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Quantization of Projective Homogeneous Spaces and Duality Principle

We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.


Introduction
A projective variety can be described via its homogeneous graded coordinate ring. This ring is not an invariant associated to the variety, but depends on a chosen embedding of the variety into some projective space. Different embeddings will, in general, produce non isomorphic graded rings.
When a projective variety is homogeneous, i.e. endowed with a transitive action of an (affine) algebraic group on it, it can be realized as quotient of affine algebraic groups G H . In this case a projective embedding can be obtained via sections of a line bundle on G H, uniquely given once a character of H is specified.
If one approaches a quantization of this picture in the context of quantum groups the problem immediately arising is that standard quantum groups have a very limited set of quantum subgroups. This explains why usually the preferred approach goes through representation theoretic techniques.
An explanation of the lack of quantum subgroups, together with a way to circumvent this problem, is suggested by considering the semiclassical picture, i.e. in the context of algebraic Poisson groups. In such setting algebraic Poisson subgroups are quite rare too; however there is no need of an algebraic Poisson subgroup to cook up a Poisson quotient. The existence of a surjective Poisson map G → G H is guaranteed simply by requiring H to be a coisotropic subgroup of G . This condition can be expressed by saying that the defining ideal of H , in the function algebra of G , is required to be a Poisson subalgebra rather than a Poisson ideal, as required for Poisson subgroups.
Let O q (G) be a quantization of the affine algebraic Poisson groups G. At the quantum level, a quantization O q (H) of its coisotropic subgroup H can be defined through conditions on the projection π : O q (G) −→ O q (H) . We will see this in full detail in the sections 2, 3.
Our first aim is to build a quantum deformation O q G H of the projective variety G H , i.e. of its graded ring O G H , subject to the following requirements: (1) there exists a one dimensional corepresentation of the quantum coisotropic subgroup O q (H) which is a deformation of the corepresentation of O(H) corresponding to the character of H which defines the line bundle giving the projective embedding of G H ; (2) a quantum analogue O q G H to O G H is defined as the subset -inside O q (G) -of "semi-invariant functions" with respect to the given corepresentation of O q (H) ; (3) the subset O q G H is a graded subalgebra of O q (G) ; (4) the graded subalgebra O q G H is a graded left coideal of O q (G) , so the coproduct in O q (G) induces a (left) O q (G)-coaction on O q G H , and the latter can be thought of as a quantum homogeneous space. In other words, a quantum deformation of a projective homogeneous space, embedded into some projective space, consists of the deformation of the graded algebra associated to the embedding, in such a way that the action of the group on the homogeneous space is also naturally quantized.
We will work out the details of the construction for the case of the Grassmannian and its Plücker embedding, that is when G is the special linear group and H = P is a maximal parabolic subgroup and we will sketch it in the more general case of quantum flag varieties of simple Lie groups.
Our main motivation to develop this point of view is to adapt to projective homogeneous spaces, the correspondence introduced by Ciccoli and Gavarini [5] for coinvariant subalgebras. This recipe allows to associate functorially to a quantum quasi-affine homogeneous space another quantum homogeneous space, through a generalization of the quantum duality principle (QDP), defined by Drinfeld for quantum groups. A part of the arguments in [5] does not directly apply to projective homogeneous spaces, since it is based on the realization of the ring of the homogeneous space as the set of coinvariant functions inside the ring of the quantum group acting on it. But this is possible -as in the classical case -if and only if the homogeneous space is quasi-affine, which is not the case of projective varieties. The coordinate ring of the homogeneous space is replaced by a graded ring inside the quantum group ring, consisting of semi-coinvariants with respect to a one-dimensional representation, which can be seen as a deformation of the line bundle that classically determines the projective embedding. The definitions introduced in section 3 will allow us to define a quantum duality functor and obtain the QDP construction in this more general setting. In the last chapter we will discuss applications to quantum flag manifolds.

The classical setting
In this section we recall some Poisson geometry (see Ref. [20] for details).

The affine case
Let k be a fixed field of characteristic 0. When doing algebraic geometry construction, we tacitly assume that k be algebraically closed. However, this assumption is not needed for our quantum constructions.
Let G be an affine algebraic group over k . We denote by O(G) the algebra of regular functions of G, in short its "function algebra", which is naturally a Hopf algebra (over k ). We denote g := Lie(G) the tangent Lie algebra of G , and similarly h := Lie(H) for any closed algebraic subgroup H of G .
Assume that G is a Poisson group: this means that O(G) is a Poisson Hopf algebra, i.e. we have a Poisson bracket { , } : O(G) ⊗ O(G) −→ O(G) which is compatible with the Hopf algebra structure. Moreover, g is a Lie bialgebra, for some Lie cobracket δ : g −→ g ⊗ g , and the same holds for its dual space g * , these two Lie bialgebras structures being dual to each (PQ-i) there existsm ∈ M whose stabilizer Gm is a coisotropic subgroup of G ; (PQ-ii) there existsm ∈ M such that φm : G −→ M , g → φ(g,m) , is a Poisson map, that is to say M is a Poisson quotient; (PQ-iii) there ism ∈ M such that {m} is a symplectic leaf of M .
For anym ∈ M with stabilizer Gm one has M ∼ G Gm as affine Gvarieties. As M is affine, this is equivalent to O(M) ∼ = O G/Gm . Finally, O G Gm ∼ = O(G) Gm , the subalgebra of Gm-invariants in O(G). The same holds with Gm replaced by any subgroup H whose coset space G H is affine. We have then an additional characterization of (affine) Poisson quotients.

