U_{q}(sl(n))-invariant quantization of symmetric coadjoint orbits via reflection equation algebra

J. Donin

Max-Planck-Istitute für Mathematik, Vivatsgasse 7, 53111 Bonn, Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel;

e-mail: donin@mpim-bonn.mpg.de, donin@macs.biu.ac.il

A. Mudrov

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel; e-mail: mudrova@macs.biu.ac.il

Abstract

We study relations between the two-parameter U_{q}(sl(n))-invariant deformation quantization on sl^{∗}(n) and the reflection equation algebra. The latter is described by a quantum permutation on End(ℂ^{n}) given explicitly. The reflection equation algebra is used for constructing the oneparameter quantization on coadjoint orbits, including symmetric and certain bisymmetric and nilpotent ones. Our approach is based on embedding the quantized function algebras on orbits into the algebra of functions on the quantum group SL_{q}(n) via reflection equation algebra characters.

1 Introduction

This paper is devoted to a particular case of the following problem. Let M be a manifold with a Poisson structure invariant under an action of a Lie group G. Let 𝔤 be the Lie algebra of the group G and U_{h}(𝔤) the corresponding quantized universal enveloping algebra, [3]. The problem is to construct a quantization A_{h} of the function algebra A on M invariant under the action of U_{h}(𝔤). In this paper, we consider the case 𝔤 = sl(n) or gl(n) and M is either 𝔤^{∗} or an orbit in 𝔤^{∗}. We quantize the algebra of polynomial functions on M and present it as a quotient of the corresponding tensor algebra by very natural relations, which are similar to those in the classical case. We prove the flatness of the deformations obtained. In this way, we quantize all quantum symmetric orbits and some bisymmetric ones, including the twistor space. Note that the analogous relations can be written for all semisimple orbits but there arises the question whether the quotient algebras by those relations are flat.

Let us recall some facts about the quantization on 𝔤^{∗} and its orbits. It was shown in [4] that in the case 𝔤 = sl(n) there exists a two-parameter deformation of the polynomial algebra S(𝔤) on 𝔤^{∗} which can be viewed as a U_{h}(𝔤)-invariant quantization of the Lie-Poisson bracket on 𝔤^{∗}. Recall that in the classical case a natural one-parameter U(𝔤)-invariant quantization of S(𝔤) is given by the family S_{t}(𝔤) = T(𝔤)[t]/J_{t}, where T(𝔤) is the tensor algebra of the vector space 𝔤 and the ideal J_{t} is generated by the elements x ⊗ y − τ(x ⊗ y) − t[x, y], x, y ∈ 𝔤. Here, τ is the flip operator on 𝔤^{⊗2}. So, S_{t}(𝔤) is a quadratic-linear algebra. By the Poincaré-Birkhoff-Witt theorem, S_{t}(𝔤) is a free module over ℂ[t]. As was proven in [4], [5], that picture can be extended to the quantum case for 𝔤 = sl(n). Namely, there exist deformations τ_{h} and [ ·, · ]_{h} of both maps τ and [ ·, ·] such that the ideal J_{h,t} generated by the elements x ⊗ y − τ_{h}(x ⊗ y) − t[x, y]_{h}, x, y ∈ 𝔤, gives the two-parameter U_{h}(𝔤)-invariant quantization S_{h,t}(𝔤) = T(𝔤)[[h]][t]/J_{h,t} of the Lie-Poisson bracket on 𝔤^{∗}. It was also shown that such a family does not exist for 𝔤 ≠ sl(n).

It is possible to prove that S_{h,t}(g) can be restricted to any semisimple orbit to give a two-parameter U_{h}(𝔤)-invariant quantization on it. On the other hand, one can obtain a oneparameter invariant quantization on a semisimple orbit M as the subalgebra of quantized functions on G invariant under the action of the quantized stabilizer, [6]. Note that these statements do not give any explicit form of the quantized algebras. In this paper, we explicitly describe S_{h,t}(𝔤) and the quantum symmetric orbits in 𝔤^{∗}. We also describe quantizations of certain nilpotent and bisymmetric orbits. We conjecture that all bisymmetric orbits can be quantized within our approach.

The paper is organized as follows. Sections 2 and 3 contain some basic material about the quantum group U_{q}(𝔤), 𝔤 = sl(n), and its defining representation in the vector space V = ℂ^{n}. In this paper, we work with the quantum group U_{q}(𝔤) in the sense of Lusztig, [17], instead of U_{h}(𝔤). The latter may be considered as the completion of U_{q}(𝔤) at the point q = 1. Correspondingly, instead of S_{h,t}(𝔤) we deal with the two-parameter family S_{q,t}(𝔤) as a free module over ℂ(q). In Section 4, we study two different algebra structures on T(V) ⊗ T(V^{∗}) invariant with respect to the actions of either U_{q}^{⊗2}(𝔤) or U_{q}(𝔤). As a result, we come to two inequivalent embeddings of T(V ⊗ V^{∗}) into T(V) ⊗ T(V^{∗}). That, in its turn, leads to two different quantizations of the polynomial algebra on V ⊗ V^{∗} studied in Sections 5 and 6. We prove that they are the reflection equation (RE) algebra (see [14]) and the algebra of functions on the quantum group known as the FRT algebra (after Faddeev, Reshetikhin, and Takhtajan), [8]. It turns out that both algebras can be described uniformly by quantum permutations on (V ⊗ V^{∗})^{⊗2} but within different categories. In Section 7 we evaluate the lowest weight vectors of the irreducible U_{q}(𝔤)- modules in M^{⊗2}, where M is the algebra of n × n complex matrices. Using this information, we derive the involutive permutation τ_{RE} giving the commutation relations in the reflection equation algebra. That is done in Section 8. In Section 9, we reduce this permutation to the submodule 𝔤^{⊗2} ⊂ M^{⊗2}, 𝔤 = sl(n).

In Section 7, we formulate a relation between S_{q,t}(𝔤) and the RE algebra, following [5]. That relation involves the reduction of τ_{RE} from M to 𝔤, and a quantum deformation of the Lie commutator 𝔤^{⊗2} → 𝔤 determined by τ_{RE} up to a factor. Specifying values of the quantum trace in the RE algebra, we obtain one-parameter subfamilies of S_{q,t}(𝔤) as quotients of the RE algebra.

Section 11 is devoted to the quantization on coadjoint orbits in 𝔤^{∗}. We work, actually, with the adjoint orbits in the matrix space using the isomorphism between adjoint and coadjoint modules and employing the relation between the RE algebra and the two-parameter family S_{q,t}(𝔤).

The RE algebra appeared as an abstraction of algebraic constructions, [1], [14], arising from the theory of integrable models, [2]. It is also related to the braid group for a solid handlebody, [13]. In this paper, we consider the RE algebra as the U_{q}(𝔤)-invariant quantization of the polynomial algebra on the matrix space. It is generated by elements arranged in the matrix L and subject to a set of quadratic relations called the reflection equation.

The conjugation transformation of L with T, the generating matrix of the FRT algebra whose entries commute with the entries of L, is again an RE matrix, [15]. It follows from this observation that any character of the RE algebra specifies a homomorphism to the FRT algebra. This is done in the same way as in the classical geometry, where maximal ideals of the function algebra on M correspond to the points of M. Each point defines an embedding of the function algebra on the orbit into that on the group G. We use this idea for constructing quantized manifolds as quotient spaces of the quantized group. Characters of the RE algebra are exactly solutions to the matrix reflection equation, [16]. We find such solutions in the form of projectors of rank k < n. Thus, we come to presenting the one-parameter quantizations on symmetric orbits as quotients of the RE algebra. There are also solutions other than projectors, corresponding to different paths in the parameter space of the two-parameter quantization S_{h,t}(𝔤).

