conditions on 𝔨 ⊆ 𝔤:

(C-i) δ(𝔨) ⊆ 𝔨 ∧ 𝔤 , that is 𝔨 is a Lie coideal in 𝔤 ,

(C-ii)  𝔨^{⊥} is a Lie subalgebra of 𝔤^{∗} ,

while analogous characterizations of Poisson subgroups correspond to 𝔨 being a Lie subcoalgebra or 𝔨^{⊥} being a Lie ideal. The most important features of coisotropic subgroups, in this setting, is the fact that G/K naturally inherits a Poisson structure from that of G. Actually, a Poisson manifold (M , ω_{M}) is called a Poisson homogeneous G–space if there exists a smooth, homogeneous G–action ϕ : G × M ⟶ M which is a Poisson map (w.r. to the product Poisson structure on the domain). In particular, we will say that (M, ω_{M}) is a Poisson quotient if it verifies one of the following equivalent conditions (cf. [26]):

(P-i) there exists x_{0} ∈ M whose stabilizer G_{x_{0}} is coisotropic in G ;

(P-ii) there exists x_{0} ∈ M such that ϕ_{x_{0}} : G → M , ϕ(x_{0}, g) = ϕ(x, g) ,

is a Poisson map ; (P-iii) there exists x_{0} ∈ M such that ω_{M}(x_{0}) = 0 .

It is important to remark here that inside the same conjugacy class of subgroups of G there may be subgroups which are Poisson, coisotropic, or non coisotropic. Therefore, on the same homogeneous space there may exist many Poisson homogeneous structures, some of which make it into a Poisson quotient while some others do not. For a fixed connected subgroup K of a Poisson group G, with Lie algebra 𝔨, one can consider the following descriptions in terms of the Poisson Hopf algebra F [G] or of the co-Poisson Hopf algebra U(𝔤) :

I ≤_{P} F [G] ,  C ≤_{P} F [G]  (2.6)
ℑ ˙⊴_{P} U(𝔤) ,  ℭ ˙⊴_{P} U(𝔤)  (2.7)

where on first line we have global conditions and on second line local ones. Conversely each one of these conditions imply coisotropy of G with the exception of the condition on C, which implies only that the observable hull K̂ is coisotropic. Therefore a connected, observable, coisotropic subgroup of G is identified by one of the following algebraic objects:

I ⊴_{H} ≤_{P} F [G] ,  C ≤^{1} ˙⊴_{ℓ} ≤_{P} F [G]  (2.8)
ℑ ⊴_{l} ˙⊴ ˙⊴_{P} U(𝔤) ,  ℭ ≤_{H} ˙⊴_{P} U(𝔤)  (2.9)

(still with the usual, overall restriction on the use of C, which in general only describes the observable hull K̂). Thanks to self-duality in the notion of Lie bialgebra, with any Poisson group there is associated a natural Poisson dual, which is fundamental in the QDP; note that a priori many such dual groups are available, but when dealing with the QDP such an (apparent) ambiguity will be solved. As we aim to extend the QDP to coisotropic subgroups, we need to introduce a suitable notion of (Poisson) duality for coisotropic subgroups as well.

Definition 2.1. Let G be a Poisson group and G^{∗} a fixed Poisson dual.

1. If K is coisotropic in G we call complementary dual of K the unique connected subgroup K^{⊥} in G^{∗} such that Lie(K^{⊥}) = 𝔨^{⊥}.

2. If M is a Poisson quotient and M ≃ G/K_{M} we call complementary dual of M the Poisson G^{∗}–quotient M^{⊥} := G^{∗}/K_{M}^{⊥} .

3. For any subgroup H of G we call coisotropic interior of H the unique

maximal, closed, connected, coisotropic subgroup H^{∘} of G contained in H.

Remarks:

1. The complementary dual of a coisotropic subgroup is, trivially, a coisotropic subgroup whose complementary dual is the connected component of the one we started with:. Similarly, the complementary dual of a Poisson quotient is a Poisson quotient, and if we start with a Poisson quotient whose coisotropy subgroup (w.r. to any point) is connected then taking twice the complementary dual brings back to the original Poisson quotient.

2. The coisotropic interior may be characterized, at an algebraic level, as the unique closed subgroup whose Lie algebra is maximal between Lie subalgebras of 𝔥 which are Lie coideals in 𝔤.

Proposition 2.2. Let K be any subgroup of G and let K^{⟨⊥⟩} := ⟨exp(𝔨^{⊥})⟩ be the closed, connected, subgroup of G^{∗} generated by exp(𝔨^{⊥}) . Then:

(a) the Lie algebra 𝔨^{⟨⊥⟩} of K^{⟨⊥⟩} is the Lie subalgebra of 𝔤^{∗} generated by 𝔨^{⊥};

(b) 𝔨^{⟨⊥⟩} is a Lie coideal of 𝔤^{∗}, hence K^{⟨⊥⟩} is a coisotropic subgroup of G^{∗};

(c) K^{⟨⊥⟩} = (K^{∘})^{⊥}; in particular if Kis coisotropic then K^{⟨⊥⟩} = K^{⊥};

(d) (K^{⟨⊥⟩})^{⟨⊥⟩} = K^{∘} and K is coisotropic if and only if (K^{⟨⊥⟩})^{⟨⊥⟩} = K.

Proof. Part (a) is trivial. As for (b), since 𝔨 = (𝔨^{⊥})^{⊥} is a Lie subalgebra of 𝔤, we have that 𝔨^{⊥} is a Lie coideal in 𝔤^{∗}: therefore, due to the identity

δ([x, y]) = ∑_{[y]}([x, y_{[1]}] ⊗ y_{[2]} + y_{[1]} ⊗ [x, y_{[2]}]) + ∑_{[x]}([x_{[1]}, y] ⊗ x_{[2]} + x_{[1]} ⊗ [x_{[2]}, y])

(where δ(z) = ∑_{[z]} z_{[1]}⊗z_{[2]} for z ∈ 𝔤^{∗}), the Lie subalgebra ⟨𝔨^{⊥}⟩ of 𝔤^{∗} generated by 𝔨^{⊥} is a Lie coideal too. It follows then by claim (a) that K^{⟨⊥⟩} is coisotropic. Thus (b) is proved.

As for part (c) we have

(𝔨^{⟨⊥⟩})^{⊥} = ⟨𝔨^{⊥}⟩^{⊥} = ( ⋂_{\begin{matrix}𝔥≤_{L} 𝔤^{∗} \\ 𝔥⊇𝔨^{⊥} \end{matrix}}𝔥)^{⊥} = ∑_{\begin{matrix}𝔥≤_{L} 𝔤^{∗} \\ 𝔥⊇𝔨^{⊥} \end{matrix}}𝔥 = ∑_{\begin{matrix}𝔣 ˙⊴_{L}𝔤 \\ 𝔣⊇𝔨 \end{matrix}}𝔣 = 𝔨^{∘}

(with ≤_{L} meaning “Lie subalgebra” and ˙⊴_{L} meaning “Lie coideal”) where 𝔨^{∘} is exactly the maximal Lie subalgebra and Lie coideal of 𝔤 contained in 𝔨. To be precise, this last statement follows from the above formula for δ([x, y]), since that formula implies that the Lie subalgebra generated by a family of Lie coideals is still a Lie coideal.

Now 𝔨^{∘} = Lie(K^{∘}) , so Lie(K^{⟨⊥⟩}) = 𝔨^{⟨⊥⟩} = ((𝔨^{⟨⊥⟩})^{⊥})^{⊥} = ( 𝔨^{∘} )^{⊥} = Lie(K^{∘})^{⊥}

implies K^{⟨⊥⟩} = (𝔨^{∘})^{⊥} as we wished to prove. If, in addition, K is coisotropic then, obviously, K^{⟨⊥⟩} = K. All other statements follow easily.

3 Strict, proper, weak quantizations

The purpose of this section is to fix some terminology concerning the meaning of the word “quantization” and to describe some possible ways of quantizing a (closed) subgroup, or a homogeneous space. We set the algebraic machinery needed for talking of “quantization” and “specialization”: these notions must be carefully specified before approaching the construction of Drinfeld’s functors. Let q be an indeterminate, ℂ[q, q^{−1}] the ring of complex-valued Laurent polynomials in q, and ℂ(q) the field of complex-valued rational functions in q. Denote by HA the category of all Hopf algebras over ℂ[q, q^{−1}] which are torsion-free as ℂ[q, q^{−1}]–modules. Given a Hopf algebra H over the field ℂ(q) , a subset \overline{H} ⊆ H is called a ℂ[q, q^{−1}]–integral form (or simply a ℂ[q, q^{−1}]–form) if it is a ℂ[q, q^{−1}]–Hopf subalgebra of H and H_{F} := ℂ(q) ⊗_{ℂ[q,q^{−1}]} \overline{H} = H. Then \overline{H} is torsion-free as a ℂ[q, q^{−1}]–module, hence \overline{H} ∈ HA. For any ℂ[q, q^{−1}]–module M, we set M_{1} := M/(q − 1)M = ℂ ⊗_{ℂ[q,q^{−1}]} M :

this is a ℂ–module (via ℂ[q, q^{−1}] → ℂ[q, q^{−1}]/(q − 1) = ℂ), called specialization of M at q = 1 . Given two ℂ(q)–modules A and B and a ℂ(q)–bilinear pairing A × B ⟶ F , for any ℂ[q, q^{−1}]–submodule A_{×} ⊆ A we set:

A_{×}^{•} := { b ∈ B | ⟨A_{×}, b⟩ ⊆ ℂ[q, q^{−1}]}  (3.1)

In such a setting, we call A_{×}^{•} the ℂ[q, q^{−1}]–dual of A_{×}.

We will call quantized universal enveloping algebra (or, in short, QUEA) any U_{q} ∈ HA such that U_{1} := (U_{q})_{1} is isomorphic to U(𝔤) for some Lie algebra 𝔤, and we will call quantized function algebra (or, in short, QFA) any F_{q} ∈ HA such that F_{1} := (F_{q})_{1} is isomorphic to F [G] for some connected algebraic group G and, in addition, the following technical condition holds:

⋂_{n≥0}(q − 1)^{n}F_{q} = ⋂_{n≥0}((q − 1)F_{q} + Ker(ϵ_{Fq}))^{n}

We will add the specification that such quantum algebras are real whenever the starting object is a ∗–Hopf algebra. As a matter of notation, we write

𝕌_{q} := ℂ(q) ⊗_{ℂ[q,q^{−1}]} U_{q} , 𝔽_{q} := ℂ(q) ⊗_{ℂ[q,q^{−1}]} F_{q} .

When U_{q} is a (real) QUEA, its specialization U_{1} is a (real) co–Poisson Hopf algebra so that 𝔤 is in fact a (real) Lie bialgebra. Similarly, for any (real) QFA F_{q} the specialization F_{1} is a (real) Poisson-Hopf algebra and therefore G is a (real) Poisson group (see [4] for details). On occasions it is useful to consider simultaneous quantizations of both the universal enveloping algebra and the function algebra, or, in a larger generality, of a pair of dual Hopf algebra. Let H , K ∈ HA and assume that there exists a pairing of Hopf algebras ⟨ , ⟩ : H × K ⟶ ℂ[q, q^{−1}]. If the pairing is such that

(a) H = K^{•}, K = H^{•} (notation of (3.1)) w.r.t. the pairing ℍ × 𝕂 → ℂ(q) , for ℍ := ℂ(q)⊗_{ℂ[q,q^{−1}]}H, 𝕂 := ℂ(q)⊗_{ℂ[q,q^{−1}]}K, induced from H ×K → ℂ(q)

(b) the Hopf pairing H_{1} × K_{1} → ℂ given by specialization at q = 1 is perfect (i.e. non-degenerate)

then we will say that H and K are dual to each other. Note that all these assumptions imply that the initial pairing between H and K is perfect. When H = U_{q}(𝔤) is a QUEA and K = F_{q}[G] is a QFA, if the specialized pairing at 1 is the natural pairing between U(𝔤) and F [G] we will say that the pair (U_{q}(𝔤) , F_{q}[G]) is a double quantization of (G, 𝔤) .

Let us now move to the case in which G is a Poisson group and K a subgroup. We want to define a reasonable notion of “quantization” of K and of the corresponding homogeneous space G/K. There is a standard way to implement this, which actually implies — cf. Lemma 3.3 and Proposition 3.5 later on — the additional constraint that K be coisotropic.

Definition 3.1. Let F_{q}[G] and U_{q}(𝔤) be a QFA and a QUEA for G and 𝔤 and let

π_{F_{q}}: F_{q}[G] −↠ F_{q}[G]/(q−1)F_{q}[G] ≅ F [G]

π_{U_{q}}: U_{q}(𝔤) −↠ U_{q}(𝔤)/(q −1) U_{q}(𝔤) ≅ U(𝔤) be the specialization maps. Let I, C, ℑ and ℭ be the algebraic objects associated with the subgroup K of G (see 2.1). We call “strict quantization” (and sometimes we shall drop the adjective “strict”) of each of them any object I_{q}, C_{q}, ℑ_{q} or ℭ_{q} respectively, such that

\begin{matrix}(a) I_{q} ⊴_{ℓ} ˙⊴ F_{q}[G] , & π_{F_{q}}(I_{q}) = I , & π_{F_{q}}(I_{q}) ≅ I_{q}/(q−1) I_{q} \\ (b) C_{q} ≤^{1} ˙⊴_{ℓ} F_{q}[G] , & π_{F_{q}}(C_{q}) = C , & π_{F_{q}}(C_{q}) ≅ C_{q}/(q−1) C_{q} \\ (c) ℑ_{q} ⊴_{ℓ} ˙⊴ U_{q}(𝔤) , & π_{U_{q}}(ℑ_{q}) = ℑ , & π_{U_{q}}(ℑ_{q}) ≅ ℑ_{q}/(q−1) ℑ_{q} \\ (d) ℭ_{q} ≤^{1} ˙⊴_{ℓ} U_{q}(𝔤) , & π_{U_{q}}(ℭ_{q}) = ℭ , & π_{U_{q}}(ℭ_{q}) ≅ ℭ_{q}/(q−1) ℭ_{q} \end{matrix} (3.2)

In order to explain this definition let us start by considering the first two conditions in each line of (3.2).

a) A left ideal and two-sided coideal in a QFA quantizes the Hopf ideal of functions which are zero on a (closed) subgroup;

b) a left coideal subalgebra in a QFA quantizes the algebra of invariant functions on a homogeneous space;

c) a left ideal and two-sided coideal in a QUEA quantizes the infinitesimal algebra on a homogeneous space;

d) a left coideal subalgebra in a QUEA quantizes the universal enveloping subalgebra of a subgroup.

Once again, we must stress the fact that C_{q}, as was explained in Proposition 2.4, has to be seen as a quantization of the observable hull K̂ rather than of K itself.

Let us now be more precise about the last condition in the previous definition. By asking I_{q}/(q − 1)I_{q} ≅ π_{F_{q}}(I_{q}) = I we mean the following: the specialization map sends I_{q} inside F [G]. This map factors through I_{q}/(q − 1)I_{q}; in addition, we require that the induced map I_{q}/(q − 1)I_{q} ⟶ F [G] be a bijection on I. Of course this bijection will respect the whole Hopf structure, since π_{F_{q}} does. Now, since

π_{F_{q}}(I_{q}) = I_{q}/(I_{q} ∩ (q − 1)F_{q}[G])

this property may be equivalently rephrased by saying that I_{q} ∩(q−1) F_{q}[G] = (q−1) I_{q} as well. The previous discussions may be repeated unaltered for all four algebraic objects under consideration. An equivalent definition of strict quantizations is therefore the following:

\begin{matrix}(a) I_{q} ⊴_{ℓ} ˙⊴ F_{q}[G] , & π_{F_{q}}(I_{q}) = I , & I_{q} ∩ (q−1)F_{q}[G] = (q−1) I_{q} \\ (b) C_{q} ≤^{1} ˙⊴_{ℓ} F_{q}[G] , & π_{F_{q}}(C_{q}) = C , & C_{q} ∩ (q−1)F_{q}[G] = (q−1) C_{q} \\ (c) ℑ_{q} ⊴_{ℓ} ˙⊴ U_{q}(𝔤) , & π_{U_{q}}(ℑ_{q}) = ℑ , & ℑ_{q} ∩ (q−1)U_{q}(𝔤) = (q−1) ℑ_{q} \\ (d) ℭ_{q} ≤^{1} ˙⊴_{ℓ} U_{q}(𝔤) , & π_{U_{q}}(ℭ_{q}) = ℭ , & ℭ_{q} ∩ (q−1)U_{q}(𝔤) = (q−1) ℭ_{q} \end{matrix} (3.3)

The purpose of the last condition — which is often mentioned by saying that ℭ_{q} is a flat quantization (typically, in the literature on deformation quantization)

— should be clear: indeed, removing it means losing any control on what is contained, in quantization, inside the kernel of the specialization map. Although the just mentioned notion of quantization appears to be, in many respect, the “correct” one — and indeed is typically the one considered in literature — another notion of quantization naturally appears when one has to deal with quantum duality principle.