The projective case
We are now interested in the case when the homogeneous G-variety G H is projective, i.e. H is parabolic. To describe it in algebraic terms (the setting we need for quantum deformations), we require a specific realization, namely an embedding into a projective space.
Given a representation ρ of H on some vector space V , we can construct a vector bundle associated to it, namely The space of global sections of this bundle is identified with the induced module (see, e.g., [18] for more details) Definition 2.7. Let χ : H −→ k * be a character of H, i.e. a one dimensional representation of H on L ∼ = k . Then L ⊗n is again a one dimensional representation of H with character χ n . Let L n : , whose elements are called semiinvariants. Note that now the notation O G H has not the same meaning as when G H is affine.
Assume now the bundle L to be very ample. In the present context, this is the same (cf. [18], §II.7) as saying that L is generated by a set of global sections f 0 , f 1 , . . . , f N ∈ O G H 1 -in particular, the algebra O G H is graded, generated in degree 1 (by the f i 's). Then O G H is the homogeneous coordinate ring of the projective variety G H , with respect to the embedding given via the global sections of L (see [17], p. 176).
We want to reformulate this classical construction in purely Hopf algebraic terms, more suited to the quantum setting we shall presently deal with.
Remark 2.8. In algebraic terms, saying that χ : H → k * is a character is the same as saying that it is a group-like element in the coalgebra O(H) . The same holds for all powers χ n (n ∈ N ). In fact if χ is group-like, then the same is true for all its powers χ n since O(H) is a Hopf algebra. As the χ n 's are group-like, if they are pairwise different they also are linearly independent, which ensures that the sum Moreover, once the embedding is given, each summand O G H n can be described in purely Hopf algebraic terms as To simplify notation, we set λ := S(χ) -the character of H which maps h ∈ H to λ(h) = χ h −1 -and we set ∆ π := (id ⊗ π) • ∆ , so that and O G H is generated in degree 1, namely by O G H 1 .
Proof. If f ∈ O G H n , then Now, by assumption there exists a non-zero global section of the line bundle on G H , i.e. a regular function t ∈ O G H 1 \ {0} on G and ǫ(t) = 0. By the above (for n = 1 ), up to dividing out by ǫ(t), we can assume that π(t) = λ . The result follows immediately.
Notice that while λ = π(t) is group-like, t instead is something less, yet still has an "almost group-like property", given by (2.2). This element t and its quantization will turn to be crucial for the quantum setting.
Remark 2.10. We point out that O G H is a unital subalgebra, and also a (left) coideal of O(G) ; the latter reflects the fact that G H is a (left) G-space. Thus, the restriction of the comultiplication of O(G) , namely As to Poisson geometrical properties in this projective setup, the following characterization -which might be used to define the notion of Poisson quotient structure for the projective G-space G H -holds: Proposition 2.11. Let G be a Poisson algebraic group, H a closed parabolic subgroup, and t ∈ O(G) as in Remark 2.8. The following are equivalent: Proof. To simplify notation, we set I H,n := I(H) O G H n for n ∈ N .
This along with (2.6) gives Then, for any f ∈ O G H r and ℓ ∈ O G H s , we have because any Poisson group structure is zero at the identity. Eventually, this yields t , O G H ⊆ I(H) , q.e.d.
To prove that I(H) , I(H) ⊆ I(H) , we need some additional tools.
. This is the stalk at the point 1 G of the structure sheaf of G , and the Poisson bracket of O(G) canonically (and uniquely) extends -for f 1 , f 2 ∈ O(G) , y 1 , y 2 ∈ J , n 1 , n 2 ∈ N -via the identity to the local algebra O(G) 1 G . The counit ǫ of O(G) uniquely extends to an algebra morphism from O(G) 1 G to k , again denoted by ǫ , whose kernel is Second, let X t := t = 0 be the zero locus in G H defined by the vanishing of the divisor t , and let Γ := G H \ X t . This is an affine open dense subset of G H , whose algebra of regular functions is the graded localisation of O G H by the multiplicative subset t n n∈N , that is : by construction, this is the stalk at 1 G = 1 G H of the structure sheaf of G H , and the Poisson bracket of O G H [t] uniquely extends to O G H 1 G . Now, the maximal ideal J 1 G in the local algebra O(G) 1 G can be generated by a local system of parameters on G at the point 1 G , say {y 1 , . . . , y n } , with n := dim (G) . As H is a closed subgroup of G , we can choose this system of parameters in such a way that: . . , y h } is a local system of parameters on H at the point 1 H = 1 G ; (2) y h = t − 1 ; (3) {y h+1 , . . . , y n } is a local system of parameters on X t at 1 G ; in partic- As a direct consequence of the above assumptions, the elements y h+1 , . . . , y n generate the ideal . . , y n = I(H) 1 G , for all i, j . This together with Leibnitz' rule implies that k i y i , k j y j ∈ y h+1 , . . . , y n = I(H) 1 G for any 3 Quantum bundles and quantum homogeneous spaces

Quantum groups
We want to translate all the framework of section 2 into the quantum setup. The first step is to introduce quantum groups, in the form of quantum (or "quantized") function algebras, as follows.
Let G be an algebraic Poisson group, O(G) its function algebra.
Definition 3.1. By quantization of O(G), we mean a Hopf algebra O q (G) over the ground ring k q := k q, q −1 , where q is an indeterminate, such that: We call O q (G) quantum (or quantized) function algebra over G , or quantum deformation of G , or even simply quantum group. It is standard terminology to say that the Poisson Hopf algebra O(G) is the semiclassical limit of O q (G) . Similarly, we say that a k q -algebra O q (X) is a quantization of the commu- Then O(X) is also a Poisson algebra, called semiclassical limit of O q (X) .

Remark 3.2.
(1) The technical requirement in (c) corresponds, in the context of formal deformations, to require that the algebra is separated; we also point out that it is satisfied by all quantum function algebras usually considered in literature. In any case, it will not be necessary till §5. Moreover, both (b) and (c) above are automatically satisfied when O q (G) is free as a k q -module.
(2) The classical algebra O(G) inherits from O q (G) a Poisson bracket, for any lifts x ′ , y ′ ∈ O q (G) of x and y respectively. One checks that this bracket is well-defined, and makes O(G) into a Poisson Hopf algebra, so that G is a Poisson group. But G already had, by assumption, a Poisson group structure; then, the requirement in (a) above that O q (G) (q−1) O q (G) ∼ = O(G) as Poisson Hopf algebra amounts to say, in particular, that the two Poisson group structures of G are isomorphic.
On the other hand, if we start without asking G to have a Poisson group structure, then the previous analysis tells that if a quantization O q (G) exists, then it automatically endows O(G) with a Poisson algebra structure. And similarly for a quantization O q (X) of a commutative algebra O(X) .

