Explicit equivariant quantization on coadjoint orbits of

GL(n, ℂ)

Joseph Donin, Andrey Mudrov

Abstract

We present an explicit U_{h}(gl(n, ℂ))-equivariant quantization on coadjoint orbits of GL(n, ℂ). It forms a two-parameter family quantizing the Poisson pair of the reflection equation and Kirillov-Kostant-Souriau brackets.

1 Introduction

Let G be a reductive complex connected Lie group with Lie algebra 𝔤. Let U(𝔤) be the universal enveloping algebra of 𝔤 and U_{h}(𝔤) the corresponding quantum group, [Dr]. We consider the problem of U_{h}(𝔤)-equivariant quantization on 𝔤^{∗} and on coadjoint orbits of G.

By a U_{h}(𝔤)-equivariant quantization on a G-manifold M we mean a two-parameter family A_{t,h} of U_{h}(𝔤)-module algebras such that A_{0,0} = A(M), the function algebra on M, and the U_{h}(𝔤)-action on A_{t,h} is an extension of the U(𝔤)-action on A(M). Briefly, we refer to such a quantization as (t, h)-quantization while to its (0, h)- and (t, 0)-subfamilies as h- and tquantizations. This definition implies that the family A_{t,0} is G-equivariant.

At the infinitesimal level, a (t, h)-quantization gives rise to a pair of compatible Poisson brackets. The one along the parameter h is represented as a difference of bivector fields, f − r_{M}, [D2], where f is a G-invariant one and r_{M} is induced on M via the group action by the r-matrix r ∈ ∧^{2}𝔤 corresponding to U_{h}(𝔤). The Poisson bracket along the parameter t is G-invariant.

In the paper, as G-manifolds we consider 𝔤^{∗} and coadjoint orbits, as a G-invariant Poisson bracket along t we take the Poisson-Lie bracket on 𝔤^{∗} and its restriction to an orbit; the latter is known as the Kirillov-Kostant-Souriau (KKS) bracket. As was shown in [D1], a (t, h)- quantization on 𝔤^{∗} exists only for 𝔤 with simple components being sl(n, ℂ). So we assume 𝔤 = gl(n, ℂ) in the paper.

Firstly, we show that the quantization on 𝔤^{∗} is given by the so-called extended reflection equation (ERE) algebra, L_{t,h}. At (t, h) = (0, 0), it is the polynomial algebra on 𝔤^{∗} (or symmetric algebra of 𝔤). The ordinary or quadratic reflection equation (RE) algebra L_{h} was studied in [KSkl, KS]. The ERE algebra naturally appears in non-commutative differential calculus related to quantum groups, see e.g. [IP].

Secondly, using the generalized Verma module over U(𝔤), we show that the ERE algebra can be restricted to any semisimple orbit in 𝔤^{∗}. By the restriction we mean the following. Let {f} ⊂ A(𝔤^{∗}) be a set of functions generating the ideal of the orbit O. Then, there exist their extensions {f_{t,h}} ⊂ L_{t,h} generating an ideal in L_{t,h}; the quotient of L_{t,h} by that ideal gives a flat deformation over ℂ[[t, h]].

Thirdly, we give explicit formulas for the generators {f_{t,h}}.

Let us describe our approach in more detail. In the classical case, an orbit of rank k − 1 is specified by a matrix polynomial equation,

(X − μ_{1}) … (X − μ_{k}) = 0,  (1)

and the conditions on traces

Tr(X^{m}) = ∑_{i=1}^{k} n_{i}μ_{i}^{m}, m = 1, …, k − 1,

where n_{i} are non-negative integers such that ∑_{i=1}^{k} n_{i} = n. This orbit consists of matrices with eigenvalues μ_{i} of multiplicities n_{i}, i = 1, …, k. The ideal of the orbit is generated by the n × n entries of the matrix polynomial and k − 1 functions involving traces. Matrix polynomial equation (1) defines a G-invariant subvariety M_{μ}^{k}, which is a finite collection of rank < k orbits at generic μ, namely, when μ_{i} ≠ μ_{j} if i ≠ j. The condition on traces defines a character of the subalgebra S of invariants restricted to M_{μ}^{k}. This restriction is a finite dimensional semisimple algebra and it has a finite spectrum whose points correspond to orbits.

We extend this picture to the quantum case. It turns out that the quotient of the algebra L_{t,h} by the relations

(L − μ_{1}) … (L − μ_{k}) = 0,

where L is the matrix whose entries are generators of L_{t,h}, gives a (t, h)-quantization, A_{t,h}(M_{μ}^{k}), on M_{μ}^{k}. Moreover, the quotient of L_{t,h} by the ideal generated by entries of any polynomial in L is a flat algebra over ℂ[[t, h]]. The first part of the paper is devoted to the proof of this statement.

With the matrix polynomial condition imposed, the equations on traces have to be deformed in a consistent way. The quantum version of these equations involves the quantum trace Tr_{q}, where q = e^{h}, that is the quantum analog of the ordinary trace, [FRT]. It is defined in such a way that the elements Tr_{q}(L^{m}), m ∈ ℕ, are U_{h}(𝔤)-invariant. The first n traces Tr_{q}(L^{m}), m = 1, …, n, generate the center of the algebra L_{t,h}. The problem of restricting the quantization A_{t,h}(M_{μ}^{k}) to particular orbits in M_{μ}^{k} reduces to the problem of computing the values of quantum traces on them. The second part of the paper is devoted to solution of this problem.

Let us illustrate our results on the simplest example of symmetric (rank one) orbits. The quantized orbit of matrices with eigenvalues μ_{1}, μ_{2} of multiplicities n_{1}, n_{2} is the quotient of the ERE algebra L_{t,h} by the relations

(L − μ_{1})(L − μ_{2}) = 0,
Tr_{q}(L) = n̂_{1}μ_{1} + n̂_{2}μ_{2} + tn̂_{1}n̂_{2}.

In the paper, we define m̂ as \frac{1−q^{−2m}}{1−q^{−2}} for m ∈ ℤ and normalize the quantum trace so that Tr_{q}(1) = n̂. Note that these particular formulas for symmetric orbits were derived in our paper [DM2] by different methods.

As a limit case of our quantization, we come, putting h = 0, to an explicit quantization of the KKS bracket. The problem of quantizing this Poisson structure was put forward in [BFFLS]. It is interesting that the G-invariant quantization of the KKS bracket is obtained with the substantial use of quantum groups.