Definition 3.2. Let F_{q}[G] and U_{q}(𝔤) be a QFA and a QUEA for G and 𝔤 and let

π_{F_{q}}: F_{q}[G] −↠ F_{q}[G]/(q−1)F_{q}[G] ≅ F [G]
π_{U_{q}}: U_{q}(𝔤) −↠ U_{q}(𝔤)/(q −1) U_{q}(𝔤) ≅ U(𝔤)

be the specialization maps. Let ∇ := Δ − Δ^{op}. Let I, C, ℑ and ℭ be the algebraic objects associated with the subgroup K of G (see 2.1). We call “proper quantization” of each of them any object I_{q}, C_{q}, ℑ_{q} or ℭ_{q} respectively, such that

\begin{matrix}(a) I_{q} ⊴_{ℓ} ˙⊴ F_{q}[G] , & π_{F_{q}}(I_{q}) = I , & [I_{q} , I_{q}] ⊆ (q − 1) I_{q} \\ (b) C_{q} ≤^{1} ˙⊴_{ℓ} F_{q}[G] , & π_{F_{q}}(C_{q}) = C , & [C_{q} , C_{q}] ⊆ (q − 1) C_{q} \\ (c) ℑ_{q} ⊴_{ℓ} ˙⊴ U_{q}(𝔤) , & π_{U_{q}}(ℑ_{q}) = ℑ , & ∇( ℑ_{q}) ⊆ (q−1) U_{q}(𝔤) ∧ ℑ_{q} \\ (d) ℭ_{q} ≤^{1} ˙⊴_{ℓ} U_{q}(𝔤) , & π_{U_{q}}(ℭ_{q}) = ℭ , & ∇(ℭ_{q}) ⊆ (q−1) U_{q}(𝔤) ∧ ℭ_{q} \end{matrix} (3.4)

The link between these two notions of quantization is the following:

Lemma 3.3. Any strict quantization is a proper quantization.

Proof. This is an easy consequence of definitions. Indeed, let K be a subgroup of G. If I_{q} := I(K̂) is any strict quantization of I(K), we have

I_{q} ∩ (q − 1) F_{q} = (q − 1) I_{q}

by assumption, and moreover [F_{q} , F_{q}] ⊆ (q − 1) F_{q} . Then

[I_{q} , I_{q}] ⊆ I_{q} ∩ [F_{q} , F_{q}] ⊆ I_{q} ∩ (q − 1) F_{q} = (q − 1) I_{q}

thus [I_{q} , I_{q}] ⊆ (q − 1) I_{q}, i.e. I_{q} is proper. A similar argument works for quantizations of type C_{q}(K) . Also, if ℑ_{q}(K) is any strict quantization of ℑ(K) , then we have ℑ_{q} ∩ (q − 1) U_{q} = (q − 1) ℑ_{q} by assumption, and moreover ∇(U_{q}) ⊆ (q − 1) U_{q}^{∧2} . Then

∇( ℑ_{q}) ⊆ (U_{q} ∧ ℑ_{q}) ∩ ∇(U_{q}) ⊆ (U_{q} ∧ ℑ_{q}) ∩ (q − 1) U_{q}^{∧2} ⊆ (q − 1) U_{q} ∧ ℑ_{q}

so that ℑ_{q} is proper. A similar argument works for quantizations of type ℭ_{q}(K) as well.

Remark 3.4. The converse to Lemma 3.3 here above is false. Indeed, there exist quantizations (of subgroups / homogeneous spaces) which are proper but not strict: we present an explicit example — of type C_{q} — in Subsection 6.3 later on. This means that giving two different versions of “quantization” does make sense, in that they actually capture two inequivalent notions — hierarchically related via Lemma 3.3.

The following statement clarifies why such definitions actually apply only to the (restricted) case of coisotropic subgroups (this result can be traced back to [18], where it is mentioned as coisotropic creed).

Proposition 3.5. Let K be a subgroup of G and assume a proper quantization of it exists. Then K is coisotropic or, in case the quantization is C_{q}, its observable hull K̂ is coisotropic.

Proof. Assume I_{q} exists. Let f, g ∈ I, and let φ, γ ∈ I_{q} with π_{F_{q}}(φ) = f , π_{F_{q}}(γ) = g. Then by definition {f, g} = π_{F_{q}}((q − 1)^{−1}[φ, γ]). But

[φ, γ] ∈ [I_{q} , I_{q}] ⊆ (q − 1) I_{q}

by assumption, hence (q−1)^{−1}[φ, γ] ∈ I_{q}, thus {f, g} = π_{F_{q}}((q−1)^{−1}[φ, γ]) ∈ π_{F_{q}}(I_{q}) = I, which means that I is closed for the Poisson bracket. Thus (see (2.6)) K is coisotropic. Similar arguments work when dealing with C_{q}, ℑ_{q} or ℭ_{q}. We shall only remark that working with C_{q} we end up with C(K̂) = C(K) ≤_{P} F [G] , whence K̂ is coisotropic.

Since we would like to show also what happens in the non coisotropic case, we will consider, also, the weakest possible — naïve — version of quantization.

Definition 3.6. Let F_{q}[G] and U_{q}(𝔤) be a QFA and a QUEA for G and 𝔤 and let

π_{F_{q}}: F_{q}[G] −↠ F_{q}[G]/(q−1)F_{q}[G] ≅ F [G]
π_{U_{q}}: U_{q}(𝔤) −↠ U_{q}(𝔤)/(q −1) U_{q}(𝔤) ≅ U(𝔤)

be the specialization maps. Let I, C, ℑ and ℭ be the algebraic objects associated with the subgroup K of G (see 2.1). We call “weak quantization” of each of them any object I_{q}, C_{q}, ℑ_{q} or ℭ_{q} respectively, such that

\begin{matrix}(a)  I_{q} ⊴_{ℓ} ˙⊴ F_{q}[G] , & π_{F_{q}}(I_{q}) = I \\ (b)  C_{q} ≤^{1} ˙⊴_{ℓ} F_{q}[G] , & π_{F_{q}}(C_{q}) = C \\ (c)  ℑ_{q} ⊴_{ℓ} ˙⊴ U_{q}(𝔤) , & π_{U_{q}}(ℑ_{q}) = ℑ \\ (d)  ℭ_{q} ≤^{1} ˙⊴_{ℓ} U_{q}(𝔤) , & π_{U_{q}}(ℭ_{q}) = ℭ \end{matrix}  (3.5)

It is obvious that strict or proper quantizations are weak. Let us remark that every subgroup of G is quantizable in the weak sense, since we may just consider e.g. I_{q} := π_{F_{q}}^{−1}(I) to be a quantization of I. As naïf as it may seem, this remark will play a rôle in what follows.

Let us lastly remark how the real case should be treated.

Definition 3.7. Let (F_{q}[G] , ∗) and (U_{q}(𝔤) , ∗) be a real QFA and a real QUEA for G and 𝔤. Let I_{q}, C_{q}, ℑ_{q} and ℭ_{q} be subgroup quantizations (either strict, proper or weak). Then such quantizations are called real if

(S(I_{q}))^{⋆} = I_{q} ,  C_{q}^{⋆} = C_{q} ,  (S(ℑ_{q}))^{⋆} = ℑ_{q} ,  ℭ_{q}^{⋆} = ℭ_{q} (3.6)

3.8. The formal quantization approach. In the present work we are dealing with global quantizations. In [5] instead we treated formal quantizations: these are topological Hopf ℂ[[h]]–algebras which for h = 0 yield back the (formal) Hopf algebras associated with a (formal) group. In this case, such objects as I_{q}, C_{q}, ℑ_{q} and ℭ_{q} are defined in the parallel way. However, in [5] we did not consider the notions of proper nor weak quantizations but only dealt with strict quantizations. Actually, one can consider the notions of proper or weak quantizations in the formal quantization setup as well; then the relation between these and strict quantizations will be again the same as we showed here above. We point out also that the semiclassical limits of formal quantizations are just formal Poisson groups, or their universal enveloping algebras, or subgroups, homogeneous spaces, etc. In any case, this means — see the end of Subsection 2.1 — that no restrictions on subgroups apply (all are “observable”) nor on homogeneous spaces (all are “quasi-affine”).

4 Quantum duality principle

Drinfeld’s quantum duality principle (cf. [7], §7; see also [10] for a proof) has a stronger version (see [12]) best suited for our quantum groups — in the sense of Section 3.

Let H be any Hopf algebra in HA and let

I := Ker(H ↠^{ϵ} ℂ[q, q^{−1}] −↠^{ev_{1}} ℂ ) = Ker(H −↠^{ev_{1}} H/(q−1) H ↠^{ϵ̄} ℂ ) (4.1)

Then I is a Hopf ideal of H. We define

H^{∨} := ∑_{n≥0} (q −1)^{−n}I^{n} = ⋃_{n≥0} ((q −1)^{−1}I )^{n} ( ⊆ ℂ(q) ⊗_{ℂ[q,q^{−1}]} H ) (4.2)

Notice that, setting J := Ker (H −↠^{ϵ} ℂ[q, q^{−1}]), one has I = (q − 1) · 1_{H} + J , so that

H^{∨} = _{∑n≥0} (q − 1)^{−n}J^{n} = _{∑n≥0} ((q − 1)^{−1}J )^{n}  (4.3)

Consider, now, for every n ∈ ℕ the iterated coproduct Δ^{n}: H → H^{⊗n} where

Δ^{0} := ϵ  Δ^{1} := id_{H}  Δ^{n} := (Δ ⊗ id_{H}^{⊗(n−2)}) ∘ Δ^{n−1} if n ≥ 2 .

For any ordered subset Σ = {i_{1}, …, i_{k}} ⊆ {1, …, n} with i_{1} < ⋯ < i_{k}, define the morphism j_{Σ} : H^{⊗k} ⟶ H^{⊗n} by

j_{Σ}(a_{1} ⊗ ⋯ ⊗ a_{k}) := b_{1} ⊗ ⋯ ⊗ b_{n} where \{ \begin{matrix}b_{i} := 1 & if & i ∉ Σ \\ b_{i_{m}} := a_{m} & if & 1 ≤ m ≤ k \end{matrix}

then set Δ_{Σ} := j_{Σ} ∘ Δ^{k}, Δ_{∅} := Δ^{0} , and δ_{Σ} := ∑_{Σ′⊂Σ} (−1)^{n−|Σ′|}Δ_{Σ′} , δ_{∅} := ϵ. By the inclusion-exclusion principle, the inverse formula Δ_{Σ} = ∑_{Ψ⊆Σ} δ_{Ψ} holds. We shall use notation δ_{0} := δ_{∅}, δ_{n} := δ_{{1,2,...,n}}, and the key identity δ_{n} = (id_{H} − ϵ)^{⊗n} ∘ Δ^{n}, for all n ∈ ℕ_{+} . Given H ∈ H, we define

H′ := { a ∈ H | δ_{n}(a) ∈ (q − 1)^{n}H^{⊗n}, ∀ n ∈ ℕ } (⊆ H ) .  (4.4)

Theorem 4.1 (Global Quantum Duality Principle). (cf. [12]) For any H ∈ HA one has:

(a) H^{∨} is a QUEA and H′ is a QFA. Moreover the following inclusions hold:

H ⊆ (H^{∨})′ , H ⊇ (H′)^{∨} , H^{∨} =((H^{∨})′ )^{∨} , H′ =((H′)^{∨})′ (4.5)

(b) H = (H^{∨})′ ⟺ H is a QFA, and H = (H′)^{∨} ⟺ H is a QUEA;

(c) If G is a Poisson group with Lie bialgebra 𝔤, then

F_{q}[G]^{∨}/(q − 1)F_{q}[G]^{∨} = U(𝔤^{∗}) U_{q}(𝔤)′/(q − 1)U_{q}(𝔤)′ = F [G^{∗}]

where G^{∗} is some connected Poisson group dual to G;

(d) Let F_{q}[G] and U_{q}(𝔤) be dual to each other w.r. to some perfect Hopf pairing. Then F_{q}[G]^{∨} and U_{q}(𝔤)′ are dual to each other w.r. to the same pairing.

A number of remarks are due, at this point:

1. The Poisson group G^{∗} dual to G appearing in (c) of Theorem 4.1 does depend on U_{q}(𝔤) which is given as a data. Different choices of U_{q}(𝔤), though associated with the same Lie bialgebra 𝔤 may give rise to a different connected Poisson dual group G^{∗}.

2. For all Hopf ℂ(q)–algebra ℍ the existence of a ℂ[q, q^{−1}]-integral form H_{f} which is a QUEA at q = 1 is equivalent to the existence of a ℂ[q, q^{−1}]– integer form H_{u} which is a QFA at q = 1 .

3. All claims above have obvious analogues in the real case.

4. If H is a Hopf algebra and Φ ⊆ ℕ is a finite subset, then ([16], Lemma

3.2)

δ_{Φ}(ab) = ∑_{Λ∪Y =Φ}δ_{Λ}(a) δ_{Y} (b)  ∀ a, b ∈ H  (4.6)

furthermore, if Φ ≠ ∅ we have

δ_{Φ}(ab − ba) = ∑_{\begin{matrix}Λ∪Y =Φ \\ Λ∩Y ≠∅ \end{matrix}}(δ_{Λ}(a) δ_{Y} (b) − δ_{Y} (b) δ_{Λ}(a))  ∀ a, b ∈ H (4.7)

The above formulas will be used frequently in what follows

Having clarified the exact statement of quantum duality principle that we have in mind, let us extend it to objects of subgroup type as in Definition 3.6, i.e. to left coideal subalgebras and to left ideals and two-sided coideals — either in F_{q}[G] or in U_{q}(𝔤) . This was already done in [5] where we only considered local (i.e. over ℂ[[h]]) quantizations. Let us remark that the quantum duality principle we have in mind not only exchanges the rôle of algebras of functions with that of universal enveloping algebras, but also exchanges the rôle of subgroups with that of homogeneous spaces. At the semiclassical level, the pair of dual objects is given by a coisotropic subgroup H and a Poisson quotient G^{∗}/H^{⊥}. When H is a Poisson subgroup, its orthogonal H^{⊥} turns out to be normal in G^{∗} and G^{∗}/H^{⊥} ≅ H^{∗} as a Poisson group, thus recovering the usual quantum duality principle. In particular, we will consider a process moving along the following draft:

(a) I (⊆F [G] ) ⟶^{(1)} I_{q} (⊆F_{q}[G] ) ⟶^{(2)} I_{q}^{⋎} (⊆F_{q}[G]^{∨} ) ⟶^{(3)} I_{1}^{⋎} (⊆U(𝔤^{∗}))

(b) C (⊆ F [G] ) ⟶^{(1)} C_{q} (⊆F_{q}[G] ) ⟶^{(2)} C_{q}^{▽} (⊆F_{q}[G]^{∨} ) ⟶^{(3)} C_{1}^{▽} (⊆U(𝔤^{∗}))

(c) ℑ (⊆U(𝔤) ) ⟶^{(1)} ℑ_{q} (⊆U_{q}(𝔤) ) ⟶^{(2)} ℑ_{q}^{!} (⊆U_{q}(𝔤)′ ) ⟶^{(3)} ℑ_{1}^{!} (⊆F [G^{∗}])

(d) ℭ (⊆U(𝔤) ) ⟶^{(1)} ℭ_{q} (⊆U_{q}(𝔤) ) ⟶^{(2)} ℭ_{q}^{↰} (⊆U_{q}(𝔤)′ ) ⟶^{(3)} ℭ_{1}^{↰} (⊆F [G^{∗}])

where arrows (1) are quantizations, arrows (3) are specializations at q = 1 and the definition of arrows (2) will be the core of what follows. It will turn out that:

1. each one of the right-hand-side objects above is one of the four algebraic objects which describe a closed connected subgroup of G^{∗}: namely, the correspondence is

(a) ⟹ (c) , (b) ⟹ (d) , (c) ⟹ (a) , (d) ⟹ (b) .

2. the four quantizations of subgroups of G^{∗} so obtained are always proper — hence the subgroups of G^{∗} associated with them are coisotropic.

3. if we begin with strict quantizations, and we start from a subgroup K, then the quantization of the unique coisotropic closed connected subgroup of G^{∗} mentioned above is strict as well, and the subgroup itself is K^{⊥} (cf. Definition 2.1), with some care in case (b), i.e. if we start from C(K). This will partially generalize to weak quantizations, for which, starting from a subgroup K of G, the unique coisotropic closed connected subgroup of G^{∗} obtained above is K^{⟨⊥⟩} (cf. Proposition 2.2).

Let us fix, in what follows, quantizations U_{q}(𝔤) and F_{q}[G] as in Section 3. Unless explicitly mentioned we will not assume that this is a double quantization. To simplify notations, let us set

𝕌_{q} := 𝕌_{q}(𝔤) ,  U_{q} := U_{q}(𝔤) ,  U_{q}′ := U_{q}(𝔤)′
𝔽_{q} := 𝔽_{q}[G] ,  F_{q} := F_{q}[G] ,  F_{q}^{∨} := F_{q}[G]^{∨}

As mentioned in the first remark after Theorem 4.1, this implies that a specific connected Poisson dual G^{∗} of G is selected (it depends on the choice of U_{q} := U_{Q}(𝔤) , not only on 𝔤 itself). Let us consider quantum subgroups I_{q}, C_{q}, ℑ_{q} and ℭ_{q} as defined in 3.6.

