Orbits for the adjoint coaction on quantum matrices

M. Domokos^{∗}, R. Fioresi^{†}, and T. H. Lenagan

Abstract

Conjugation coactions of the quantum general linear group on the algebra of quantum matrices have been introduced in an earlier paper and the coinvariants have been determined. In this paper the notion of orbit is considered via co-orbit maps associated with ℂ-points of the space of quantum matrices, mapping the coordinate ring of quantum matrices into the coordinate ring of the quantum general linear group. The co-orbit maps are calculated explicitly for 2 × 2 quantum matrices. For quantum matrices of arbitrary size, it is shown that when the deformation parameter is transcendental over the base field, then the kernel of the co-orbit map associated with a ℂ-point ξ is a right ideal generated by coinvariants, provided that the classical adjoint orbit of ξ is maximal. If ξ is diagonal with pairwise different eigenvalues, then the image of the co-orbit map coincides with the subalgebra of coinvariants with respect to the left coaction of the diagonal quantum subgroup of the quantum general linear group.

2000 Mathematics Subject Classification. 16W35, 16W30, 20G42, 17B37, 81R50 Keywords: coinvariant, quantum general linear group, quantum matrices, orbit map, quantum homogeneous space, quantum sphere

1 Introduction

Consider O(M_{q}), the coordinate ring of N × N quantum matrices over ℂ, where q is a non-zero element of ℂ. Denote by O(GL_{q}) the coordinate ring of the quantum GL(N, ℂ) (see [12]). Define the map

β : O(GL_{q}) → O(GL_{q}) ⊗ O(GL_{q}), β(h) = ∑ h_{2} ⊗ S(h_{1})h_{3},  (1)

where we use Sweedler’s convention for the Hopf algebra O(GL_{q}) (and S denotes the antipode).

∗This research was supported through a European Community Marie Curie Fellowship. Partially supported by OTKA No. F 32325 and T 34530.

†Investigation supported by the University of Bologna, funds for selected research topics.

Similarly, set

α : O(GL_{q}) → O(GL_{q}) ⊗ O(GL_{q}), α(h) = ∑ h_{2} ⊗ h_{3}S(h_{1}).

Both α and β are right coactions of the Hopf algebra O(GL_{q}) on itself. Being the formal dual of the right adjoint action, β is usually called the right adjoint coaction of O(GL_{q}). Obviously O(M_{q}) is a subcomodule of O(GL_{q}) with respect to α and β, and we use the same symbols to denote the restrictions α : O(M_{q}) → O(M_{q}) ⊗ O(GL_{q}), β : O(M_{q}) → O(M_{q}) ⊗ O(GL_{q}). These coactions can be viewed as quantum analogues of the adjoint action of GL(N, ℂ) on its Lie algebra M(N, ℂ). The coinvariants with respect to these coactions were described in [4].