Among other works relevant to our study, we would like to mention [Ast, Kar], where the Karabegov quantization with separation of variables is applied to building the quantum moment map. That construction establishes relations between G-equivariant quantizations and generalized Verma modules. A construction of the G-equivariant quantization on semisimple orbits by the generalized Verma modules is presented in [DGS].

None of the mentioned methods gives explicit formulas for quantized orbits. In [DGK], explicit formulas was given for quantizing ℂP^{n} type orbits. In our papers [DM1, DM2], we developed a method of quantization on orbits of small ranks and built an explicit quantization on symmetric and bisymmetric orbits of gl(n, ℂ). In the present paper, we generalize those formulas to all semisimple orbits.

Let us remark that our quantized orbits are relevant to the so-called fuzzy spaces considered in the physics literature (see, for example, [DolJ, PawSt] and references therein).

The paper is organized as follows. Section 2 deals with the t-quantization of the KKS bracket. There, we prove that the natural G-equivariant quantization on 𝔤^{∗} can be restricted to semisimple orbits. In Section 3 we generalize that approach to the (t, h)-quantization. Therein, we prove that the quotient of L_{t,h} by any matrix polynomial equation is a flat deformation over ℂ[[t, h]]. In Section 4, we study the subalgebra of invariants in L_{t,h} restricted to M_{μ}^{k}. We prove that it is a finite dimensional semisimple commutative algebra and compute its characters. The explicit formulas for the equivariant quantization on orbits are given in Section 5.

Acknowledgment. We are grateful to J. Bernstein and S. Shnider for helpful discussions and valuable comments. The first author thanks Max-Planck-Institute für Mathematik for hospitality and very stimulating working atmosphere.

2 G-equivariant quantization on orbits

2.1 Quantization on 𝔤^{∗}

Let U(𝔤) be the universal enveloping algebra of a complex Lie algebra 𝔤 with the Lie bracket [ . , .]. Let G be a connected Lie group corresponding to 𝔤. We consider 𝔤[t] as a Lie algebra over ℂ[t] with respect to the bracket [x, y]_{t} = t[x, y]. Let L_{t} denote the universal enveloping algebra of 𝔤[t]. By definition, it is a quotient of the tensor algebra T(𝔤)[t] by the ideal generated by relations x ⊗ y − y ⊗ x − t[x, y], x, y ∈ 𝔤. The algebra L_{t} is a G-equivariant quantization of the symmetric algebra of 𝔤 (considered as the polynomial algebra on 𝔤^{∗}) with the Poisson-Lie bracket induced by [ . , .]. The assignment x ↦ tx, x ∈ 𝔤, defines a ℂ[t]-algebra morphism,

ϕ_{t}: L_{t} → U(𝔤)[t],  (2)

which is obviously an embedding of free algebras over ℂ[t].

2.2 Levi and parabolic subalgebras

Let 𝔤 be a complex reductive Lie algebra with the Cartan subalgebra 𝔥 and 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} its polarization with respect to 𝔥.

We fix a Levi subalgebra 𝔩, which is, by definition, the centralizer of an element in 𝔥. The algebra 𝔩 is reductive, so it is decomposed into the direct sum of its center and the semisimple part, 𝔩 = 𝔠 ⊕ [𝔩, 𝔩]. Also, there exists a decomposition

𝔤 = 𝔫_{𝔩}^{−} ⊕ 𝔩 ⊕ 𝔫_{𝔩}^{+},  (3)

where 𝔫_{𝔩}^{±} are subalgebras in 𝔫^{±}.

It is clear that 𝔠 ⊂ 𝔥 and 𝔥 = 𝔠 ⊕ 𝔥_{𝔩}, where 𝔥_{𝔩} = 𝔥 ∩ [𝔩, 𝔩]. So, we have a projection 𝔥 → 𝔠. On the other hand, the Cartan decomposition defines a projection

π: 𝔤 → 𝔥.  (4)

Composition of these maps gives the natural projection 𝔤 → 𝔠 which defines an embedding 𝔠^{∗} → 𝔤^{∗}.

Taking an element μ ∈ 𝔠^{∗}, we consider the coadjoint orbit of G in 𝔤^{∗} passing through μ. For generic μ, this orbit is semisimple with 𝔩 being the Lie algebra of the stabilizer at the point μ. Denote by O^{k}, k = dim 𝔠, the closure in 𝔤^{∗} × 𝔠^{∗} of this family of semisimple orbits and by O_{μ}^{k} the fiber over μ. It is known that the family O^{k} is flat over 𝔠^{∗}. For generic μ ∈ 𝔠^{∗} the variety O_{μ}^{k} is a semisimple orbit in 𝔤^{∗}.

Let 𝔭 denote the parabolic subalgebra 𝔩 ⊕ 𝔫_{𝔩}^{+}. Decomposition (3) turns into the decomposition

𝔤 = 𝔫_{𝔩}^{−} ⊕ 𝔭.  (5)

It is clear that characters (one-dimensional representations) of 𝔭 (as well as of 𝔩) are generated by linear forms from 𝔠^{∗}.

2.3 Generalized Verma modules

For any μ ∈ 𝔠^{∗}, projection (4) defines a representation π_{μ} of 𝔭 on ℂ: π_{μ}(x) = μ(π(x)), x ∈ 𝔭. It extends to a representation of U(𝔭), which we still denote by π_{μ}.

The generalized Verma U(𝔤)-module corresponding to μ ∈ 𝔠^{∗} is the left module V_{μ} = U(𝔤) ⊗_{U(𝔭)} ℂ. In the tensor product, ℂ is a U(𝔭)-module with respect to π_{μ}, while U(𝔤) is considered as a right U(𝔭)-module. The element v_{0} = 1⊗1 ∈ V_{μ} is the highest weight vector. In particular, xv_{0} = π_{μ}(x)v_{0} for x ∈ U(𝔭). Moreover, the map U(𝔫_{𝔩}^{−}) → V_{μ}, x ↦ xv_{0}, is an isomorphism of vector spaces. So, all V_{μ} are canonically isomorphic to V = U(𝔫_{𝔩}^{−}) as vector spaces. The representation V_{μ} is given by a homomorphism of algebras

φ_{μ}: U(𝔤) → End(V).  (6)

We treat V_{μ} as belonging to the family of Verma modules parameterized by 𝔠^{∗}. Namely, we consider trivial bundles over 𝔠^{∗} with the fibers U(𝔤) and End(V); then φ_{μ} is a map of their polynomial sections, which we denote by U(𝔤)[μ] and End(V)[μ]. One can think of μ as a collection of formal coordinates on 𝔠^{∗}, μ = (μ_{1}, ..., μ_{k}). Then, U(𝔤)[μ] and End(V)[μ] are polynomials in μ with values in U(𝔤) and End(V), respectively.