We prove that there is an epimorphism from the RE algebra defined on n × n matrices onto the RE algebra defined on k × k matrices for k < n. That enables us to build new solutions to the matrix RE by embedding a given solution into a bigger matrix as the left upper block and extending it to the whole matrix with zeros. In particular, starting from a semisimple non-degenerate RE matrix we gain an additional, zero eigenvalue. We use this method for constructing examples of quantized bisymmetric orbits. The complete classification of solutions to the matrix RE is unknown. In particular, it is interesting to find nilpotent matrices providing quantization of nilpotent orbits. We managed to build such solutions among the matrices with the zero square.

2 The quantum universal enveloping algebra U_{q}(sl(n))

By 𝔤 we mean the complex Lie algebra sl(n), and U_{q}(𝔤) is the corresponding quantum group. The latter is understood in the sense of Lusztig [17], i.e., a free module over the field of rational functions in q. In the classical limit q → 1, U_{q}(𝔤) turns into the universal enveloping algebra U(𝔤). The quantum group U_{q}(𝔤) is generated by the elements H_{i}, X_{i}^{±}, i = 1, …, n − 1, satisfying the commutation relations

[H_{i}, X_{i}^{±}] = ±2X_{i}^{±}, [H_{i}, X_{i±1}^{±}] = ∓X_{i±1}^{±},

[H_{i}, X_{j}^{±}] = 0, |i − j| > 1,

[X_{i}^{+}, X_{j}^{−}] = δ_{ij} \frac{q^{H_{i}} − q^{−H_{i}}}{q − q^{−1}} .

Besides, the Serre relations hold:

(X_{i})^{2}X_{i±1} − 2_{q}X_{i}X_{i±1}X_{i} + X_{i±1}(X_{i})^{2} = 0, where X_{k} = X_{k}^{±}.

Here 2_{q} ≡ q + q^{−1}. In general, the quantum integer numbers are defined as n_{q} ≡ \frac{q^{n}−q^{−n}}{q−q^{−1}} . The coproduct Δ, counit ε, and antipode γ are

Δ(H_{i}) = H_{i} ⊗ 1 + 1 ⊗ H_{i}, Δ(X_{i}^{±}) = X_{i}^{±} ⊗ q^{− \frac{H_{i}}{2}} + q^{\frac{H_{i}}{2}} ⊗ X_{i}^{±}.
ε(H_{i}) = ε(X_{i}^{±}) = 0, γ(H_{i}) = −H_{i}, γ(X_{i}^{±}) = −q^{∓1}X_{i}^{±}.

Remark 2.1. The conjugation of the Lie algebra 𝔤 by the skew-diagonal matrix J = ∑ e_{α}^{n+1−α} extends to an automorphism of U_{q}(𝔤). The subset of fixed points with respect to this automorphism is a quantum subgroup. It is the stabilizer of J in the representation on matrices. We use this fact in the sequel.

In the tensor square of U_{q}(𝔤), there is an element R called the universal R-matrix. It satisfies the conditions

(Δ ⊗ id)(R) = R_{13}R_{23},  (id ⊗ Δ)(R) = R_{13}R_{12},  (1)
RΔ(x) = Δ′(x)R, x ∈ U_{q}(𝔤).  (2)

The subscripts distinguish the legs of tensor objects and the prime stands for the opposite coproduct. We adopt the Sweedler symbolic notations Δ(x) = x_{(1)} ⊗ x_{(2)} for the coproduct and denote the antipode by the bar: γ(x) = x̄, x ∈ U_{q}(𝔤). The exact expression for R can be found in [20, 12, 11]. We do not use it in the sequel, rather the image of R in the fundamental representation on ℂ^{n}.

3 Fundamental representation of U_{q}(sl(n))

By M we denote the algebra of n × n complex matrices with the multiplication e_{j}^{i}e_{l}^{k} = e_{l}^{i}δ_{j}^{k} defined on the matrix units e_{j}^{i}. Here δ_{j}^{k} stand for the Kronekker symbol. The elements of M are considered as right endomorphisms of V = ℂ^{n} with the action e_{i}e_{l}^{k} = e_{l}δ_{i}^{k} on the basis elements e_{i} ∈ V. Formulas

ρ(H_{i}) = e_{i}^{i} − e_{i+1}^{i+1}, ρ(X_{i}^{+}) = e_{i+1}^{i}, ρ(X_{i}^{−}) = e_{i}^{i+1}, i = 1, …, n − 1.

define a homomorphism ρ of the algebras U_{q}(𝔤) into M. On the Chevalley generators, it is given by the same formulas as in the classical limit q → 1.

The homomorphism ρ defines the right adjoint and left coadjoint actions of U_{q}(𝔤) on M:

A ◁ ad_{ρ}(x) = ρ(\overline{x_{(1)}})Aρ(x_{(2)}),  (3)
ad_{ρ}^{∗}(x) ▷ A = ρ(x_{(2)})Aρ(\overline{x_{(1)}})  (4)

for x ∈ U_{q}(𝔤) and A ∈ M. These actions are conjugate via the trace pairing on M:

Tr((A ◁ ad_{ρ}(x)) B) = Tr(A (ad_{ρ}^{∗}(x) ▷ B)).

The left coadjoint module M equipped with action (4) will be denoted M^{∗}.

The element

R = q^{− \frac{1}{n}} (q ∑_{i} e_{i}^{i} ⊗ e_{i}^{i} + ∑_{i≠j} e_{i}^{i} ⊗ e_{j}^{j} + ω ∑_{i<k} e_{i}^{k} ⊗ e_{k}^{i}) ∈ M ⊗ M,  (5)

where ω ≡ q − q^{−1}, satisfies the Yang-Baxter equation

R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}.

It differs from the R-matrix used in [8] by the overall factor q^{− \frac{1}{n}}. In this form, this is the image of the universal R-matrix R under the homomorphism ρ^{⊗2}.

Consider the braid matrix S = q^{\frac{1}{n}} σ_{V ⊗V} R, where σ_{V ⊗V} is the ordinary (classical) permutation on V^{⊗2}. The matrix S satisfies the Hecke condition S^{2} = ωS + 1. The projectors

P_{q}^{+} = \frac{1}{2_{q}} (q − S),  P_{q}^{−} = \frac{1}{2_{q}} (q^{−1} + S),  (6)

determine a U_{q}(𝔤)-invariant decomposition of the tensor square V^{⊗2} into the sum of two submodules. They are deformations of the subspaces of symmetric and antisymmetric 2-tensors.

Let V^{∗} be the space of linear functionals on V. We choose a basis {e_{i}} in V and denote {f^{i}} its dual with respect to the canonical pairing between V and V^{∗}.

Proposition 3.1. The homomorphism ρ: U_{q}(𝔤) → M induces the right and left actions of U_{q}(𝔤) on the tensor algebra T(V ⊕ V^{∗}). They are given on the basis elements e_{i} ∈ V and f^{i} ∈ V^{∗} by

e_{i} ◁ x = ∑_{α=1}^{n} e_{α}ρ(x)_{i}^{α},  f^{i} ◁ x = ∑_{α=1}^{n} f^{α}ρ(x̄)_{α}^{i},  (7)

x ▷ e_{i} = ∑_{α=1}^{n} e_{α}ρ(x̄)_{i}^{α},  x ▷ f^{i} = ∑_{α=1}^{n} f^{α}ρ(x)_{α}^{i},  (8)

for x ∈ U_{q}(𝔤). Formula ⟨e_{i}, f^{j}⟩ = δ_{i}^{j} defines a U_{q}(𝔤)-invariant right pairing between V and V^{∗}. The right module V^{∗} ⊗ V with the action induced by (7) is isomorphic to the adjoint module M via the correspondence f^{i} ⊗ e_{j} → e_{j}^{i}. The left module V ⊗ V^{∗} with the action induced by (8) is isomorphic to the coadjoint module M^{∗} via the correspondence e_{j} ⊗ f^{i} → e_{j}^{i}.

Proof. Direct verification.

One can introduce the following U_{q}(𝔤)-invariant involutive permutations among the elements of V and V^{∗}.

