# Star product on complex sphere \(\mathbb {S}^{2n}\)

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## Abstract

We construct a \(U_q\bigl (\mathfrak {s}\mathfrak {o}(2n+1)\bigr )\)-equivariant local star product on the complex sphere \(\mathbb {S}^{2n}\) as a non-Levi conjugacy class \(SO(2n+1)/SO(2n)\).

## Keywords

Quantum groups Star product Even spheres Verma modules## Mathematics Subject Classification

17B10 17B37 53D55## 1 Introduction

In this paper, we incorporate an example of homogeneous space with non-Levi stabilizer into a uniform quantization scheme for closed conjugacy classes of simple algebraic groups. Originally, 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 \(\mathbb {S}^4\) was pointed out without proof in [6]. Here we give a solution for all even-dimensional spheres.

Sphere is a relatively simple curved space endowed with a rich structure that has numerous applications. An interest to its quantum version started to grow with the invention of quantum groups [7], and \(\mathbb {S}^{2}_q\) was the first quantum *G*-space [8] after Manin’s \(\mathbb {C}^2_q\) [9]. A review of various constructions of the q-sphere in small dimensions and some references to its applications can be found in [10].

An even sphere admits several independent although isomorphic equivariant quantizations: a subvariety of the quantum Euclidean plane [11], an induced representation of a quantum symmetric pair (cf. [12]), and a subalgebra of linear operators on a highest weight module of the orthogonal quantum group [13]. Each particular reincarnation has its pluses that help tackling hard issues arising in other approaches. For instance, the operator realization of \(\mathbb {C}_q[\mathbb {S}^{2n}]\) allows to study representations of the coideal subalgebra in the corresponding symmetric pair [12]. All realizations of \(\mathbb {C}_q[\mathbb {S}^{2n}]\) known to date appeal to generators and relations. At the same time, a local formulation may be of interest for some applications, like Fedosov’s star product approach to the index theorem [14]. The present work fills that gap. Note that, like in the Levi case [1], this problem can be placed in a more general context of quantum vector bundles addressed in [12]. This is also a part of the Gelfand–Zetlin reduction for orthogonal quantum groups, that is open by now. It turns out that local quantization of the function algebra on \(\mathbb {S}^{2n}\) (the trivial bundle) can be done with elementary means and deserves a special consideration.

The original approach to the star product on Levi classes was as follows. Let \(\mathfrak {k}\subset \mathfrak {g}\) be the isotropy Levi subalgebra of a point *t* and \(\mathfrak {p}_\pm \subset \mathfrak {g}\) its parabolic extensions. One associates with *t* a certain weight \(\lambda \in \mathfrak {h}^*\) and a pair of modules \(M_\lambda , N_\lambda \) of, respectively, highest and lowest weights \(\lambda \) and \(-\,\lambda \). There is a (essentially unique) \(U_q(\mathfrak {g})\)-invariant form \(M_\lambda \otimes N_\lambda \rightarrow \mathbb {C}\), which is non-degenerate if and only if the modules are irreducible. In that case, there exists the inverse form \(\mathbb {C}\rightarrow N_\lambda \otimes M_\lambda \) and its lift assigning \(1\mapsto \mathcal {F}\in U_q(\mathfrak {p}_+)\otimes U_q(\mathfrak {p}_-)\) (a completed tensor product). The element \(\mathcal {F}\) gives rise to a “bidifferential” operator via the left coregular action on the Hopf dual \(\mathcal {A}=U_q^*(\mathfrak {g})\). With this operator, the multiplication in \(\mathcal {A}\) is twisted to a non-associative operation invariant under the right coregular action of \(U_q(\mathfrak {g})\). The key observation is that the new multiplication becomes associative when restricted to the subspace \(\mathcal {A}^\mathfrak {k}\) of \(U_q(\mathfrak {k})\)-invariants in \(\mathcal {A}\). As a (right) \(U_q(\mathfrak {g})\)-module, \(\mathcal {A}^\mathfrak {k}\) has the same structure as the \(U(\mathfrak {g})\)-module \(\mathbb {C}[G/K]\), where \(K\subset G\) is the centralizer subgroup of the point *t*. Hence, \(\mathcal {A}^\mathfrak {k}\) is a flat deformation of \(\mathbb {C}[G/K]\). It is known that the initial star product on \(\mathcal {A}\) is local [15]; therefore, the resulting multiplication is local as well.