Quantum subgroups and quantum coisotropic subgroups
Our second step is to introduce the notions of quantum coisotropic subgroup and of quantum subgroup, the former being weaker than the latter.
If, in addition, O q (H) is a Hopf algebra and π is a Hopf algebra morphism, and for I H : For later use, we introduce also the notation I q (H) := Ker (π) .  In the Hopf algebra language, conditions (b) and (c) are expressed by saying that O q (H) is an O q (G)-module coalgebra, that is a coalgebra and O q (G)-module such that ∆ Oq(H) and ǫ Oq(H) are O q (G)-module morphisms.
(3) Assumptions at the quantum level imply properties for specializations. So, the semiclassical specialization of a quantum coisotropic subgroup is (the function algebra of) a coisotropic subgroup, because I 1 (H) = Ker (π 1 ) is a Poisson subalgebra of O(G) . On the other hand, the specialization of a quantum subgroup instead is (the function algebra of) a Poisson subgroup.
At the semiclassical level there are many examples of coisotropic subgroups, among which only a few are Poisson subgroups. This is a key motivation to focus on the more general setting of quantum coisotropic subgroups.
(4) A quantum coisotropic subgroup O q (H) is by no means a "quantum group", in the sense of Definition 3.1, unless it is a quantum subgroup.

Quantum line bundles
We now want to carry across to the quantum setting the notion of embedding G H ֒−→ P N associated to a line bundle L that we assume to be very ample. The idea is to transfer to this framework the description (2.4) of O G H n given in terms of an element t ∈ O(G) as in Remark 2.8 and Proposition 2.9. Thus, the starting point will be a quantization of such an element t that we will call a pre-quantum section.
Given G and H as in §2.2, we assume that quantizations of them be given, i.e. we are given O q (G), O q (H) and π : O q (G) −։ O q (H) as in Definitions 3.1, 3.3. To simplify notation, hereafter we shall also write ℓ : Moreover, we assume that an element t ∈ O(G) as in Remark 2.8, and the corresponding closed embedding G H ֒−→ P N , be given as in Proposition 2.9 (so, in particular, t is a section of the line bundle L on G H ).
We define a quantization of the latter setup as follows: Definition 3.5. We define pre-quantization of t , or pre-quantum section of the line bundle L on G H (given by t ), any d ∈ O q (G) such that corresponding to the character λ = π(t) of the group H we started from.
(b) In the classical setup, having the character λ is equivalent to having a Hopf algebra morphism k x, Indeed, this occurs because the powers λ z do exist in O(H), and are grouplike because λ is. In fact, each one of them corresponds to a one-dimensional corepresentation, namely the z-th tensor power of ρ λ On the other hand, in the quantum setup there is no natural analogue, since the powers d z are not even defined in O q (H) -which is not an algebra! -nor we can assume (would we define them in some way) that they are group-like. This means that we miss somehow the "tensor powers" of ρ d .
In [1] one can find an example of a countable family of group-like elements in a quantum coisotropic subgroups, which are not obtained by projecting powers of the same element, but quantize a classical character. However, when O q (H) is a quantum subgroup instead, it is a Hopf algebra, hence the group-like d is invertible, and all powers d z exist, and are group-like in O q (H). So we do have all "tensor power corepresentations" Moreover, notice also that d z = d z for all z ≥ 0 , so that ρ d ⊗z for z ≥ 0 can be directly recovered from the element d z in O q (G) ; thus in the end we can handle everything working with the elements d n ∈ O q (G) , n ∈ N .
(a) We call d-semi-invariants of degree n the elements of the set , hence the same holds for O q G H . We shall now see some further properties of these modules, which eventually will tell us that -under suitable, additional assumptions -we can take O q G H as a quantization of O G H .
Proof. The only statement which needs a proof here is (b), which quickly follows applying (ǫ ⊗ id ) to both sides of ∆ π (ℓ) = ℓ ⊗ d n .
Remark 3.9. The semi-invariants have a good arithmetic property, which ensures that the specialization of O q G H n ⊂ O q (G) (and of O q G H ) at q = 1 will be consistent with that of O q (G) itself. Namely, given n ∈ N , since O q G H n is defined by k q -linear conditions we find at once n are torsion free as k q -modules. Using this embedding, given any ℓ ∈ O q G H n we can always write ∆(ℓ ) s are all linearly independent, and similarly for the h ′ i 's. Then (taking a common denominator) there exists c ∈ k q such that We shall now prove that By coassociativity of ∆ , the latter map coincides with and now the last map applied to c ℓ gives = follows from the assumption ℓ ∈ O q G H n , which implies c ℓ ∈ O q G H n as well. This eventually gives the right-hand side of (3.1), q.e.d. Now, because of the linear independence of the h i 's, the identity (3.1) implies that all the h i 's satisfy the g i 's, resp. the h i 's, being linearly independent, and also