Our aim here is to find a counterpart in the quantum setting of the notion of orbits of the classical adjoint action. We summarize the results concerning β, the case of α being similar. There is a natural way to associate with a “ℂ-point ξ of the space of N × N quantum matrices” a morphism β^{ξ} : O(M_{q}) → O(GL_{q}) of right O(GL_{q})-comodules (the right coaction on O(M_{q}) is given by β, whereas the right coaction on O(GL_{q}) comes from the comultiplication of O(GL_{q})). When q is specialized to 1, the kernel of β^{ξ} is the vanishing ideal of the closure of the orbit of ξ, so the image of β^{ξ} can be identified with the coordinate ring of the closure of the orbit of ξ. The q-deformed map β^{ξ} is not an algebra homomorphism since the β^{ξ} is not an algebra homomorphism (see [4]). However, it turns out that the kernel of β^{ξ} contains the right ideal of O(M_{q}) generated by the elements τ_{i} − τ_{i}(ξ), i = 1, …, N, where τ_{1}, …,τ_{N} are the basic coinvariants for β introduced in [4]. When ξ is diagonal, the image of β^{ξ} is contained in the quantum quotient space O(D\GL_{q}), the subalgebra of coinvariants of the left coaction of the diagonal quantum subgroup O(D) on O(GL_{q}) (note that in the classical case q = 1, D is the stabilizer of ξ, provided that ξ has pairwise different diagonal entries). In the special case when N = 2 and q is not a root of unity, we are able to show that although β^{ξ} is not an algebra homomorphism, it has further nice properties. Namely, if ξ is diagonal, and ξ is not a scalar power of any of the non-negative powers of diag(q^{2}, 1), then β^{ξ} is surjective onto O(D \ GL_{q}). Moreover, for such ξ, the kernel of β^{ξ} is the right ideal of O(M_{q}) generated by the elements τ_{1} − τ_{1}(ξ), τ_{2} − τ_{2}(ξ), where τ_{1}, τ_{2} are the basic β-coinvariants from [4]; that is, τ_{1} = q^{−2}x_{11} + q^{−4}x_{22}, a weighted trace, and τ_{2} = x_{11}x_{22} − qx_{12}x_{21}, the quantum determinant. When ξ is a scalar multiple of some non-negative power of diag(q^{2}, 1), then the kernel of β^{ξ} is larger, and the image is finite dimensional. In the final section we change the setup, and treat q as an indeterminate over a subfield of ℂ. In this generic situation, both the kernel and the image of the co-orbit map coincide with the subset predicted by the classical theory, for a ξ ∈ M(N, ℂ) whose classical adjoint orbit is maximal. To be more precise, the kernel of the co-orbit map is a right ideal generated by coinvariants, and if ξ is diagonal, then the image of the co-orbit map is the subalgebra of coinvariants with respect to the left coaction of the diagonal quantum subgroup. The present work can be viewed as a quantum version of some of the basic results of Kostant [7] on the classical adjoint orbits of the general linear group acting on its Lie algebra (which we use in our proof).

Recall that O(M_{q}) is the ℂ-algebra generated by N^{2} indeterminates x_{ij}, for i = 1, …, N, subject to the following relations.

x_{ij}x_{il} = qx_{il}x_{ij},  (2)
x_{ij}x_{kj} = qx_{kj}x_{ij},
x_{il}x_{kj} = x_{kj}x_{il},
x_{ij}x_{kl} − x_{kl}x_{ij} = (q − q^{−1})x_{il}x_{kj},

for 1 ≤ i < k ≤ N and 1 ≤ j < l ≤ N, where q ∈ ℂ^{∗}. The algebra O(M_{q}) is an iterated Ore extension, and so a noetherian domain. The quantum determinant, det_{q}, is the element

det_{q} := ∑_{σ∈S_{N}}(−q)^{l(σ)}x_{1,σ(1)} … x_{N,σ(N)}.

It is known that det_{q} is a central element in O(M_{q}) (see [10, Theorem 4.6.1]), and by adjoining its inverse we get the quantum general linear group

O(GL_{q}) := O(M_{q})[det_{q}^{−1}].

The algebra O(GL_{q}) is a Hopf algebra with comultiplication Δ and antipode S. Recall that Δ(x_{ij}) = ∑_{k=1}^{N} x_{ik}⊗x_{kj}. We shall need the explicit form of S only in the special case N = 2, when S(x_{11}) = x_{22}/det_{q}, S(x_{22}) = x_{11}/det_{q}, S(x_{12}) = −q^{−1}x_{12}/det_{q}, S(x_{21}) = −qx_{21}/det_{q}. The reader should be aware that in many papers the rôles of q and q^{−1} are interchanged, and so one has to be careful in translating results from one paper to another. In the special case q = 1, the algebra O(M_{q}) becomes the coordinate ring O(M) of M(N, ℂ), and O(GL_{q}) becomes the coordinate ring O(GL) of GL(N, ℂ).