Definition 4.2. Using notations as in (4.1) we define:

(a) I_{q}^{⋎} := ∑_{n=1}^{∞} (q − 1)^{−n} · I^{n−1} · I_{q} = ∑_{n=1}^{∞} (q − 1)^{−n} · J^{n−1} · I_{q}

(b) C_{q}^{▽} := ∑_{n=0}^{∞}(q − 1)^{−n} · (C_{q} ∩ I )^{n} = ∑_{n=0}^{∞} (q − 1)^{−n} · (C_{q} ∩ J)^{n}

(c) ℑ_{q}^{!} := { x ∈ ℑ_{q} | δ_{n}(x) ∈ (q−1)^{n} ∑_{s=1}^{n}U_{q}^{⊗(s−1)}⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}, ∀ n∈ℕ_{+}}

(d) ℭ_{q}^{↰} := { x ∈ ℭ_{q} | δ_{n}(x) ∈ (q − 1)^{n} U_{q}^{⊗(n−1)}⊗ ℭ_{q} , ∀ n ∈ ℕ_{+} }

Let us remark that the following inclusions hold directly by definitions:

(i) I_{q}^{⋎} ⊇ I_{q} , (ii) C_{q}^{▽} ⊇ C_{q} , (iii) ℑ_{q}^{!} ⊆ ℑ_{q} , (iv) ℭ_{q}^{↰} ⊆ ℭ_{q} . (4.8)

5 Duality maps

In the present section we will prove properties of the four Drinfeld–type maps defined in the previous section, namely the maps I_{q} ↦ I_{q}^{⋎}, C_{q} ↦ C_{q}^{▽}, ℑ_{q} ↦ ℑ_{q}^{!} and ℭ_{q} ↦ ℭ_{q}^{↰}. Let us recall that such maps do not change, as we will see, the algebraic properties of subobjects, but interchanges quantized function algebra with quantum enveloping algebra and therefore quantizations of coisotropic subgroups will be sent to quantizations of (embeddable) homogeneous spaces — of the dual quantum group — and viceversa. Let us start by considering the map I_{q} ↦ I_{q}^{⋎}.

Proposition 5.1. Let I_{q} = I_{q}(K) be a left ideal and two-sided coideal in F_{q}[G] , that is a weak quantization (of type I) of some subgroup K of G. Then

1. I_{q}^{⋎} is a left ideal and two-sided coideal in F_{q}[G]^{∨};

2. if I_{q} is strict, then I_{q}^{⋎} is strict too, i.e. I_{q}^{⋎}⋂ (q − 1) F_{q}[G]^{∨} = (q − 1) I_{q}^{⋎} ;

3. there exists a coisotropic subgroup L of G^{∗} such that I_{q}(K)^{⋎} = ℑ_{q}(L) : namely, I_{q}(K)^{⋎} is a proper quantization, of type ℑ, of some coisotropic subgroup L of G^{∗};

4. in the real case, i.e. if the quantization I_{q} is a real one, I_{q}^{⋎} is real too, i.e. (S(I_{q}^{⋎}))^{∗} = I_{q}^{⋎}. Therefore claims (1–3) still hold in the framework of real quantum subgroups.

Proof. (1) Consider that I_{q}^{⋎} is the left ideal of F_{q}^{∨} generated by (q − 1)^{−1} I_{q} ; therefore, in order to prove I_{q}^{⋎} ˙⊴ F_{q}^{∨} it is enough to show that Δ((q−1)^{−1} I_{q}) ⊆ F_{q}^{∨}⊗ I_{q}^{⋎} + I_{q}^{⋎}⊗ F_{q}^{∨}. Since I_{q} is a coideal of F_{q}, we have

\begin{matrix}Δ((q − 1)^{−1} I_{q} ) ⊆ \\ ⊆ F_{q} ⊗ (q−1)^{−1} & I_{q} + (q−1)^{−1} I_{q} ⊗ F_{q} ⊆ F_{q}^{∨}⊗ I_{q}^{⋎} + I_{q}^{⋎}⊗ F_{q}^{∨} \end{matrix}  (5.1)

whence I_{q}^{⋎} ˙⊴ F_{q}^{∨} follows, and the first claim is proved. (2) Assume I_{q} to be a strict quantization, so that I_{q} ⋂ (q − 1) F_{q} = (q − 1) I_{q} . Let J := Ker (ϵ : F_{q} ⟶ ℂ[q, q^{−1}] ) . Then

J mod (q−1)F_{q} = Ker (ϵ)|_{F [G]} = 𝔪_{e}

and 𝔪_{e}/𝔪_{e}^{2} = 𝔤^{∗}, the cotangent Lie bialgebra of G. Let {y_{1}, …, y_{n}} be a subset of 𝔪_{e} whose image in the local ring of G at the identity e is a local system of parameters, and pull it back to a subset {j_{1}, …, j_{n}} of J. Let F̂_{q} be the J–adic completion of F_{q}. From [12], Lemma 4.1, we know that the set of ordered monomials \{j^{\underline{e}} | \underline{e} ∈ ℕ^{n}\} (where hereafter j^{\underline{e}} := ∏_{s=1}^{n} j_{s}^{\underline{e}(i)} , for all \underline{e} ∈ ℕ^{n}) is a ℂ[q, q^{−1}]–pseudobasis of F̂_{q}, which means that each element of F̂_{q} has a unique expansion as a formal infinite linear combination of the j^{\underline{e}}’s. In a similar way, the (q−1)–adic completion of F_{q}^{∨} admits \{(q − 1)^{−|\underline{e}|}j^{\underline{e}} | \underline{e} ∈ ℕ^{n}\} as a ℂ[q, q^{−1}]–pseudobasis, where |\underline{e}| := ∑_{i=1}^{n} \underline{e}(i) .

For our purposes we need a special choice of the set {j_{1}, …, j_{n}} adapted to the smooth subvariety K of G. By general theory we can choose {y_{1}, …, y_{n}} so that y_{1}, …, y_{k} ∈ 𝔪_{e} and y_{k+1}, …, y_{n} ∈ I(K) , where k = dim (K) . We can also choose the lift {j_{1}, …, j_{n}} of {y_{1}, …, y_{n}} inside J so that j_{s} is a lift of y_{s}, for all s = 1, …, k, and j_{k+1}, …, j_{n} ∈ I_{q}. With these assumptions, it’s easy to see that

φ ∈ I_{q}^{⋎} ⋂ (q − 1) F_{q}^{∨} ⟹ (q − 1)^{n} φ ∈ (J^{n−1} · I_{q}) ⋂ (q − 1) J^{n}

for some n ∈ ℕ, which in turn yields (q − 1)^{n}φ ∈ J^{n−1} ·(I_{q} ⋂ (q −1) J). Since

I_{q} ⋂ (q − 1) J ⊆ I_{q} ⋂ (q − 1) F_{q} = (q − 1) I_{q}

we conclude that (q − 1)^{n} φ ∈ (q − 1) J^{n−1} · I_{q}, whence φ ∈ (q − 1) I_{q}^{⋎}. The converse inclusion I_{q}^{⋎} ⋂ (q − 1) F_{q}^{∨} ⊇ (q − 1) I_{q}^{⋎} is obvious, hence claim (2) is proved. (3) It is an obvious statement that I_{q}^{⋎} is a weak quantization of its image π_{F_{q}}^{∨}(I_{q}^{⋎}): in particular, π_{F_{q}}^{∨}(I_{q}^{⋎}) ⊴_{ℓ} ˙⊴ π_{F_{q}}^{∨}(F_{q}^{∨}) = U(𝔤^{∗}) implies that π_{F_{q}}^{∨}(I_{q}^{⋎}) = ℑ(L) for some subgroup L of G^{∗}. Thus I_{q}^{⋎} is a weak quantization, to be called ℑ_{q}(L), of ℑ(L) , and it is even strict if I_{q} itself is strict, as we’ve just seen. Now we show that such quantization ℑ_{q}(L) turns out to be always proper. In fact, (5.1) implies ∇((q − 1)^{−1} I_{q}) ⊆ (q − 1)^{−1} (F_{q} ∧ I_{q}). On the other hand F_{q} ∧ I_{q} ⊆ J ∧ I_{q} ⊆ (q − 1)^{2} F_{q}^{∨} ∧ I_{q}^{⋎}, thus, finally, ∇(I_{q}^{⋎}) ∈ (q − 1) F_{q}^{∨} ∧ I_{q}^{⋎}, which means that I_{q}^{⋎} is proper and (3) holds. (4) This is an obvious consequence of definitions.

Remark 5.2. In functorial language we may say that the map I_{q} ↦ I_{q}^{⋎} establishes a functor between quantizations of coisotropic subgroups of G and quantizations of (embeddable) homogeneous spaces of G^{∗}, moving from a global to a local description, sending each type of quantization in a proper one and preserving strictness. Indeed, we should make precise what are the “arrows” in our categories of “quantum subgroups” or “quantum homogeneous spaces”, and how the functor acts on these: we leave these details to the interested reader.

Let us move on to properties of the map C_{q} ↦ C_{q}^{▽} .

Proposition 5.3. Let C_{q} = C_{q}(K) be a left coideal subalgebra in F_{q}[G]. Then

1. C_{q}^{▽} is a left coideal subalgebra in F_{q}[G]^{∨};

2. if C_{q} is strict, then C_{q}^{▽} is strict too, i.e. C_{q}^{▽} ⋂ (q − 1) F_{q}[G]^{∨} = (q − 1) C_{q}^{▽} .

3. there exists a coisotropic subgroup L of G^{∗} such that C_{q}(K)^{▽} = ℭ_{q}(L) : namely, C_{q}(K)^{▽} is a proper quantization, of type ℭ, of some coisotropic subgroup L of G^{∗}; 4. in the real case, i.e. if the quantization C_{q} is a real one, C_{q}(K)^{▽} is real

too, i.e. (C_{q}^{▽})^{∗} = C_{q}^{▽}. Therefore claims (1–3) still hold in the framework of real quantum subgroups.

Proof. The proof uses essentially the same arguments as the previous one.

(1) By the very definitions C_{q}^{▽} ≤^{1} F_{q}^{∨} := F_{q}[G]^{∨}. More precisely, C_{q}^{▽} is (by construction) the unital ℂ[q, q^{−1}]–subalgebra of F_{q}^{∨} generated by (q − 1)^{−1} (C_{q})^{+} , where (C_{q})^{+} := C_{q} ⋂ J . So to get C_{q}^{▽} ˙⊴_{ℓ} F_{q}^{∨} we must only prove Δ((q−1)^{−1}(C_{q})^{+})⊆ F_{q}^{∨}⊗ C_{q}^{▽}. But C_{q} ˙⊴_{ℓ} F_{q}, so:

Δ((q − 1)^{−1}(C_{q})^{+}) ⊆ F_{q} ⊗ (q − 1)^{−1}(C_{q})^{+} ⊆ F_{q}^{∨}⊗ C_{q}^{▽}  (5.2)

therefore C_{q}^{▽} ˙⊴_{ℓ} F_{q}^{∨}, and claim (1) is proved. (2) Now suppose C_{q} to be a strict quantization, i.e. C_{q} ⋂ (q −1) F_{q} = (q −1) C_{q} . We need an explicit description of F_{q}^{∨} and of C_{q}^{▽}. This goes along the same lines followed to describe I_{q}^{⋎} in the proof of Proposition 5.1: but now the choice of the subset {j_{1}, …, j_{n}} of J is different. First, since C(K) = C(K̂) we can assume that K = K̂, i.e. K is observable. Then we can choose {j_{1}, …, j_{n}} so that j_{k+1}, …, j_{n} ∈ J ⋂ C_{q} = C_{q}^{+} (where again k = dim (K) ) and, letting y_{s} := j_{s} mod (q −1) F_{q} , the set {y_{1}, …, y_{n}} yields a local system of parameters at e ∈ G (in the localized ring), as before; now in addition we have y_{k+1}, …, y_{n} ∈ 𝔪_{e} ⋂ C(K) =: C(K)^{+} . With these assumptions, the (q − 1)–adic completion of F_{q}^{∨} admits \{(q − 1)^{−|\underline{e}|}j^{\underline{e}} | \underline{e} ∈ ℕ^{n}} as a ℂ[q, q^{−1}]–pseudobasis, like before, but in addition the same analysis can be done for the (q − 1)–adic completion of C_{q}^{▽} (just because C_{q} is strict), which then has ℂ[q, q^{−1}]–pseudobasis { ∏_{s=k+1}^{n} j_{s}^{e_{s}} | (e_{k+1}, …, e_{n}) ∈ ℕ^{n−k}}. From these description of the completions, and comparing the former with F_{q}^{∨} and C_{q}, we easily see that C_{q}^{▽} ⋂ (q − 1) F_{q}^{∨} ⊆ (q − 1) C_{q}^{▽}. The converse is trivial, hence claim (1) is proved. (3) It follows directly from (1) that C_{q}^{▽} is a weak quantization of its image π_{F_{q}}^{∨}(C_{q}^{▽}): in particular, π_{F_{q}}^{∨}(C_{q}^{▽}) ≤^{1} ˙⊴_{ℓ} π_{F_{q}}^{∨}(F_{q}^{∨}) = U(𝔤^{∗}) means that π_{F_{q}}^{∨}(C_{q}^{▽}) = ℭ(L) for some subgroup L of G^{∗}. Thus C_{q}^{▽} is a weak quantization — to be called ℭ_{q}(L) — of ℭ(L), and it is even strict if C_{q} itself is strict, by claim (1). Now in addition we show that, in any case, such a quantization ℭ_{q}(L) is always proper. From (5.2) we have

∇((q − 1)^{−1} (C_{q})^{+}) ⊆ (q − 1)^{−1}J ∧ (C_{q})^{+} ⊆

⊆ (q − 1)^{−1+2} F_{q}^{∨}∧ C_{q}^{▽} = (q − 1) F_{q}^{∨}∧ C_{q}^{▽}

which implies exactly that C_{q}^{▽} — which by definition is the unital subalgebra generated by (q − 1)^{−1} (C_{q})^{+} — is proper. (4) This follows directly from definitions and from C_{q}^{∗} = C_{q}, which holds by assumption.

Remark 5.4. In functorial language we may say that the map C_{q} ↦ C_{q}^{▽} establishes a functor between quantized homogeneous spaces of G and quantizations of coisotropic subgroups of G^{∗}, moving from a global to a local description, sending each type of quantization in a proper one and preserving strictness. Again, to be precise, several details need to be fixed, and are left to the reader.

The third step copes with the map ℑ_{q} ↦ ℑ_{q}^{!} .

Proposition 5.5. Let ℑ_{q} = ℑ_{q}(K) be a left ideal and two-sided coideal in U_{q}(𝔤) , weak quantization (of type ℑ) of some coisotropic subgroup K of G. Then:

1. ℑ_{q}^{!} is a left ideal and two-sided coideal in U_{q}(𝔤)′;

2. if ℑ_{q} is strict, then ℑ_{q}^{!} is strict too, i.e. ℑ_{q}^{!} ⋂ (q−1) U_{q}(𝔤)′ = (q−1) ℑ_{q}^{!} ;

3. there exists a coisotropic subgroup L in G^{∗} such that ℑ_{q}(K)^{!} = I_{q}(L) : namely, ℑ_{q}(K)^{!} is a proper quantization, of type I, of some coisotropic subgroup L of G^{∗};

4. in the real case, i.e. if the quantization ℑ_{q} is a real one, ℑ_{q}^{!} is real too, i.e. (S(ℑ_{q}^{!}))^{∗} = ℑ_{q}^{!}. Therefore claims (1–3) still hold in the framework of real quantum subgroups.

Proof. (1) Let a ∈ U_{q}′ and b ∈ ℑ_{q}^{!} : by definition of ℑ_{q}^{!}, from ℑ_{q} ⊴_{ℓ} U_{q} and from (4.6) we get

δ_{n}(ab) ∈ (q − 1)^{n} ∑_{s=1}^{n}U_{q}^{⊗(s−1)}⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}

so a b ∈ ℑ_{q}^{!} , thus ℑ_{q}^{!} ⊴_{ℓ} U_{q}′. As to the coideal property, it is proven resorting to (q − 1)–adic completions, arguing as in the proof of Proposition 3.5 in [12], and basing on the fact that ℑ_{q} ˙⊴ U_{q}. Details are left to the reader. (2) Assume now ℑ_{q} to be strict. The inclusion

ℑ_{q}^{!} ⋂ (q − 1) U_{q}(𝔤)′ ⊇ (q − 1) ℑ_{q}^{!}

is trivially true, and we must prove the converse. Let η ∈ ℑ_{q}^{!} ⋂ (q − 1) U_{q}(𝔤)′. We have

δ_{n}(η) ∈ (q − 1)^{n} ((∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}) ⋂ (q − 1) U_{q}^{⊗n})

for all n ∈ ℕ_{+} . But then our assumption gives

( ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}) ⋂ (q − 1) U_{q}^{⊗n} =

= ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ (ℑ_{q} ⋂ (q−1) U_{q}) ⊗ U_{q}^{⊗(n−s)} =

= (q−1)^{n+1} ∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}

which, in turn, means η ∈ (q−1) ℑ_{q}^{!} . Thus ℑ_{q}^{!} ⋂ (q−1) U_{q}(𝔤)′ ⊆ (q−1) ℑ_{q}^{!} , as expected. (3) Claim (1) implies that ℑ_{q}^{!} is a weak quantization of its image, therefore there exists a subgroup L of G^{∗} such that π_{U_{q}}′(ℑ_{q}^{!} ) = I(L) . This quantization is even strict if ℑ_{q} itself is strict, by the previous. Now we show that this quantization I_{q}(L) is always proper — hence the subgroup L is coisotropic, by Lemma 3.5. Recall that, by definition, I_{q}(L) is proper if and only if [x, y] ∈ (q − 1) ℑ_{q}^{!} for all x, y ∈ ℑ_{q}^{!} . From definitions we have

[x, y] ∈ (q−1) ℑ_{q}^{!} ⟺ δ_{n}([x, y]) ∈ (q − 1)^{n+1} ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}

for all n ∈ ℕ. Then by formula (4.7) we have (for all n ∈ ℕ)

δ_{n}([x, y]) = _{∑ \begin{matrix}Λ∪Y =\{1,...,n\} \\ Λ∩Y ≠∅ \end{matrix}}( δ_{Λ}(x) δ_{Y} (y) − δ_{Y} (y) δ_{Λ}(x) )  (5.3)

while (with notation of §4)

δ_{Λ}(x) ∈ (q − 1)^{|Λ|} · j_{Λ} (∑_{s=1}^{|Λ|} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(|Λ|−s)}) ,
δ_{Y} (y) ∈ (q − 1)^{|Y |} · j_{Y} (∑_{s=1}^{|Y |} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(|Y |−s)}) ;

since Λ ∪ Y = {1, …, n} and Λ ∩ Y ≠ ∅ we have |Λ| + |Y | ≥ n + 1 ; moreover, for each index i ∈ {1, …, n} we have i ∈ Λ (and otherwise Im (j_{Λ}) has 1 in the i–th spot) or i ∈ Y (with the like remark on Im (j_{Y}) if not). As ℑ_{q} is a left ideal of U_{q}, we conclude

δ_{Λ}(x) · δ_{Y} (y) , δ_{Y} (y) · δ_{Λ}(x) ∈ (q − 1)^{|Λ|+|Y |} ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}

⊆ (q − 1)^{n+1} ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}

so that (5.3) gives δ_{n}([x, y]) ∈ (q−1)^{n+1} ∑_{s=1}^{n} U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}, as expected. (4) In the real case, (S(ℑ_{q}^{!} ))^{∗} = ℑ_{q}^{!} follows at once from definitions and from the identity (S(ℑ_{q}))^{∗} = ℑ_{q} .

