Star-product on complex sphere $\mathbb{S}^{2n}$

We construct a $U_q(\mathrm{so}(2n+1))$-equivariant local star-product on the complex sphere $\mathbb{S}^{2n}$ as a Non-Levi conjugacy class $SO(2n+1)/SO(2n)$.


Introduction
In this paper, we incorporate a simple example of homogeneous space with non-Levi stabilizer into a uniform quantization scheme for closed conjugacy classes of simple algebraic groups.
This approach was developed in 2003 for Levi classes and utilized the presence of quantum isotropy subgroup in the total quantum group, [1,2]. The key distinction of non-Levi classes is the absence of a natural candidate for such a subgroup because its root basis cannot be made a part of the total root basis. Still the coordinate ring of the class can be quantized by an operator realization on certain modules, [3]. Such a quantization is formulated in terms of generators and relations and is not apparently local. On the other hand, a dynamical twist constructed from the Shapovalov form yields a local version of the star product on Levi classes, [1,2] (see also [4,5] for coadjoint orbits with the Kirillov bracket). It is natural to extend that approach to all closed conjugacy classes. Such a possibility for S 4 was pointed out without proof in [6]. Here we give a solution for all even dimensional spheres.
The original approach to the star product on Levi classes was as follows. Let k ⊂ g be the isotropy Levi subalgebra of a point t and p ± ⊂ g its parabolic extensions. The point t is associated with a certain weight λ ∈ h * and a pair of modules M λ , N λ of, respectively, highest and lowest weights λ and −λ. There is a unique U q (g)-invariant form M λ ⊗ N λ → C, which is non-degenerate if and only if the modules are irreducible. In that case, there exists the inverse form C → N λ ⊗ M λ and its lift 1 → F ∈ U q (p + ) ⊗ U q (p − ). The element F gives rise to a "bidifferential" operator via the left co-regular action on the Hopf dual A = U * q (g). With this operator, the multiplication in A is twisted to a non-associative operation invariant under the right co-regular action of U q (g). The key observation is that the new multiplication becomes associative when restricted to the subspace A k of U q (k)invariants in A. As a (right) U q (g)-module, A has the same structure as the U(g)-module It is known that the initial star product on A is local, [7], therefore the resulting multiplication is local as well.
In the non-Levi case, one can go along those lines and define A k as the joint kernel of certain operators that deform generators of k. Then the new product will be associative on A k as in the Levi case, [8]. However, those operators do not close up to a deformation of U q (k) so one cannot be sure that A k has the proper size (observe that kernel can decrease under deformation). Therefore the problem is to check the size of A k . We do it for even dimensional spheres regarded as conjugacy classes of SO(2n+ 1). Note that odd dimensional spheres belong to the second connected component of the orthogonal group O(2n), and the current methods are not directly applicable.
The paper consists of five sections. After the introduction we recall quantization of C[S 2n ] via operator realization on a highest weight module M λ in Section 2. In the next section we construct a system of vectors that spans M λ . We prove it to be a basis in Section 4 by computing the Shapovalov form on M λ . This way we show that M λ is irreducible and the form is invertible. In the final section we show that for locally finite U q (g)-module V q , the dimension of V k q is equal to dim V k of the classical k-invariants. We do it via realization of finite dimensional module V q with dim V k q > 0 in the coordinate ring of the quantum Euclidean plane C 2n+1 q . This way we complete the task.