It is easy to check that the map φ_{μ/t} ∘ ϕ_{t}: L_{t} → End(V)[[t]], where ϕ_{t} is given by (2), defines a map of polynomial sections over 𝔠^{∗},

Φ_{t,μ}: L_{t}[μ] → End(V)[[t]][μ].  (7)

Theorem 2.1 ([Ast, DGS]). The image A_{t,μ} of L_{t,μ} in End(V)[[t]][μ] with respect to Φ_{t,μ} is a free module over ℂ[[t]][μ]. For all μ ∈ 𝔠^{∗}, the algebra A_{t,μ} specified to the fiber O_{μ}^{k} gives a G-equivariant quantization of the KKS bracket on it.

In particular, this theorem gives G-equivariant quantizations on all semisimple orbits.

3 U_{h}(𝔤)-equivariant quantization on orbits

3.1 Extended reflection equation algebra and (t, h) quantization on gl(n)^{∗}

We specialize our further considerations to the case 𝔤 = gl(n, ℂ). The G-module 𝔤^{∗} is identified with End(ℂ^{n}) using the trace pairing. The multiplication on matrix units {e_{j}^{i}}_{i,j=1}^{n} ⊂ End(ℂ^{n}) is given by e_{j}^{i}e_{k}^{l} = δ_{j}^{l}e_{k}^{i}, where δ_{j}^{l} is the Kronecker symbol. Let S ∈ End^{⊗2}(ℂ^{n}) be the Hecke symmetry related to the representation of U_{h}(𝔤) on ℂ^{n}. It satisfies the relations

S_{12}S_{23}S_{12} = S_{23}S_{12}S_{23},  (8)
S^{2} − (q − q^{−1})S = 1 ⊗ 1, q = e^{h}.  (9)

The symmetry S is expressed through the image R of the universal R-matrix of U_{h}(𝔤) and the flip operator P on ℂ^{n} ⊗ ℂ^{n}, S = P R. Let {L_{j}^{i}} ⊂ End^{∗}(ℂ^{n}) be the dual basis to {e_{i}^{j}}. Consider an associative algebra, L_{t,h}, over ℂ[[t, h]] generated by L_{j}^{i} subject to the quadratic-linear relations

SL_{2}SL_{2} − L_{2}SL_{2}S = qt(L_{2}S − SL_{2}).  (10)

Here, L is the matrix ∑_{i,j} L_{j}^{i}e_{i}^{j} and L_{2} = 1 ⊗ L. Note that at h = 0 these relations give the algebra L_{t} defined in Subsection 2.1.

Let R be the universal R-matrix of the quantum group U_{h}(𝔤). Consider the element Q̄ = qt/(q − q^{−1})(Q − 1 ⊗ 1), where Q = R_{21}R. It is easy to check that the map L_{t,h} → U_{h}(𝔤)[[t]], L_{j}^{i} ↦ L_{j}^{i}(Q̄_{1})Q̄_{2} respects relations (10). So we have the homomorphism

ϕ_{t,h}: L_{t,h} → U_{h}(𝔤)[[t]].  (11)

Proposition 3.1. The map ϕ_{t,h} is an embedding of free ℂ[[t, h]]-modules.

Proof. At the point h = 0, the quotient of the tensor algebra T(End^{∗}(ℂ^{n})) by relations (10) coincides with L_{t} and the map ϕ_{t,0} coincides with (2), so the proposition is a corollary of the following lemma.

Lemma 3.2. Let E and F be vector spaces over ℂ and λ a set of formal parameters. Let ψ_{λ}: E[[λ]] → F[[λ]], ψ_{λ} = ψ_{0} mod λ, be a map of free ℂ[[λ]]-modules and K_{λ} a submodule in E[[λ]] satisfying the conditions: a) K_{λ} ⊂ ker ψ_{λ}, b) K_{0} = ker ψ_{0}, where K_{0} = {a ∈ E|∃f = a + λb ∈ K_{λ}}. Then, the ℂ[[λ]]-module E[[λ]]/K_{λ} is free and the map E[[λ]]/K_{λ} → F[[λ]] is an embedding.

Proof. Clear.

Corollary 3.3. The algebra L_{t,h} is a U_{h}(𝔤)-equivariant quantization on gl^{∗}(n, ℂ).

3.2 Double quantization on orbits in gl^{∗}(n, ℂ)

Let 𝔤 = End(ℂ^{n}). As a Cartan subalgebra in 𝔤 we take diagonal matrices. As a Levi subalgebra 𝔩 we take the subspace of matrices commuting with a diagonal matrix with eigenvalues μ_{1}, ..., μ_{k}. Then 𝔠 consists of matrices commuting with 𝔩. Abusing notations, we will consider (μ_{1}, ..., μ_{k}) as coordinates in 𝔠^{∗} ≃ ℂ^{k}.

Since U_{h}(𝔤) ≃ U(𝔤)[[h]] as associative algebras, there is an extension,

φ_{h,μ}: U_{h}(𝔤) → End(V)[[h]],

of map (6). This algebra homomorphism is automatically equivariant with respect to the adjoint actions of the Hopf algebra U_{h}(𝔤). It is easy to check that the map φ_{h,μ/t}∘ϕ_{t,h}: L_{t,h} → End(V)[[t, h]], where ϕ_{t,h} is given by (11), defines a map of polynomial sections in μ,

Φ̃_{t,h,μ}: L_{t,h}[μ] → End(V)[[t, h]][μ].  (12)

Theorem 3.4. The image Ã_{t,h,μ} of L_{t,h}[μ] in End(V)[[t, h]][μ] with respect to Φ̃_{t,h,μ} is flat over ℂ[[t, h]][μ]. For a fixed μ ∈ 𝔠^{∗}, Ã_{t,h,μ} gives a (t, h)-quantization on O_{μ}^{k}.

Proof. The statement follows from Theorem 2.1 and Lemma 3.2.

In the classical case, elements of O_{μ}^{k} satisfy the matrix equation

(X − μ_{1})...(X − μ_{k}) = X^{k} + ξ_{k−1}(μ)X^{k−1} + ... + ξ_{0}(μ) = 0,  (13)

where coefficients ξ_{i} are symmetric functions of μ. Entries of the matrix X^{m}, m ∈ ℕ, generate a U(𝔤)-submodule in Ã_{0,0,μ} of End^{∗}(ℂ^{n})-type, which contains a one-dimensional trivial module and a submodule of sl(n, ℂ)-type. Moreover, invariants form a one dimensional submodule in Ã_{0,0,μ}, while the sl(n, ℂ)-isotypic component has multiplicity k − 1 and is generated by X̄^{1}, …, X̄^{k−1}, where X̄^{m} is the sl(n, ℂ)-type component in X^{m}.