Remark 5.6. In functorial language we may say that the map ℑ_{q} ↦ ℑ_{q}^{!} establishes a functor between quantized homogeneous spaces of G and quantizations of coisotropic subgroups of G^{∗}, moving from a local to a global description, sending each type of quantization in a proper one and preserving strictness. Once more, details are left to the interested reader.

The fourth and last step is devoted to the map ℭ_{q} ↦ ℭ_{q}^{↰}.

Proposition 5.7. Let ℭ_{q} = ℭ_{q}(K) be a subalgebra and left coideal in U_{q}(𝔤) , weak quantization (of type ℭ) of some subgroup K of G. Then:

1. ℭ_{q}^{↰} is a subalgebra and left coideal in U_{q}(𝔤)′; 2. if ℭ_{q} is strict, then ℑ_{q}^{!} is strict too, i.e. ℭ_{q}^{↰} ⋂ (q−1) U_{q}(𝔤)′ = (q−1) ℭ_{q}^{↰} ;

3. there exists a coisotropic subgroup L in G^{∗} such that ℭ_{q}(K)^{↰} = C_{q}(L) : namely, ℭ_{q}(K)^{↰} is a proper quantization, of type C, of some coisotropic subgroup L of G^{∗};

4. in the real case, i.e. if the quantization ℭ_{q} is a real one, ℭ_{q}(K)^{↰} is real too, i.e. (ℭ_{q}^{↰} )^{∗} = ℭ_{q}^{↰}. Therefore claims (1–3) still hold in the framework of real quantum subgroups.

Proof. The whole proof is very similar to that of Proposition 5.5. (1) By definitions, 1 ∈ ℭ_{q} and δ_{n}(1) = 0 for all n ∈ ℕ, so 1 ∈ ℭ_{q}^{↰}. Let x, y ∈ ℭ_{q}^{↰} and n ∈ ℕ; by (4.6) we have δ_{n}(xy) = ∑_{Λ∪Y ={1,...,n}} δ_{Λ}(x) δ_{Y} (y) . Each of

the factors δ_{Λ}(x) belongs to a module (q − 1)^{|Λ|} U_{q}^{⊗(|Λ|−1)}⊗ X where the last tensor factor is either X = ℭ_{q} (if n ∈ Λ ) or X = {1} ⊂ ℭ_{q} (if n ∉ Λ ), and similarly for δ_{Y} (y); in addition Λ∪Y = {1, …, n} implies |Λ|+|Y | ≥ n, and summing up δ_{n}(xy) ∈ (q−1)^{n}U_{q}^{⊗(n−1)}⊗ ℭ_{q}, whence x y ∈ ℭ_{q}^{↰}. Thus ℭ_{q}^{↰} is a subalgebra of U_{q}′. In order to prove that ℭ_{q}^{↰} is a left coideal in U_{q}′, one can again resort to (q − 1)–adic completions, with exactly the same arguments as in the proof of Proposition 3.5 in [5], starting from the fact that ℭ_{q} ˙⊴_{ℓ} U_{q}. Details are left to the reader. (2) Assume, now, that ℭ_{q} is a strict quantization, i.e. ℭ_{q} ⋂ (q − 1) F_{q} = (q − 1) ℭ_{q}. Then clearly ℭ_{q}^{↰} ⋂ (q − 1) U_{q}(𝔤)′ ⊇ (q − 1) ℭ_{q}^{↰}, and we must prove the converse inclusion. Let κ ∈ ℭ_{q}^{↰} ⋂ (q − 1) U_{q}(𝔤)′. Then:

δ_{n}(κ) ∈ (q − 1)^{n} ((U_{q}^{⊗(n−1)} ⊗ ℭ_{q}) ⋂ (q − 1) U_{q}^{⊗n}) =

= (q − 1)^{n}(U_{q}^{⊗(n−1)} ⊗ (ℭ_{q} ⋂ (q−1) U_{q})) = (q − 1)^{n+1} · U_{q}^{⊗(n−1)} ⊗ ℭ_{q}

which means κ ∈ (q − 1) ℭ_{q}^{↰}. Therefore ℭ_{q}^{↰} ⋂ (q − 1) U_{q}(𝔤)′ ⊆ (q − 1) ℭ_{q}^{↰}, as claimed. (3) The above algebraic properties show that ℭ_{q}^{↰} is a weak quantization of its image π_{U_{q}}′(ℭ_{q}^{↰} ); thus there exists a coisotropic subgroup L of G^{∗} such that: π_{U_{q}}′(ℭ_{q}^{↰} ) = C(L). Thus ℑ_{q}^{!} is a weak quantization — to be called I_{q}(L) — of I(L) , and it is even strict if ℑ_{q} itself is strict, by the previous. Now we show first that this quantization I_{q}(L) is always proper — hence the subgroup L is coisotropic, by Lemma 3.5. Proving that I_{q}(L) is proper amounts to show that [x, y] ∈ (q − 1) ℭ_{q}^{↰} for all x, y ∈ ℭ_{q}^{↰}. By definition we have

[x, y] ∈ (q−1) ℭ_{q}^{↰} ⟺ δ_{n}([x, y]) ∈ (q−1)^{n+1}U_{q}^{⊗(n−1)} ⊗ ℭ_{q} ∀ n∈ℕ

and formula (4.7) gives, for all n ∈ ℕ,

δ_{n}([x, y]) = _{∑\begin{matrix}Λ∪Y =\{1,...,n\} \\ Λ∩Y ≠∅ \end{matrix}}( δ_{Λ}(x) δ_{Y} (y) − δ_{Y} (y) δ_{Λ}(x) )  (5.4)

while

δ_{Λ}(x) ∈ (q−1)^{|Λ|} j_{Λ}(U_{q}^{⊗(|Λ|−1)}⊗ℭ_{q}) , δ_{Y} (y) ∈ (q−1)^{|Y |} j_{Y}(U_{q}^{⊗(|Y |−1)}⊗ℭ_{q})

Now, Λ ∪ Y = {1, …, n} and Λ ∩ Y ≠ ∅ give |Λ| + |Y | ≥ n + 1, and since ℭ_{q} is a subalgebra of U_{q} we get

δ_{Λ}(x) δ_{Y} (y) , δ_{Y} (y) δ_{Λ}(x) ∈ (q−1)^{|Λ|+|Y |} U_{q}^{⊗(n−1)} ⊗ ℭ_{q} ⊆

⊆ (q−1)^{n+1}U_{q}^{⊗(n−1)} ⊗ ℭ_{q}

so that (5.4) yields

δ_{n}([x, y]) ∈ (q−1)^{n+1}U_{q}^{⊗(n−1)} ⊗ ℭ_{q}

thus [x, y] ∈ (q − 1) ℭ_{q}^{↰} . (4) In the real case (ℭ_{q})^{∗} = ℭ_{q}: this and the very definitions imply the claim.

Remark 5.8. In functorial language we may say that the map ℭ_{q} ↦ ℭ_{q}^{↰} establishes a functor between quantization of coisotropic subgroups of G and quantizations of Poisson homogeneous spaces of G^{∗}, moving from a local to a global description, sending each type of quantization in a proper one and preserving strictness. We leave to the interested reader all details which still need to be fixed.

We now move to connectedness properties of the coisotropic subgroup L identified in Propositions 5.5 and 5.7.

Proposition 5.9.

1. Let ℑ_{q}(K) be a strict quantization (of type ℑ) of a (coisotropic) subgroup K in G. Then the subgroup L of G^{∗} such that ℑ_{q}(K)^{!} = I_{q}(L) is connected.

2. Let ℭ_{q}(K) be a strict quantization of type ℭ of a (coisotropic) subgroup K of G. Then the subgroup L of G^{∗} such that ℭ_{q}(K)^{!} = C_{q}(L) is connected.

Proof. (1) Saying that the (closed) subgroup L is connected is equivalent to saying that its function algebra F [L] = F [G^{∗}]/I(L) has no non-trivial

idempotents. Note that, since F [G^{∗}] is the specialization of U_{q}′ at q = 1 and

I(L) is the similar specialization of ℑ_{q}^{!} , the quotient F [L] = F [G^{∗}]/I(L) is

canonically isomorphic to the specialization at q = 1 of U_{q}′/ ℑ_{q}^{!} . Let \overline{a} be an

idempotent in F [L]: if we take any lift of it in U_{q}′/ ℑ_{q}^{!} , i.e. any a ∈ U_{q}′/ ℑ_{q}^{!}

such that \overline{a} = a mod (q−1) U_{q}′/ ℑ_{q}^{!} . We must prove:

a^{2} ≡ a mod (q−1) U_{q}′/ ℑ_{q}^{!} ⟹ a mod (q−1) U_{q}′/ ℑ_{q}^{!} ∈ {0, 1} (5.5)

We can clearly reduce to the case when ϵ(\overline{a}) = 0: in fact, if \overline{a}^{2} = \overline{a} then ϵ(\overline{a}) is necessarily 0 or 1 (for it is unipotent too), and in the latter case we then find that \overline{a}_{0} := 1 − \overline{a} is idempotent and ϵ(\overline{a}_{0}) = 0. Also the lift a ∈ U_{q}′/ℑ_{q}^{!} can be chosen, in this case, such that: ϵ(a) = 0. To simplify notation, we set H := U_{q}/ℑ_{q} and H’ := U_{q}′/ℑ_{q}^{!}. We shall prove that, if a ∈ H’, ϵ(a) = 0 and

a^{2} ≡ a mod (q−1) H’, then a ≡ 0 mod (q−1) H’, i.e. a ∈ (q−1) H’ ; in fact, this will give (5.5). Having assumed that ℑ_{q} to be strict, H’ identifies with a ℂ[q, q^{−1}]–submodule of H given in terms of the coalgebra structure of the latter: the embedding is the one canonically induced by the maps U_{q}′ ⟶ U_{q} −↠ U_{q}/ ℑ_{q}. In fact, the

kernel of the latter map is U_{q}′ ⋂ ℑ_{q} (by strictness assumption). It is easy to see from definitions that U_{q}′ ⋂ ℑ_{q} = ℑ_{q}^{!}. Thus H’ does embed into H:

H’ = { η ∈ H | δ_{n}(η) ∈ (q−1)^{n} H^{⊗ n} , ∀ n ∈ ℕ } .  (5.6)

Now, a^{2} ≡ a mod (q−1)H’ means a = a^{2} + (q−1) c for some c ∈ H’; since ϵ(a) = 0, we have ϵ(c) = 0 as well. Applying δ_{n} to the identity a = a^{2} + (q−1) c and using formula (4.6) we get

δ_{n}(a) = δ_{n}(a^{2} ) + (q − 1) δ_{n}(c) = ∑_{Λ∪Y ={1,...,n}} δ_{Λ}(a) δ_{Y} (a) + (q − 1) δ_{n}(c)

for all n ∈ ℕ, which — noting that δ_{0}(a) := ϵ(a) = 0 yields:

δ_{n}(a) = ∑_{\begin{matrix}Λ∪Y =\{1,...,n\} \\ Λ,Y ≠∅ \end{matrix}}δ_{Λ}(a) δ_{Y} (a) + (q − 1) δ_{n}(c)  (5.7)

Since c ∈ H’ , the last summand (q−1) δ_{n}(c) in right-hand side of (5.7) belongs to (q −1)^{n+1} H^{⊗n}, thanks to (5.6). Similarly, since a ∈ H’ we have δ_{k}(a) ∈ (q−1)^{k} H^{⊗k} for all k ∈ ℕ, by (5.6) again: therefore each summand δ_{Λ}(a) δ_{Y} (a) in right-hand side of (5.7) belongs to (q − 1)^{n+1} H^{⊗n} as well. But then (5.7) yields δ_{n}(a) ∈ (q − 1)^{n+1}H^{⊗n} for all n ∈ ℕ, which, again by (5.6), means exactly that a ∈ (q −1) H’. This ends the proof of the first claim. (2) We will use similar arguments to show this claim: F [L] = F [G^{∗}]/I(L) has no

non-trivial idempotents. Since ℭ_{q}^{↰} = C_{q}(L) and C(L) = C(L̂ ), we can assume L = L̂, i.e. L is observable. This implies I(L) = Ψ(C(L)), which is clearly the

specialization at q = 1 of Ψ(C(L)) = U_{q}′ ℭ_{q}^{↰}; therefore, F [L] = F [G^{∗}]/I(L) is

canonically isomorphic to the specialization at q = 1 of U_{q}′/ U_{q}′ ℭ_{q}^{↰} . From now on, one can mimic step by step the proof of part (1). The only detail to modify is that one must take U_{q} ℭ_{q}^{+} =: Ψ(ℭ_{q}) in place of ℑ_{q}, and U_{q}′ (ℭ_{q}^{↰})^{+} =: Ψ(ℭ_{q}^{↰}) in place of ℑ_{q}^{!} . Letting H := U_{q}/Ψ(ℭ_{q}) , and H’ := U_{q}′/Ψ(ℭ_{q}^{↰}), the thesis amounts to prove that

a ∈ H’ , a^{2} ≡ a mod (q−1) H’ ⇒ a ≡ 0 mod (q−1) H’

(In fact also a ≡ 1 mod (q−1) H’ would be ok, but, arguing as before, we’ll restrict to the case ϵ(a) = 0). As ℭ_{q} is strict, it is easy to see from definitions that ℭ_{q}^{↰} = U_{q}′ ⋂ ℭ_{q}, hence Ψ(ℭ_{q}^{↰}) := U_{q}′ (ℭ_{q}^{↰})^{+} = U_{q}′(U_{q}′ ∩ ℭ_{q})^{+}: the latter is the kernel of the map

U_{q}′ ⟶ U_{q} −↠ U_{q}/U_{q} ℭ_{q}^{+}, so H’ embeds as a ℂ[q, q^{−1}]–submodule of H, namely

H’ = { η ∈ H | δ_{n}(η) ∈ (q−1)^{n} H^{⊗ n} , ∀ n ∈ ℕ } .

With this description at hand, computations are as in the proof of claim (1).

Our next results are about the behavior of quantum subgroups under composition of Drinfeld-like maps.

Proposition 5.10. Let I_{q}, C_{q}, ℑ_{q}, ℭ_{q} be weak quantizations of a subgroup K of G. Then:

1. I_{q} ⊆ (I_{q}^{⋎})^{!} , C_{q} ⊆ (C_{q}^{▽} )^{↰} ;

2. ℭ_{q} ⊇ (ℭ_{q}^{↰})^{▽} , ℑ_{q} ⊇ (ℑ_{q}^{!} )^{⋎} .

Proof. (1) By the very definitions, for any n ∈ ℕ we have

δ_{n}(I_{q}) ⊆ J_{Fq}^{⊗n} ⋂ ( ∑_{s=0}^{n} F_{q}^{⊗s} ⊗ I_{q} ⊗ F_{q}^{⊗(n−s−1)}) =

= ∑_{s=0}^{n} J_{Fq}^{⊗s} ⊗ I_{q} ⊗ J_{Fq}^{⊗(n−s−1)} ⊆ (q − 1)^{n} · ∑_{s=0}^{n} (F_{q}^{∨})^{⊗s}⊗ I_{q}^{⋎}⊗ (F_{q}^{∨})^{⊗(n−s−1)}

which means exactly I_{q} ⊆ (I_{q}^{⋎})^{!} . Similarly we can remark that:

δ_{n}(C_{q}) ⊆ J_{Fq}^{⊗n} ⋂ (F_{q}^{⊗(n−1)} ⊗ C_{q} ) =

= J_{Fq}^{⊗(n−1)} ⊗ (C_{q} ⋂ J_{Fq}) ⊆ (q−1)^{n}(F_{q}^{∨})^{⊗(n−1)}⊗ C_{q}^{▽}

which means C_{q} ⊆ (C_{q}^{▽})^{↰}. Therefore claim (1) is proved. (2) As (ℭ_{q}^{↰})^{▽} is generated — as an algebra — by (q−1)^{−1}ℭ_{q}^{↰} ⋂ J_{Uq}′, it is enough to show that the latter space is contained in ℭ_{q}. Let, then, x′ ∈ ℭ_{q}^{↰} ⋂ J_{Uq}′ . Surely δ_{1}(x′) ∈ (q − 1) ℭ_{q}, hence x′ = δ_{1}(x′) + ϵ(x′) ∈ (q − 1) ℭ_{q}. Therefore (q − 1)^{−1}x′ ∈ ℭ_{q}, q.e.d. Similarly, (ℑ_{q}^{!})^{⋎} is the left ideal of U_{q}′ generated by (q−1)^{−1} ℑ_{q}^{!} ⋂ J_{Uq}′, thus — since U_{q}′ ⊆ U_{q} — we must only prove that (q−1)^{−1} ℑ_{q}^{!} ⋂ J_{Uq}′ is contained in U_{q}. Again, if y′ ∈ ℑ_{q}^{!} ⋂ J_{Uq}′ then y′ = δ_{1}(y′) + ϵ(y′) ∈ (q − 1) ℑ_{q}. Thus we get (q − 1)^{−1}y′ ∈ ℑ_{q}, and (2) is proved.