2 Orbits of classical points

Start from the classical situation when we are given a (right) action

X × G → X, (x, g) ↦ xg

of an affine algebraic group G on an affine algebraic variety X. The orbit of some x ∈ X is the image of the composition of the morphisms

G → X × G → X, g ↦ (x, g) ↦ xg.

Passing to coordinate rings, the action is encoded in the comorphism μ : O(X) → O(X) ⊗ O(G), which makes O(X) a right O(G)-comodule algebra. The point x corresponds to the ℂ-algebra homomorphism ev_{x} : O(X) → ℂ, f ↦ f(x). The comorphism of the orbit map G → xG ⊂ X, g ↦ xg is the composition

μ_{x} : O(X) ⟶^{μ} O(X) ⊗ O(G) ⟶^{ev_{x}⊗id} O(G).

The kernel of μ_{x} is the ideal of the (Zariski) closure of the orbit xG, and so the image of μ_{x} can be identified with the coordinate ring of the closure of xG. Note that μ_{x} : O(X) → O(G) is an algebra homomorphism, as well as a morphism of right O(G)- comodules (where the right coaction of O(G) on itself comes from the action of G on itself by right translations).

Now assume in addition that the orbit of x is closed in the Zariski topology of X, and denote by H the stabilizer of x. Then H acts by left translation on G, and the orbit map G → xG factors through the quotient variety H \ G, and induces an isomorphism of the G-varieties H \ G ≅ xG. In terms of coordinate rings this means that im(μ_{x}) is the subalgebra of O(H)-coinvariants in O(G) (the action of H on G by left translations induces a left coaction of O(H) on O(G)).

Furthermore, if G is reductive, then O(X)^{G}, the algebra of polynomial invariants (which coincides with the algebra of μ-coinvariants) is finitely generated as an algebra, say f_{1}, …, f_{s} is a generating system. Assume in addition that the orbit xG is closed and maximal; that is, x is not contained in the closure of another orbit (note that for the adjoint action of GL(N, ℂ) on its Lie algebra M(N, ℂ) this holds for the general x). Then ker(μ_{x}) can be described in terms of polynomial invariants. In this case the common zero locus of f_{1} − f_{1}(x), …, f_{s} − f_{s}(x) is the orbit xG, so ker(μ_{x}) is the radical of the ideal generated by f_{1} − f_{1}(x), …, f_{s} − f_{s}(x).

Recall that

β = (id ⊗ p) ∘ (τ ⊗ id) ∘ (S ⊗ id ⊗ id) ∘ Δ^{(2)} : O(GL_{q}) → O(GL_{q}) ⊗ O(GL_{q}),

where Δ^{(2)} = (Δ ⊗ id) ∘ Δ = (id ⊗ Δ) ∘ Δ, τ(h ⊗ k) = k ⊗ h, and p : O(GL_{q}) ⊗ O(GL_{q}) → O(GL_{q}) is the multiplication map. The corresponding formula for α is

α = (id ⊗ p) ∘ τ_{(132)} ∘ (S ⊗ id ⊗ id) ∘ Δ^{(2)},

where τ_{(132)}(h⊗k⊗l) = k⊗l⊗h. The above discussion on algebraic group actions motivates the following definition. Take a surjective ℂ-algebra homomorphism ev_{ξ} : O(M_{q}) → ℂ. The ξ in the notation refers to the N × N matrix with complex entries obtained by evaluating the given homomorphism on the generators x_{ij} of O(M_{q}). Note that for ξ ∈ M(N, ℂ) there is a corresponding homomorphism ev_{ξ} if and only if ξ is a “quantum matrix” in the sense of [8]; that is, if the entries of ξ satisfy the relations defining O(M_{q}). We say that ξ is a ℂ-point of M_{q} in this case. If q ≠ 1, then ξ is a ℂ-point of M_{q} if and only if there is at most one non-zero entry in each column and each row of ξ, and ξ_{ij}ξ_{kl} ≠ 0 with i < k implies that j < l. For example, a diagonal ξ satisfies this condition.