In the non-Levi case, one can go along those lines and define \(\mathcal {A}^\mathfrak {k}\) as the joint kernel of certain operators that deform generators of \(\mathfrak {k}\). Then the new product will be associative on \(\mathcal {A}^\mathfrak {k}\) as in the Levi case [16]. However, those operators do not close up to a deformation of \(U_q(\mathfrak {k})\), so one cannot be sure that \(\mathcal {A}^\mathfrak {k}\) has the proper size. (Observe that kernel can decrease under deformation.) Therefore, the problem is to check the size of \(\mathcal {A}^\mathfrak {k}\). We do it for \(\mathbb {S}^{2n}\) via a harmonic analysis relative to quantized \(SO(2n+1)\). Note that odd-dimensional spheres belong to the second connected component of the orthogonal group *O*(2*n*), and the current methods are not directly applicable.

The paper consists of five sections: after Introduction, we recall quantization of \(\mathbb {C}[\mathbb {S}^{2n}]\) via operator realization on a highest weight module \(M_\lambda \) in Sect. 2. In Sect. 3 we construct a system of vectors that spans \(M_\lambda \). We prove it to be a basis in Sect. 4 by computing the Shapovalov form on \(M_\lambda \). This way, we show that \(M_\lambda \) is irreducible and the form is invertible. In the final section we show that for finite \(U_q(\mathfrak {g})\)-module \(V_q\), the dimension of \(V^\mathfrak {k}_q\) is equal to \(\dim V^\mathfrak {k}\) of the classical \(\mathfrak {k}\)-invariants. We do it via realization of \(V_q\) with \(\dim V_q^\mathfrak {k}>0\) in the coordinate ring of the quantum Euclidean plane \(\mathbb {C}^{2n+1}_q\).

## 2 Operator realization of \(\mathbb {C}_q[\mathbb {S}^{2n}]\)

*G*supports the Semenov–Tian–Shansky bivector field

*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

*f*is a smooth function on

*G*. This bivector field (2.1) is tangent to every conjugacy class of

*G*. In particular, the sphere \(\mathbb {S}^{2n}\) becomes a homogeneous Poisson manifold over the Poisson group

*G*[17].

Quantization of \(\mathbb {C}[G]\) along (2.1) gives rise to the reflection equation dual of \(U_q(\mathfrak {g})\) [18]. Accordingly, the algebra \(\mathbb {C}_q[\mathbb {S}^{2n}]\) can be presented as its quotient. Here we recall that construction.

Let \(\mathfrak {h}\subset \mathfrak {g}\) denote the Cartan subalgebra equipped with the inner product restricted from an \(\mathrm {ad}\)-invariant form on \(\mathfrak {g}\). We endow the dual space \(\mathfrak {h}^*\) with the inverse form (., .) and normalize it so that short roots have length 1. For any \(\mu \in \mathfrak {h}^*\) we denote by \(h_\mu \in \mathfrak {h}\) the vector such that \(\nu (h_\mu )=(\nu ,\mu )\) for all \(\nu \in \mathfrak {h}^*\).

The root system \(\mathrm {R}\) contains an orthonormal basis \(\Lambda =\{\varepsilon _i\}_{i=1}^n\subset \mathfrak {h}^*\) of short roots. We choose the basis of simple positive roots \(\Pi \) as \(\alpha _1=\varepsilon _1, \alpha _{i}=\varepsilon _{i}-\varepsilon _{i-1}, i=2,\ldots , n\). We define the subalgebra \(\mathfrak {l}\simeq \mathfrak {g}\mathfrak {l}(n)\subset \mathfrak {g}\) of maximal rank with the root basis \(\Pi _\mathfrak {l}=\{\alpha _i\}_{i=2}^n\).

The subset \(\Pi _\mathfrak {k}=\{\delta ,\alpha _1,\ldots ,\alpha _n\}\subset \mathrm {R}^+\) forms a root basis for a subalgebra \(\mathfrak {k}\subset \mathfrak {g}\) isomorphic to \(\mathfrak {s}\mathfrak {o}(2n)\). Although \(e_{\pm \delta }\) are deformations of classical root vectors, they do not generate an \(\mathfrak {s}\mathfrak {l}(2)\)-subalgebra in \(U_q(\mathfrak {g})\), so we have no natural subalgebra \(U_q(\mathfrak {k})\) in \(U_q(\mathfrak {g})\). Still \(e_{\pm \delta }\) play a role in what follows.

By \(U_q(\mathfrak {h})\subset U_q(\mathfrak {g})\) we denote the subalgebra generated by \(\{q^{\pm h_\alpha }\}_{\alpha \in \Pi }\). We use the notation \(\mathfrak {g}_\pm \subset \mathfrak {g}\) for the Lie subalgebras generated by \(\{e_{\pm \alpha }\}_{\alpha \in \Pi }\). They generate subalgebras \(U_q(\mathfrak {g}_\pm )\subset U_q(\mathfrak {g})\).