Remarks:

(a) By repeated applications of the previous proposition it is easily proved that:

I_{q}^{⋎} = ((I_{q}^{⋎})^{!})^{⋎} , C_{q}^{▽} = ((C_{q}^{▽})^{↰})^{▽} , ℭ_{q}^{↰} = ((ℭ_{q}^{↰} )^{▽})^{↰} , ℑ_{q}^{!} = ((ℑ_{q}^{!} )^{⋎})^{!}

(b) Since we proved that Drinfeld-like maps always produce proper quantizations, and that proper quantizations specialize to coisotropic subgroups (cf. Proposition 3.5), the following holds:

1. if I_{q} = (I_{q}^{⋎})^{!} then I_{q} is a proper quantization (of type I) of a coisotropic subgroup of G;

2. if C_{q} = (C_{q}^{▽})^{↰} then C_{q} is a proper quantization (of type C) of a coisotropic subgroup of G;

3. if ℑ_{q} = (ℑ_{q}^{!})^{⋎} then ℑ_{q} is a proper quantization (of type ℑ) of a coisotropic subgroup of G;

4. if ℭ_{q} = (ℭ_{q}^{↰} )^{▽} then ℭ_{q} is a proper quantization (of type ℭ) of a coisotropic subgroup of G.

(c) Since the whole construction is independent of the existence of real structures all the above claims hold true in the real framework as well.

Next result reads as a converse of the previous one, holding for Drinfeld maps applied to strict quantizations:

Theorem 5.11.

(a) if I_{q} is a strict quantization of a coisotropic subgroup of G then one has I_{q} = (I_{q}^{⋎})^{!} ;

(b) if C_{q} is a strict quantization of a coisotropic subgroup of G then one has C_{q} = (C_{q}^{▽})^{↰};

(c) if ℑ_{q} is a strict quantization of a coisotropic subgroup of G then one has ℑ_{q} = (ℑ_{q}^{!})^{⋎};

(d) if ℭ_{q} is a strict quantization of a coisotropic subgroup of G then one has ℭ_{q} = (ℭ_{q}^{↰} )^{▽};

(e) The above claims hold true in the real framework as well.

Proof. (a) Let I_{q} be a strict quantization; by Proposition 5.10(1), it is enough to prove I_{q} ⊇ (I_{q}^{⋎})^{!} . For this we apply the argument used in [12], Proposition 4.3, to prove that F_{q} ⊇ (F_{q}^{∨})′. We denote by L the closed, coisotropic, connected subgroup of G^{∗} such that I_{q}^{⋎} = ℑ_{q}(L) , as in Proposition 5.1, and with 𝔩 its Lie algebra.

Let y′ ∈ (I_{q}^{⋎})^{!} . Then there is n ∈ ℕ and y^{∨} ∈ I_{q}^{⋎} \ (q − 1) I_{q}^{⋎} such that y′ = (q − 1)^{n}y^{∨}. As we have seen strictness of I_{q} implies strictness of I_{q}^{⋎} and therefore y^{∨} ∉ (q − 1) F_{q}^{∨}, and so for \overline{y^{∨}} := y^{∨} mod (q − 1) F_{q}^{∨} we have \overline{y^{∨}} ≠ 0 ∈ F_{q}^{∨}|_{q=1}= U(𝔤^{∗}).

As F_{q}^{∨} is a quantization of U(𝔤^{∗}), we can pick an ordered basis {b_{λ}}_{λ∈Λ} of 𝔤^{∗}, and a subset {x_{λ}^{∨}}_{λ∈Λ} of (q − 1)^{−1}J_{Fq} so that x_{λ}^{∨} mod (q − 1) F_{q}^{∨} = b_{λ} for all λ ∈ Λ ; therefore x_{λ}^{∨} = (q − 1)^{−1}x_{λ} for some x_{λ} ∈ J_{Fq} , for all λ (like in the proof of [12] Proposition 4.3). In addition, we choose now the basis and its lift so that a subset {b_{θ}}_{θ∈Θ} (for some suitable Θ ⊆ Λ ) is a basis of 𝔩, and, correspondingly, \{x_{θ}^{∨}\}_{θ∈Θ} ⊆ I_{q}^{⋎}. Since \overline{y^{∨}} ≠ 0 ∈ F_{q}^{∨}|_{q=1}= U(𝔤^{∗}), by the Poincaré-Birkhoff-Witt theorem there is a non-zero polynomial P ({b_{θ}}_{θ∈Θ}) in the b_{θ}’s such that \overline{y^{∨}} = P (\{b_{θ}\}_{θ∈Θ}), hence

y^{∨} − P ({x_{θ}^{∨}}_{θ∈Θ}) ∈ I_{q}^{⋎} ⋂ (q − 1) F_{q}^{∨} = (q − 1) I_{q}^{⋎} .

This implies y^{∨} = P ({x_{θ}^{∨}}_{θ∈Θ}) + (q − 1)^{ν} y_{1}^{∨} for some ν ∈ ℕ_{+} where y_{1}^{∨} ∈ I_{q}^{⋎} \ (q − 1) I_{q}^{⋎}. One can see, like in [9], Lemma 4.12, that the polynomial P has degree not greater than n. Thus y′ = (q−1)^{n}y^{∨} = (q−1)^{n}P ({x_{θ}^{∨}}_{θ∈Θ}) + (q−1)^{n+ν}y_{1}^{∨}, and

(q−1)^{n}P ({x_{θ}^{∨}}_{θ∈Θ}) = (q−1)^{n}P ({(q−1)^{−1} x_{θ}}_{θ∈Θ}) ∈ I_{q}

by a degree argument. But now, Proposition 5.10 gives I_{q} ⊆ (I_{q}^{⋎})^{!} . Then

y′_{1} := y′−(q−1)^{n}P ({x_{μ}^{∨}}_{θ∈Θ}) ∈ (I_{q}^{⋎})^{!} and y′_{1} = (q−1)^{n+ν}y_{1}^{∨} = (q−1)^{n_{1}}y_{1}^{∨}

where n_{1} := n + ν > n, and y_{1}^{∨} ∈ I_{q}^{⋎} \ (q − 1) I_{q}^{⋎}. We can then repeat the construction, with y′_{1} instead of y′, n_{1} instead of n, etc.: iterating, we find an increasing sequence of numbers {n_{s}}_{s∈ℕ} (with n_{0} := n) and a sequence of polynomials {P_{s}({X_{θ}}_{θ∈Θ})}_{s∈ℕ} (again P_{0} := P) such that the degree of P_{s}({X_{θ}}_{θ∈Θ}) is at most n_{s}, and the formal identity y′ = ∑_{s∈ℕ} (q − 1)^{n_{s}}P_{s}({x_{θ}^{∨}}_{θ∈Θ}) holds.

Now set I_{n} := ∑_{k=1}^{n} (q − 1)^{n−k} I_{q}^{k} (for all n ∈ ℕ), and let Î_{q} be the topological completion of I_{q} with respect to the filtration provided by the I_{n}’s. Then, by construction, (q−1)^{n_{s}}P_{s}({x_{θ}^{∨}}_{θ∈Θ}) ∈ I_{n} for all s∈ℕ. This yields

∑_{s∈ℕ}(q−1)^{n_{s}}P_{s}({x_{θ}^{∨}}_{θ∈Θ}) ∈ Î_{q}  and  y′ = ∑_{s∈ℕ}(q−1)^{n_{s}}P_{s}({x_{θ}^{∨}}_{θ∈Θ})

where the last is an identity in Î_{q}. Thus y′ ∈ (I_{q}^{⋎})^{!} ⋂ Î_{q}. Again with the same arguments as in [12], we see that I_{q} ⋂ (q−1)^{ℓ} Î_{q} = (q−1)^{ℓ} I_{q} for any ℓ ∈ ℕ. This together with y′ ∈ (I_{q}^{⋎})^{!} ⋂ Î_{q} give y′ = (q − 1)^{−m}η for some m ∈ ℕ and η ∈ I_{q}; thus

η = (q−1)^{m} y′ ∈ I_{q} ⋂ (q−1)^{m} Î_{q} = (q−1)^{m} I_{q} ,

whence y′ ∈ I_{q}, q.e.d.

(b) Assume that C_{q} is a strict quantization; by Proposition 5.10(2), it is enough to prove C_{q} ⊇ (C_{q}^{▽})^{↰}. To do that, we resume the argument used in [12], Proposition 4.3, to show that F_{q} ⊇ (F_{q}^{∨})′. We denote by L the closed, coisotropic, connected subgroup of G^{∗} such that C_{q}^{▽} = ℭ_{q}(L) and with 𝔩 its Lie algebra.

Let c′ ∈ (C_{q}^{▽})^{↰}. Then there exist n ∈ ℕ and c^{∨} ∈ C_{q}^{▽} \ (q − 1) C_{q}^{▽} such that c′ = (q − 1)^{n}c^{∨}. Note that strictness of C_{q} implies strictness of C_{q}^{▽}; hence c^{∨} ∉ (q − 1) F_{q}^{∨}, so that for \overline{c^{∨}} := c^{∨} mod (q − 1) F_{q}^{∨} we have \overline{c^{∨}} ≠ 0 ∈ F_{q}^{∨}|_{q=1}= U(𝔤^{∗}). Moreover, \overline{c^{∨}} ∈ C_{q}^{▽}|_{q=1} = ℭ(L) = U(𝔩) ⊆ U(𝔤^{∗}). Since F_{q}^{∨} is a quantization of U(𝔤^{∗}), we can fix an ordered basis {b_{λ}}_{λ∈Λ} of 𝔤^{∗}, and a subset {x_{λ}^{∨}}_{λ∈Λ} of (q − 1)^{−1}J_{Fq} such that x_{λ}^{∨} mod (q − 1) F_{q}^{∨} = b_{λ} for all λ ∈ Λ; so x_{λ}^{∨} = (q − 1)^{−1}x_{λ} for some x_{λ} ∈ J_{Fq}, for all λ (as in the proof of [12] Proposition 4.3). We can choose both the basis and its lift so that a subset {b_{μ}}_{μ∈M} is a basis of 𝔩 (here M ⊆ Λ), and, correspondingly, {x_{μ}^{∨}}_{μ∈M} ⊆

(q−1)^{−1}J_{Fq} ⋂ C_{q}^{▽}. Since \overline{c^{∨}} ≠ 0 ∈ F_{q}^{∨}|_{q=1}= U(𝔤^{∗}), by the Poincaré-Birkhoff-

Witt theorem there exists a non-zero polynomial P ({b_{μ}}_{μ∈M}) in variables b_{μ}’s such that \overline{c^{∨}} = P (\{b_{μ}\}_{μ∈M}), hence:

c^{∨} − P ({x_{μ}^{∨}}_{μ∈M}) ∈ C_{q}^{▽} ⋂ (q − 1) F_{q}^{∨} = (q − 1) C_{q}^{▽} .

Therefore, c^{∨} = P ({x_{μ}^{∨}}_{μ∈M}) + (q − 1)^{ν} c_{1}^{∨} for some ν ∈ ℕ_{+} where c_{1}^{∨} ∈ C_{q}^{▽} \ (q − 1) C_{q}^{▽}. Now, we can see — like in [9], Lemma 4.12 — that the degree of P is not greater than n. Then

c′ = (q − 1)^{n}c^{∨} = (q − 1)^{n}P ({x_{μ}^{∨}}_{μ∈M}) + (q − 1)^{n+ν}c_{1}^{∨}

with (q − 1)^{n}P ({x_{μ}^{∨}}_{μ∈M}) = (q − 1)^{n}P ({(q − 1)^{−1} x_{μ}}_{μ∈M}) ∈ C_{q} because P

has degree bounded (from above) by n. As C_{q} ⊆ (C_{q}^{▽})^{↰}, by Proposition 5.10, we get

c′_{1} := c′ −(q−1)^{n}P ({x_{μ}^{∨}}_{μ∈M}) ∈ (C_{q}^{▽})^{↰} and c′_{1} = (q−1)^{n+ν}c_{1}^{∨} = (q−1)^{n_{1}}c_{1}^{∨}

with n_{1} := n + ν > n, and c_{1}^{∨} ∈ C_{q}^{▽} \ (q − 1) C_{q}^{▽}. We can repeat this construction with c′_{1} in place of c′, n_{1} in place of n, etc.. Iterating, we get an increasing sequence of numbers {n_{s}}_{s∈ℕ} (n_{0} := n) and a sequence of polynomials {P_{s}({X_{μ}}_{μ∈M})}_{s∈ℕ} (P_{0} := P) such that the degree of P_{s}({X_{μ}}_{μ∈M}) is at most n_{s}, and c′ = ∑_{s∈ℕ} (q − 1)^{n_{s}}P_{s}({x_{μ}^{∨}}_{μ∈M}). Consider

I_{C_{q}} := Ker (C_{q} −↠^{ϵ} ℂ[q, q^{−1}] −↠^{ev_{1}} ℂ) = Ker (C_{q} −↠^{ev_{1}} C_{q}/(q − 1) C_{q} −↠^{ϵ̄} ℂ)

By construction, we have (q − 1)^{n_{s}}P_{s}({x_{μ}^{∨}}_{μ∈M}) ∈ I_{C_{q}}^{n_{s}} for all s ∈ ℕ;

in turn, this means that ∑_{s∈ℕ} (q − 1)^{n_{s}}P_{s}({x_{μ}^{∨}}_{μ∈M}) ∈ Ĉ_{q}, the latter being the I_{C_{q}}–adic completion of C_{q}, and the formal expression c′ = ∑_{s∈ℕ} (q − 1)^{n_{s}}P_{s}({x_{μ}^{∨}}_{μ∈M}) is an identity in Ĉ_{q}: therefore c′ ∈ (C_{q}^{▽})^{↰} ⋂ Ĉ_{q}.

Acting as in [12], again, we see that C_{q} ⋂ (q − 1)^{ℓ} Ĉ_{q} = (q − 1)^{ℓ} C_{q} for all ℓ ∈ ℕ. Getting back to c′ ∈ (C_{q}^{▽})^{↰} ⋂ Ĉ_{q}, we have c′ = (q − 1)^{−m}κ for some m ∈ ℕ and κ ∈ C_{q} ; thus κ = (q−1)^{m} c′ ∈ C_{q} ⋂ (q−1)^{m} Ĉ_{q} = (q−1)^{m} C_{q}, whence c′ ∈ C_{q}, q.e.d.

(c) Let ℑ_{q} be a strict quantization: by Proposition 5.10(2) it is enough to prove ℑ_{q} ⊆ (ℑ_{q}^{!})^{⋎}; so given y ∈ ℑ_{q}, we must prove that y ∈ (ℑ_{q}^{!})^{⋎}. Recall that ℑ_{q} ⊆ U_{q} = (U_{q}′)^{∨}, the last identity following from Theorem 4.1. By construction,

(U_{q}′)^{∨} = _{∑n≥0} (q − 1)^{−n} I n_{Uq′} , I_{Uq}′ := (U_{q}′)^{+} + (q − 1) U_{q}′

so for y ∈ ℑ_{q} ⊆ U_{q} = (U_{q}′)^{∨} there exists N ∈ ℕ such that

y_{+} := (q − 1)^{N} y ∈ I N_{Uq′} ⊆ U_{q}′  (5.8)

Strictness of ℑ_{q}, i.e. ℑ_{q} ⋂ (q − 1) U_{q} = (q − 1) ℑ_{q}, implies

( ∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}) ⋂ ((q−1)^{n} U_{q}^{⊗n}) =

= (q−1)^{n} ( ∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)})

for all n ∈ ℕ_{+} ; then, by the very definitions, the latter yields ℑ_{q}^{!} = ℑ_{q} ⋂ U_{q}′ . If in (5.8) N = 1, then y_{+} = y ∈ U_{q}′, thus y ∈ ℑ_{q} ⋂ U_{q}′ = ℑ_{q}^{!} , q.e.d. If N > 1 instead, then formula (5.8), along with ℑ_{q} ˙⊴ U_{q}, yields

δ_{n}(y_{+}) ∈ ((q − 1)^{N} · ∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}) ⋂ ((q − 1)^{n} U_{q}^{⊗n}) (5.9)

for all n ∈ ℕ_{+} , and since ℑ_{q} is strict, from (5.9) one gets

δ_{n}(y_{+}) ∈ (q−1)^{n} ∑_{s=1}^{n}U_{q}^{⊗(s−1)} ⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)} ∀ n ∈ ℕ

which means y_{+} ∈ ℑ_{q}^{!} . Eventually, we have found y_{+} ∈ ℑ_{q}^{!} ⋂ I N_{Uq′} . Now look at I_{ℑ^{!}_{q}} := I_{Uq}′ ⋂ ℑ_{q}^{!} . Using the fact that U_{q}′ = U_{q}(𝔤)′ = F [G^{∗}] —

from Theorem 4.1 — and ℑ_{q}^{!} = ℑ_{q}(K)^{!} = I_{q}(L) for some coisotropic subgroup L in G^{∗} — as granted by Proposition 5.5 — and still taking into account strictness, by an easy geometrical argument (via specialization at q = 1) we see that

I n_{Uq′} ⋂ ℑ_{q}^{!} ≡ I n_{ℑ^{!}_{q}} mod (q − 1) U_{q}′ ∀ n ∈ ℕ_{+} .