Proposition 3.2. Formulas

τ_{V ⊗V} (e_{i} ⊗ e_{j}) = qe_{j} ⊗ e_{i}, i > j,  (9)
τ_{V^{∗}⊗V^{∗}}(f^{i} ⊗ f^{j}) = qf^{j} ⊗ f^{i}, i < j,  (10)
τ_{V ⊗V^{∗}}(e_{j} ⊗ f^{i}) = q^{− \frac{1}{n}} ((1 − δ_{j}^{i})f^{i} ⊗ e_{j} + δ_{j}^{i}(qf^{i} ⊗ e_{i} + ω ∑_{k>i} f^{k} ⊗ e_{k})).  (11)

define a U_{q}(𝔤)-invariant involutive permutation on (V ⊕ V^{∗})^{⊗2}.

Proof. Operations (9) and (10) are determined by projectors (6). Operation (11) is readily obtained from the action of the universal R-matrix. To compute it, one needs to know only the image of R in M^{⊗2}, which is given by (5).

4 Algebra T(V) ⊗ T(V^{∗}) in braided categories

In this section, we consider the tensor algebras T(V) and T(V^{∗}) from the two points of view: as objects from the category of either U_{q}(𝔤)- or U_{q}^{⊗2}(𝔤)-representations. Correspondingly, the algebra structure on T(V) ⊗ T(V^{∗}) may be introduced in two ways, as the tensor product of algebras in the two categories. The particular choice of the category determines embedding of T(V ⊗ V^{∗}), the tensor algebra of the space V ⊗ V^{∗}, into T(V) ⊗ T(V^{∗}). That leads to two different quantizations of the polynomial algebra on V ⊗ V^{∗}.

Let us remind the construction of the tensor product of algebras in the category of a quantum group representation. Let H be a quantum group with an R-matrix R. Recall that a (left) H-module algebra A for a quantum group H is an associative algebra with unity in the category of H-representations:

x ▷ (ab) = (x_{(1)} ▷ a)(x_{(2)} ▷ b), x ▷ 1 = ε(x)1, x ∈ U_{q}(𝔤), a, b ∈ A.  (12)

Given two module algebras A and B, one can introduce their (braided) tensor product A ˇ⊗B, which is again an H-module algebra.

Proposition 4.1. Let R_{1} ⊗ R_{2} denote the symbolic decomposition of the universal R-matrix into the two tensor factors. The formula

(a_{1} ˇ⊗b_{1})(a_{2} ˇ⊗b_{2}) = a_{1}(R_{2} ▷ a_{2}) ˇ⊗(R_{1} ▷ b_{1})b_{2}, a_{i} ∈ A, b_{i} ∈ B  (13)

defines an associative multiplication on A ˇ⊗B turning it into an H-module algebra. Embeddings A → A ˇ⊗1 and B → 1 ˇ⊗B are homomorphisms of algebras.

Proof. Associativity of multiplication (13) follows from (1). Compatibility with the action of H is a consequence of (2). The last statement of the proposition is immediate due to the equality (ε ⊗ id)(R) = (id ⊗ ε)(R) = 1 ⊗ 1 following from the definition of the universal R-matrix.

One sees that multiplication (13) is characterized by the permutation relation

ba = (R_{2} ▷ a)(R_{1} ▷ b)  (14)

between the elements a ∈ A and b ∈ B.

Proposition 4.2. Let V and U be H-modules. The subalgebra in T(V) ˇ⊗T(U) generated by the submodule V ˇ⊗U is isomorphic to the tensor algebra T(V ⊗ U). It is an algebra in the category of H-modules.

Proof. Embedding V ⊗ U → T(V) ˇ⊗T(U) is extended to a homomorphism of the free algebra T(V ⊗ U). It is an monomorphism due to invertibility of permutation (14). The subalgebra generated by V ⊗ U is H-invariant, so it is a module algebra.

We will apply this proposition when U = V^{∗}.

Let us apply the construction above to the algebras A = T(V), B = T(V^{∗}) considered as U_{q}^{⊗2}(𝔤)-module algebras in the following way. The representation of the quantum group U_{q}(𝔤) in V generates the actions of U_{q}^{⊗2}(𝔤) on V and V^{∗}:

(x ⊗ y) ▷ v ≡ ε(y)x ▷ v,  (x ⊗ y) ▷ ϕ ≡ ε(x)y ▷ ϕ,  (15)

where x ⊗ y ∈ U_{q}^{⊗2}(𝔤), v ∈ V, and ϕ ∈ V^{∗}. These actions are extended to the actions on T(V), T(V^{∗}), and on the tensor product T(V) ⊗ T(V^{∗}).

Note that the multiplications in the algebras T(V) and T(V^{∗}) considered as U_{q}(𝔤)- or U_{q}^{⊗2}(𝔤)-module algebras coincide with the usual classical multiplication.

Proposition 4.3. The algebra T(V) ˇ⊗T(V^{∗}) in the category of U_{q}^{⊗2}(𝔤)-modules has the classical multiplication:

(a_{1} ˇ⊗b_{1})(a_{2} ˇ⊗b_{2}) = a_{1}a_{2} ˇ⊗b_{1}b_{2}, a_{i} ∈ T(V), b_{i} ∈ T(V^{∗}).  (16)

Proof. The universal R-matrix of U_{q}^{⊗2}(𝔤) is the product of two R-s supported in the two different copies of U_{q}(𝔤). By construction (15), each copy acts trivially either on T(V) or on T(V^{∗}), so the universal R-matrix turns into the identity operation in (13).

Note that being considered in the category of U_{q}(𝔤)-modules, the algebra T(V) ˇ⊗T(V^{∗}) has a non-classical multiplication characterized by the permutation relation (14), where R is the R-matrix for U_{q}(𝔤).

Due to Proposition 4.2, the tensor algebra T(V ⊗ V^{∗}) has two different realizations in the categories of either U_{q}^{⊗2}(𝔤)- or U_{q}(𝔤)-representations. One can introduce additional relations in T(V ⊗ V^{∗}) resulting in two different quantizations of the polynomial algebra on V ⊗ V^{∗}; they are the FRT and RE algebras. We consider them in Sections 5 and 6 and present their defining commutation relations in terms of involutive permutations τ_{F RT} and τ_{RE}.

Taking into account Proposition 4.3, we reserve the symbol ˇ⊗ only for the tensor product in the category of U_{q}(𝔤)-modules where the permutation (14) between V and V^{∗} is non-classical. So, by T(V) ˇ⊗T(V^{∗}) ( T(V) ⊗ T(V^{∗}) ) we will denote the tensor product considered as U_{q}(𝔤)- ( U_{q}^{⊗2}(𝔤)-) module.

5 Algebra A_{F RT}(M)

Let A_{F RT} (M) be the associative unital algebra over ℂ(q) generated by the matrix elements T_{j}^{i} subject to relations

ST_{1}T_{2} = T_{1}T_{2}S.  (17)

Here, S is the Hecke matrix introduced in Section 3. The algebra A_{F RT} (M) was introduced in [8] as a quantized polynomial algebra on the space of matrices. It is endowed with the U_{q}^{⊗2}(𝔤)- module structure coming from that on V ⊗ V^{∗}, since relations (17) are U_{q}^{⊗2}(𝔤)-invariant. They are given by a quantum permutation τ_{F RT} on the space V ⊗ V^{∗}, which we are going to present explicitly.

Consider T(V ⊗ V^{∗}) as a subalgebra of T(V) ⊗ T(V^{∗}) in the tensor category of U_{q}^{⊗2}(𝔤)- modules, along the line of Proposition 4.2. Consider the permutation

τ_{F RT} = σ_{V ⊗V^{∗}} ∘ τ_{V ⊗V} ∘ τ_{V^{∗}⊗V^{∗}} ∘ σ_{V ⊗V^{∗}}^{−1},  (18)

which is the composition of elementary permutations (9), (10) and the classical flip σ_{V ⊗V^{∗}}. This permutation is U_{q}^{⊗2}(𝔤)-invariant. Let (V ⊗ V^{∗})^{⊗2} = I_{F RT}^{+} ⊕ I_{F RT}^{−} be the decomposition into symmetric and skew-symmetric submodules with respect to τ_{F RT}. It is easy to see that I_{F RT}^{−} equals (P_{q}^{+} ⊗ P_{q}^{−} + P_{q}^{−} ⊗ P_{q}^{+})(V^{⊗2} ⊗ V^{∗⊗2}), where P_{q}^{±} are projectors (6), and is a deformation of the exterior square (V ⊗ V^{∗})^{∧2}.