Since Ã_{t,h,μ} is a U_{h}(𝔤)-equivariant ℂ[[t, h]][μ]-flat quotient of L_{t,h}[μ], we conclude that Ã_{t,h,μ} has the same decomposition into irreducible components as Ã_{0,0,μ}, the classical algebra of polynomial functions on O_{μ}^{k}, and (13) extends to a matrix polynomial equation,

L^{k} + ξ_{k−1}(t, h, μ)L^{k−1} + ... + ξ_{0}(t, h, μ) = 0.  (14)

Let us fix μ such that O_{μ}^{k} is a semisimple orbit. Let U ⊂ ℂ^{k} be a neighborhood of μ such that O_{μ′}^{k} is a semisimple orbit for μ′ ∈ U. Then, we can think of elementary symmetric functions ξ_{i}(μ′) of μ′ as local coordinates on U. Let A(U) denote the algebra of analytic functions on U. Coefficients of the polynomial in the left hand side of (14) define an algebra map ψ: A(U)[[t, h]] → A(U)[[t, h]], (ξ_{k−1}(μ′), ..., ξ_{0}(μ′), t, h) ↦ (ξ_{k−1}(t, h, μ′), ..., ξ_{0}(t, h, μ′), t, h). Since ψ is identical at (t, h) = (0, 0), it is an algebra automorphism. Therefore, ψ defines a transformation of parameters (t, h, μ′) in the neighborhood U of μ. These arguments lead to the following

Proposition 3.5. For every semisimple orbit O_{μ}^{k}, there exists a (t, h)-quantization, A_{t,h,μ}, on O_{μ}^{k} and an epimorphism

Φ_{t,h,μ}: L_{t,h} → A_{t,h,μ}  (15)

factored through the ideal J_{t,h,μ} ⊂ L_{t,h} generated by the relations

(L − μ_{1})...(L − μ_{k}) = L^{k} + ξ_{k−1}(μ)L^{k−1} + ... + ξ_{0}(μ) = 0.  (16)

Proof. Let ℑ̃_{t,h,μ′} be the kernel of the map Φ̃_{t,h,μ′} from (12), where μ′ are considered as local coordinates on the neighborhood U of the point μ. Let ℑ_{t,h,μ} = ℑ̃_{ψ^{−1}(t,h,μ)} and A_{t,h,μ} = L_{t,h}/ℑ_{t,h,μ}. Since ψ^{−1} transforms the coefficients of polynomial (14) to those of polynomial (16), the natural map L_{t,h} → A_{t,h,μ} satisfies the proposition.

Remark 3.6. Polynomial (16) is minimal for the semisimple orbit O_{μ}^{k}. Proposition 3.5 obviously holds for any polynomial divided by (16).

In the next subsection, we show that the quotient of the algebra L_{t,h} by the ideal generated by entries of any matrix polynomial is a flat deformation.

3.3 Flatness of matrix polynomial relations

Let a = (a_{m−1}, ..., a_{0}) be coordinates in ℂ^{m}. Let us consider the subvariety M^{m} in 𝔤^{∗} × ℂ^{m} generated by the matrix equation

X^{m} + a_{m−1}X^{m−1} + ... + a_{0} = 0.  (17)

It is clear that if the polynomial f_{a}(x) = x^{m} + a_{m−1}x^{m−1} + ... + a_{0} has no multiple roots, the fiber M_{a}^{m} is the union of semisimple orbits passing through diagonal matrices with eigenvalues from the set {λ_{i}}, i = 1, ..., m, of simple roots of f(x). It is easy to see that the variety M^{m} is flat over ℂ^{m}.

Theorem 3.7. Let J_{a} be the ideal in the ERE algebra L_{t,h} generated by entries of a matrix polynomial relation,

f_{a}(L) = L^{m} + a_{m−1}L^{m−1} + ... + a_{0} = 0.  (18)

Then the quotient algebra L_{t,h}/J_{a} is flat over ℂ[[t, h]][a] and for any a ∈ ℂ^{m} gives a (t, h)- quantization on M_{a}^{m}.

Proof. First, we suppose that f_{a}(x) has only simple roots (λ_{1}, …, λ_{m}). Let the fiber M_{a}^{k} be the union of semisimple orbits O_{ℓ}, ℓ = 1, ..., K. As pointed out in Remark 3.6, for every ℓ there exists an epimorphism Φ_{t,h,}^{ℓ}: L_{t,h}/J_{a} → A_{t,h}^{ℓ}, where A_{t,h}^{ℓ} is a (t, h)-quantization on O_{ℓ}. Consider the algebra homomorphism

⊕_{ℓ=1}^{K}Φ_{t,h}^{ℓ}: L_{t,h}/J_{a} → ⊕_{ℓ=1}^{K}A_{t,h}^{ℓ}.  (19)

This map is an isomorphism at the classical point (t, h) = (0, 0). Therefore, as follows from Lemma 3.2, it is an isomorphism in the quantum case, too. This proves flatness of L_{t,h}/J_{a} over neighborhoods of all a ∈ ℂ^{m} corresponding to polynomials with simple roots.

Let f_{a}(x) be an arbitrary polynomial. Then, it is a limit of polynomials with simple roots. Now, the flatness of L_{t,h}/J_{a} over ℂ[[t, h]][a] follows from the above part of the proof and from the fact that the quotient algebra L_{0,0}/J_{a} is flat over ℂ[a].

The orbits in M_{a}^{m} correspond to points of the spectrum of the subalgebra of invariants in the function algebra on M_{a}^{m}. In order to pass to the quantization on a particular orbit, we should study the subalgebra of invariants in L_{t,h} restricted to M_{a}^{m}. That is done in the next section.

4 Algebra of invariants on M^{k}

4.1 Center of the ERE algebra

The subalgebra S_{t,h} ⊂ L_{t,h} of U_{h}(gl(n, ℂ))-invariants coincides with the center of L_{t,h}. It is generated by elements s_{1}, …, s_{n}, where s_{m} = Tr_{q}(L^{m}), m ∈ ℕ. The quantum trace Tr_{q} of a matrix A is defined as, [FRT],

Tr_{q}(A) = Tr(DA), q = e^{h},  (20)

with the weight matrix D normalized by the condition (id ⊗ Tr_{q})(S) = q 1. In this normalization, Tr_{q}(1) = n̂. We use the notation m̂ for quantum integers, m̂ = \frac{1−q^{−2m}}{1−q^{−2}} , m ∈ ℤ.