This, together with ℑ_{q} ⋂ (q − 1) U_{q} = (q − 1) ℑ_{q}, yields also

I n_{Uq′} ⋂ ℑ_{q}^{!} ≡ I n_{ℑ^{!}_{q}} mod (q − 1) ℑ_{q}^{!} ∀ n ∈ ℕ_{+}

Finally, by suitable, iterated cancelation of factors (q − 1), which is possible because of the condition ℑ_{q} ⋂ (q − 1)U_{q} = (q − 1) ℑ_{q}, we eventually obtain

I n_{Uq′} ⋂ ℑ_{q}^{!} ≡ I n_{ℑ^{!}_{q}} mod (q − 1)^{n} ℑ_{q}^{!} ∀ n ∈ ℕ_{+} .

To sum up, we have y_{+} ∈ I N_{Uq′} ⋂ ℑ_{q}^{!} = I N_{ℑ^{!}_{q}} ; therefore, by definitions,

y = (q − 1)^{−N} y_{+} ∈ (q − 1)^{−N} I N_{ℑ ^{!}_{q}} ⊆ (ℑ_{q}^{!})^{⋎} .

(d) Let ℭ_{q} be a strict quantization: by Proposition 5.10(2) it is enough to prove ℭ_{q} ⊆ (ℭ_{q}^{↰} )^{▽}. We follow the same arguments used for claim (c). Let c ∈ ℭ_{q}, since

ℭ_{q} ⊆ U_{q} = (U_{q}′)^{∨} — from Theorem 4.1 — and (U_{q}′)^{∨} = _{∑n≥0} (q − 1)^{−n} I n_{Uq′},

(notation as above) for c ∈ ℭ_{q} ⊆ U_{q} = (U_{q}′)^{∨} there exists N ∈ ℕ such that

c_{+} := (q − 1)^{N} c ∈ I N_{Uq′} ⊆ U_{q}′ .

Now, strictness of ℭ_{q} implies

(U_{q}^{⊗(n−1)} ⊗ ℭ_{q}) ⋂ (q − 1)^{n} U_{q}^{⊗n} = (q − 1)^{n} (U_{q}^{⊗(n−1)} ⊗ ℭ_{q}) ∀ n ∈ ℕ_{+}

hence ℭ_{q}^{↰} = ℭ_{q} ⋂ U_{q}′. If the above N is 1, then c_{+} = c ∈ U_{q}′, thus c ∈ ℭ_{q} ⋂ U_{q}′ = ℭ_{q}^{↰} , q.e.d. If instead N > 1 , then

δ_{n}(c_{+}) ∈ ((q − 1)^{N} · U_{q}^{⊗n−1} ⊗ ℭ_{q}) ⋂ ((q − 1)^{n} U_{q}^{⊗n}) ∀ n ∈ ℕ_{+}

and, since ℭ_{q} is strict, δ_{n}(c_{+}) ∈ (q − 1)^{n} · U_{q}^{⊗n−1} ⊗ ℭ_{q} for all n ∈ ℕ_{+} , which means c_{+} ∈ ℭ_{q}^{↰} . Thus, eventually, we have c_{+} ∈ ℭ_{q}^{↰} ⋂ I N_{Uq′} . Let us look, now, at I_{ℭ^{↰}_{q}} := I_{Uq}′ ⋂ ℭ_{q}^{↰} . Again in force of strictness of ℭ_{q}, a geometrical argument (at q = 1) as before leads us to

I n_{Uq′} ⋂ ℭ_{q}^{↰} ≡ I n_{ℭ^{↰}_{q}} mod (q − 1)^{n} ℭ_{q}^{↰} , ∀ n ∈ ℕ_{+}

from which we conclude that c_{+} ∈ I N_{Uq′} ⋂ ℭ_{q}^{↰} = I N_{ℭ^{↰}_{q}} . Therefore, by the very definitions,

c = (q − 1)^{−N} c_{+} ∈ (q − 1)^{−N} I N_{ℭ^{↰}_{q}} ⊆ (ℭ_{q}^{↰} )^{▽} , q.e.d.

(e) This is a direct consequence of claims from (a) through (d). (f) Once again, this is true because the whole construction is independent of the existence of real structures.

It is now time to clarify how the coisotropic subgroup L of G^{∗} is linked to the coisotropic subgroup K of G. We will give this relation in the weak quantization case first, and show how it improves under stronger hypothesis.

Theorem 5.12. Let K be a subgroup of G, and let I_{q}(K) , C_{q}(K) , ℑ_{q}(K) and ℭ_{q}(K) be weak quantizations as in Definition 3.6. Then (with notation of Proposition 2.2)

(a) I_{q}(K)^{⋎} = ℑ_{q}(K^{⟨⊥⟩}) ;

(b) C_{q}(K)^{▽} = ℭ_{q}(K^{⟨⊥⟩}) ;

(c) if ℑ_{q}(K) = (ℑ_{q}(K)^{!} )^{⋎}, then ℑ_{q}(K)^{!} = I_{q}(K^{⟨⊥⟩}); in particular, this holds if the quantization ℑ_{q}(K) is strict;

(d) if ℭ_{q} = (ℭ_{q}(K)^{↰} )^{▽}, then ℭ_{q}(K)^{↰} = C_{q}(K^{⟨⊥⟩}); in particular, this holds if the quantization ℭ_{q}(K) is strict;

(e) claims (a–d) hold as well in the framework of real quantum subgroups.

Proof. (a) By Proposition 5.1 we already have I_{q}(K)^{⋎} = ℑ_{q}(L) for some subgroup L ⊆ G^{∗}. In order to show that L = K^{⟨⊥⟩}, we will proceed much like in the proof of F_{q}^{∨}/(q − 1) F_{q}^{∨} ≅ U(𝔤^{∗}), as given in [12], Theorem 4.7. Let us fix a subset {j_{1}, …, j_{n}} of J adapted to K as in the proof of Proposition 5.1. Let J^{∨} := (q−1)^{−1}J ⊂ F_{q}^{∨} and j^{∨} := (q − 1)^{−1}j for all j ∈ J. From the discussion in that proof, we argue also that \{(q − 1)^{−|\underline{e}|}j^{\underline{e}} mod (q−1) F_{q}^{∨} | \underline{e} ∈ ℕ^{n} \}, where j^{\underline{e}} = ∏_{s=1}^{n} j_{s}^{\underline{e}(i)}, is a ℂ–basis of F_{1}^{∨}, and \{j_{1}^{∨}, …, j_{n}^{∨}\} is a ℂ–basis of 𝔱 = J^{∨} mod (q − 1) F_{q}^{∨}. Now, j_{μ} j_{ν} − j_{ν} j_{μ} ∈ (q−1) J (for μ, ν ∈ {1, …, n}) implies that:

j_{μ} j_{ν} − j_{ν} j_{μ} = (q − 1) ∑_{s=1}^{n} c_{s} j_{s} + (q − 1)^{2} γ_{1} + (q − 1) γ_{2}

for some c_{s} ∈ ℂ[q, q^{−1}], γ_{1} ∈ J and γ_{2} ∈ J^{2}. Therefore

[j_{μ}^{∨}, j_{ν}^{∨} ] := j_{μ}^{∨} j_{ν}^{∨} − j_{ν}^{∨} j_{μ}^{∨} = ∑_{s=1}^{n} c_{s} j_{s}^{∨} + γ_{1} + (q−1) γ_{2}^{∨} ≡

≡ ∑_{s=1}^{n} c_{s} j_{s}^{∨} mod (q−1) F_{q}^{∨}

(where we set γ_{2}^{∨} := (q − 1)^{−2}γ_{2} ∈ (q − 1)^{−2}(J^{∨})^{2} ⊆ F_{q}^{∨}) thus the subspace 𝔱 := J^{∨} mod (q−1) F_{q}^{∨} is a Lie subalgebra of F_{1}^{∨}. But then it should be F_{1}^{∨} ≅ U(𝔱) as Hopf algebras, by the above description of F_{1}^{∨} and PBW theorem. Now for the second step. The specialization map π^{∨}: F_{q}^{∨} −↠ F_{1}^{∨} = U(𝔱) actually restricts to η : J^{∨} ↠ 𝔱 = J^{∨}/J^{∨} ⋂ ((q−1) F_{q}^{∨}) = J^{∨}/(J + J^{∨}J), because J^{∨} ⋂ ((q−1) F_{q}^{∨}) = J^{∨} ⋂ (q−1)^{−1}I_{Fq}^{2} = J + J^{∨}J. Also, multiplica-

tion by (q−1)^{−1} yields a ℂ[q, q^{−1}]–module isomorphism μ : J −↠^{≅} J^{∨}. Let ρ : 𝔪_{e} −↠ 𝔪_{e}/𝔪_{e}^{2} = 𝔤^{∗} be the natural projection map, and ν : 𝔤^{∗} ⟶ 𝔪_{e} a section of ρ. The specialization map π : F_{q} −↠ F_{1} restricts to a map π′: J −↠ J/(J ⋂ (q − 1) F_{q}) = 𝔪_{e}. Let’s fix a section γ : 𝔪_{e} ⟶ J of π′ and consider the composition σ := η ∘ μ ∘ γ ∘ ν : 𝔤^{∗} ⟶ 𝔱: this is a well-defined Lie bialgebra morphism, independent of the choice of ν and γ. In the proof of Proposition 5.1 we made a particular choice for the subset {j_{1}, …, j_{n}}. As a consequence, the above analysis to prove that σ : 𝔤^{∗} ≅ 𝔱 shows also that the left ideal I_{1}^{⋎} := I_{q}^{⋎} mod (q−1) F_{q}^{∨} of U(𝔱) is generated by

η(I_{q}^{⋎}) = (η ∘ μ)(I_{q}) = (σ ∘ ρ ∘ π)(I_{q}) = σ(ρ(I)) = σ(𝔨^{⊥}) .

So I_{1}^{⋎} = U(𝔤^{∗}) · 𝔨^{⊥} = U(𝔤^{∗}) · ⟨𝔨^{⊥}⟩ = ℑ(K^{⟨⊥⟩}) — where we are identifying 𝔤^{∗} with its image via σ — which eventually means 𝔩 = ⟨𝔨^{⊥}⟩. (b) By Proposition 5.3 we have C_{q}(K)^{▽} = ℭ_{q}(L) for some coisotropic subgroup L in G^{∗}. We must prove that L = K^{⟨⊥⟩}. Once again, we mimic the procedure of the proof of Proposition 5.3, and we fix a subset {j_{1}, …, j_{n}} of J as in the proof of such Proposition. Then, tracking the analysis we did there to prove that σ : 𝔤^{∗} ≅ 𝔱, we see also that the unital subalgebra C_{1}^{▽} := C_{q}^{▽} mod (q−1) F_{q}^{∨} of U(𝔤^{∗}) is generated by η(C_{q}^{▽}) = (μ ∘ η)(C_{q}) = (σ ∘ ρ ∘ π)(C_{q}) = σ(ρ(C)) = σ(𝔨^{⊥}). Thus C_{1}^{▽} is the subalgebra of U(𝔤^{∗}) generated by 𝔨^{⊥}, hence C_{1}^{▽} = ⟨𝔨^{⊥}⟩_{Alg} = U(⟨𝔨^{⊥}⟩_{Lie}) = U(𝔨^{⟨⊥⟩}) = ℭ(K^{⟨⊥⟩}), which means 𝔩 = ⟨𝔨^{⊥}⟩, q.e.d. (c) Thanks to Proposition 5.5 we already know that ℑ_{q}(K)^{!} = I_{q}(L) for some coisotropic subgroup L in G^{∗}. Again, we must prove that L = K^{⟨⊥⟩}. Note that we can assume K to be connected, as its relationship with ℑ_{q}(K) passes through 𝔨 alone; thus in the end we simply have to prove that 𝔩 := Lie (L) = 𝔨^{⟨⊥⟩} = 𝔨^{⊥}, taking into account that 𝔨^{⟨⊥⟩} = 𝔨^{⊥} because 𝔨 is coisotropic, by a remark following Proposition 5.10. By assumption ℑ_{q}(K) = (ℑ_{q}(K)^{!} )^{⋎}; this and (a) together give

ℑ_{q}(K) = (ℑ_{q}(K)^{!} )^{⋎} = I_{q}(L)^{⋎} = ℑ_{q}(L^{⟨⊥⟩}) = ℑ_{q}(L^{⊥})

where L^{⟨⊥⟩} = L^{⊥} because L is coisotropic as well: at q = 1 this implies 𝔨 = 𝔩^{⊥}, q.e.d. (d) We must prove that L = K^{⟨⊥⟩}: as above we can assume K to be connected, so we only have to prove that 𝔩 := Lie (L) = 𝔨^{⟨⊥⟩} = 𝔨^{⊥} (as 𝔨 is coisotropic, by Proposition 5.11. By assumption ℭ_{q} = (ℭ_{q}(K)^{↰} )^{▽}; this along with (c) gives

ℭ_{q}(K) = (ℭ_{q}(K)^{↰} )^{▽} = C_{q}(L)^{▽} = ℭ_{q}(L^{⟨⊥⟩}) = ℭ_{q}(L^{⊥})

with L^{⟨⊥⟩} = L^{⊥} since L is coisotropic too: specializing at q = 1, this eventually yields 𝔨 = 𝔩^{⊥}. (e) This is clear again since all arguments pass through unchanged in the real setup.

Corollary 5.13. Let I_{q}(K) and C_{q}(K) be weak quantizations of a (not necessarily) coisotropic subgroup K of G, of type I and C respectively. Then, with notation of Definition 2.1, we have

(I_{q}(K)^{⋎})^{!} = I_{q}( K^{∘} ) , (C_{q}(K)^{▽})^{↰} = C_{q}( K^{∘} ) .

Proof. Theorem 5.12(a) gives I_{q}(K)^{⋎} = ℑ_{q}(K^{⟨⊥⟩}), and Proposition 5.10 yields

(ℑ_{q}(K^{⟨⊥⟩})^{!})^{⋎} = ((I_{q}(K)^{⋎})^{!})^{⋎} = I_{q}(K)^{⋎} = ℑ_{q}(K^{⟨⊥⟩})

so that (ℑ_{q}(K^{⟨⊥⟩})^{!})^{⋎} = ℑ_{q}(K^{⟨⊥⟩}). Then Theorem 5.12 gives

ℑ_{q}(K^{⟨⊥⟩})^{!} = I_{q}((K^{⟨⊥⟩})^{⟨⊥⟩}) = I_{q}(K^{∘} )

by Proposition 2.2. Therefore (I_{q}(K)^{⋎})^{!} = ℑ_{q}(K^{⟨⊥⟩})^{!} = I_{q}(K^{∘} ) as claimed. Similarly, Theorem 5.12(b) gives C_{q}(K)^{▽} = ℭ_{q}(K^{⟨⊥⟩}), and the first remark after Proposition 5.10 yields

(ℭ_{q}(K^{⟨⊥⟩})^{↰} )^{▽} = ((C_{q}(K)^{▽})^{↰} )^{▽} = C_{q}(K)^{▽} = ℭ_{q}(K^{⟨⊥⟩})

so that (ℭ_{q}(K^{⟨⊥⟩})^{↰} )^{▽} = ℭ_{q}(K^{⟨⊥⟩}) . Then again by Theorem 5.12(d) we get

ℭ_{q}(K^{⟨⊥⟩})^{↰} = C_{q}((K^{⟨⊥⟩})^{⟨⊥⟩}) = C_{q}(K^{∘} )

still by Proposition 2.2. Thus (C_{q}(K)^{▽})^{↰} = ℭ_{q}(K^{⟨⊥⟩})^{↰} = C_{q}( K^{∘} ) as claimed.

Remark 5.14. One might guess that the analogue to this Corollary holds true for weak quantizations of type ℑ and ℭ as well: actually, we have no clue about that, in either sense.

We now consider the “compatibility” among different Drinfeld-like maps acting on quantizations of different types over a single pair (subgroup, space). Indeed, we show that Drinfeld’s functors preserve the subgroup-space correspondence — Proposition 5.15 — and the orthogonality correspondence — Proposition 5.17 — (if either occurs at the beginning) between different quantizations as mentioned.

Proposition 5.15. Let K be a closed subgroup of G, and let Ψ and Φ be the map mentioned in §2.1. Then the following holds:

(a) Let C_{q} and I_{q} be as in Section 3. If Ψ(C_{q}) = I_{q}, then Ψ(C_{q}^{▽}) = I_{q}^{⋎} .

(b) Let I_{q} and C_{q} be as in Section 3. If Φ(I_{q}) = C_{q}, then Φ(I_{q}^{⋎}) = C_{q}^{▽} .

(c) Let ℭ_{q} and ℑ_{q} be as in Section 3. If Ψ(ℭ_{q}) = ℑ_{q}, then Ψ(ℭ_{q}^{↰}) ⊆ ℑ_{q}^{!} .

(d) Let ℑ_{q} and ℭ_{q} be as in Section 3. If Φ(ℑ_{q}) = ℭ_{q}, then Φ(ℑ_{q}^{!}) = ℭ_{q}^{↰} .