Theorem 5.1. Let J_{F RT} be the ideal in T(V ⊗ V^{∗}) generated by I_{F RT}^{−}. The quotient algebra T(V ⊗ V^{∗})/J_{F RT} is isomorphic to A_{F RT} (M).

Proof. Let us take a basis {e_{i}} ⊂ V and its right dual {f^{i}} ⊂ V^{∗}, i. e., ⟨e_{i}, f^{j}⟩ = δ_{i}^{j}. The elements e_{i} ⊗ f^{j} transform as entries of the matrix T, under the action of U_{q}^{⊗2}(𝔤), so we can identify them. The product of two matrix entries can be rewritten as T_{j}^{i}T_{n}^{m} = e_{j} ⊗ f^{i}e_{n} ⊗ f^{m} = e_{j}e_{n} ⊗ f^{i}f^{m}, since the elements of V and V^{∗} commute. This implies that the ideal determined by (17) turns into J_{F RT}.

6 Algebra A_{RE}(M)

The reflection equation algebra A_{RE}(M) is generated by the entries of the matrix L subject to the quadratic relations, [1, 14],

L_{2}SL_{2}S = SL_{2}SL_{2}.  (19)

Here, S is the Hecke matrix introduced in Section 3. The generators L_{j}^{i} form the coadjoint module M^{∗} via the correspondence L_{j}^{i} → e_{j}^{i}. Action (4) is extended over A_{RE}(M) turning it into a U_{q}(𝔤)-module algebra;

The element Q = R_{21}R defines a morphism of U_{q}(𝔤)-modules A_{RE}(M) → U_{q}(𝔤) by the correspondence L → Q_{2} ⊗ ρ(Q_{1}), [5], where U_{q}(𝔤) is considered as a left U_{q}(𝔤)-module via the adjoint action. This allows to prove that the algebra A_{RE}(M) is a flat deformation of the polynomial algebra on the matrix space M, (see also [18]).

The algebra A_{RE}(M) can be described in the same way as A_{F RT} (M) by a quantum permutation τ_{RE} . Following Proposition 4.2, consider the tensor algebra T(V ⊗V^{∗}) as the subalgebra T(V ˇ⊗V^{∗}) ⊂ T(V) ˇ⊗T(V^{∗}) in the category of U_{q}(𝔤)-modules. Let

τ_{RE} = τ_{V ⊗V^{∗}} ∘ τ_{V ⊗V} ∘ τ_{V^{∗}⊗V^{∗}} ∘ τ_{V ⊗V^{∗}}^{−1},  (20)

which is the composition of elementary permutations (9–11). The permutation τ_{RE} is U_{q}(𝔤)- invariant. Consider the decomposition (V ⊗ V^{∗})^{⊗2} = I_{RE}^{+} ⊕ I_{RE}^{−} into the symmetric and skewsymmetric submodules with respect to τ_{RE}. It is easy to see that the submodule I_{RE}^{−} is equal to (P_{q}^{+} ⊗ P_{q}^{−} + P_{q}^{−} ⊗ P_{q}^{+})(V^{⊗2} ˇ⊗V^{∗⊗2}), where P_{q}^{±} are projectors (6), and is a deformation of the exterior square (V ⊗ V^{∗})^{∧2}.

Theorem 6.1. Let J_{RE} be the ideal in T(V ⊗ V^{∗}) generated by I_{RE}^{−}. The quotient algebra T(V ˇ⊗V^{∗})/J_{RE} is isomorphic to A_{RE}(M).

Proof. As in the proof of Theorem 5.1, let {f^{i}} ⊂ V^{∗} be the right dual to the basis {e_{i}} ⊂ V. Due to Proposition 3.1, the elements e_{i} ˇ⊗f^{j} form the coadjoint module M^{∗} for U_{q}(𝔤), so we can put L_{i}^{j} = e_{i} ˇ⊗f^{j}. By Proposition 4.2, the algebra T(V ˇ⊗V^{∗}) is isomorphic to the free associative algebra generated by L_{j}^{i}. It remains to show that the quadratic submodule generating the ideal J_{RE} is isomorphic to the submodule specified by the reflection equation. They are conjugate by the τ_{V ⊗V^{∗}} which is the restriction to V ⊗ V^{∗} of permutation (14). The permutation τ_{V ⊗V^{∗}} is given by the formula (11), and it acts by the rule matrix is exactly q^{− \frac{1}{n}} S, e_{i}e_{l} ˇ⊗f^{k}f^{n} = q^{− \frac{1}{n}} S_{lj}^{km}(e_{i} ˇ⊗f^{j})(e_{m} ˇ⊗f^{n}).

Remark 6.2. In the classical case, the polynomial algebra on V ⊗ V^{∗} can be built equivalently as a U(𝔤)- or U^{⊗2}(𝔤)-module. In the quantum case, dependingly of the point of view, we come to either FRT or RE algebras, which are different.

7 Irreducible submodules of M^{⊗2}

This section is technical, it contains information about submodules of M^{⊗2} and its dual in terms of highest and lowest weight vectors. They will be used in computing the quantum permutation τ_{RE} on (V ⊗ V^{∗})^{⊗2} ∼ M^{∗⊗2}. It is convenient to evaluate the conjugate operator τ_{RE}^{∗} in the dual space first. For that reason we consider three U_{q}(𝔤)-module structures on M^{⊗2}:

(A ⊗ B) ◁ x = ρ(\overline{x_{(2)}})Aρ(x_{(3)}) ⊗ ρ(\overline{x_{(1)}})Bρ(x_{(4)}),  (21)
(A ⊗ B) ◁ x = ρ(\overline{x_{(1)}})Aρ(x_{(2)}) ⊗ ρ(\overline{x_{(3)}})Bρ(x_{(4)}),  (22)
x ▷ (A ⊗ B) = ρ(x_{(2)})Aρ(\overline{x_{(1)}}) ⊗ ρ(x_{(4)})Bρ(\overline{x_{(2)}}).  (23)

Actions (21) and (22) are nothing else than ad_{ρ⊗ρ} and ad_{ρ} ⊗ ad_{ρ}, correspondingly. The left action (23) is conjugate to (22) with respect to the ordinary trace pairing and it is just ad_{ρ}^{∗}⊗ad_{ρ}^{∗}.

A basis of the lowest weight vectors of irreducible U_{q}(𝔤)-submodules with respect to the action (23) may be calculated directly. It is:

δ_{1} = ∑_{i=1}^{n} q^{−2i+1}e_{i}^{i} ⊗ e_{1}^{n},

δ_{2} = ∑_{i=1}^{n} e_{1}^{n} ⊗ q^{−2i+1}e_{i}^{i},

δ_{3} = ∑_{i=1}^{n} e_{1}^{i} ⊗ e_{i}^{n},

δ_{4} = ∑_{i=1}^{n} a_{i}e_{i}^{n} ⊗ e_{1}^{i}, a_{i} = 1, q^{−3}, …, q^{−2n+3}, q^{−2n},

δ_{5} = ∑_{i,k=1}^{n} q^{−(2i−1)}q^{−(2k−1)}e_{i}^{i} ⊗ e_{k}^{k},

δ_{6} = ∑_{i,k=1}^{n} a_{i}e_{i}^{k} ⊗ e_{k}^{i}, a_{i} = q^{−(2n−1)}, q^{−(2n−3)}, …, q^{−3}, q^{−1},
δ_{7} = qe_{1}^{n} ⊗ e_{2}^{n−1} + q^{−1}e_{2}^{n−1} ⊗ e_{1}^{n} − e_{1}^{n−1} ⊗ e_{2}^{n} − e_{2}^{n} ⊗ e_{1}^{n−1},
δ_{8} = e_{1}^{n} ⊗ e_{1}^{n},
δ_{9} = e_{1}^{n} ⊗ e_{1}^{n−1} − q^{−1}e_{1}^{n−1} ⊗ e_{1}^{n},
δ_{10} = e_{1}^{n} ⊗ e_{2}^{n} − q^{−1}e_{2}^{n} ⊗ e_{1}^{n},

The coefficients a_{i} entering δ_{4} and δ_{6} decrease by q^{−2} each step within the interval i = 1, …, n. Here, we assume n > 2. The case n = 2 is simple and left to the reader.