The algebra S_{t,h} is isomorphic to the polynomial algebra ℂ[[h]][σ_{1}, …, σ_{n}], where the elements σ_{i} are coefficients of the characteristic polynomial in the quantum Cayley-Hamilton identity

L^{n} − σ_{1}L^{n−1} + … + (−1)^{n}σ_{n} = 0  (21)

which follows from the same representation theory arguments as in the proof of (14).

We substantially rely on relations between generators {σ_{i}} and {s_{i}} calculated by Pyatov and Saponov, [PS], for the algebra S_{h} = S_{0,h}. In our normalization of the quantum trace, their result is formulated as follows.

Theorem 4.1 ([PS]). Elements {s_{i}}_{i=1}^{∞} and {σ_{i}}_{i=1}^{∞}, where σ_{i} = 0 for i > n, of the algebra S_{h} satisfy the quantum Newton identities

m̂σ_{m} − s_{1}σ_{m−1} + … + (−1)^{m}s_{m} = 0, m = 1, 2, ….  (22)

Given a sequence x⃗ = {x_{m}}_{m=0}^{∞} with values in a ℂ[μ]-module, where μ = (μ_{1}, ..., μ_{k}) ∈ ℂ^{k}, k ≤ n, let us define the sequence {r_{m}(x⃗)}_{m=k}^{∞},

r_{m}(x⃗) = x_{m} − σ_{1}(μ)x_{m−1} + … + (−1)^{k}σ_{k}(μ)x_{m−k},  (23)

where σ_{i}(μ) are the elementary symmetric functions in μ, σ_{i}(μ) = ∑_{1≤j_{1}<...<j_{i}≤k} μ_{j_{1}} … μ_{j_{i}}. The following proposition describes the intersection of S_{t,h} with the ideal J_{t,h,μ} ⊂ L_{t,h} generated by relations (16).

Proposition 4.2. Let s⃗ = {s_{m}}_{m=0}^{∞} be the sequence of q-traces, s_{m} = Tr_{q}(L^{m}). Then, the set {r_{m}(s⃗)}_{m=k}^{∞} generates the ideal J_{t,h,μ}^{S} = S_{t,h} ∩ J_{t,h,μ} in S_{t,h}.

Proof. Denote by p the polynomial on the left-hand side of (16) whose entries p_{j}^{i} generate the ideal J_{t,h,μ}. Clearly r_{m}(s⃗) = Tr_{q}(L^{m−k}p), so the elements r_{m}(s⃗) belong to J_{t,h,μ}^{S} for all m ≥ k. Let us prove that they generate J_{t,h,μ}^{S}. Suppose y ∈ J_{t,h,μ}; then it is representable as y = ∑_{i,j=1}^{n} y_{j}^{i}p_{i}^{j}, where {y_{j}^{i}} ⊂ L_{t,h}. If y ∈ S_{t,h}, the elements y_{j}^{i} generate a module of End(ℂ^{n})-type. But the End(ℂ^{n})-type component in L_{t,h} is an S_{t,h}-module spanned by entries of the matrix powers L^{l}, l = 0, …, n − 1. Therefore y_{j}^{i} are entries of a polynomial ∑_{l=0}^{n−1} y_{l} L^{l} with coefficients y_{l} ∈ S_{t,h}, so y = ∑_{l=0}^{n−1} y_{l} r_{l+k}(s⃗).

4.2 Special polynomials

Characters of the algebra S_{t,h} restricted to M^{k} are described by means of special polynomials, which are introduced in this subsection.

Let us define the set {n : k} ⊂ ℤ^{k} of k-tuples n = (n_{1}, …, n_{k}) such that 0 ≤ n_{i} and n_{1} + … + n_{k} = n; the subset {n:k}_{+} ⊂ {n:k} consists of n with all n_{i} > 0. We denote by |n:k| the number of elements in {n:k}.

Given n ∈ {n:k} let us introduce polynomials in μ of degree m = 0, 1, … setting

ϑ_{m}(n, q^{−2}, μ) = ∑_{ℓ=1}^{k}(1 − q^{−2})^{ℓ−1} ∑_{d∈{m:ℓ}_{+}} ∑_{1≤i_{1}<...<i_{ℓ}≤k} n̂_{i_{1}} … n̂_{i_{ℓ}} μ_{i_{1}}^{d_{1}} … μ_{i_{ℓ}}^{d_{ℓ}}  (24)

for m > 0 and ϑ_{0}(n, q^{−2}, μ) = n̂. At the classical point q = 1, the polynomial ϑ_{m}(n, q^{−2}, μ) is equal to ∑_{i=1}^{k} n_{i}μ_{i}^{m}, i.e., the trace Tr(A^{m}) of a matrix A with eigenvalues (μ_{1}, …, μ_{k}) of multiplicities (n_{1}, …, n_{k}).

The following proposition will be important for our consideration.

Proposition 4.3. The polynomials ϑ_{m}(n, q^{−2}, μ), m ≥ 0, can be represented as

ϑ_{m}(n, q^{−2}, μ) = ∑_{j=1}^{k} C_{j}(n, q^{−2}, μ)μ_{j}^{m},  (25)

where C_{j}(n, q^{−2}, μ) = n̂_{j} + n̂_{j} ∑_{ℓ=1}^{k−1}(1 − q^{−2})^{ℓ} ∑_{{1≤i_{1}<...<i_{ℓ}≤k \atop i_{1},...,i_{ℓ}≠j}} \frac{n̂_{i_{1}}μ_{i_{1}}}{μ_{j} − μ_{i_{1}}} … \frac{n̂_{i_{ℓ}}μ_{i_{ℓ}}}{μ_{j} − μ_{i_{ℓ}}} . (26)

Proof. It follows from definition (24) that the polynomials ϑ_{m}(n, q^{−2}, μ) satisfy the identity

ϑ_{m}(n, q^{−2}, μ) = ϑ_{m}(n′, q^{−2}, μ′) + n̂_{k} ∑_{i=1}^{m−1} ϑ_{m−i}(n′, q^{−2}, μ′)μ_{k}^{i} + n̂_{k}μ_{k}^{m},  (27)

where μ′ = (μ_{1}, …, μ_{k−1}) and n′ = (n_{1}, …, n_{k−1}). Equation (27) allows to apply induction on k.