Operator realization of C q [S 2n ]
Throughout the paper, g stands for the Lie algebra sp(2n + 1). We are looking for quantization of the polynomial ring C[S 2n ] that is invariant under an action of the quantized universal enveloping algebra U q (g). We regard S 2n as a conjugacy class of the Poisson group G = SO(2n + 1) equipped with the Drinfeld-Sklyanin bracket corresponding to the standard solution r ∈ g ⊗ g of the classical Yang-baxter equation, [9]. The group G supports the Semenov-Tian-Shansky bivector field making it a Poisson G-space with respect to conjugation. Here r − and r + are, respectively, the skew-symmetric and invariant symmetric parts of r, and the superscripts designate the vector fields where ξ ∈ g and f is a smooth function G. This bivecftor field (2.1) is tangent to every conjugacy class of G. In particular, the sphere S 2n becomes a homogeneous Poisson-Lie manifold over G.
Quantization of C[G] along (2.1) gives rise to the reflection equation dual of U q (g), [10]. Accordingly, the algebra C[S 2n ] can be presented as its quotient. Here we recall that construction.
Let h ⊂ g denote the Cartan subalgebra equipped with the inner product restricted from an ad-invariant form on g. We endow the dual space h * with the inverse form, (., .). For any µ ∈ h * we denote by h µ ∈ h the vector such that ν(h µ ) = (ν, µ) for all ν ∈ h * . We normalize the inner product so that the short roots have length 1.
The subset Π k = {δ, α 1 , . . . , α n } ⊂ R + forms a root basis for a subalgebra k ⊂ g isomorphic to so(2n). Although e ±δ are deformations of classical root vectors, they do not generate an sl(2)-subalgebra in U q (g), so we have no natural subalgebra U q (k) in U q (g). Still e ±δ play a role in what follows.
Fix the weight λ ∈ h * by the conditions q 2(λ,ε i ) = −q −1 for all i = 1, . . . , n, and (α i , λ) = 0 and by zero on the generators on non-zero weight. Extend them to representations of U q (p ± ) by zero on e ±α for all α ∈ Π g . Then set Denote by 1 λ ∈ M λ and 1 * λ ∈ N λ their highes/lowest weight generators. Due to the special choice of λ, the vectors e −δ 1 λ ∈M λ and e δ 1 * λ ∈N λ are killed by e α and, respectively, e −α for all α ∈ Π. They generate The module M λ supports quantization of C[S 2n ] in the following sense. The sphere S 2n is isomorphic a subvariety in G of orthogonal matrices with eigenvalues ±1, where 1 is

Spanning M λ
In this section we introduce a set of vectors in M λ which is proved to be a basis in the subsequent section. Here we prove that it spans M λ . Put f α = e −α for all simple roots and The elements f ε i can be included in the set of composite root vectors generating a Poincare-Birkhoff-Witt basis in U q (g − ), [12]. By deformation arguments, the set of monomials B = , where = log q, see [11] and references therein. We prove that B is a C-basis once q is not a root of unity.
Lemma 3.1. For all 1 < i m, the elements f ε i belong to the normalizer of the left ideal The element f δ commutes with f ε 2 , which completes the proof for m = 2. Suppose that where the left equality means f δ ∈ U q (g)k − 3 , and the last two equalities are obtained from it and from [f δ , f α 1 ] = 0. Furthermore, Serre relations along with (3.2) yield Multiply the first equality by a and subtract from the second: This completes the case m = 3. For m i > 3, put Corollary 3.2. The set B spans M λ . The action of U q (g − ) on M λ is given by . . . f mn εn 1 λ , i > 1.

Proof. First let us show that
The first summand vanishes since f ε i −ε 2 commutes with f ε i+1 , by Lemma A.1. The internal as well.
Now we can complete the proof. The linear span CB is invariant under the obvious action of f α 1 = f ε 1 . For i > 0, we push f α i+1 to the right in the product Thanks to Lemma 3.1, we replace the product For any form Φ in n − i variables, the ideal U q (g)k − j+1 kills Φ(f ε i+1 , . . . , f εn )1 λ by Lemma 3.1. This yields the action of f α i+1 on CB and proves its U q (g − )-invariance. Since CB ∋ 1 λ , it coincides with M λ . The elements f ε i , e ε i are known to generate U q (sl(2))-subalgebras in U q (g) with the commu-
Proof. The δ-symbols are due to orthogonality of weight subspaces M λ [µ] and N λ [ν] unless µ = −ν. Now we prove factorization of the matrix coefficients on setting k i = m i for all i.
Observe that There is also ω-contravariant form on M λ defined by x1 λ ⊗ y1 λ → ω(x)y , for all x, y ∈ U q (g). It is called the Shapovalov form and related with the invariant form in the obvious way. 2. The modules M λ and N λ are irreducible.