Definition 2.1 Let ξ be a ℂ-point of M_{q}. The co-orbit map β^{ξ} of ξ with respect to the right coaction β is the composition (ev_{ξ} ⊗ id) ∘ β, that is,

β^{ξ} : O(M_{q}) ⟶^{β} O(M_{q}) ⊗ O(GL_{q}) ⟶^{ev_{ξ}⊗id} O(GL_{q}).

In the same manner, α^{ξ} is defined to be (ev_{ξ} ⊗ id) ∘ α.

The above discussion suggests that the kernel of the co-orbit map should contain information about the “embedding of the orbit into M_{q}”, and the image of the co-orbit map should determine the isomorphism type of (the closure of) the “orbit” as a quantum space. The main point of this paper is that we treat the co-orbit map (and not only its image) as our central object, and demonstrate that it has certain nice properties, even though we started with a coaction which was not an algebra homomorphism. In particular, the coinvariants are used to study the kernel of the co-orbit map.

Remark 2.2 Note that Definition 2.1 makes sense if β is replaced by any right coaction ν of a Hopf algebra A on an associative ℂ-algebra R, and ev_{ξ} is an algebra homomorphism of R to ℂ. Proposition 2.3 below clearly holds in this general setting. The map ν^{ξ} (and Proposition 2.3) appears in [2], [3] in the situation when R is an A- comodule algebra. In this case ν^{ξ} is an algebra homomorphism as well, and im(ν^{ξ}) is a right coideal subalgebra of A. A right coideal subalgebra of a quantum group is called a quantum homogeneous space, see [6, 11.6.1]. So if A is a quantum group and R is a quantum A-space (that is, R is an A-comodule algebra), then im(ν^{ξ}) is a quantum homogeneous space. Moreover, any quantum homogeneous A-space can be obtained as the image of ν^{ξ} for some classical point ξ in some quantum A-space, see [3, Proposition 1.1] or [2, Proposition 3.2].

Proposition 2.3 The co-orbit map β^{ξ} : O(M_{q}) → O(GL_{q}) is a morphism of right O(GL_{q})- comodules (given by β and Δ, respectively). Similarly, α^{ξ} is a morphism of O(GL_{q})- comodules (given by α and Δ, respectively).

Proof. The claim is the equality Δ∘β^{ξ} = (β^{ξ} ⊗id)∘β. Observe that Δ∘(ev_{ξ} ⊗id) = (ev_{ξ} ⊗ id⊗id)∘(id⊗Δ), because both are equal to ev_{ξ}⊗Δ : O(M_{q})⊗O(GL_{q}) → O(GL_{q})⊗O(GL_{q}). Since β is a right coaction we have (id ⊗ Δ) ∘ β = (β ⊗ id) ∘ β. Using these two equalities one gets

Δ ∘ β^{ξ} = Δ ∘ (ev_{ξ} ⊗ id) ∘ β
= (ev_{ξ} ⊗ id ⊗ id) ∘ (id ⊗ Δ) ∘ β
= (ev_{ξ} ⊗ id ⊗ id) ∘ (β ⊗ id) ∘ β
= (β^{ξ} ⊗ id) ∘ β.
□

The spaces of coinvariants

O(M_{q})^{α−GL_{q}} := {f ∈ O(M_{q}) | α(f) = f ⊗ 1}
O(M_{q})^{β−GL_{q}} := {f ∈ O(M_{q}) | β(f) = f ⊗ 1}

are studied in [4]. It is shown there that although α and β are not algebra homomorphisms, nevertheless

α(fh) = α(f)α(h) if h ∈ O(M_{q})^{α−GL_{q}}  (3)

and similarly

β(fh) = β(f)β(h) if f ∈ O(M_{q})^{β−GL_{q}}.  (4)