Proof. Claims (a) and (c) both follow trivially from definitions. As to claim (b), let η ∈ C_{q}^{+} = Φ(I_{q})^{+} , so that Δ(η) ∈ η ⊗ 1 + F_{q} ⊗ I_{q}. Then η^{∨} := (q − 1)^{−1}η enjoys

Δ(η^{∨}) ∈ η^{∨}⊗ 1 + F_{q} ⊗ (q − 1)^{−1} I_{q} ⊆ η^{∨}⊗ 1 + F_{q}^{∨}⊗ I_{q}^{⋎}

whence η^{∨} ∈ (F_{q}^{∨})^{coI_{q}^{⋎}} =: Φ(I_{q}^{⋎}). Since C_{q}^{▽} is generated (as a subalgebra) by (q − 1)^{−1} C_{q}^{+} , we conclude that C_{q}^{▽} ⊆ Φ(I_{q}^{⋎}). Conversely, let φ ∈ Φ(I_{q}^{⋎}). Then Δ(φ) ∈ φ ⊗ 1 + F_{q}^{∨}⊗ I_{q}^{⋎}, and there exists n ∈ ℕ such that φ_{+} := (q − 1)^{n}φ ∈ I_{q}, so that Δ(φ_{+}) ∈ F_{q} ⊗ I_{q} + I_{q} ⊗ F_{q} (since I_{q} ˙⊴ F_{q}). Then

Δ(φ_{+}) ∈ (φ_{+} ⊗ 1 + (q − 1)^{n}F_{q}^{∨}⊗ I_{q}^{⋎}) ⋂ (F_{q} ⊗ I_{q} + I_{q} ⊗ F_{q})

or equivalently

Δ(φ_{+}) − φ_{+} ⊗ 1 ∈ ((q − 1)^{n}F_{q}^{∨}⊗ I_{q}^{⋎}) ⋂ (F_{q} ⊗ I_{q} + I_{q} ⊗ F_{q})  (5.10)

Now, the description of I_{q}^{⋎} given in the proof of Proposition 5.1 implies that

((q − 1)^{n}F_{q}^{∨}⊗ I_{q}^{⋎}) ⋂ (F_{q} ⊗ I_{q} + I_{q} ⊗ F_{q}) = F_{q} ⊗ I_{q}

this together with (5.10) yields Δ(φ_{+}) ∈ φ_{+}⊗1+F_{q}⊗I_{q}, hence φ_{+} ∈ F_{q}^{coI_{q}} =: Φ(I_{q}) = C_{q} and so φ ∈ (q − 1)^{n} C_{q} ⋂ F_{q}^{∨}. On the other hand, the description of C_{q}^{▽} in the proof of Proposition 5.3 implies that (q − 1)^{−n} C_{q} ⋂ F_{q}^{∨} ⊆ C_{q}^{▽}, hence we get φ ∈ C_{q}^{▽}, q.e.d. We finish with claim (d). For the inclusion Φ(ℑ_{q}^{!}) ⊇ ℭ_{q}^{↰}, let κ ∈ ℭ_{q}^{↰}. Since Φ(ℑ_{q}^{!}) contains the scalars, we may assume that κ ∈ Ker (ϵ), thus Δ(κ) = κ⊗1+1⊗κ+δ_{2}(κ) . By Proposition 5.7, we have ℭ_{q}^{↰} ˙⊴_{ℓ} U_{q}′; thus Δ(κ)−κ⊗1 = 1 ⊗ κ + δ_{2}(κ) ∈ U_{q}′ ⊗ ℭ_{q}^{↰}, and more precisely

Δ(κ) − κ ⊗ 1 = 1 ⊗ κ + δ_{2}(κ) ∈ U_{q}′ ⊗ (ℭ_{q}^{↰} )^{+} .

Since ℭ_{q}^{↰} ⊆ Ψ(ℭ_{q}^{↰}) ⊆ ℑ_{q}^{!} , by claim (c), we get Δ(κ) − κ ⊗ 1 ∈ U_{q}′ ⊗ ℑ_{q}^{!} , so κ ∈ (U_{q}′)^{coℑ_{q}^{!}} =: Φ(ℑ_{q}^{!}). Thus ℭ_{q}^{↰} ⊆ Φ(ℑ_{q}^{!}). For the converse inclusion, let η ∈ Φ(ℑ_{q}^{!}); again, we can assume η ∈ Ker (ϵ) too. As ℑ_{q}^{!} ⊆ ℑ_{q}, we get η ∈ Φ(ℑ_{q}^{!}) ⊆ Φ(ℑ_{q}) = ℭ_{q}. Then δ_{n}(η) ∈ U_{q}^{⊗n}⊗ ℭ_{q} for all n ∈ ℕ_{+} , so

δ_{n}(η) ∈ (q−1)^{n}( ∑_{s=1}^{n−1} U_{q}^{⊗(s−1)}⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}) ⋂ (U_{q}^{⊗(n−1)}⊗ ℭ_{q}) ⊆

⊆ (q−1)^{n} U_{q}^{⊗(n−1)}⊗ ℭ_{q}

hence δ_{n}(η) ∈ (q−1)^{n} U_{q}^{⊗(n−1)}⊗ ℭ_{q} ( n ∈ ℕ_{+} ) and η ∈ ℭ_{q}, which means that η ∈ ℭ_{q}^{↰}.

Remark 5.16. The inclusion Ψ(ℭ_{q}^{↰}) ⊆ ℑ_{q}^{!} of Proposition 5.15(c) is not an identity in general — indeed, counterexamples do exist.

Finally, we look at what happens when our Drinfeld-like recipes are applied to a pair of quantizations associated with a same subgroup / homogeneous spaces with respect to some fixed double quantization (in the sense of Section 3). The result reads as follows:

Proposition 5.17. Let (F_{q}[G] , U_{q}(𝔤)) be a double quantization of (G, 𝔤) . Then:

(a) Let C_{q} and ℑ_{q} be weak quantizations and assume that C_{q} = ℑ_{q}^{⊥} and ℑ_{q} = C_{q}^{⊥}. Then ℑ_{q}^{!} = (C_{q}^{▽})^{⊥} and C_{q}^{▽} ⊆ (ℑ_{q}^{!} )^{⊥}. If, in addition, either

one of C_{q} or ℑ_{q} is strict, then also C_{q}^{▽} = (ℑ_{q}^{!} )^{⊥}.

(b) Let ℭ_{q} and I_{q} be weak quantizations and assume that I_{q} = ℭ_{q}^{⊥} and ℭ_{q} = I_{q}^{⊥}. Then ℭ_{q}^{↰} = (I_{q}^{⋎})^{⊥} and I_{q}^{⋎} ⊆ (ℭ_{q}^{↰} )^{⊥}. If, in addition,

either one of ℭ_{q} or I_{q} is strict, then also I_{q}^{⋎} = (ℭ_{q}^{↰} )^{⊥}.

Proof. Both in claim (a) and in claim (b) the orthogonality relations between ℭ_{q} and I_{q} and between C_{q} and ℑ_{q} are considered w.r.t. the pairing between F_{q}[G] and U_{q}(𝔤), and the subsequent orthogonality relations are meant w.r.t. the pairing between F_{q}[G]^{∨} and U_{q}(𝔤)′. Indeed, by Theorem 4.1, (U_{q}(𝔤)′ , F_{q}[G]^{∨})

is a double quantization of (G^{∗}, 𝔤^{∗}). (a) First, ϵ(ℑ_{q}) = 0 because ℑ_{q} is a coideal. Then x = δ_{1}(x) ∈ (q − 1) U_{q} for all x ∈ ℑ_{q}^{!} , hence ℑ_{q}^{!} ⊆ (q − 1) U_{q}. Thus we have

⟨C_{q}, ℑ_{q}^{!} ⟩ ⊆ (q − 1) ℂ[q, q^{−1}] .

Now let J = J_{Fq} be the ideal of F_{q}, and take c_{i} ∈ C_{q} ∩ J (i = 1, …, n) ; then ⟨c_{i}, 1⟩ = ϵ(c_{i}) = 0 (i = 1, …, n) . Given y ∈ ℑ_{q}^{!} , look at

⟨ ∏_{i=1}^{n}c_{i} , y⟩ = ⟨ ⊗_{i=1}^{n} c_{i} , Δ^{n}(y)⟩ = ⟨ ⊗_{i=1}^{n} c_{i} , ∑_{Ψ⊆{1,...,n}}δ_{Ψ}(y)⟩ =

= ∑_{Ψ⊆{1,...,n}}⟨ ⊗_{i=1}^{n} c_{i} , δ_{Ψ}(y)⟩

Consider the summands in the last term of the above formula. Let |Ψ| = t (t ≤ n) , then

⟨ ⊗_{i=1}^{n} c_{i} , δ_{Ψ}(y)⟩ = ⟨ ⊗_{i∈Ψ} c_{i} , δ_{t}(y)⟩ · ∏_{j∉Ψ}⟨ c_{j} , 1 ⟩

by definition of δ_{Ψ} . Thanks to the previous analysis, we have ∏_{j∉Ψ}⟨c_{j} , 1⟩ = 0 unless Ψ = {1, …, n}, and in the latter case

δ_{Ψ}(y) = δ_{n}(y) ∈ (q − 1)^{n} ∑_{s=1}^{n} U_{q}^{⊗(s−1)}⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)} .

The outcome is

⟨ ⊗_{i=1}^{n} c_{i} , y⟩ = ⟨ ⊗_{i=1}^{n} c_{i} , δ_{n}(y)⟩ ∈

∈ ⟨ ⊗_{i=1}^{n} c_{i} , (q − 1)^{n} ∑_{s=1}^{n} U_{q}^{⊗(s−1)}⊗ ℑ_{q} ⊗ U_{q}^{⊗(n−s)}⟩ = 0

because y ∈ ℑ_{q}^{!} and ℑ_{q} = C_{q}^{⊥} by assumption. Therefore one has ⟨(q−1)^{−n}(C_{q} ⋂ J )^{n}, ℑ_{q}^{!} ⟩ = 0 , for all n ∈ ℕ_{+} . In addition, ⟨1 , ℑ_{q}^{!} ⟩ = ϵ(ℑ_{q}^{!}) = 0 . The outcome is ⟨C_{q}^{▽}, ℑ_{q}^{!} ⟩ = 0 , whence ℑ_{q}^{!} ⊆ (C_{q}^{▽})^{⊥} and C_{q}^{▽} ⊆ (ℑ_{q}^{!} )^{⊥}. Now we prove also (C_{q}^{▽})^{⊥} ⊆ ℑ_{q}^{!} . Notice that C_{q}^{▽} ⊇ C_{q}, whence (C_{q}^{▽})^{⊥} ⊆ C_{q}^{⊥} = ℑ_{q}; therefore (C_{q}^{▽})^{⊥} ⊆ ℑ_{q}. Pick now η ∈ (C_{q}^{▽})^{⊥} (inside U_{q}′). Since η ∈ U_{q}′, for all n ∈ ℕ_{+} we have δ_{n}(η) ∈ (q − 1)^{n}U_{q}^{⊗n}, and from η ∈ (C_{q}^{▽})^{⊥} we get also that η_{+} := (q − 1)^{−n}δ_{n}(η) enjoys ⟨( C_{q} ⋂ J_{F_{q}} )^{⊗n}, η_{+}⟩ = 0 — acting as before — so that

η_{+} ∈ (( C_{q} ⋂ J_{F_{q}} )^{⊗n})^{⊥} = ∑_{r+s=n−1}U_{q}^{⊗r} ⊗ ( C_{q} ⋂ J_{F_{q}} )^{⊥} ⊗ U_{q}^{⊗s} .

Moreover δ_{n}(η) ∈ J_{U_{q}}^{⊗n}, hence δ_{n}(η) ∈ ((q−1)^{n}U_{q}^{⊗n}) ⋂ J_{U_{q}}^{⊗n} = (q−1)^{n}J_{U_{q}}^{⊗n}, so

η_{+} ∈ (( C_{q} ⋂ J_{F_{q}})^{⊗n})^{⊥} ⋂ J_{U_{q}}^{⊗n} =

= ( ∑_{r+s=n−1} U_{q}^{⊗r} ⊗ ( C_{q} ⋂ J_{F_{q}} )^{⊥} ⊗ U_{q}^{⊗s}) ⋂ J_{U_{q}}^{⊗n} =

= ∑_{r+s=n−1} J_{U_{q}}^{⊗r} ⊗ (( C_{q} ⋂ J_{F_{q}} )^{⊥} ⋂ J_{U_{q}}) ⊗ J_{U_{q}}^{⊗s} .

Since ( C_{q} ⋂ J_{F_{q}} )^{⊥} ⋂ J_{U_{q}} = C_{q}^{⊥} ⋂ J_{U_{q}} = ℑ_{q} ⋂ J_{U_{q}} = ℑ_{q}, we have

η_{+} ∈ ∑_{r+s=n−1}J_{U_{q}}^{⊗r}⊗ ℑ_{q} ⊗ J_{U_{q}}^{⊗s}

whence

δ_{n}(η) ∈ (q−1)^{n} ∑_{r+s=n−1}U_{q}^{⊗r}⊗ ℑ_{q} ⊗ U_{q}^{⊗s} ∀ n ∈ ℕ_{+} .

Being, in addition, η ∈ ℑ_{q}, for we proved that (C_{q}^{▽})^{⊥} ⊆ ℑ_{q}, we get η ∈ ℑ_{q}^{!} . Therefore (C_{q}^{▽})^{⊥} ⊆ ℑ_{q}^{!} , q.e.d. Finally, assume that C_{q} or ℑ_{q} are strict quantizations. Then we must still prove that C_{q}^{▽} = (ℑ_{q}^{!} )^{⊥}. Since C_{q} = ℑ_{q}^{⊥} and ℑ_{q} = C_{q}^{⊥}, it is easy to check that C_{q} is strict if and only if ℑ_{q} is; therefore, we can assume that ℑ_{q} is strict.

The assumptions and Theorem 5.11 (b) give ℑ_{q} = (ℑ_{q}^{!} )^{⋎}; moreover, I_{q} := ℑ_{q}^{!} is strict. Then we can apply the first part of claim (b) — which is proved, later on, in a way independent of the present proof of claim (a) itself — and get (I_{q}^{⋎})^{⊥} = (I_{q}^{⊥})^{↰}. Therefore

C_{q}^{▽} = (ℑ_{q}^{⊥})^{▽} = (((ℑ_{q}^{!} )^{⋎})^{⊥} )^{▽} = ((I_{q}^{⋎})^{⊥})^{▽} = ((I_{q}^{⊥})^{↰} )^{▽} . (5.11)

Now, it is straightforward to prove that I_{q} strict implies that I_{q}^{⊥} is strict as

well. Then Proposition 5.11(d) ensures ((I_{q}^{⊥})^{↰} )^{▽} = I_{q}^{⊥}. This along with

(5.11) yields C_{q}^{▽} = ((I_{q}^{⊥})^{↰} )^{▽} = I_{q}^{⊥} = (ℑ_{q}^{!} )^{⊥}, ending the proof of (a). (b) With much the same arguments as for (a), we find as well that

⟨I_{q}^{⋎}, ℭ_{q}^{↰} ⟩ ∈ ⟨J^{⊗(n−1)}⊗ I_{q} , U_{q}^{⊗(n−1)}⊗ ℭ_{q}⟩ ⊆ ⟨I_{q} , ℭ_{q}⟩ = 0

because I_{q} = ℭ_{q}^{⊥}; this means that

I_{q}^{⋎} ⊆ (ℭ_{q}^{↰} )^{⊥} ,  ℭ_{q}^{↰} ⊆ (I_{q}^{⋎})^{⊥} .  (5.12)

Let now κ ∈ (I_{q}^{⋎})_{q}^{⊥} ( ⊆ U_{q}′ ). Since κ ∈ U_{q}′, we have δ_{n}(κ) ∈

(q − 1)^{n}U_{q}^{⊗n} for all n ∈ ℕ; moreover, from κ ∈ (I_{q}^{⋎})^{⊥} it follows that

κ_{+} := (q − 1)^{−n}δ_{n}(κ) ∈ U_{q}^{⊗n} enjoys ⟨J^{⊗(n−1)} ⊗ I_{q} , κ_{+}⟩ = 0 , so that

κ_{+} ∈ (J^{⊗(n−1)} ⊗ I_{q})^{⊥} = _{∑r+s=n−2} U_{q}^{⊗r}⊗J^{⊥}⊗U_{q}^{⊗s}⊗U_{q} + U_{q}^{⊗(n−1)}⊗I_{q}^{⊥} .

In addition, δ_{n}(κ) ∈ J_{U_{q}}^{⊗n}, where J_{U_{q}} := Ker ( ϵ: U_{q} ⟶ ℂ[q, q^{−1}]); therefore δ_{n}(κ) ∈ ((q − 1)^{n}U_{q}^{⊗n}) ⋂ J_{U_{q}}^{⊗n} = (q − 1)^{n}J_{U_{q}}^{⊗n}, which together with the above formula yields

κ_{+} ∈ (J^{⊗(n−1)} ⊗ I_{q})^{⊥}⋂ J_{U_{q}}^{⊗n} =

= ( ∑_{r+s=n−2}U_{q}^{⊗r}⊗J^{⊥}⊗U_{q}^{⊗s}⊗U_{q}) ⋂ J_{U_{q}}^{⊗n} + (U_{q}^{⊗(n−1)}⊗I_{q}^{⊥}) ⋂ J_{U_{q}}^{⊗n} =

= ∑_{r+s=n−2}J_{U_{q}}^{⊗r} ⊗(J^{⊥} ⋂ J_{U_{q}})⊗J_{U_{q}}^{⊗s}⊗J_{U_{q}} + J_{U_{q}}^{⊗(n−1)}⊗(I_{q}^{⊥} ⋂ J_{U_{q}}) =

= J_{U_{q}}^{⊗(n−1)} ⊗ (I_{q}^{⊥} ⋂ J_{U_{q}}) = J_{U_{q}}^{⊗(n−1)} ⊗ (ℭ_{q} ⋂ J_{U_{q}}) ⊆ U_{q}^{⊗(n−1)} ⊗ ℭ_{q}

where in the third equality we used the fact that J^{⊥} ⋂ J_{U_{q}} = {0}. So κ_{+} ∈ U_{q}^{⊗(n−1)} ⊗ ℭ_{q}, hence δ_{n}(κ) ∈ (q − 1)^{n}U_{q}^{⊗(n−1)} ⊗ ℭ_{q} for all n ∈ ℕ_{+} : thus κ ∈ ℭ_{q}^{↰}. Therefore (I_{q}^{⋎})^{⊥} ⊆ ℭ_{q}^{↰}, which together with the right-hand side inequality in (5.12) gives ℭ_{q}^{↰} = (I_{q}^{⋎})^{⊥}. In the end, suppose also that one between ℭ_{q} and I_{q} is strict. As I_{q} = ℭ_{q}^{⊥} and ℭ_{q} = I_{q}^{⊥}, one sees easily that I_{q} is strict if and only if ℭ_{q} is; then we can assume that ℭ_{q} is strict. We want to show that I_{q}^{⋎} = (ℭ_{q}^{↰} )^{⊥}. The assumptions and Theorem 5.11(d) give ℭ_{q} = (ℭ_{q}^{↰} )^{▽}. Moreover, we have that C_{q} is strict by Proposition 5.3(3) and Proposition 5.7 (3). Then we can apply the first part of claim (a), thus getting (C_{q}^{▽})^{⊥} = (C_{q}^{⊥})^{!} . Therefore

I_{q}^{⋎} = (ℭ_{q}^{⊥})^{⋎} = (((ℭ_{q}^{↰} )^{▽})^{⊥} )^{⋎} = ((C_{q}^{▽})^{⊥})^{⋎} = ((C_{q}^{⊥})^{!} )^{⋎}  (5.13)

Now, one proves easily that C_{q} strict implies C_{q}^{⊥} strict. Then Theorem 5.11(c)

yields ((C_{q}^{⊥})^{!} )^{⋎} = C_{q}^{⊥}. This and (5.13) give I_{q}^{⋎} = ((C_{q}^{⊥})^{!} )^{⋎}= C_{q}^{⊥} =(ℭ_{q}^{↰} )^{⊥}, which eventually ends the proof of (b).