The vectors δ_{5} and δ_{6} are invariant elements under action (23). They are symmetric with respect to the permutation τ_{RE}^{∗}, since that is so for the invariant elements in the classical situation. The vectors δ_{7} and δ_{8} belong to the symmetric part, too, while δ_{9} and δ_{10} are antisymmetric with respect to τ_{RE}^{∗}. Indeed, they generate submodules of multiplicity one in M^{⊗2} and are deformations of classical submodules belonging to symmetric and antisymmetric parts, respectively. The situations is more complicated in what concerns vectors δ_{1}, …, δ_{4} because they generate the isotypic component of type 𝔤, and in the classical limit q = 1 the corresponding classical component intersects the symmetric and skew-symmetric parts of M^{⊗2} with multiplicity two. So the problem of evaluating the permutation τ_{RE} reduces to calculating its restriction to the 𝔤-isotypic component of M^{∗⊗2}.

First, we compute the dual conjugate operation to τ_{RE} on the right module M^{⊗2} with action (22). A basis of lowest weight vectors has the dual consisting of highest weight ones; by that reason we also consider the highest weight vectors α^{i} and β^{i} of type 𝔤 with respect to actions (21) and (22), respectively:

α^{1} = 1 ⊗ e_{n}^{1},
α^{2} = ∑_{i=1}^{n} a_{i}e_{n}^{1} ⊗ e_{i}^{i}, a_{i} = q, 1, …, 1, q^{−1},

α^{3} = ∑_{i=1}^{n} a_{i}e_{i}^{1} ⊗ e_{n}^{i}, a_{i} = q, 1, …, 1, q^{−1},

α^{4} = ∑_{i=1}^{n} a_{i}e_{n}^{i} ⊗ e_{i}^{1}, a_{i} = 1, …, 1, q^{−1},
β^{1} = 1 ⊗ e_{n}^{1},
β^{2} = e_{n}^{1} ⊗ 1,
β^{3} = ∑_{i=1}^{n} a_{i}e_{i}^{1} ⊗ e_{n}^{i}, a_{i} = q^{−(2n−1)}, q^{−(2n−3)}, …, q^{−3}, q^{−1},

β^{4} = ∑_{i=1}^{n} a_{i}e_{n}^{i} ⊗ e_{i}^{1}, a_{i} = q, 1, …, 1, q^{−1}.  (24)

Let us denote the isotypic 𝔤-type components with respect to actions (21), (22), and (23) by M_{α}^{⊗2}, M_{β}^{⊗2}, and M_{δ}^{∗⊗2}. They are generated by the vectors α^{i}, β^{i}, and δ_{i}, i = 1, …, 4, respectively.

8 The quantum permutation τ_{RE}

In this section, we calculate the permutation τ_{RE} in terms of the lowest weight vectors introduced in the previous section. The permutations τ_{F RT} and τ_{RE} define quadratic relations in the deformed algebras of functions on matrices. In both cases, relations (17) and (19) are formulated using the algebra structure on M^{⊗2}, i. e., in the dual setting. Thus, the FRT and RE algebra permutations are introduced through the dual operations τ_{F RT}^{∗} and τ_{RE}^{∗} conjugate to τ_{F RT} and τ_{RE} by the trace pairing. The involutions τ_{F RT}^{∗} and τ_{RE}^{∗} are invariant with respect to right actions (21) and (22). It is natural to compute them first, using the algebra structure on M^{⊗2}, and then evaluate τ_{RE} by duality.

The subspace of Ω ∈ M^{⊗2} such that SΩ = ΩS is the annulator of the submodule defining FRT relations (17). Therefore, the involution τ_{F RT}^{∗} is determined by the conjugation transformation with the Hecke matrix S. Solutions to the equation Sσ(Ω) = σ(Ω)S, where the map σ is defined as Ω → (1⊗Ω_{1})S(1⊗Ω_{2}), Ω = Ω_{1} ⊗Ω_{2} ∈ M^{⊗2}, form the submodule annihilating the RE relations. Hence, the involutions τ_{RE}^{∗} and τ_{F RT}^{∗} are σ-conjugate: τ_{RE}^{∗} = σ^{−1}τ_{F RT}^{∗} σ. Since the Hecke matrix S is invariant under the adjoint action ad_{ρ⊗ρ}, the map σ intertwines actions (22) and (21). It is an isomorphism of modules, being a deformation of the classical flip σ_{M⊗M}.

As was noted in Section 7, only the restriction of τ_{RE}^{∗} to the isotypic 𝔤-type component requires an additional consideration. Restricted to the 𝔤-component M_{α}^{⊗2}, the conjugation operation with the matrix S has eigenvalues +1 and −q^{±2}. The eigenspace M_{α;+}^{⊗2} corresponding to the eigenvalue +1 consists of symmetric tensors, in the sense of the permutation τ_{F RT}^{∗}. Eigenvectors of the eigenvalues −q^{±2} may be called τ_{F RT}^{∗} -antisymmetric. We denote this submodule M_{α;−}^{⊗2}. Thus, the conjugation with the matrix S determines decomposition

M_{α}^{⊗2} = M_{α;+}^{⊗2} ⊕ M_{α;−}^{⊗2},  (25)

where both M_{α;±}^{⊗2} contain irreducible submodule 𝔤 with multiplicity two. Decomposition (25) induces decompositions

M_{β}^{⊗2} = M_{β;+}^{⊗2} ⊕ M_{β;−}^{⊗2},  (26)
M_{δ}^{∗⊗2} = M_{δ;+}^{∗⊗2} ⊕ M_{δ;−}^{∗⊗2}.  (27)

into the symmetric and antisymmetric parts. The subspaces M_{β;±}^{⊗2} are the images of M_{α;±}^{⊗2} via the inverse transformation σ^{−1}. They yield the involution τ_{RE}^{∗}. The subspaces M_{δ;±}^{∗⊗2} are the annulators of M_{β;∓}^{⊗2} with respect to the trace pairing between M^{⊗2} and M^{∗⊗2}. The permutation τ_{RE} is determined by the submodules M_{δ;±}^{∗⊗2}.

Proposition 8.1. The 𝔤-type submodules of symmetric and antisymmetric tensors are generated by the following highest and lowest weight vectors:

\begin{aligned}M_{α;+}^{⊗2} & : & α^{3} + α^{4}, & α^{1} + α^{2} + ωα^{3}, & M_{α;−}^{⊗2} & : & q^{±1}α^{1} − q^{∓1}α^{2} ∓ α^{3} ± α^{4}, \\ M_{β;+}^{⊗2} & : & β^{1} + β^{2} − ωβ^{3}, & β^{3} + β^{4} + ωβ^{1}, & M_{β;−}^{⊗2} & : & β^{1} − β^{2}, β^{3} − β^{4}, \\ M_{δ;+}^{∗⊗2} & : & δ_{3} + δ_{4} − ωδ_{2}, & δ_{1} + δ_{2} + ωδ_{3}, & M_{δ;−}^{∗⊗2} & : & δ_{1} − δ_{2}, δ_{3} − δ_{4}. \end{aligned}

Proof. The eigenvectors of the conjugation by S restricted to M_{α}^{⊗2} are computed directly. Thus we obtain (25). The transformation σ acts on the highest weight vectors of the 𝔤-type components as

σ: ( \begin{matrix}β^{1} \\ β^{2} \\ β^{3} \\ β^{4} \end{matrix} ) → ( \begin{matrix}α^{3} \\ α^{4} + ωα^{1} \\ α^{1} \\ α^{2} \end{matrix} ) .