The tensor
Proof. 1) Corollary 3.2 with Lemma 4.3 prove completeness of B and independence. Since

Star-product on S 2n
Denote by A q the RTT dual of U q (g) with multiplication • and the Hopf paring (., .). It is equipped with the two-sided action (here a (1) ⊗ a (2) = ∆(a) in the Sweedler notation) x ⊲ a = a (1) (a (2) , x), a ⊳ x = (a (1) , x)a (2) , ∀x ∈ U q (g), a ∈ A q , making it a U q (g)-bimodule algebra. The multiplication • is known to be local, [7]. We define a new operation ⋆ by where ⊲ stands for the left co-regular action. It is obviously equivariant with respect to the right co-regular action of U q (g). However, ⋆ is not associative on the entire A q .
For every U q (g)-module V we define V k ⊂ V to be the intersection of the space V l of U q (l)-invariants with the joint kernel of the operators e δ and f δ . For q = 1, this definition coincides with the subspace of U(k)-invariants.
Proposition 5.1. A k q is an associative U q (g)-algebra with respect to ⋆.
Proof. Identify M * λ with N * * λ and M λ ⊗ M * λ with the module of locally finite endomorphisms End • C (M λ ). For every completely reducible module V , there is a uniqueφ ∈ Hom(V * , End • C (V )) for each φ ∈ Hom(M λ ⊗ N λ , V ), due to the natural isomorphism of the Hom-sets.
LetM λ andN λ denote the Verma modules, i. e. induced from the U q (b ± )-modules C ∓λ . Every homomorphism M λ ⊗ N λ → V amounts to a homomorphismM λ ⊗N λ → V vanishing on α∈Π kM λ−α ⊗N λ + α∈Π kM λ ⊗N λ−α . Therefore φ corresponds to a unique zero weight element Φ(φ) ∈ ∩ α∈±Π k ker e α = V k . Given also ψ ∈ Hom(M λ ⊗ N λ , W ) there is Now take V = W = A q and f, g ∈ A k q (with respect to the ⊲-action). Then f ⋆ g is the image of f ⊛ g ∈ (A q ⊗ A q ) k under the multiplication • : A q ⊗ A q → A q , which is again in Associativity of ⋆ follows from associativity of • and •.
Proof. We only need to make sure that It is done by Proposition 6.2 below.
Remark, that despite A k q goes over to C[G] k at q = 1, the fact A k q ≃ C[G] k ⊗ C[q, q −1 ] needs a proof because ker e δ and ker f δ may decrease under deformation. That is done in the next section.

Quantum Euclidian plane
To complete the proof of Theorem 5.2, it is sufficient to check dim V k q = dim V k for all finite dimensional modules V that appear in C[S 2n ]. They all can be realized in the polynomial ring of the Euclidian plane C 2n+1 , [14]. So we going to look at its quantum version.
Choose a basis {x i } n i=−n ⊂ C N , N = 2n + 1, and define a representation of U q (g) on C N by the assignment . . , n. Then x i carry weights ε i subject to ε i = −ε −i . The quantum Euclidian plane C q [C N ] is an associative algebra generated by {x i } n i=−n with relations These relations are equivalent to those presented in [13].
The representation on C N extends to an action ⊲ on C q [C N ] making it a U q (g)-module algebra. Let θ denote the involutive algebra and anti-coalgebra linear automorphism of U q (g) determined by the assignment e α → −f α , q hα → q −hα . Define also an anti-algebra linear involution on C q [C N ] by ι(x i ) = (x −i ). They are compatible with the action ⊲, that is, Lemma 6.1. For all k ∈ Z + the monomials x k 0 are killed by e δ and f δ .
The q-version of the quadratic invariant is C q = 1 1+q  Let P m and H m denote their classical counterparts.
Proposition 6.2. For any finite dimensional U q (g)-module V q , dim V k q is equal to dim V k of the classical k-invariants.
Proof. It is sufficient to show that dim(P m q ) k = dim(P m ) k . In the classical limit, the trivial ksubmodule in H m−2l is multiplicity free, so its dimension in P m is [ m 2 ]+1. On the other hand, the subspace of U q (l)-invariants is spanned by l=0 and has the same dimension. Since all U q (l)-invariants are killed by e δ , f δ , in view of Lemma 6.1, this proves the statement.