These two conditions imply
The above construction provides us with a reasonable candidate for a quantum analogue of O G H , namely the space of the d-semi-invariants O q G H , which we proved has many important properties. Nevertheless, we still would like O q G H to verify three more key properties, namely: Indeed, we are still missing these properties so far. In order to have them, an additional property must be required to the pre-quantum section d we started from. This is provided by the following result.
This, along with assumption (b) to get the equality ⊛ = , yields the chain of identities : This is just a matter of rephrasing. The following result gives a criterion to detect quantum sections, and shows that for quantum subgroups they are just pre-quantum sections. gives Therefore, for the product f g we have which means exactly f g ∈ O q G H r+s , by Definition 3.7 again. Thus condition (a) of Proposition 3.11 holds, hence we conclude by Definition 3.12.
The following result records yet another feature of quantum sections: Moreover, Span kq d n n∈N is a k q -subcoalgebra of O q (H) , and Proof. By Definition 3.12 and condition (a) in Proposition 3.11 we have d n ∈ O q G H n , for all n ∈ N . This means thus the d n 's are group-like, and different from each other because the t n = d n q=1 are. Finally, this implies that Span kq d n n∈N is a k q -subcoalgebra of O q (H) , and also that the d n 's are linearly independent, which eventually Gathering all together the previous results, we can now show that semiinvariants built out of a quantum section satisfy all properties we look for: Proof. (a) We must simply prove that the sum But this is an easy consequence of Lemma 3.14.
Indeed, let n∈N c n f n = 0 a linear dependence relation, with f n ∈ O q G H n and c n ∈ k q (almost all zero) for every n ∈ N . Applying ∆ π to this relation we get n∈N c n f n ⊗ d n = 0 . But the d n 's, by Lemma 3.14, are linearly independent; thus c n = 0 for all n , q.e.d.
(b) This follows directly from Definition 3.12 and Proposition 3.11.
(c) This follows (again) from Definition 3.12 and Proposition 3.11, from Proposition 3.10, and from the Hopf algebra axioms.
(d) This easily follows from the identity O q G H = n∈N O q G H n , given by claim (a), and from Remark 3.9.
Proof. By Lemma 3.8 we know that π ′ is a well-defined epimorphism of graded k q -modules. The rest follows from O q G H being a subalgebra of O q (G) , and π being a morphism of O q (G)-module coalgebras.
Last aspect to take into account is the behavior of O q G H under specialization at q = 1 . The last part of claim (d) in Theorem 3.15 ensures that such specialization is consistent with that of O q (G): in other words, the embedding O q G H ֒−→ O q (G) gets down under specialization to an Proof. For all n ∈ N , the restriction to O q G H n of the specialization map Remark 3.9. This in turn ensures also that the restriction of where the right-hand side is a subalgebra of p 1 O q (G) = O(G) . Moreover, we have also that the specialization maps preserves the grading, namely , while the right-hand side, by (2.4), is But all specialization maps commute with the coproducts ∆ and with the (quantum and classical) maps π , and the specialization of each d n is nothing but t n . Therefore, we conclude that (3.2) holds.
Finally, we can define our "quantum projective homogeneous spaces".
, where t is a non-zero section of the very ample line bundle on G H giving the embedding into some projectve space.

Then given
as graded O(G)-module algebras and as Poisson algebras over k . We will then refer to O q G H as quantum projective homogeneous space.
In particular, we have seen that any such O q G H has the following properties (Theorem 3.15): This requirement might be seen as the quantum analogue of the requirement -at the semiclassical level, see §2.

The QDP philosophy
The quantum duality principle (QDP) is a two-fold recipe which allows to obtain a quantum group dual, in an appropriate sense, to a given one.
In [5] Ciccoli and Gavarini extended this principle to quantum formal homogeneous spaces. Their result goes as follows.
Given a Poisson group G , we consider pairs H, G/H where H is a coisotropic subgroup of G and G H is the corresponding homogeneous space.
At a local level, i.e. in the setup of formal geometry, any such pair can be described in algebraic terms by any one of the following four objects: where hereafter h := Lie (H) and g := Lie (G) , and the notation is standard, but for O G H which here denotes the algebra of regular functions on the formal algebraic variety G H . The main result in [5] is a four-fold functorial recipe which, from a quantization of each object in (4.1), constructs a quantization of one of the four object of the similar quadruple which describes the "dual" pair H ⊥ , G * H ⊥ , where "dual" refers to Poisson duality.
If we try to do the same at a global level (cf. [6]), i.e. not restricting ourselves to the framework of formal geometry, then something changes when handling the algebra O G H . Namely, the latter is meaningful -that is, it permits to get back the pair H, G/H -only if G H is a quasi-affine variety. This is the case, in particular, if G H is affine, and instead it is not if G H is projective. Therefore, in the latter case one describes G H taking, instead of the algebra of regular functions, the algebra of (algebraic) sections of a line bundle on G H realizing an embedding in a projective space, i.e. what is denoted O G H in §2. Once this is settled, one can consider a quantization O q G H and try to cook up a suitable analogue of the recipe of [5], [6].
With this program in mind, we want to associate to any quantum homogeneous G-space O q G H -in the sense of Definition 3.18 -a (local) quantization U q h ⊥ of the dual G * -space (actually, of the dual coisotropic subgroup), right in the spirit of the QDP.
Warning: In order to make our statements simpler, from now on we take as ground ring the local ring k ′ q := k q (q−1) = localization of k q at the ideal generated by (q−1) .
Therefore, hereafter we shall tacitly extend scalars from k q to k ′ q for all k qmodules and k q -algebras we have considered so far, with no further mention.
where the localization is a projective localization, i.e. we take the elements of degree zero in the ring O q G H d −1 , where d −1 is given degree −1 .

The QDP functor
We now recall the definition of the functor O q (G) → O q (G) ∨ , which sends quantized function algebras (of Poisson groups) to quantized universal enveloping algebras (of Lie bialgebras). More precisely, O q (G) ∨ is a quantization of U g * , where g * is the Lie bialgebra dual to g . For more details and proofs, we refer the reader to [14], Theorem 2.2 and Theorem 4.7.
Remark: the overall assumption in [14] for G is to be connected; nevertheless, this condition is not needed in the proof of Theorem 4.7 therein.
This is a well defined k ′ q -subalgebra of k(q) ⊗ k ′ q O q (G) . Notice also that The results of [14] -in particular, Theorem 4.7 therein -tell us that  In order to study O q G H ∨ , we need a rather explicit description of it.