It follows that O(M_{q})^{α−GL_{q}} and O(M_{q})^{β−GL_{q}} are subalgebras. Let us recall their generators. Fix an integer t with 1 ≤ t ≤ N. Let I and J be subsets of {1, …, N} with |I| = |J| = t. The subalgebra of O(M_{q}) generated by x_{ij} with i ∈ I and j ∈ J can be regarded as an algebra of t × t quantum matrices, and so we can calculate its quantum determinant - this is a t × t quantum minor and we denote it by [I|J]. The quantum minor [I|I] is said to be a principal quantum minor. We denote the sum of all the principal quantum minors of a given size i by σ_{i}. Note that σ_{1} = x_{11} + ⋯ + x_{NN} and that σ_{N} = det_{q}. It is shown in [4] that the σ_{i} are α-coinvariants that pairwise commute, and if q is not a root of unity, then O(M_{q})^{α−GL_{q}} = ℂ[σ_{1}, …, σ_{N}], an N-variable commutative polynomial subalgebra of O(M_{q}). For the coaction β we have that the weighted sums of principal minors τ_{i} := ∑_{I} q^{−2w(I)}[I|I] (here w(I) denotes the sum of the elements of I, and the summation ranges over all subsets I of size i), i = 1, …, N, are pairwise commuting β-coinvariants. Moreover, assuming again that q is not a root of unity, O(M_{q})^{β−GL_{q}} = ℂ[τ_{1}, …, τ_{N}], the N-variable commutative polynomial subalgebra generated by the τ_{i}.

We shall write f(ξ) for ev_{ξ}(f), where f ∈ O(M_{q}) and ξ ∈ M_{q}(ℂ).

Proposition 2.4 The kernel of β^{ξ} contains the right ideals

∑_{f∈O(M_{q})^{β−GLq}}(f − f(ξ))O(M_{q}) ⊇ ∑_{i=1}^{N}(τ_{i} − τ_{i}(ξ))O(M_{q}).

If q is not a root of unity (or q = 1), then the latter two right ideals are equal. The kernel of α^{ξ} contains the left ideals

∑_{f∈O(M_{q})^{α−GLq}} O(M_{q})(f − f(ξ)) ⊇ ∑_{i=1}^{N} O(M_{q})(σ_{i} − σ_{i}(ξ)).

If q is not a root of unity (or q = 1), then the latter two left ideals are equal.

Proof. If f ∈ O(M_{q})^{β−GL_{q}}, then β^{ξ}(f) = (ev_{ξ} ⊗ id)(β(f)) = (ev_{ξ} ⊗ id)(f ⊗ 1) = f(ξ) · 1 ∈ O(M_{q}), hence f − f(ξ) ∈ ker(β^{ξ}). By formula (4) then (f − f(ξ))O(M_{q}) ⊆ ker(β^{ξ}). Note that τ_{i} are β-coinvariants, and if q is not a root of unity, they generate O(M_{q})^{β−GL_{q}} by [4]. Hence in the latter case the elements τ_{i} − τ_{i}(ξ) generate the same right ideal within O(M_{q})^{β−GL_{q}} as all the f − f(ξ) with f ∈ O(M_{q})^{β−GL_{q}}.

The same argument works for α. □

Remark 2.5 More is known in the classical case q = 1. If the adjoint orbit of ξ ∈ M(N, ℂ) is maximal, then ker(β^{ξ}), the vanishing ideal of the closure of the orbit of ξ, is generated by τ_{i} − τ_{i}(ξ), (i = 1, …, N); this is proved in [7, Theorem 10], considering more generally the adjoint action on reductive Lie algebras.

The coordinate Hopf algebra O(D) of the diagonal subgroup D of GL(N, ℂ) is the commutative algebra ℂ[t_{1}, t_{1}^{−1}, …, t_{N}, t_{N}^{−1}] of Laurent polynomials with comultiplication Δ(t_{i}) = t_{i} ⊗ t_{i} and counit ε(t_{i}) = 1. There exists a surjective Hopf algebra homomorphism π_{D} : O(GL_{q}) → O(D) determined by π_{D}(x_{ij}) = δ_{ij}t_{i}, i, j = 1, …, N. Therefore we say that D is a quantum subgroup of GL_{q}, called the diagonal subgroup.