The module \(M_\lambda \) supports quantization of \(\mathbb {C}[\mathbb {S}^{2n}]\) in the following sense. The sphere \(\mathbb {S}^{2n}\) is isomorphic to a subvariety in *G* of orthogonal matrices with eigenvalues \(\pm 1\), where 1 is multiplicity-free. It is a conjugacy class with a unique point of intersection with the maximal torus relative to \(\mathfrak {h}\). The isotropy subalgebra of this point is \(\mathfrak {k}\). Quantization of \(\mathbb {C}[G]\) along the Poisson bracket (2.1) can be realized as a subalgebra \(\mathbb {C}_q[G]\subset U_q(\mathfrak {g})\) invariant under the adjoint action. The image of \(\mathbb {C}_q[G]\) in \(\mathrm {End}(M_\lambda )\) is an equivariant quantization of \(\mathbb {C}[\mathbb {S}^{2n}]\), see [13] for details.

## 3 Spanning \(M_\lambda \)

*q*is not a root of unity.

Let \(\mathfrak {k}_m^-\) denote the subspace \(\mathbb {C}f_\delta +\mathrm {Span}\{f_{\alpha _2},\ldots ,f_{\alpha _m}\}\subset U_q(\mathfrak {g}_-)\) assuming \(m\geqslant 2\).

## Lemma 3.1

For all \(1<i\leqslant m\), the elements \(f_{\varepsilon _i}\) belong to the normalizer of the left ideal \( U_q(\mathfrak {g})\mathfrak {k}_m^-\).

## Proof

The Serre relations readily yield \([f_{\alpha _m}, f_{\varepsilon _m}]_{\bar{q}}=0\), for \(m>1\). For \(1<i<m\), the identity \([f_{\alpha _i}, f_{\varepsilon _m}]=0\) follows from Lemma A.1. So we are left to study how \(f_{\varepsilon _i}\) commute with \(f_\delta \).

*a*and subtract from the second:

For all \(i>k\geqslant 1\), the elements \(f_{\varepsilon _i-\varepsilon _k}=[f_{\alpha _{k+1}},\ldots [f_{\alpha _{i-1}},f_{\alpha _i}]_{\bar{q}}\ldots ]_{\bar{q}}\) belong to \(\in U_q(\mathfrak {g})\mathfrak {k}_m^-\) with \(m\geqslant i\). Then \( f_\delta f_{\varepsilon _i}=-q^{-1}f_\delta f_{\varepsilon _i-\varepsilon _3}f_{\varepsilon _3}=-q^{-1}f_{\varepsilon _i-\varepsilon _3}f_\delta f_{\varepsilon _3}=0 \> \mathrm {mod}\> U_q(\mathfrak {g})\mathfrak {k}_m^-, \) for all \(m\geqslant i>3\), as required. \(\square \)

## Corollary 3.2

## Proof

## 4 Invariant bilinear form \(M_\lambda \otimes N_\lambda \rightarrow \mathbb {C}\)

## Lemma 4.1

## Proof

*i*. Observe that

*s*completes the proof. \(\square \)

There is also an \(\omega \)-contravariant form on \(M_\lambda \) defined by \(x1_\lambda \otimes y1_\lambda \mapsto \langle \omega (x)y\rangle \), for all \(x,y\in U_q(\mathfrak {g})\). It is called the Shapovalov form and related to the invariant form in the obvious way.

## Proposition 4.2

*q*is not a root of unity. Then

- 1.
\(\mathcal {B}\subset M_\lambda \) is an orthogonal (non-normalized) basis with respect to the Shapovalov form.

- 2.
The modules \(M_\lambda \) and \(N_\lambda \) are irreducible.

- 3.The tensoris a lift of the inverse invariant form, \(\mathbb {C}\rightarrow N_\lambda \otimes M_\lambda \rightarrow U_q(\mathfrak {g}_+)\otimes U_q(\mathfrak {g}_-), 1\mapsto \mathcal {F}\).$$\begin{aligned} \mathcal {F}=\sum _{m_1,\ldots ,m_n=0}^{\infty }(-\theta )^{\sum _{i=1}^{n}m_i}\> \frac{\prod _{i=1}^{n}q^{-\frac{m_i^2}{2}+2m_i(i-1)}}{\prod _{i=1}^n[m_i]_{q}!} \tilde{e}_{\varepsilon _1}^{m_1}\ldots \tilde{e}_{\varepsilon _n}^{m_n} \otimes f_{\varepsilon _1}^{m_1}\ldots f_{\varepsilon _n}^{m_n} \end{aligned}$$