Using representation (25), let us introduce the functions

ϑ_{m}(n, q^{−2}, μ, t) = ∑_{j=1}^{k} C_{j}(n, q^{−2}, μ + \frac{t}{1 − q^{−2}} )μ_{j}^{m},

where we put μ+ \frac{t}{1−q^{−2}} = (μ_{1}+ \frac{t}{1−q^{−2}} , …, μ_{k} + \frac{t}{1−q^{−2}} ). Although the coefficients C_{j}(n, q^{−2}, μ) are rational, the functions ϑ_{m}(n, q^{−2}, μ, t) are, in fact, polynomials in t, q^{−2}, and μ. Evidently, ϑ_{m}(n, q^{−2}, μ) = ϑ_{m}(n, q^{−2}, μ, t)|_{t=0}.

4.3 Characters of the subalgebra of invariants in L_{0,h}

Let us reserve the notation λ = μ for the case k = n and consider polynomials ϑ_{m}(q^{−2}, λ); by definition,

ϑ_{m}(q^{−2}, λ) = ϑ_{m}(n, q^{−2}, λ)|_{n=(1,...,1)} = ∑_{ℓ=1}^{n}(1 − q^{−2})^{ℓ−1} ∑_{d∈{m:ℓ}_{+}} ∑_{1≤j_{1}<...<j_{ℓ}≤n} λ_{j_{1}}^{d_{1}} … λ_{j_{ℓ}}^{d_{ℓ}}. (28)

Lemma 4.4. Substitution λ_{i} = q^{−2(i−1)}, i = 1, …, n, to ϑ_{m}(q^{−2}, λ) returns n̂.

Proof. We apply induction on n. For n = 1 we obviously have ϑ_{m}(q^{−2}, λ) = 1. Suppose the theorem holds for n = l. Using the induction assumption and identity (27) for the case n = (1, …, 1) we find, for n = l + 1,

ϑ_{m}(q^{−2}, λ)|_{λ=(1,q^{−2},...,q^{−2l})} = l̂+ (q^{−2l})^{m} + (1 − q^{−2})l̂∑_{d=1}^{m−1}(q^{−2l})^{d}
= l̂+ (q^{−2l})^{m} + (1 − q^{−2})l̂(q^{−2l})\frac{1 − (q^{−2l})^{m−1}}{1 − q^{−2l}} .

After elementary transformations the last expression is brought to { \atop l + 1}.

Given n = (n_{1}, …, n_{k}) ∈ {n:k} let us consider non-intersecting intervals I_{i} ⊂ {1, …, n}, i = 1, …, k, some of them may be empty, satisfying the following requirements: 1) if a ∈ I_{i} and b ∈ I_{j}, then a < b whenever i < j, 2) #I_{i} = n_{i}. Clearly, I_{1} ∪ … ∪ I_{k} = {1, …, n}. Let us introduce double indexing λ_{j,i}, j = 1, …, k, i = 1, …, n_{j}, of the indeterminates λ_{1}, …, λ_{n}: if m ∈ I_{j} and i is the relative position of m within the interval I_{j}, then we identify λ_{j,i} = λ_{m}. Double indices are ordered by the lexicographic ordering, which coincides with that induced from ℕ. We shall consider the vectors λ̄_{j} = (λ_{j,1}, …, λ_{j,n_{j}}) and polynomials ϑ_{m}(q^{−2}, λ̄_{j}), j = 1, …, k.

Proposition 4.5. Substitution λ_{j,i} = μ_{j}q^{−2(i−1)} to ϑ_{m}(q^{−2}, λ), m ∈ ℕ, gives ϑ_{m}(n, q^{−2}, μ).

Proof. The case k = 1 follows from Lemma 4.4, because the polynomials ϑ_{m}(q^{−2}, λ), m ∈ ℕ, are homogeneous in λ of degree m. The general situation is reduced to the case k = 1 if one observes that, upon rearranging summation in (28), the polynomial ϑ_{m}(q, λ) may be rewritten as

ϑ_{m}(q^{−2}, λ) = ∑_{ℓ=1}^{k}(1 − q^{−2})^{ℓ−1} ∑_{d∈{m:ℓ}_{+}} ∑_{1≤j_{1}<...<j_{ℓ}≤k} ϑ_{d_{1}}(q^{−2}, λ̄_{j_{1}}) … ϑ_{d_{ℓ}}(q^{−2}, λ̄_{j_{ℓ}}), (29)

Now we apply Lemma 4.4 to each factor ϑ_{d_{1}}(q^{−2}, λ̄_{j_{1}}), …, ϑ_{d_{ℓ}}(q^{−2}, λ̄_{j_{ℓ}}) on the right-hand side of (29).

Recall that the subalgebra of invariants S_{h} = S_{0,h} of the quadratic RE algebra L_{h} = L_{0,h} is a polynomial algebra in n variables σ_{i}, i = 1 …, n, which are the coefficients of the Cayley-Hamilton identity (21). Let us assign to λ ∈ ℂ^{n}[[h]] a character χ^{λ}: S_{h} → ℂ[[h]] setting on the generators χ^{λ}(σ_{m}) = ∑_{i_{1}<...<i_{m}} λ_{i_{1}} … λ_{i_{m}} = σ_{m}(λ), the elementary symmetric polynomial of degree m ∈ {1, …, n}. The correspondence λ → χ^{λ} defines an epimorphism from ℂ^{n}[[h]] to ch S_{h}, the set of characters of S_{h}.

Proposition 4.6. One has χ^{λ}(s_{m}) = ϑ_{m}(q^{−2}, λ), where s_{m} = Tr_{q}(L^{m}), m ∈ ℕ.

Proof. Assuming σ_{m} = 0 and σ_{m}(λ) = 0 for m > n we must check that the substitution σ_{m} → σ_{m}(λ), s_{m} → ϑ_{m}(q^{−2}, λ) satisfies Newton identities (22). Since representation (25) is valid for ϑ_{m}(q^{−2}, λ) = ϑ_{m}(n, q^{−2}, λ)|_{n=(1,...,1)}, it is enough to check identities (22) for m ≤ n only. Let i and j be non-negative integer numbers and integers d_{1}, …, d_{j} > 1 be such that i + d_{1} + … + d_{j} = m ≤ n. Consider the coefficient before the monomial p = p′p′′ in equality (22), where p′ = λ_{1} … λ_{i} and p′′ = λ_{i+1}^{d_{1}} … λ_{i+j}^{d_{j}}. Let us show that this coefficient is equal to zero. This will prove the statement because all the polynomials involved are symmetric in λ. Let us compute contributions coming, say, from the term (−1)^{l}σ_{l}(λ)s_{m−l}(λ), where l ≥ 0 (we assume σ_{0} = 1). We should consider all the monomials \frac{p}{r}, where r = λ_{α_{1}} … λ_{α_{l}}, α_{1} < … < α_{l} ≤ i + j, enters σ_{l}(λ) and take the coefficients before \frac{p}{r} entering s_{m−l}(λ), with the sign (−1)^{l}. We can represent r as the product r = r′r′′, where r′ contains only λ_{l}, 1 ≤ l ≤ i, while r′′ involves only λ_{l}, i < l ≤ i + j; so r′ and r′′ divide p′ and p′′, respectively. Observe that the coefficients before \frac{p}{r} in s_{m}(λ) do not depend on r′′ and are equal to (1 − q^{−2})^{i+j−deg(r′)−1}.