There is a natural left coaction λ_{D} := (π_{D} ⊗ id) ∘ Δ of O(D) on O(GL_{q}). The subset of λ_{D}-coinvariants is denoted by

O(D \ GL_{q}) = {f ∈ O(GL_{q}) | λ_{D}(f) = 1 ⊗ f}.

This is a subalgebra (since O(GL_{q}) is a comodule algebra with respect to λ_{D}), as well as a right O(GL_{q})-subcomodule (since λ_{D} commutes with the right coaction of O(GL_{q}) on itself). So O(D \ GL_{q}) is the coordinate algebra of a quantum homogeneous space in the sense of [6, 11.6.1.]. This quantum homogeneous space is often called the quantum quotient space D \ G_{q}, because it arises as the subalgebra of coinvariants with respect to the coaction of a quantum subgroup.

Recall that in the classical case q = 1, the subgroup D is the stabilizer of ξ, provided that ξ is a diagonal matrix with pairwise different eigenvalues, and the orbit of a diagonal ξ is closed in O(M(N, ℂ)). Hence for such ξ, the coordinate ring of the orbit of ξ is O(D \ G_{1}). A partial analogue of this holds for any q.

Proposition 2.6 Let ξ be a diagonal N ×N matrix with complex entries. Then the image of β^{ξ} is contained in O(D \ GL_{q}). Similarly we have im(α^{ξ}) ⊆ O(D \ GL_{q}).

Proof. The restriction of π_{D} to O(M_{q}) is denoted by π_{C}, it maps O(M_{q}) onto O(C) = ℂ[t_{1}, …, t_{N}]. By our assumption on ξ the homomorphism ev_{ξ} : O(M_{q}) → ℂ factors through π_{C}, we write ev_{ξ} also for the homomorphism O(C) → ℂ, t_{i} ↦ ξ_{ii}.

By definition β^{ξ} is the composition

β^{ξ} : O(M_{q}) ⟶^{β} O(M_{q}) ⊗ O(GL_{q}) ⟶^{π_{C}⊗id} O(C) ⊗ O(GL_{q}) ⟶^{ev_{ξ}⊗id} O(GL_{q}),  (5)

and λ_{D} is the composition

λ_{D} : O(GL_{q}) ⟶^{Δ} O(GL_{q}) ⊗ O(GL_{q}) ⟶^{π_{D}⊗id} O(D) ⊗ O(GL_{q}).

By linearity of (π_{D} ⊗ id) ⊗ Δ we have

(π_{D} ⊗ id) ∘ Δ ∘ (ev_{ξ} ⊗ id) = (ev_{ξ} ⊗ id ⊗ id) ∘ (id ⊗ π_{D} ⊗ id) ∘ (id ⊗ Δ)

as maps O(C) ⊗ O(GL_{q}) → O(D) ⊗ O(GL_{q}). Thus

(π_{D} ⊗ id) ∘ Δ ∘ (ev_{ξ} ⊗ id) ∘ (π_{C} ⊗ id) = (ev_{ξ} ⊗ id ⊗ id) ∘ (π_{C} ⊗ π_{D} ⊗ id) ∘ (id ⊗ Δ)

as maps O(M_{q}) ⊗ O(GL_{q}) → O(D) ⊗ O(GL_{q}). Therefore we have

λ_{D} ∘ β^{ξ} = (ev_{ξ} ⊗ id ⊗ id) ∘ (π_{C} ⊗ π_{D} ⊗ id) ∘ (id ⊗ Δ) ∘ β
= (ev_{ξ} ⊗ id ⊗ id) ∘ (π_{C} ⊗ π_{D} ⊗ id) ∘ (β ⊗ id) ∘ β