6 Examples

In this last section we will give some examples showing how our general constructions may be explicitly implemented. Some of the examples may look rather singular, but our aim here is mainly to draw the reader’s attention on how even badly behaved cases can produce reasonable results. It has to be remarked that a wealth of new examples of coisotropic subgroups of Poisson groups have been recently produced ([25]), to which our recipes could be interestedly applied.

N.B.: for the last two examples — Subsections 6.2 and 6.3 — one can perform the explicit computations (that we just sketch) using definitions, formulas and notations as in [5], §6, and in [11], §7.

6.1 Quantization of Stokes matrices as a GL _{n}^{∗}–space

As a first example, we mention the following. A well-known structure of Poisson group, typically known as the standard one, is defined on SL_{n}; then one can consider its (connected) dual Poisson group SL _{n}^{∗}, which in turn is a Poisson group as well. The set of Stokes matrices — i.e. upper triangular, unipotent matrices — of size n bears a natural structure of Poisson homogeneous space, and even Poisson quotient, for SL _{n}^{∗}. In [5], Section 6, it was shown that one can find an explicit quantization, of formal type, of this Poisson quotient by a suitable application of the QDP procedure for formal quantizations developed in that paper. Now, let us look at the explicit presentation of the formal quantization U_{ℏ}(𝔰𝔩_{n}) considered in [loc. cit.]. One sees easily that this can be turned into a presentation of a global quantization (of 𝔰𝔩_{n} again), i.e. a QUEA U_{q}(𝔰𝔩_{n}) in the sense of Section 3. Similarly, Drinfeld’s QDP (for quantum groups) applied to U_{ℏ}(𝔰𝔩_{n}) provides a formal quantization F_{ℏ}[[SL _{n}^{∗}]] := U_{ℏ}(𝔰𝔩_{n})′ of the function algebra over the formal group SL _{n}^{∗}; but then the analogous functor for the global version of QDP yields (cf. Theorem 4.1) a global quantization F_{q}[SL _{n}^{∗}] := U_{q}(𝔰𝔩_{n})′ of the function algebra over SL _{n}^{∗}. In a nutshell, F_{q}[SL _{n}^{∗}] is nothing but (a suitable renormalization of) an obvious ℂ[q, q^{−1}]–integral form of F_{ℏ}[[SL _{n}^{∗}]] . Carrying further on this comparison, one can easily see that the whole analysis performed in [5] can be converted into a similar analysis for the global context, yielding parallel results; in particular, one ends up with a global quantization — of type C, in the sense of Section 3 — of the space of Stokes matrices. More in detail, this quantization is a strict one, as such is the quantum subobject one starts with. Since all this does not require more than a word by word translation, we refrain from filling in details.

6.2 A parametrized family of real coisotropic subgroups

Coisotropic subgroups may come in families, in some cases inside the same conjugacy class (which is responsible for different Poisson homogeneous bivectors on the same underlying manifold). An example in the real case was described in detail in [2]. The setting is the one of standard Poisson SL_{2}(ℝ) , which contains a two parameter family of 1– dimensional coisotropic subgroups described, globally, by the right ideal and two-sided ideal

I_{μ,ν} := \{ a − d + 2 q^{\frac{1}{2}} μb , q νb + c \} · F_{q}[SL_{2}(ℝ)]  (6.1)

where a, b, c, d are the usual matrix elements generating F_{q}[SL_{2}(ℝ)], with ∗– structure in which they are all real (thus q^{∗} = q^{−1} ) and μ, ν ∈ ℝ. The corresponding family of coisotropic subgroups of classical SL_{2}(ℝ) may be described as

K_{μ,ν} := \{( \begin{matrix}d − 2μb & b \\ −νb & d \end{matrix} ) | b, d ∈ ℝ , d^{2} + νb^{2} = 1 \}

(adapting our main text arguments to the case of right quantum coisotropic subgroups, this is quite trivial and we will do it without further comments). The corresponding SL_{2}(ℝ)–quantum homogeneous spaces have local description given as follows: C_{μ,ν} is the subalgebra generated by

\begin{matrix}z_{1} = q^{− \frac{1}{2}} (ac + νbd) + 2μbc , z_{2} = c^{2} + νd^{2} + 2μq^{− \frac{1}{2}} cd , \\ z_{3} = a^{2} + νb^{2} + 2μq^{− \frac{1}{2}} ab . \end{matrix}  (6.2)

Using commutation relations — see (12) in [3] — it is easily seen that C_{μ,ν} has a linear basis given by { z_{1}^{p}z_{2}^{q} , z_{1}^{p}z_{3}^{r} | p, q, r ∈ ℕ } .

Proposition 6.1. The subalgebra C_{μ,ν} is a right coideal of F_{q}[SL_{2}(ℝ)] and is a strict quantization — of type C — of K_{μ,ν} .

Proof. The first statement is proven in [3]. As for the second we will first show that z_{1}^{p}z_{2}^{q} , z_{1}^{p}z_{3}^{r} ∉ (q − 1)F_{q}[SL_{2}(ℝ)] for any p, q, r ∈ ℕ. This may done by considering their expression in terms of the usual basis { a^{p}b^{r}c^{s} , b^{h}c^{k}d^{i} } of F_{q}[SL_{2}(ℝ)]. In fact we do not need a full expression of monomials z_{1}^{p}z_{2}^{r} or z_{1}^{p}z_{3}^{r} in terms of this basis, which would lead to quite heavy computations. It is enough to remark that, for example, since

z_{1}^{p}z_{2}^{r} = (q^{− \frac{1}{2}} ac + b(νd + 2μc))^{p} (c^{2} + (νd + 2μq^{− \frac{1}{2}} c)d)^{r}

we can get an element multiple of a^{p}c^{p+2r} only from (ac) ⋯ (ac) · c ⋯ c, which is of the form q^{h}a^{p}c^{p+2r} ∉ F_{q}[SL_{2}(ℝ)]. Since no other elements may add up with this one, we have z_{1}^{p}z_{2}^{r} ∉ (q − 1)F_{q}[SL_{2}(ℝ)]. A similar argument works for z_{1}^{p}z_{3}^{r}. In a similar way we prove that any ℂ[q, q^{−1}]–linear combination of the z_{1}^{p}z_{2}^{q}’s and the z_{1}^{s}z_{3}^{r}’s is in (q − 1)F_{q}[SL_{2}(ℝ)] if and only if all coefficients are in (q − 1)ℂ[q, q^{−1}]. Therefore C_{q} is strict, q.e.d.

It makes therefore sense to compute C_{μ,ν}^{▽} ; to this end, we can resume a detailed description of U_{q}(𝔰𝔩_{2}^{∗}) := F_{q}[SL_{2}(ℝ)]^{∨} — apart for the real structure, which is not really relevant here — from [11], §7.7. From our PBW-type basis we have that C_{μ,ν}^{▽} is the subalgebra of F_{q}[SL_{2}(ℝ)]^{∨} generated by the elements ζ_{i} := \frac{1}{q−1} (z_{i} − ε(z_{i})) ∈ F_{q}[SL_{2}(ℝ)]^{∨} (i = 1, 2, 3) . Since we know that

H_{+} := \frac{a − 1}{q − 1} ,  E := \frac{b}{q − 1} ,  F := \frac{c}{q − 1} ,  H_{−} := \frac{d − 1}{q − 1}

are algebra generators of U_{q}(𝔰𝔩_{2}^{∗}) := F_{q}[SL_{2}(ℝ)]^{∨}, we deduce that

\begin{matrix}\frac{ζ_{1}}{q − 1} = & q^{− \frac{1}{2}} (F + νE) + (q − 1) (q^{− \frac{1}{2}} H_{+}F + q^{− \frac{1}{2}} νEH_{−} + 2μEF ) \\ \frac{ζ_{2} − ν}{q − 1} = & 2 (νH_{−} + μq^{− \frac{1}{2}} F) + (q − 1) (F^{2} + νH_{−}^{2} + 2μq^{− \frac{1}{2}} F H_{−}) \\ \frac{ζ_{3} − 1}{q − 1} = & 2 (H_{+} + μq^{− \frac{1}{2}} E) + (q − 1) (H_{+}^{2} + νE^{2} + 2μq^{− \frac{1}{2}} H_{+}E) \end{matrix} (6.3)

In the semiclassical specialization U_{q}(𝔰𝔩_{2}^{∗}) ⟶^{q⟶1} U_{q}(𝔰𝔩_{2}^{∗})/(q − 1)U_{q}(𝔰𝔩_{2}^{∗}) one has that E ↦ e , F ↦ f , H_{±} ↦ ±h , where h, e, f are Lie algebra generators of 𝔰𝔩_{2}^{∗}; therefore the semiclassical limit of the right hand side of (6.3) is the Lie subalgebra generated by f + ν e , −ν h + μ e , h + μ e , or, equivalently, the 2– dimensional Lie subalgebra generated by f+ν e and h+μ e (the three elements above being linearly dependent) with relation [ h + μ e , f + ν e ] = f + ν e . The quantization of this coisotropic subalgebra of 𝔰𝔩_{2}^{∗} is therefore the subalgebra generated inside U_{q}(𝔰𝔩_{2}^{∗}) by the quadratic elements (6.3). Similar computations can be performed starting from I_{μ,ν} . The transformed I_{μ,ν}^{⋎} is the right ideal generated by the image of a − d + 2 q^{\frac{1}{2}} μb and qνb + c, i.e. the right ideal generated by H_{+} − H_{−} + 2 q^{\frac{1}{2}} μE and q νE + F ; also, from its semiclassical limit it is easily seen that this again corresponds to the same coisotropic subgroup of the dual Poisson group SL_{2}(ℝ)^{∗}. All this gives a local — i.e., infinitesimal — description of the (2–dimensional) coisotropic subgroups K_{μ,ν}^{⊥} in SL_{2}(ℝ)^{∗}.

6.3 The non coisotropic case

Let us finally consider the case of a non coisotropic subgroup. We will consider the embedding of SL_{2}(ℂ) into SL_{3}(ℂ) corresponding to a non simple root, which easily generalizes to higher dimensions. Computations will only be sketched. Let 𝔥 be the subalgebra of 𝔰𝔩_{3}(ℂ) spanned by E_{1,3} , F_{1,3} , H_{1,3} = H_{1} + H_{2} . Easy computations show that the standard cobracket values are

\begin{matrix}δ(E_{13}) = & E_{13} ∧ (H_{1} + H_{2}) + 2E_{23} ∧ E_{12} \\ δ(F_{13}) = & F_{13} ∧ (H_{1} + H_{2}) − 2F_{23} ∧ F_{12} \\ δ(H_{1} + H_{2}) = & 0 \end{matrix}  (6.4)

and, therefore, the corresponding embedding SL_{2}(ℂ) ⟶ SL_{3}(ℂ) is not

coisotropic. To compute the coisotropic interior 𝔥^{∘} of 𝔥, consider that ⟨H_{1}+H_{2}⟩

is, trivially, a subbialgebra of 𝔥, thus contained in 𝔥^{∘}. Let X := (H_{1} + H_{2}) + αE_{13} + βF_{13} : then

δ(X) = X ∧ (H_{1} + H_{2}) + 2 (αE_{23} ∧ E_{12} − βF_{23} ∧ F_{12})

shows that no such X is in 𝔥^{∘}, unless α = 0 = β. The outcome is that we have

H^{∘} = ( \begin{matrix}γ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & γ^{−1} \end{matrix} ) ⊆ SL_{3}(ℂ)

with γ ∈ ℂ^{∗}. Correspondingly

𝔥^{⟨⊥⟩} = ( 𝔥^{∘} )^{⊥} = ⟨ e_{1,2} , e_{1,3} , e_{2,3} , f_{1,2} , f_{1,3} , f_{2,3} , h_{2.2} ⟩ ( ⊆ 𝔰𝔩_{3}(ℂ)^{∗} )

and, thus SL_{3}(ℂ)^{∗}/H^{⟨⊥⟩} is a 1– dimensional Poisson homogeneous space — with, of course, zero Poisson bracket. Let us consider now any weak quantization ℭ_{q}(H) of H. It should certainly contain the subalgebra of U_{q}(𝔰𝔩_{3}) generated by the root vectors E_{1,3} , F_{1,3} , together with K_{1}K_{3}^{−1} and Ĥ_{1,3} := (K_{1}K_{3}^{−1} − 1)/(q − 1) . The equality

Δ(E_{1,3}) = E_{1,3} ⊗ K_{1}K_{3}^{−1} + 1 ⊗ E_{1,3} + (q − 1)E_{1,2} ⊗ E_{2,3}

tells us that, in order to be a left coideal, such a quantization should also contain either (q − 1)E_{1,2} or (q − 1)E_{2,3} (and thus, as expected, it cannot be strict). Let us try to compute some elements in ℭ_{q}(H)^{↰}. Certainly, since

δ_{2}(Ĥ_{1,3}) = Ĥ_{1,3} ⊗ (K_{1}K_{3}^{−1} − 1) = (q − 1) Ĥ_{1,3} ⊗ Ĥ_{1,3}

we can conclude that (q − 1)Ĥ_{1,3} ∈ ℭ_{q}(H)^{↰}. On the other hand,

δ_{2}(E_{1,3}) = (q − 1)E_{1,3} ⊗ Ĥ_{1,3} + (q − 1)E_{1,2} ⊗ E_{2,3}

implies that (q − 1)E_{1,3} ∉ ℭ_{q}(H)^{↰}, while (q − 1)^{2}E_{1,3} ∈ ℭ_{q}(H)^{↰}. All this means the following. Within ℭ_{q}(H)^{↰} we find a non-diagonal matrix element of the form (q−1) t_{1,3} : it belong to (q−1)U_{q}(𝔰𝔩_{3})′ but not to (q−1)ℭ_{q}(H)^{↰}, so that

ℭ_{q}(H)^{↰} ⋂ (q−1) U_{q}(𝔰𝔩_{3})′ ⫌ (q−1) ℭ_{q}(H)^{↰}

which means that the quantization ℭ_{q}(H)^{↰} is not strict. On the other hand, we know by Proposition 5.7(3) that ℭ_{q}(H)^{↰} is proper. Therefore, we have an example of a quantization (of type C_{q}, still by Proposition 5.7(3)) which is proper, yet it is not strict. In addition, in the specialization map π : U_{q}(𝔰𝔩_{3})′ −↠ U_{q}(𝔰𝔩_{3})′/(q−1)U_{q}(𝔰𝔩_{3})′ the element (q−1) t_{1,3} is mapped to zero, i.e. it yields a

trivial contribution to the semiclassical limit of ℭ_{q}(H)^{↰} — which here is meant as being π(ℭ_{q}(H)^{↰} ) = ℭ_{q}(H)^{↰}/ℭ_{q}(H)^{↰} ⋂ (q − 1) U_{q}(𝔰𝔩_{3})′ . With similar computations it is possible to prove, in fact, that the only generating element in ℭ(H)^{↰} having a non-trivial semiclassical limit is (q−1)Ĥ_{1,3} . Therefore, through specialization at q = 1 , from ℭ(H)^{↰} one gets only π(ℭ_{q}(H)^{↰}) = ℂ[t_{2,2}]: indeed, this in turn tells us exactly that ℭ_{q}(H)^{↰} is a quantization, of proper type, of the homogeneous SL_{3}(ℂ)^{∗}–space SL_{3}(ℂ)^{∗}/H^{⟨⊥⟩} (whose Poisson bracket is trivial). Remark. It is worth stressing that this example — no matter how rephrased — could not be developed in the language of formal quantizations as a direct application of the construction in [5], for only strict quantizations were taken into account there.

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Nicola Ciccoli Dipartimento di Matematica e Informatica Università di Perugia via Vanvitelli 1 I-06123 Perugia Italy ciccoli@dipmat.unipg.it Fabio Gavarini Dipartimento di Matematica Università di Roma “Tor Vergata” via della ricerca scientifica 1 I-00133 Roma Italy gavarini@mat.uniroma2.it