Definition 4.2. We define
In turn, this requires a description of O q (G) ∨ , which we take from [14]. Let J 1 be the augmentation ideal of O(G) , namely so that J 1 J 2 1 = g * , the cotangent Lie bialgebra of G . Let {y 1 , . . . , y n } be a subset of J 1 whose image in the local ring of G at e (the unit element of the group G ) is a local system of parameters; in particular, n = dim (G) . Define {j 1 , . . . , j n } as a pull-back of {y 1 , . . . , y n } to J G .
Proof. Observe that ǫ(d) = 1 implies that In both cases, the explicit formula for d −1 follows by taking the limit of the geometric series, namely ( which both are extensions of the embedding O q G H ֒−−→ O q (G) . Moreover, via these embeddings the pseudobases for the (topological) algebras on G H identify with subsets of the corresponding ones for the (topological) algebras on G .
Proof. By construction, we have l k = j s d cs for some j s ∈ J G/H , c s ∈ N ( s = 1, . . . , n − h ). Since d is invertible (in both cases), the previous analysis tells us that we can replace the l k 's with the j k 's ( k = 1, . . . , n − h ) in the descriptions of O q G H ∨ and O q G H ∨ given above (i.e. in the k ′ qpseudobases and as topological generators). But then, as we can always complete j s s = 1, . . . , n−h to a set j r r = 1, . . . , n such that y r := j r mod (q −1) O q (G) r = 1, . . . , n yield, in the localization of O(G) at e ∈ G , a local system of parameters at e . Thus using the latter we can describe O q (G) and O q (G) as explained above.
From now on we shall use these embeddings to identify O q G H and O q G H with a subalgebra of O q (G) and of O q (G) respectively. Next result is that O q G H ∨ is a quantization of U h ⊥ : Proof. By assumption, H is coisotropic in G . Therefore, h = Lie(H) is a Lie coideal (and subalgebra) of g = Lie(G) , and h ⊥ is a Lie subalgebra (and coideal) of g * (see Proposition 2.2). Thus the claim does make sense. In order to prove the statement, we proceed much like in the proof of the fact that O q (G) ∨ (q − 1) O q (G) ∨ ∼ = U(g * ) -cf. [14], Theorem 4.7. The arguments being the same, we briefly recall them.
Fix again a special subset {j 1 , . . . , j n } of J G as we did in the proof of Proposition 4.6, in particular with j 1 , . . . , j n−h ∈ J G/H . Also, set notation: Taking into account that the specializations at q = 1 of any k ′ q -module and of its (q − 1)-adic completion are the same, the above discussion gives that as Hopf algebras, by the above description of O 1 (G) ∨ and PBW theorem.
So far we did not exploit our special choice of the subset {j 1 , . . . , j n } : we do it now to prove that t = h ⊥ . In fact, the analysis above to prove that σ : g * ∼ = t shows also that the unital subalgebra where y s := y s mod J 2 1 ( s = 1, . . . , n − h ). Therefore O 1 G H ∨ is the subalgebra of U g * generated by the k-span of y 1 , . . . , y n−h } . Finally, the k-span of y 1 , . . . , y n−h } coincides with the subspace h ⊥ of g * . Indeed, as O G H is the algebra of semi-invariant functions on G , every y s is a (H-)semi-invariant function on G : but it also vanishes at e ∈ H, hence by H-semi-invariance it vanishes on all of H . When mapping y s to y s ∈ J 1 , then, it is mapped into h ⊥ . Thus the whole k-span of y 1 , . . . , y n−h } is contained in h ⊥ , hence coincides with it by dimension equality.
The outcome is that We now wish to explore the nature of O q G/H ∨ as a "quantum homogeneous space". We start with an important observation on the extensions of the comultiplication ∆ in O q (G) to the new algebras we have defined.
. Then ∆ can be uniquely (and canonically) extended to a coassociative morphism of topological algebras where again O q (G) is the I G -adic completion of O q (G), and ⊗ is the I G⊗adic completion of O q (G)⊗O q (G) , with I G⊗ := I G ⊗O q (G) + O q (G)⊗I G . Even more, such a ∆ actually restricts to a coassociative algebra morphism (we use the same symbol to denote it): In fact, as d is a quantum section we have (see Definition 3.12) This in turn implies where the bottom term does belong to O q (G) d −1 ⊗ O q (G) d −1 , as expected, because k i ∈ I q (H) ⊂ I G (for every i ), hence the last formal series above is convergent in the I G⊗ -adic topology.
Let us now turn our attention to the algebra O q (G) ∨ and its (q − 1)-adic completion O q (G) ∨ . By construction (cf. [14]), the coproduct of O q (G) ∨ , hence of O q (G) ∨ too, is induced by the coproduct ∆ of O q (G) . Note that the coproduct ∆ of O q (G) ∨ takes values in the topological tensor product O q (G) ∨ ⊗ O q (G) ∨ , which by definition is the (q −1)-adic completion of the algebraic tensor product O q (G) ∨ ⊗ O q (G) ∨ -and coincides, moreover, with We are ready to move another key step.
Proof. We want to show that the coproduct We first observe that

This is because
Now, we described above the completions of the algebras A and A ∨ -for A ∈ O q G/H , O q (G) -w.r.t. the I G -adic or the (q − 1)-adic topology. Using that, or an entirely similar analysis, we see also that In short, this is because Also, it is easily seen that this along with (4.5) immediately implies Finally, taking (q − 1)-adic completions on both sides, and also noting that In the end, we get the main result of this section: as an easy consequence of Lemma 4.7. Then, by Theorem 4.8 and by the fact , we have that the specialization of Moreover, Proposition 4.10 proves that the subalgebra Then, one defines ( ) ∨ on morphisms by scalar extension followed by restriction; proving the functoriality is a matter of bookkeeping. More in general, one might decide not to fix O q (G) , nor even G . Then morphisms φ : O q (G ′ ) −→ O q (G ′′ ) take the place of the endomorphisms of (the single) O q (G) in the recipe above, yet ( ) ∨ is defined again on morphisms via scalar extension and restriction -and one has to exploit the functoriality of ( ) ∨ over quantum groups O q (G) .

Examples: Quantum Grassmannians and quantum flag varieties
In this section we want to examine in detail some examples of quantum homogeneous spaces and apply the quantum duality principle recipe to them. We start with the quantum Grassmannian.