Decomposition (26) is the σ-preimage of (25). Decomposition (27) is computed by pairing δ_{i} with β^{k} via the ordinary trace.

Corollary 8.2. The submodule of τ_{RE}-symmetric tensors in M^{∗⊗2} is generated by the lowest weight vectors

{δ_{1} + δ_{2} + ωδ_{3}, δ_{3} + δ_{4} − ωδ_{2}, δ_{5}, δ_{6}, δ_{7}, δ_{8}}

The submodule of τ_{RE}-antisymmetric tensors in M^{∗⊗2} is generated by the lowest weight vectors

{δ_{1} − δ_{2}, δ_{3} − δ_{4}, δ_{9}, δ_{10}}

Proof. Immediate from the last proposition.

The permutation τ_{RE} can be reduced to the submodule 𝔤^{⊗2} ⊂ M^{∗⊗2}. We study this problem in the next section.

9 Reducing τ_{RE} to 𝔤^{⊗2}

The left coadjoint module M^{∗} contains a one-dimensional submodule 𝔪_{0} spanned by the invariant element ∑_{i=1}^{n} q^{−2i+2}e_{i}^{i} and the 𝔤-type submodule of q-traceless matrices. The decomposition M^{∗} = 𝔪_{0} ⊕ 𝔤 leads to the decomposition

M^{∗} ⊗ M^{∗} = 𝔪_{0} ⊗ 𝔪_{0} ⊕ 𝔪_{0} ⊗ 𝔤 ⊕ 𝔤 ⊗ 𝔪_{0} ⊕ 𝔤 ⊗ 𝔤.

The lowest weight vectors δ̃_{3} and δ̃_{4} belonging to 𝔤⊗𝔤 are obtained by subtracting proper linear combinations of δ_{1} and δ_{2} from δ_{3} and δ_{4}:

δ̃_{3} = δ_{3} − \frac{ω}{1 − q^{−2n}} δ_{1} − \frac{ω}{1 − q^{−2n}} δ_{2}, δ̃_{4} = δ_{4} − \frac{ω}{q^{2n} − 1}δ_{1} − \frac{ω}{1 − q^{−2n}} δ_{2}.  (28)

Now we can evaluate the reduction τ̃_{RE} of the permutation τ_{RE} to the submodule 𝔤^{⊗2}.

Proposition 9.1. The composition

𝔤 ⊗ 𝔤 → M^{∗} ⊗ M^{∗} ⟶^{τ_{RE}} M^{∗} ⊗ M^{∗} → 𝔤 ⊗ 𝔤,  (29)

where the left arrow is embedding and the right one is the projection along 𝔪_{0} ⊗M^{∗} +M^{∗} ⊗𝔪_{0}, defines an involutive U_{q}(𝔤)-invariant permutation τ̃_{RE} on 𝔤^{⊗2}. The lowest weight vectors

δ̃_{+} ≡ (1 − \frac{ωq^{−n}}{n_{q}})δ̃_{3} + (1 + \frac{ωq^{n}}{n_{q}})δ̃_{4},  δ̃_{−} ≡ δ̃_{4} − δ̃_{3},  (30)

where δ̃_{i} are introduced by (28), generate the 𝔤-type submodules in 𝔤^{⊗2} of symmetric and antisymmetric tensors with respect to τ̃_{RE}, correspondingly.

Proof. The first statement is immediate. It easy to check that the vector δ̃_{+} ∈ 𝔤^{⊗2} ⊂ M^{⊗2} is τ_{RE}-symmetric. As to the vector δ̃_{−}, it is the image of τ_{RE}-antisymmetric vector δ_{4} − δ_{3} under the projection M^{∗⊗2} → 𝔤^{⊗2}.

Corollary 9.2. The submodules of symmetric and antisymmetric tensors are generated by the sets

\{δ_{6} − \frac{ω}{1 − q^{−2n}} δ_{5}, δ_{7}, δ̃_{+}\}, and \{δ_{9}, δ_{10}, δ̃_{−}\},

of lowest weight vectors respectively.

Proof. The vector δ_{5} is the square of the invariant element ∑_{i=1}^{n} q^{−2i+2}e_{i}^{i} spanning 𝔪_{0}. The combination δ_{6} − \frac{ω}{1−q^{−2n}} δ_{5} is the Casimir element of 𝔤^{⊗2} ⊂ M^{∗⊗2}. The 𝔤-type vectors δ̃_{±} were considered in the above proposition. Concerning the other lowest weight vectors, they already lye in 𝔤^{⊗2} ⊂ M^{∗⊗2}, so the proof is an immediate consequence of Corollary 8.2.

Remark 9.3. Unlike in the classical case, the submodule 𝔤^{⊗2} in M^{∗⊗2} is not preserved by the permutation τ_{RE}. By this reason, τ̃_{RE} is not a simple restriction of τ_{RE} to 𝔤^{⊗2}.

As a left U_{q}(𝔤)-module, the tensor square M^{∗⊗2} is endowed with another permutation coming from the universal R-matrix representation: A⊗B → R_{2}▷B ⊗R_{1}▷A, A⊗B ∈ M^{∗⊗2}. It can be shown that its restriction to the 𝔤-component M_{δ}^{∗⊗2} ⊂ M^{∗⊗2} acts according to the rule

( δ_{1} ; δ_{2} ; δ̃_{3} ; δ̃_{4} ) → ( δ_{2} ; δ_{1} ; δ̃_{4} ; q^{−2n}δ̃_{3} ) .

It is easy to see, that this operator has eigensubspaces distinct from those of τ_{RE}. With this remark, we complete the study of the RE algebra. In the remainder of the paper, we investigate relations between A_{RE}(M) and the quantization on 𝔤^{∗} and its orbits.

10 Algebra A_{RE} and the quantization on 𝔤^{∗}

It was shown in [5] that a two-parameter U_{q}(𝔤)-invariant quantization S_{q,t}(𝔤) of the Lie-Poisson bracket on 𝔤^{∗} exists only in the sl(n) case. It is realized as the quotient algebra of T(𝔤)[t](q) by the quadratic-linear relations

x ⊗ y − τ̃_{RE}(x ⊗ y) = t[x, y]_{q},  (31)

where [ ·, · ]_{q}: 𝔤^{⊗2} → 𝔤 is a deformed Lie bracket. This is a U_{q}(𝔤)-invariant map 𝔤^{⊗2} → 𝔤 sending the submodule of τ̃_{RE}-symmetric tensors to zero . The commutator [ ·, ·]_{q} is uniquely defined up to a factor, since the module of τ̃_{RE}-antisymmetric tensors contains the submodule isomorphic to 𝔤 with multiplicity one. In the limit q → 1, the commutator turns into the classical Lie bracket. In this way, one recovers the algebra U(𝔤)[t] viewed as a quantization of the Lie-Poisson bracket. Another limit t → 0 yields a one-parameter family S_{q,0}(𝔤), which can be interpreted as a τ̃_{RE}-commutative algebra of polynomials on 𝔤^{∗}.

The RE algebra is closely related to S_{q,t}(𝔤). To formulate that relation, let us introduce the quantum trace of the RE matrix L:

Tr_{q}(L) ≡ ∑_{i=1}^{n} q^{−2i+2}L_{i}^{i} = Tr(DL), where D ≡ ∑_{i=1}^{n} q^{−2i+2}e_{i}^{i}.  (32)

Remark that this definition makes sense for matrices with entries being elements of any associative algebra A over ℂ(q). The quantum trace is the map id ⊗ Tr_{q}: A ⊗ M → A ⊗ ℂ(q) ∼ A. For the generating matrix L of the RE algebra, the elements Tr_{q}(L^{k}), k ∈ ℕ, are invariant. It follows from the commutation relations (19) that they belong to the center of A_{RE}(M).