Consider separately two cases: p′′ = 1 and p′′ ≠ 1. If p′′ = 1, then the coefficient before the term λ_{1} … λ_{m} entering (22) is

m̂ + ∑_{l=1}^{m}(−1)^{l}\binom{l}{m} (1 − q^{−2})^{l−1} = m̂ + \frac{(1 − (1 − q^{−2}))^{m} − 1}{(1 − q^{−2})} = 0.  (30)

If p′′ ≠ 1, the first term in (22) gives no contribution to the coefficient, which we find to be

∑_{r′}(1 − q^{−2})^{i+j−deg(r′)−1} ∑_{r′′}(−1)^{deg(r)} = ∑_{r′}(1 − q^{−2})^{i+j−1} ∑_{l=0}^{j}(−1)^{l}\binom{l}{j} = 0.

4.4 Algebra of invariants on M^{k} and its restriction to orbits

In this section, we study the image of the subalgebra of invariants S_{t,h} ⊂ L_{t,h} in the quotient of L_{t,h} by the ideal J_{t,h,μ} generated by relations (16). To analyze its structure and compute its characters, we use polynomials ϑ_{m}(n, q^{−2}, μ) and ϑ_{m}(n, q^{−2}, μ, t) introduced in Subsection 4.2.

Theorem 4.7. The quotient of the algebra S_{t,h} by the ideal J_{t,h,μ}^{S} = S_{t,h} ∩ J_{t,h,μ} is generated by the images of the elements {s_{1}, …, s_{k−1}} in S_{t,h}/J_{t,h,μ}^{S}. For any n ∈ {n:k}, the formula

χ_{t,h}(n)(s_{m}) = ϑ_{m}(n, q^{−2}, μ, t), m ∈ ℕ,  (31)

defines a character χ_{t,h}(n) of the algebra S_{t,h}/J_{t,h,μ}^{S}. The correspondence n → χ_{t,h}(n) is a bijection between {n: k} and the set of characters ch S_{t,h}/J_{t,h,μ}^{S}. The algebra S_{t,h}/J_{t,h,μ}^{S} is a free ℂ[[t, h]]-module of rank |n:k|. At generic μ, it is a direct sum of |n:k| copies of ℂ[[t, h]].

Proof. First of all, the theorem holds true in the classical case (t, h) = (0, 0). Clearly the images of the elements {s_{1}, …, s_{k−1}} generate S_{t,h}/J_{t,h,μ}^{S} because other traces are related to them via recurrent relations, according to Proposition 4.2.

Let us describe characters of the algebra S_{t,h}/J_{t,h,μ}^{S}. Suppose first t = 0. An element χ from chS_{0,h}/J_{0,h,μ}^{S} is also a character for S_{0,h}, so we can assume χ = χ^{λ}, for some λ ∈ ℂ^{n}[[h]] depending on μ. We have χ(s_{m}) = ϑ_{m}(q^{−2}, λ) according to Proposition 4.6. For χ^{λ} to define a character of the quotient algebra S_{0,h}/J_{0,h,μ}^{S}, the sequence χ(s⃗) ⊂ ℂ[[h]], must satisfy the recurrent identities r_{m}(χ(s⃗)) = 0, m ≥ k, by Proposition 4.2. By Proposition 4.5, for any n ∈ {n: k} the polynomials ϑ_{m}(n, q^{−2}, μ) are obtained by a substitution to ϑ_{m}(q^{−2}, λ). The sequence a⃗ = {ϑ_{m}(n, q^{−2}, μ)}_{m=0}^{∞} satisfies the recurrent relations r_{m}(a⃗) = 0, m ≥ k, as follows from representation (25). Therefore, for any n ∈ {n: k} there exists an element χ_{h}(n) ∈ ch S_{0,h}/J_{0,h,μ}^{S} such that χ_{h}(n)(s_{m}) = ϑ_{m}(n, q^{−2}, μ), for m ≥ 0.

As follows from their definitions, the ERE and RE algebras are defined over ℂ[q, q^{−1}], q = e^{h}, and all the ℂ[[h]]-algebras under consideration are, in fact, completions of the corresponding ℂ[q, q^{−1}]-algebras at the point q = 1. Since the functions ϑ_{m}(n, q^{−2}, μ) are polynomials in q^{−2}, the character χ_{h}(n) is the extension of a character χ_{q}(n) ∈ chS_{0,q}/J_{0,q,μ}^{S}.

Let us prove formula (31) for the case t ≠ 0. Hecke condition (9) and relations (10) imply that the shift L ↦ L − \frac{t}{1−q^{−2}} , μ ↦ μ − \frac{t}{1−q^{−2}} induces a homomorphism ϱ_{t,q}: L_{t,q} → L_{0,q}[t] (here we assume that the ring ℂ[q, q^{−1}] is extended to the field ℂ(q)). Clearly ϱ_{t,q}(J_{t,q,μ}) coincides with J_{0,q,μ} because equation (16) is invariant under this transformation. Therefore, the map ϱ_{t,q} defines a homomorphism ϱ̄_{t,q}: S_{t,q}/J_{t,q,μ}^{S} → S_{0,q}/J_{0,q,μ}^{S}. Denote by χ_{t,q}(n) the composition χ_{q}(n) ∘ ϱ̄_{t,q}, which is a character of S_{t,q}/J_{t,q,μ}^{S}. It is easy to prove by induction on m, using representation (25), that χ_{q,t}(n) evaluated on s_{m} gives ϑ_{m}(n, q^{−2}, μ, t). Since the functions ϑ_{m}(n, q^{−2}, μ, t) are polynomials in q^{−2}, we can extend χ_{t,q}(n) to a character χ_{t,h}(n) of S_{t,h}/J_{t,h,μ}^{S} over ℂ[[h]].