The quantum Grassmannian as quantum projective homogenous space
Let us recall the classical setting.
Let G = GL n (k) and let H = P a (standard) maximal parabolic subgroup, say We want first to give a very ample line bundle on the homogeneous space G P -the Grassmann variety -that realizes the classical Plücker embedding into a projective space, following the recipe in §2.2.
Let I = (i 1 , . . . , i r ) , 1 ≤ i 1 < · · · < i r ≤ n . Define as the function corresponding to the determinant of the minor of a matrix g = x ij ∈ GL n (k) obtained by taking rows i 1 , . . . , i r and columns 1, . . . , r . Then d I ∈ O GL n (k) for all I , i.e. these are regular functions on GL n (k) . If I 0 := (1, . . . , r) , then d I 0 restricts to a map (with same name) which is a character of P . One checks that the line bundle L associated to d I 0 is very ample, and it provides an embedding of G P into a projective space, following the recipe in §2.2. Algebraically, this means that the graded algebra O G P is realized as embedded into O(G) as In particular, one can easily verify, for any set I of r rows, that i.e. d I is d I 0 -semi-invariant of degree 1. In addition, one proves that the d I 's form a k-basis of the space O G P 1 of semi-invariants of degree 1 (cf. [23]). On the other hand, the spaces O G P n = H 0 G P, L ⊗n are in oneto-one correspondence -up to twisting by any integral power of det (i.e., by any character of GL n (k) ) -with the irreducible representations of GL n (k) .
We will now see that this picture extends to the quantum setup.
Let O q (M n ) be the unital associative algebra over k q = k q, q −1 with generators x ij (for 1 ≤ i, j ≤ n ) and relations (the so-called "Manin relations"). This algebra bears also a structure of k q -bialgebra, whose coproduct and counit are given by Define the "quantum determinant" (of order n ) det q as One proves that det q belongs to the center of O(M n ) , and it is group-like, i.e. ∆(det q ) = det q ⊗ det q and ǫ(det q ) = 1 . More in general, for any 1 ≤ r ≤ n and for any choice of r-tuples of increasing indices I = (i 1 , . . . , i r ) and J = (j 1 , . . . , j r ) , we define the "quantum determinant of the minor (I, J)", i.e. of the minor (of the matrix with entries the x ij 's) whose sets of rows and columns are I and J, namely These satisfy (cf. [21], §9.2.2) the following quantum analogue of wellknown classical identities (e.g., the first one is analogous to Binet theorem): it is a Ore element as well, and we can consider the enlarged algebra O q (GL n ) := O q (M n ) det q −1 obtained from O q (M n ) by formally inverting det q . Then -see [21] again -the bialgebra structure of O q (M n ) uniquely extends to O q (GL n ) ; even more, the latter then is a Hopf algebra indeed. In fact, by construction O q (GL n ) is a quantum group, namely a quantization of GL n (k) , in the sense of Definition 3.1.
We shall again denote by x ij the images in O q (GL n ) of the generators x ij of O q (M n ) . Similarly, we shall again denote by D I J the images in O q (GL n ) of the "quantum minors" of O q (M n ) : then they again enjoy (5.2) and (5.3). Letting J 0 := (1, . . . , r) =: I 0 , hereafter we shall set D I := D I J 0 . The specialization (at q = 1) of any quantum minor D I J is the corresponding classical minor d I J (on the same sets of rows and columns); in particular, every D I specializes to d I -see (5.1) -and, among them, D I 0 to d I 0 .
We define the quantum parabolic subgroup O q (P ) as the quotient algebra One can easily check that this O q (P ) is in fact a Hopf algebra quotient. Thus the natural projection map π : O q (G) −−։ O q (P ) is a Hopf algebra epimorphism. Therefore, O q (P ) is a quantum Poisson subgroup of O q (G) = O q GL n (k) , in the sense of Definition 3.3, whose specialization is O(P ) .
We are now in a position to appreciate the first important fact -in the present setting -about quantum minors: This along with Lemma 5.2 proves that the natural embedding is onto. But then, as noticed in Remark 3.19, this is enough to conclude.
Actually, we can prove the following, much more precise result: Proposition 5.4. The algebra O q G P is generated by the D I 's.
Proof. By Lemma 5.2, the D I 's belong to O q G P . Therefore, we are only left to prove that, conversely, every semi-invariant is contained in the k-subalgebra of O q (G) generated by the D I 's.
To this end, Theorems 1.2 and 1.3 in [16] give us immediately the result if we take k(q) as ground ring instead of k q := k q, q −1 . Then Lemmas 3.9, 3.10, 3.11 in [12] give us our result. We now see that in detail.
We start by rewriting the Proposition 1.1 in [16] in our notation: that B j is finitely generated, and the maps φ and ψ respect the decomposition, that is φ(A j ) ⊆ B j and ψ(B j ) ⊆ C j . Then if the sequence A −→ B −→ C obtained by reduction modulo (q − 1) is exact, then so is also Let's apply this result to our special situation. The subalgebra A := k q D I generated in O q (SL n ) by quantum determinants is a k q -graded module (by the degree). This fact is non trivial and depends on the explicit form of this algebra in terms of generators and relations, see [9]. We have that an element where ∆(a) = (a) a (1) ⊗ a (2) where x denotes reduction of x modulo I q (P ) (see notation in section 3.3). So we can set up maps where φ is the inclusion and ψ is defined by One can check that all the hypothesis of the previous result, for B := O q (SL n ) and C := O q (SL n ) ⊗ O q (P ) , are satisfied, hence we obtain that k(q) ⊗ k A ∼ = Ker id ⊗ ψ . In other words, the semi-invariants coincide with the subalgebra generated by the quantum determinants over the ring k(q) .
We now obtain the result over k q by Lemma 3.11 in [10], namely If wX ∈ k q D I , w ∈ k q , w = 0 , then X ∈ k q D I . Remark 5.5. Thus, using our own recipe, we have constructed the quantum homogeneous space O q G P . It is immediate to see that this is the same as the deformation of the algebra of the classical Grassmannian, along with its classical Plücker embedding, as it is described in [9] or in [28].
Finally, for the O q (G)-comodule structure of the space of semi-invariants of degree 1, we have also the following analogue of a classical result: Proof. This is a direct calculation. Let's sketch it. By all the previous analysis, we already know that O q G P 1 has basis the set of all the D I 's, and the left O q (G)-coaction on O q G P 1 is given by Now consider the coaction of O q (G) on q k n q , given by where σ is the permutation reordering k 1 , . . . , k r and K = k o 1 , . . . , k o r are the same indices, but reordered. Now the result follows.
Remark 5.7. Similar arguments can be used to prove that any quantum flag variety is a "quantum projective homogeneous space" in the sense of Definition 3.18 (for details about quantum flag varieties, we refer to [11]).
For the flag of type (m 1 , . . . , m s ), the quantum section d to start with is where the D (m j ) 's are the principal quantum minors of size m j . The proofs of all results go over exactly as in the Grassmannian case.
We now turn to the construction of the quantum big cell ring, that will be crucial for the explicit construction of the QDP functor.
As in the commutative setting, we have the following result: Proof. In the classical setting, the analogous result is proved by this argument: one uses the so-called "straightening relations" to get rid of the extra minors (see, for example, [8], §2). Here the argument works essentially the same, using the quantum straightening (or Plücker) relations (see [9], §4, [28], formula (3.2)(c) and Note I, Note II). Remark 5.10. As before, we have that This can be easily deduced from Remark 3.9, taking into account Proposition 5.9. As a consequence, the map is injective, so that the specialization map coincides with the restriction of the specialization map The following proposition gives a description of the algebra O loc q G P : Proposition 5.11. The big cell ring is isomorphic to a matrix algebra i.e. the generators t ij 's satisfy the Manin relations.
Remark 5.12. The Grassmannian GL n P can also be realized as a similar quotient of SL n by a suitable parabolic P ′ (corresponding to P , say). Then one can also perform all related quantum constructions -the previous and the later ones -using SL n instead of GL n , and modifying each step as needed. To begin with, one considers -where det q − 1 is the (two-sided) ideal generated by det q − 1 -which is again a Hopf algebra, for the quotient structure from either O q (GL n ) or O q (M n ) . This is a quantization of SL n (k) , in the sense of Definition 3.1, for which we can consider again quantum minors and a corresponding O q (P ) as before. Then all this can be used to give an alternative definition of O q G P = O q SL n P ′ and of all was considered above. Similarly, all constructions and results of section 5.2 hereafter can be carried on using O q (SL n ) -and its related gadgets -instead of O q (GL n ) .
Finally, similar considerations hold as well for the quantum flag varieties mentioned in Remark 5.7.