## Proof

(1) Corollary 3.2 with (4.3) proves the completeness of \(\mathcal {B}\) and independence. All weight subspaces in \(M_\lambda \) have dimension 1, and the form is non-degenerate; hence, the basis \(\mathcal {B}\) is orthogonal with respect to the Shapovalov form. (2) Non-degeneracy of the form implies irreducibility of \(M_\lambda \). (3) The normalizing coefficients in \( \mathcal {F}\) are obtained from (4.3) via the equality \((x 1_\lambda ,\gamma (y)1^*_\lambda )= \langle y x \rangle \) and \(\gamma (e_{\varepsilon _i})=-\,\tilde{e}_{\varepsilon _i}q^{-h_{\varepsilon _i}+2(i-1)}\) for all \(i=1,\ldots ,n\). \(\square \)

## 5 Star product on \(\mathbb {S}^{2n}\)

As a two-sided \(U_q(\mathfrak {g})\)-module, \(\mathcal {A}_q\) is isomorphic to \(\oplus _V V^*\otimes V\), where the summation is over all equivalence classes of irreducible finite-dimensional representations of \(U_q(\mathfrak {g})\). This is a q-version of the Peter–Weyl decomposition.

For every \(U_q(\mathfrak {g})\)-module *V* we define \(V^\mathfrak {k}\subset V\) to be the intersection of the space \(V^\mathfrak {l}\) of \(U_q(\mathfrak {l})\)-invariants with the joint kernel of the operators \(e_\delta \) and \(f_\delta \). For \(q=1\), this definition gives the subspace of \(U(\mathfrak {k})\)-invariants.

## Proposition 5.1

\(\mathcal {A}^\mathfrak {k}_q\) is an associative \(U_q(\mathfrak {g})\)-algebra with respect to \(\star \).

## Proof

Identify \(M_\lambda ^*\) with \(N_\lambda ^{**}\) and the locally finite part of \(M_\lambda \otimes M_\lambda ^*\) with the locally finite part \(\mathrm {End}^\circ _\mathbb {C}(M_\lambda )\) of \(\mathrm {End}_\mathbb {C}(M_\lambda )\) regarded as a \(U_q(\mathfrak {g})\)-module. For every completely reducible module *V*, there is a unique \(\tilde{\phi }\in \mathrm {Hom}(V^*,\mathrm {End}_\mathbb {C}^\circ (V))\) for each \(\phi \in \mathrm {Hom}(M_\lambda \otimes N_\lambda ,V)\), due to the natural isomorphism of the hom-sets.

Let \(\hat{M}_\lambda \) and \(\hat{N}_\lambda \) denote the Verma modules, i.e., induced from the \(U_q(\mathfrak {b}_\pm )\)-modules \(\mathbb {C}_{\mp \lambda }\). Every homomorphism \(M_{\lambda }\otimes N_{\lambda } \rightarrow V\) amounts to a homomorphism \(\hat{M}_{\lambda }\otimes \hat{N}_{\lambda } \rightarrow V\) vanishing on \(\sum _{\alpha \in \Pi _\mathfrak {k}}\hat{M}_{\lambda -\alpha }\otimes \hat{N}_{\lambda }+\sum _{\alpha \in \Pi _\mathfrak {k}}\hat{M}_{\lambda }\otimes \hat{N}_{\lambda -\alpha }\). Therefore, \(\phi \) corresponds to a unique zero weight element \(\Phi (\phi )\in \cap _{\alpha \in \pm \Pi _\mathfrak {k}}\ker e_\alpha = V^\mathfrak {k}\). Given also \(\psi \in \mathrm {Hom}(M_\lambda \otimes N_\lambda , W)\) there is a unique element \(\phi \circledast \psi \in \mathrm {Hom}(V\otimes W,\mathrm {End}_\mathbb {C}^\circ (M))\) such that \(\widetilde{\phi \circledast \psi }=\tilde{\phi }\circ \tilde{\psi }\), where \(\circ \) is the multiplication in \(\mathrm {End}^\circ (M_\lambda )\). Define \(\Phi (\phi )\circledast \Phi (\psi )=\Phi (\phi \circledast \psi )\in (V\otimes W)^\mathfrak {k}\).