Theorem 10.1. The quotient algebra of A_{RE}(M) by the ideal (Tr_{q}(L) − λ), where λ ∈ ℂ, is isomorphic to the one-parameter sub-family t = λ\frac{q−q^{−1}}{Tr_{q}(Id)} in S_{q,t}(𝔤).

Proof. In its essential part, this theorem was proven in [5]. Here we specify the exact relation between the parameter t and the value of λ, using the explicit expression for the permutation τ_{RE} derived in Section 8, in terms of the lower weight vectors. By Proposition 8.1, the τ_{RE}- symmetric 𝔤-type submodules are generated by the vectors δ_{1} − δ_{2} and δ_{4} − δ_{3}. The first one is irrelevant because it turns zero when projected to 𝔤^{⊗2} ⊂ M^{∗⊗2}. The second vector is represented as δ_{4} − δ_{3} = δ̃_{4} − δ̃_{3} − ωδ_{1}, where δ̃_{i} belong to 𝔤^{⊗2}. Vector δ_{1} is expressed through the matrix D introduced by (32) and the lowest weight vector e_{1}^{n} of the adjoint representation in 𝔤: δ_{1} = q^{−1}D ⊗ e_{1}^{n}. Comparing this with (31), we find the value of the parameter t.

11 The quantization of coadjoint orbits

One can show that the two-parameter family S_{q,t}(𝔤) can be restricted to any semisimple orbit in 𝔤^{∗}. It this connection, there arises the problem of explicit description of the quantized manifolds in terms of ideals in S_{q,t}(𝔤). We solve this problem (in a one-parameter setting) for certain classes of orbits including the symmetric ones, using the relation between A_{RE} and S_{q,t}(𝔤) (Theorem 10.1). Let O_{A} ⊂ M be the orbit passing through the matrix A. Consider two commutative diagrams

[figure: G × M; G × {A}; M; O_{A}; Fun(G) ⊗ Fun(M); Fun(G) ⊗ ℂ; Fun(M); Fun(O_{A}),; id ⊗ χ^{A}]

where the horizontal arrows correspond to the action of the group G. The right square represents morphisms of the polynomial algebras induced by the maps of manifolds depicted on the left. The map χ^{A} is the character of the algebra Fun(M) corresponding to the point A ∈ M. We are going to quantize the right square; that will give us realization of the quantized algebra Fun_{q}(O_{A}), on the one hand, as a quotient of A_{RE}(M) and as a subalgebra in the quantized function algebra on G, on the other.

Let F_{q}(G) be the Hopf algebra of quantized polynomial functions on the group G. It is the quotient of the FRT algebra by the additional relation det_{q}(T) = 1, where det_{q} is the quantum determinant, [8]. Since U_{q}(𝔤) acts on the RE algebra A_{RE}(M), this action generates the coaction of F_{q}(G) on A_{RE}(M). Thus, we come to

Theorem 11.1. [15]. Let T^{−1} be the matrix with entries γ(T_{j}^{i}), where T is the generating matrix of the algebra F_{q}(G) and γ the antipode. The conjugation transformation L → T^{−1}LT of an RE matrix L with the FRT matrix T whose entries commute with the entries of L is again an RE matrix.

It follows that the correspondence L → T^{−1}LT extends to an homomorphism A_{RE}(M) → F_{q} ⊗ A_{RE}(M). Theorem 11.1 yields the quantization of the upper arrow of the right square on the above diagram. To quantize the other maps, we replace χ^{A} by χ_{q}^{A}, a character of the reflection equation algebra. It is determined by the correspondence L_{j}^{i} → A_{j}^{i} ∈ ℂ(q) such that the numeric matrix A satisfies the reflection equation, [16],

A_{2}SA_{2}S = SA_{2}SA_{2}  (33)

supported in M^{⊗2}. Any solution to this equation gives rise to the algebra Fun_{q}(O_{A}) closing the commutative diagram

[figure: F_{q}(G) ⊗ A_{RE}(M); F_{q}(G) ⊗ ℂ; A_{RE}(M); Fun_{q}(O_{A}),; id ⊗ χ_{q}^{A}]

so that the bottom arrow is embedding. Thus, we obtain

Theorem 11.2. Let A be a solution of the numeric RE (33). Then the algebra Fun_{q}(O_{A}) in the diagram above is the quantization of the polynomial algebra on the orbit O_{A} passing through the matrix A.

Proof. It is clear that at q = 1 the algebra Fun_{q}(O_{A}) coincides with the polynomial algebra on the orbit. The flatness of Fun_{q}(O_{A}) over q follows from the fact that it is simultaneously a quotient of the flat module A_{RE}(M) and a submodule of the flat module F_{q}(G).

The matrix L(A) = T^{−1}AT possesses the following properties:

Lemma 11.3. For any polynomial function P,

P(L(A)) = L(P(A)).  (34)

The quantum trace is invariant under the conjugation:

Tr_{q}(L(A)) = Tr_{q}(A).  (35)

Proof. The first statement is evident. The second one is checked using the commutation relations in the algebra F_{q}(G).

Note that solutions A of (33) yield quantizations which are quotients of one-parameter subfamilies in S_{q,t}(𝔤). Those subfamilies are defined by paths in the parameter space (q, t) crossing the axis q = 1 at the point t = 0. In particular, the solutions with Tr_{q}(A) = 0 correspond to the path t = 0. The limit q → 1 for t separated from 0 cannot be reached within our approach. In the last subsections, we consider quantizations of various types of orbits, along the line of Theorem 11.2.

11.1 Quantized symmetric orbits

To begin with, let us prove a statement relating reflection equation algebras in different dimensions. Let M(n) and M(k) be the matrix algebra of n × n and k × k matrices, 0 < k < n. The homomorphism M(k) → M(n) of embedding as the left upper block induces a contravariant epimorphism of the function algebras. The same holds in the quantum situation:

Proposition 11.4. The quotient algebra A_{RE}(M(n)) by the relations L_{j}^{i} = 0, i > k or j > k, is isomorphic to A_{RE}(M(k)).

Proof. Let P^{+} and P^{−} be the projectors from ℂ^{n} = ℂ^{k} ⊕ ℂ^{n−k} to the first and the second addends, respectively. Denote R^{++} the projection of the R-matrix (5) to M(k) ⊗ M(k). Up to a nonzero scalar factor, this is the R-matrix in dimension k. The equalities

(P^{+} ⊗ 1)R = R^{++} + q^{− \frac{1}{n}} P^{+} ⊗ P^{−},  R(1 ⊗ P^{+}) = R^{++} + q^{− \frac{1}{n}} P^{−} ⊗ P^{+}  (36)

follows directly from (5). The matrix L is equal to L^{+} = P^{+}LP^{+} modulo the ideal specified in the hypothesis. In terms of the matrix R instead of the Hecke matrix S, the reflection equation is rewritten as

R_{21}L_{1}^{+}RL_{2}^{+} = L_{2}^{+}R_{21}L_{1}^{+}R

modulo the relations of concern. Using (36), the matrix R can be replaced by R^{++} thus leading to the RE in dimension k.

Proposition 11.4 suggests a method of building solutions to (33) by those in smaller dimensions extending them by zero matrix elements. For example, the projectors P_{k} = ∑_{i=1}^{k} e_{i}^{i} of rank k = 1, …, n satisfy (33). Indeed, each can be obtained from the k × k unit matrix (which is apparently a solution to the RE) by extending it with zeros to the n × n matrix.