QDP for quantum Grassmannians
We now turn to the quantum duality principle applied explicitly to the quantum homogeneous spaces constructed above. We start with Grassmannians.
Let us first explicitly describe the Poisson structure of the algebraic group GL n . Starting from O q (GL n ) , as usual the classical algebra O(GL n ) inherits from the former a Poisson bracket, which makes it into a Poisson Hopf algebra, so that GL n becomes a Poisson group (see Remark 3.2(2)). We want to describe now this Poisson bracket. Recall that In terms of generators, we have As GL n is a Poisson Lie group, its Lie algebra gl n has a Lie bialgebra structure (see [3], pg. 24). To describe it, let us denote with m ij the elementary matrices, which form a basis of gl n . Define ( ∀ i = 1, . . . , n − 1 , j = 1, . . . , n ) Then e i , f i , g j i = 1, . . . , n − 1, j = 1, . . . , n is a set of Lie algebra generators of gl n , and a Lie cobracket is defined on gl n by This cobracket makes gl n itself into a Lie bialgebra: this is the so-called standard Lie bialgebra structure on gl n . It follows immediately that U(gl n ) is a co-Poisson Hopf algebra, whose co-Poisson bracket is the (unique) extension of the Lie cobracket of gl n while the Hopf structure is the standard one.
Similar constructions hold for the group SL n . One simply drops the generator d −1 , and imposes the relation d = 1 , in the description of O(SL n ) , and replaces the g s 's with the h i 's ( i = 1, . . . , n ) when describing sl n .
Since gl n is a Lie bialgebra, its dual space gl * n admits a Lie bialgebra structure, dual to the one of gl n . Let e ij := m * ij i, j = 1, . . . , n be the basis of gl * n dual to the basis of elementary matrices for gl n . As a Lie algebra, gl * n can be realized as the subset of gl n ⊕ gl n of all pairs  with its natural structure of Lie subalgebra of gl n ⊕gl n . In fact, the elements e ij correspond to elements in gl n ⊕ gl n in the following way: Then the Lie bracket of gl * n is given by are Lie algebra generators of gl * n . In terms of them, the Lie bracket reads On the other hand, the Lie cobracket structure of gl * n is given by Finally, all these formulae also provide a presentation of U gl * n as a co-Poisson Hopf algebra.
A similar description holds for sl * n = gl * n Z gl * n , where Z gl * n is the centre of gl * n , generated by l n := g 1 +· · ·+g n . The construction is immediate by looking at the embedding sl n ֒→ gl n .
We now turn to the construction of the QDP functor.
where the x ij 's are the generators of O q (G) . As Following Definition 4.2, we define We can provide a concrete description of O q G P ∨ : Proposition 5.14. We have Proof. Thanks to the previous discussion, it is enough to show that To do this, we describe the isomorphism O q (G) ∨ q=1 ∼ = U gl n * (cf. [14]).
According to Remark 5.13, the algebra O q (G) ∨ is generated by the elements . In terms of these generators, the isomorphism reads where we used notation X : whence clearly ∆ − → −(e 1,1 + · · · + e n,n ) as claimed.
Proof. This is the same as Proposition 4.10.
Hence for the quantum Grassmannian we have proved directly the following result: ∨ is a quantum homogeneous G * -space, which is an infinitesimal quantization of the homogeneous G * -space p ⊥ .