Now take \(V=W=\mathcal {A}_q\) and \(f,g\in \mathcal {A}_q^\mathfrak {k}\) (with respect to the \(\triangleright \)-action). Then \(f\star g\) is the image of \(f\circledast g \in (\mathcal {A}_q\otimes \mathcal {A}_q)^\mathfrak {k}\) under the multiplication \(\bullet :\mathcal {A}_q\otimes \mathcal {A}_q\rightarrow \mathcal {A}_q\), which is again in \(\mathcal {A}_q^\mathfrak {k}\) since \(\bullet \) is \(\triangleright \)-equivariant. Associativity of \(\star \) follows from associativity of \(\circ \) and \(\bullet \). \(\square \)

## Theorem 5.2

The right \(U(\mathfrak {g})\)-module \(\mathcal {A}^\mathfrak {k}_q\) is a deformation of the \(U(\mathfrak {g})\)-module \(\mathbb {C}[G]^\mathfrak {k}\). The multiplication \(\star \) makes \(\mathcal {A}^\mathfrak {k}_q\) an associative \(U_q(\mathfrak {g})\)-algebra, a quantization of \(\mathbb {C}[\mathbb {S}^{2n}]\).

## Proof

We only need to make sure that \(\mathcal {A}^\mathfrak {k}_q\simeq \oplus _V\> {}^* V^\mathfrak {k}\otimes \!V\) is a deformation of \(\mathbb {C}[\mathbb {S}^{2n}]\simeq \mathbb {C}[G]^\mathfrak {k}\). It is done in Proposition 6.2 below. \(\square \)

Note that, though \(\mathcal {A}_q^\mathfrak {k}\) goes over to \(\mathbb {C}[G]^\mathfrak {k}\) at \(q=1\), the fact \(\mathcal {A}_q^\mathfrak {k}\simeq \mathbb {C}[G]^\mathfrak {k}\otimes \mathbb {C}(q)\) as a \(\mathbb {C}(q)\)-vector space needs a proof because \(\ker e_\delta \) and \(\ker f_\delta \) may decrease under deformation. That is done in the next section.

## 6 Quantum Euclidian plane

To complete the proof of Theorem 5.2, it is sufficient to check \(\dim V_q^\mathfrak {k}=\dim V^\mathfrak {k}\) for all finite-dimensional modules *V* that appear in \(\mathbb {C}[\mathbb {S}^{2n}]\). They all can be realized in the polynomial ring of the Euclidian plane \(\mathbb {C}^{2n+1}\) [20]. So we are going to look at its quantum version.

The representation on \(\mathbb {C}^N\) extends to an action \(\triangleright \) on \(\mathbb {C}_q[\mathbb {C}^N]\) making it a \(U_q(\mathfrak {g})\)-module algebra. Let \(\theta \) denote the involutive algebra and anti-coalgebra linear automorphism of \(U_q(\mathfrak {g})\) determined by the assignment \(e_{\alpha }\rightarrow - f_{\alpha }\), \(q^{h_\alpha }\rightarrow q^{-h_\alpha }\). Define also an anti-algebra linear involution on \(\mathbb {C}_q[\mathbb {C}^N]\) by \(\iota (x_i)=(x_{-i})\). They are compatible with the action \(\triangleright \) in the sense that \(\iota (u\triangleright x)= \theta (u)\triangleright \iota (x)\) for all \(u\in U_q(\mathfrak {g}), x\in \mathbb {C}_q[\mathbb {C}^N]\).

## Lemma 6.1

For all \(k\in \mathbb {Z}_+\) the monomials \(x_0^k\) are killed by \(e_{\delta }\) and \(f_\delta \).

## Proof

*m*and \(\mathfrak {P}_q^m\) the irreducible submodule of harmonic polynomials of degree

*m*. Then

## Proposition 6.2

For any finite-dimensional \(U_q(\mathfrak {g})\)-module \(V_q, \dim V_q^\mathfrak {k}\) is equal to \(\dim V^\mathfrak {k}\) of the classical \(\mathfrak {k}\)-invariants.

## Proof

It is sufficient to show that \(\dim (\mathfrak {P}^m_q)^\mathfrak {k}=\dim (\mathfrak {P}^m)^\mathfrak {k}\). In the classical limit, the trivial \(\mathfrak {k}\)-submodule in \(\mathfrak {H}^{m-2l}\) is multiplicity-free, so its dimension in \(\mathfrak {P}^m\) is \([\frac{m}{2}]+1\). On the other hand, the subspace of \(U_q(\mathfrak {l})\)-invariants is spanned by \(\{C^l_qx_0^{m-2l}\}_{l=0}^{[\frac{m}{2}]}\) and has the same dimension. Since all \(U_q(\mathfrak {l})\)-invariants are killed by \(e_{\delta }, f_\delta \) by Lemma 6.1, this proves the statement. \(\square \)

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