Let us introduce the quantum integer numbers

k̂_{q} ≡ ∑_{i=1}^{k} q^{−2i+2} = Tr_{q}(P_{k}).  (37)

Theorem 11.5. The quotient algebra of A_{RE}(M) by the relations

L^{2} = L, T r_{q}(L) = k̂_{q}  (38)

yields a U_{q}(𝔤)-invariant quantization Fun_{q}(O_{P_{k}}) of the symmetric orbit O_{P_{k}} passing through the projector P_{k}

Proof. Relations (38) follow from identities (34) and (35) applied to A = P_{k}. The projectors P_{k} are stable under the right adjoint action (3) of U_{q}(sl(k))⊗U_{q}(sl(n−k))⊗U_{q}(sl(1)) as a quantum subgroup in U_{q}(sl(n)). Therefore, the subalgebra in F_{q}(G) generated by the RE matrix L(P_{k}) is invariant under the right action of U_{q}(sl(k)) ⊗ U_{q}(sl(n − k)) ⊗ U_{q}(sl(1)).

Remark 11.6. The quantum projective spaces may be obtained directly from the description of A_{RE} given in Section 6. Indeed, imposing the additional conditions V^{∧2} = 0 and V^{∗∧2} = 0 one comes to the subalgebra in Sym_{q}(V) ˇ⊗Sym_{q}(V^{∗}) generated by e_{i} ⊗ f^{j} with e_{i} and f^{j} subject to relations (9–11). The matrix elements L_{i}^{j} = e_{i}f^{j} satisfy the equality L_{i}^{l}L_{l}^{j} = e_{i}(f^{l}e_{l})f^{j} = T r_{q}(L)L_{j}^{i} following from the commutation relations. So, conditions (38) for the case k = 1 hold simultaneously in this algebra. Another description of the quantum projective space is found in [7].

As in the classical situation, we can consider solution λP_{k} with arbitrary λ ≠ 0. This will lead to the relation L^{2} = λL and the corresponding rescaling of the quantum trace. The resulting conditions give the deformation quantization of the orbit passing through λP_{k}. It may be regarded as the quantization of the same manifold but along the path t = λω \frac{k̂_{q}}{n̂_{q}} in the two parameter space of the universal family S_{q,t}(𝔤) (Theorem 10.1).

11.2 The Cayley-Hamilton identity

The goal of this subsection is to exhibit correspondence with identities in A_{RE}(M) of the Cayley-Hamiltonian type, [19]. In the algebra A_{RE}(M), the k-th powers L^{k} of the generating matrix form the left coadjoint module M^{∗}. The quantum traces Tr_{q}(L^{k}), k = 1, 2, … are U_{q}(sl(n))-invariant and central, (see, e.g., [19]).

Theorem 11.7. [19]. The generating matrix of the reflection equation algebra A_{RE}(M) obeys the relation

∑_{j=0}^{n} σ_{q}^{j}(L)(−L)^{n−j} = 0,

where σ_{q}^{j}(L) are central elements expressed through the quantum traces of powers in L by the recursive formula

k̂_{q}σ_{q}^{k}(L) = ∑_{j=0}^{k−1} σ_{q}^{j}(L)(−1)^{k−j+1}Tr_{q}(L^{k−j}), σ_{q}^{0}(L) = 1.  (39)

In particular, σ_{q}^{n+1}(L) = 0.

It follows that imposing the projector condition on the matrix L specifies the quantum trace modulo the finite set of values.

Corollary 11.8. The condition L^{2} = L on the generating matrix of the A_{RE}(M) implies the discrete set {ˆ0_{q}, ˆ1_{q}, …, n̂_{q}} of values for the quantum trace.

Proof. The statement follows from the formula (39) since, under the hypothesis made, one has

σ_{q}^{n+1}(L) = \frac{Tr_{q}(L)(Tr_{q}(L) − ˆ1_{q}) … (Tr_{q}(L) − n̂_{q})}{{ \atop (n + 1)_{q}}!} = 0.

All the possible values of Tr_{q}(L) are realized, by Theorem 11.5.

The Cayley-Hamiltonian identity in the RE algebra was used for construction of the quantum sphere 𝕊_{q}^{2} in [10].

11.3 On the quantization of non-semisimple orbits

As was already mentioned, the two-parameter quantization S_{q,t}(sl(n)) may be “restricted” to every semisimple orbits. There are no definite assertions of that kind concerning orbits passing through nilpotent elements. In this section, we prove

Theorem 11.9. There exists a one-parameter U_{q}(𝔤)-invariant quantization of nilpotent orbits in M satisfying the matrix equation A^{2} = 0. It is a restriction of the one-parameter subfamily t = 0 in the universal two-parameter quantization S_{q,t}(𝔤).

Proof. Like in the case of symmetric orbits, we seek for a solution to the matrix reflection equation, which will realize the quantized algebra as a subalgebra in F_{q}(G). Specializing to the skew-diagonal matrices,

A = ∑_{i=1}^{n} λ_{i}e_{i}^{i′}, i′ ≡ n + 1 − i,

we find it in the form λ_{i}λ_{i′} = λ^{2}, i = 1, …, n. In case of non-zero λ we come to nondegenerate matrices with two eigenvalues. These solutions were found in [16]. They lead to other realization of quantum symmetric orbits than by means of projectors. In case of λ = 0, we obtain a solution with arbitrary values of λ_{i} either above or below the principal diagonal. All these matrices being squared are zero. By re-enumerating the basis elements, they can be brought to the sum of 2 × 2 jordanian blocks yielding all such matrices. Let us fix the solution A = ∑_{i=1}^{k} e_{i′}^{i}, for k ≤ \frac{n}{2}. The stabilizer is a subalgebra in the parabolic Lie algebra with the semisimple part sl(k) ⊕ sl(n − k) ⊕ sl(k). It is selected by the identification between symmetrically allocated sl(k) via the involutive conjugation by the skew-diagonal unit of dimension k. As was noted in Remark 2.1, this automorphism extends to the quantized algebras.

11.4 On the quantization of bisymmetric orbits

To quantize the symmetric orbits in g^{∗}, we used the projectors, i. e., the semisimple elements with the eigenvalues 1 and 0. There are non-degenerate semisimple solutions to the matrix reflection equation with two eigenvalues. For example, one can take ∑_{i=1}^{n} e_{n+1−1}^{i}. Such matrices can be used for constructing solutions with three eigenvalues by the embedding method, along the line of Proposition 11.4. In this way, one gains the additional zero eigenvalue. One might have expected that this will provide a tool for quantizing all the semisimple orbits, which are classified as homogeneous spaces by the number of eigenvalues and their multiplicities. However, that is probably impossible, due to certain indications. Indeed, in [16], there were written out all non-degenerate solutions up to n = 4. They have at most two eigenvalues and lead to different quantizations for the symmetric orbits than by projectors. Among them, there are also q-traceless matrices. We conjecture that non-degenerate matrix solutions to the reflection equation with arbitrary value of the quantum trace do exist for each symmetric orbit. This implies the existence of the quantization via the RE algebra characters for every bisymmetric orbit, i. e., consisting of matrices with three eigenvalues. Let us present, following [16], the non-degenerate q-traceless RE matrices up to n = 4.

A^{1,1} = ( \begin{matrix}& 1 \\ 1 \end{matrix} ) , A^{2,1} = ( \begin{matrix}−q^{−2} & & ˆ2_{q} \\ & 1 \\ 1 \end{matrix} )

A^{2,2} = ( \begin{matrix}& & & 1 \\ & & 1 \\ & 1 \\ 1 \end{matrix} ) , A^{3,1} = ( \begin{matrix}−q^{−2}ˆ2_{q} & & & ˆ3_{q} \\ & 1 \\ & & 1 \\ 1 \end{matrix} ) .

The matrix A^{2,2} is interesting from the physical point of view because it yields the quantization of the twistor space. Using these RE matrices together with the projectors P_{k}, we can recover all the semisimple orbits for n ≤ 4, excepting the maximal orbit in dimension n = 4. But the maximal orbits can be quantized by specifying values of the Casimir elements, i. e., Tr(L^{i}), i = 1, …, n − 1. So we conclude that, at least to dimension n = 4, all the semisimple orbits can be quantized explicitly as quotients of the reflection equation algebra.

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