Thus we constructed a deformation χ_{t,h}(n) for each classical character χ(n). We have the inequality

|n:k| = #ch S/J_{μ}^{S} ≤ #ch S_{t,h}/J_{t,h,μ}^{S} ≤ rk S_{t,h}/J_{t,h,μ}^{S} ≤ N ≤ rk S/J_{μ}^{S} = |n:k|,

where S = S_{0,0} and J_{μ} = J_{0,0,μ}^{S} are the classical algebras and N the number of generators of the ℂ[[t, h]]-module S_{t,h}/J_{t,h,μ}^{S}. This inequality shows that S_{t,h}/J_{t,h,μ}^{S} is a free module over ℂ[[t, h]] of rank |n:k|.

Let us fix μ such that μ_{i} ≠ μ_{j} for i ≠ j. Since at the classical point (t, h) = (0, 0) the algebra map

⊕_{n∈{n:k}}χ_{t,h}(n): S_{t,h}/J_{t,h,μ}^{S} → ⊕_{n∈{n:k}}ℂ[[t, h]]

is an isomorphism, it is an isomorphism of ℂ[[t, h]]-algebras.

5 Quantum orbits

In this section, we present explicit formulas of the U_{h}(gl(n, ℂ))-equivariant quantization on orbits using characters of the algebra S_{t,h}/J_{t,h,μ}^{S} calculated in the previous section.

Theorem 5.1. For any element n = (n_{1}, …, n_{k}) ∈ {n:k}, the quotient of the ERE algebra (10) by the relations

(L − μ_{1}) … (L − μ_{k}) = 0,  (32)
Tr_{q}(L^{m}) = ϑ_{m}(n, q^{−2}, μ, t), m = 1, …, k − 1,  (33)

is a flat ℂ[[t, h]]-algebra. When μ_{i} ≠ μ_{j} for i ≠ j, it is a quantization on the orbit of semisimple matrices with eigenvalues (μ_{1}, … μ_{k}) of multiplicities (n_{1}, …, n_{k}).

Proof. By Theorem 3.7, the quotient algebra L_{t,h}/J_{t,h,μ} defined by relations (32) is a quantization on M^{k}. The algebra of invariants on a semisimple orbit is one-dimensional thus corresponding to a character of S_{t,h}/J_{t,h,μ}^{S}. At the classical point, characters are χ(n)(s_{m}) = ∑_{i=1}^{k} n_{i}μ_{i}^{m}, for n ∈ {n:k}. Their deformations are given by formula (31). So relations (33) should hold on the orbit of semisimple matrices with eigenvalues (μ_{1}, … μ_{k}) of multiplicities (n_{1}, …, n_{k}). Because relations (32) and (33) define this orbit at (t, h) = (0, 0), they define a (t, h)-quantization on it.

Since functions ϑ_{m}(n, q^{−2}, μ, t) are polynomials in q^{−2}, t, and μ, we obtain the following

Corollary 5.2. At q = 1, formulas (32) and (33) give G-equivariant quantizations of the Kostant-Kirillov-Souriau brackets on semisimple orbits.

Remark 5.3. One can consider the ERE algebra associated with arbitrary Hecke symmetry S and define ”quantum orbits” by formulas (32) and (33). Equations (33) are forced by condition (32) and have the same form, since Newton identities (22) hold for any Hecke symmetry, [IOP]. The question is about the module structure (supply of ”functions”) of the orbits defined in this way. In our next paper we will prove that the supply of functions on an ”orbit” for any even Hecke symmetry is rich enough.

References

[Ast] A. Astashkevich, On Karabegov’s quantizations of semisimple coadjoint orbits, Adv. Geom., 172 (1999) 1–18.

[BFFLS] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation Theory and Quantization, Ann. Physics, 111 (1978), 61–110.

[DolJ] B. P. Dolan, O. Jahn, Fuzzy Complex Grassmannian Spaces and their Star Products, hepth/0111020.

[D1] J. Donin, Double quantization on coadjoint representations of simple Lie groups and its orbits, math.QA/9909160.

[D2] J. Donin, U_{h}(𝔤)-invariant quantization of coadjoint orbits and vector bundles over them. J. Geom. Phys., 38 #1 (2001) 54–80.

[D3] J. Donin, Double quantization on the coadjoint representation of sl(n), Czech J. of Physics, 47, n.11, 1997, 1115–1122.

[Dr] V. G. Drinfeld, Quantum Groups, in Proc. Int. Congress of Mathematicians, Berkeley, 1986, ed. A. V. Gleason, AMS, Providence (1987) 798–820.

[DM1] J. Donin, A. Mudrov, U_{q}(sl(n))-invariant quantization of symmetric coadjoint orbits via reflection equation algebra, Contemp. Math., in press, math.QA/0108112.

[DM2] J. Donin, A. Mudrov, Method of quantum characters in equivariant quantization, math.QA/0204298.

[DGK] J. Donin, D. Gurevich, S. Khoroshkin, Double quantization of ℂP^{n} type by generalized Verma modules , J. Geom. Phys., 28 (1998) 384–406.

[DGS] J. Donin, D. Gurevich, S. Shnider, Quantization of function algebras on semisimple orbits in 𝔤^{∗}, q-alg/9607008.

[FRT] L. Faddeev, N. Reshetikhin, and L. Takhtajan, Quantization of Lie groups and Lie algebras Leningrad Math. J. 1 (1990) 193–226.

[FL] R. Fioresi, M. A. Lledó, On the deformation quantization of coadjoint orbits of semisimple groups, Pacific J. Math., 198 #2 (2001) 411–436.

[IOP] A. Isaev, O. Ogievetsky, P. Pyatov, On quantum matrix algebras satisfying te Cayley-Hamilton-Newton identities, J. Phys. A, 32 # (1999) L115–L121.

[IP] A. Isaev, P. Pyatov, Covariant differential complexes on quantum linear groups, J. Phys. A, 28 (1995) 2227–2246.

[Kar] A. Karabegov, Pseudo-Kähler quantization on flag manifolds, Commun. Math. Phys., 200 #2 (1999) 355–379.

[KS] P. P. Kulish, R. Sasaki, Covariance properties of reflection equation algebras, Prog. Theor. Phys. 89 #3 (1993) 741–761.

[KSkl] P. P. Kulish, E. K. Sklyanin, Algebraic structure related to the reflection equation, J. Phys. A 25 (1992) 5963–2389.

[L] M. A. Lledó, Deformation quantization of nonregular orbits of compact Lie groups, Lett. Math. Phys. 58 #1 (2001) 57–67.

[PS] P. N. Pyatov, P. A. Saponov, Characteristic relations for quantum matrices, J. Phys. A, 28 (1995) 4415–4421.

[PawSt] J. Pawelczyk, H. Steinacker, A quantum algebraic description of D-branes on group manifolds, hep-th/0203110.