Quantum generalized flag varieties for simple groups as quantum projective homogeneous spaces
We now turn to a more general example of quantum projective space: namely the quantization of a generalized flag variety for any simple group, following [19]. As before, we begin with a brief description of the classical setting.
Let G be a connected, simply connected, complex simple Lie group, and let g its Lie algebra. Let S be a subset of simple roots of g and let λ = α i ∈S ω i be a weight of g , where the ω i 's are fundamental weights. Let V (λ) be the highest weight representation of g (and of G ) associated with the weight λ , and let v λ be a non-zero highest weight vector of V (λ) . We have the following morphism of algebraic varieties: This induces a projective embedding of the flag variety G P S into the projective space P V (λ) , where P S := Stab G (v λ ) is the parabolic subgroup associated to the set S . The graded algebra of regular functions on G P S relative to this embedding is given by where the grading is given by O G P S n := V (nλ) * and the multiplication is via the Cartan multiplication (see [12] for more details).
We are now going to identify O G P S with a graded subalgebra of O(G) . Indeed, the algebra O(G) is in (Hopf) duality with U(g) and it can be thought of as the linear span inside U(g) * of the functionals c λ f,v : U(g) −→ C (the so-called "matrix coefficients") given by Then O G P S can be realized, equivalently, as the subalgebra of O(G) generated by the c λ f,v λ 's, for all f ∈ V (λ) * . This point of view carries over without changes to the quantum setting: it is considered in [22], [27] and by several others; a quick review of this construction can be found in [19]. The key point is that every finite dimensional representation V (µ) of U(g) of highest weight µ has a (standard) quantization, which is a representation for U q (g) , call it V q (µ) . In particular, one can again define matrix coefficient functionals c µ f,v -for every f ∈ V q (µ) * and v ∈ V q (µ) -for all dominant weights µ of g . Their k q -span inside U q (g) * is, by definition, the algebra O q (G) , which is a quantization of O(G) . In fact, the latter follows because V q (µ) , respectively V q (µ) * , specializes to V (µ) , respectively to V (µ) * , at q = 1 , hence "quantum" and "classical" matrix coefficients bijectively correspond to each other -via c µ f,v → c µ f,v , sayunder specialization at q = 1 .
For later use, we point out how matrix coefficient behave under the coproduct. For any dominant weight µ of g , let {v 1 , . . . , v r } be a k q -basis of V q (µ) -with r = dim V q (λ) -and let {f 1 , . . . , f r } be the dual basis of V q (µ) * . Then every matrix coefficient c µ f,v has coproduct given by ∆ c µ (just because U q (g) is acting on V q (µ) via matrices!).
From the quantization O q (G) the group G inherits a Poisson group structure -cf. Remark 3.2(2) -for which P S is a Poisson subgroup. On the other hand, let I q (P S ) be the two-sided ideal of O q (G) generated by the set of matrix coefficients c µ f,v ∀ n ∈ N , µ = nλ ∨ v ∈ k q \{0} ·v nλ . Then, using (5.5), one easily shows that I q (P S ) is a Hopf ideal of O q (G) ; therefore the quotient k q -module and the canonical map where v λ is a non-zero element of weight λ in V q (λ) . Then, as the quantum matrix coefficients specialize to the classical ones, comparing (5.7) with (5.4) we see at once that so that O q G P S ′ is a quantization, as a k q -module, of O G P S .
We are now going to show that this O q G P S ′ is in fact a quantum homogeneous space in the sense of Definition 3.18, in particular it can be realized as the space of semi-invariants inside O q (G) with respect to a suitable quantum section. Indeed, we shall find O q G P S ′ = O q G P S where the latter is the space of all semi-invariants (for a suitable quantum section) as in Definition 3.7 and Definition 3.18. First of all, let λ be the dominant weight fixed above, and let v λ be the (up to a scalar factor) uniquely determined non-zero element of weight λ in V q (λ) . Fix a k q -basis {v 1 , . . . , v r } of V q (λ) with v 1 = v λ , and let {f 1 , . . . , f r } be the dual basis of V q (µ) * , for which we set f λ := f 1 .
Proposition 5.18. The element c λ f λ ,v λ is a quantum section in O q (G) . Proof. Thanks to Proposition 3.13(b) we only need to show that c λ f λ ,v λ is a pre-quantum section, with respect to the setup of (5.6), i.e.
But this follows at once from the identity (5.5), applied to the bases chosen above, once we notice in addition that c λ f i ,v λ ∈ I q (P S ) for all i = 1 .
Proposition 5. 19. The space of c λ f λ ,v λ -semi-invariants of degree 1 inside O q (G) , that is O q G P S 1 := f ∈ O q (G) ∆ π (f ) = f ⊗ π c λ f λ ,v λ , is just the k q -span of c λ f,v λ f ∈ V q (λ) * . In other words, We are now ready for the main result of this subsection: Theorem 5.21. Let O q G P S be defined as in Definition 3.7, with respect to the quantum section d := c λ fe,v λ ∈ O q (G) . Then O q G P S is a quantum projective homogeneous space, namely, it is a quantization of O G P S , in the sense of Definition 3.18.
Proof. This follows at once by putting together the previous results, i.e. Propositions 5.18, 5.19 and 5.20, and the specialization formula (5.8).

Remark 5.22.
(1) With some extra work, one can also show that O q G P S is generated -as a graded algebra -in degree 1, i.e. by O q G P S 1 .
(2) Our last remark is that in this setup of quantum generalized flag varieties one can also apply the QDP, following the general recipe of §4.
Indeed, in [19], and we can then apply the QDP -according to §4 -to this setting.

The coisotropic case
One could argue whether the generality we are working with is necessary or not. In this section we will describe how to find families of coisotropic parabolic subgroups inside semisimple algebraic groups with the standard multiplicative Poisson structure. Such families give rise to smooth Poisson bivectors on the projective quotients which cannot be obtained as quotient by Poisson parabolic subgroups. It would be therefore interesting to investigate whether in such examples it is possible to find and quantize a graded Poisson algebra associated to them as described in section 2.
We shall start with a low-dimensional example and describe in a very explicit manner the objects we are interested in and then we will generalize such results.  [20]). We take the standard parabolic subgroup of upper triangular matrices Let G be a complex semisimple algebraic group and let K be its real compact form. Up to a factor ı , which is inessential in what follows, the standard Poisson structure on G can be identified with the one which is automatically defined on it when it is identified with the Drinfeld's double of K . Let now H be a coisotropic subgroup of K and let us consider the subgroup HK * of G (here K * = AN is the Manin dual of K inside G ). Then P = HK * is parabolic in G , H = P ∩ K and K H ≃ G P as smooth manifolds. It can be shown quite easily that the coisotropy of K implies the coisotropy of P , and furthermore, via Theorem 4.1 in [7], that K H and G P are also Poisson diffeomorphic. Thus in order to check whether P is coisotropic it is enough to check whether P ∩ K is coisotropic w.r. to the standard Poisson structure on the compact real group K . There we can rely on results in [7], where a 1-parameter family of coisotropic subgroups H ε ⊆ SU(n) was given. Such subgroups induce a 1-parameter family of homogeneous Poisson quotients on complex Grassmannians.