(in the second equality we use the fact that O(M_{q}) is a right O(GL_{q})-comodule under β). Note that the formula (1) makes sense if we replace O(GL_{q}) by any Hopf algebra, in particular, it defines a right coaction β^{D} : O(D) → O(D) ⊗ O(D). Since π_{D} is a Hopf algebra homomorphism, we have β^{D} ∘ π_{D} = (π_{D} ⊗ π_{D}) ∘ β. On the other hand, β^{D} is equal to the trivial corepresentation id ⊗ 1, since it is the comorphism of the conjugation action of D on itself, which is trivial. It follows that (π_{C} ⊗ π_{D}) ∘ β = (id ⊗ 1) ∘ π_{C} : O(M_{q}) → O(C) ⊗ O(D), and hence λ_{D} ∘ β^{ξ} can be written as the composition

O(M_{q}) ⟶^{(π_{C}⊗id)∘β} O(C) ⊗ O(GL_{q}) ⟶^{(id⊗1)⊗id}  (6)

O(C) ⊗ O(D) ⊗ O(GL_{q}) ⟶^{ev_{ξ}⊗id⊗id} O(D) ⊗ O(GL_{q}).

In order to prove our Proposition we need to show that for all f ∈ O(M_{q}) the equality λ_{D}(β^{ξ}(f)) = 1 ⊗ β^{ξ}(f) holds. Write (π_{D} ⊗ id)(β(f)) as ∑_{i} a_{i} ⊗ b_{i}, where a_{i} ∈ O(C) and b_{i} ∈ O(GL_{q}). Then β^{ξ}(f) = ∑_{i} ev_{ξ}(a_{i})b_{i} by (5). Therefore, using (6), we have λ_{D}(β^{ξ}(f)) = ∑_{i} ev_{ξ}(a_{i})1 ⊗ b_{i} = 1 ⊗ ∑_{i} ev_{ξ}(a_{i})b_{i} = 1 ⊗ β^{ξ}(f). □

3 Co-orbit maps of 2 × 2 quantum matrices

In the special case N = 2 and q not a root of unity (or q = 1) we are able to refine Propositions 2.4 and 2.6. Throughout this section we assume that N = 2, O(M_{q}), O(GL_{q}) denote the coordinate rings of 2 × 2 quantum matrices and quantum GL(2), and O(D) = ℂ[t_{1}^{±1}, t_{2}^{±1}]. We will see in the proof of Lemma 3.4 that as an algebra O(D \ GL_{q}) is generated by x_{11}x_{21}/det_{q}, x_{12}x_{21}/det_{q}, x_{12}x_{22}/det_{q}. We write O(D \ GL_{q})^{≤m} for the subspace spanned by products of length at most m in these generators.

Proposition 3.1 Assume that q is not a root of unity (or q = 1), and take ξ := [\begin{matrix}ξ_{1} & 0 \\ 0 & ξ_{2} \end{matrix}] .

(i) If ξ_{1} − q^{2k}ξ_{2} ≠ 0 for k = 0, 1, 2, …, then im(β^{ξ}) = O(D \ GL_{q}).

(ii) If ξ_{1} − ξ_{2}, ξ_{1} − q^{2}ξ_{2}, …, ξ_{1} − q^{2m−2}ξ_{2} are non-zero and ξ_{1} − q^{2m}ξ_{2} = 0, then im(β^{ξ}) = O(D \ GL_{q})^{≤m}.

(iii) If ξ_{1} − q^{2k}ξ_{2} ≠ 0 for k = 0, 1, 2, …, then ker(β^{ξ}) = ∑_{i=1}^{2}(τ_{i} − τ_{i}(ξ))O(M_{q}).

(iv) If ξ_{1}−ξ_{2}, ξ_{1}−q^{2}ξ_{2}, …, ξ_{1}−q^{2m−2}ξ_{2} are non-zero and ξ_{1}−q^{2m}ξ_{2} = 0, then ker(β^{ξ}) is the sum of ∑_{i=1}^{2}(τ_{i}−τ_{i}(ξ))O(M_{q}) and the O(GL_{q})-comodule generated by {x_{21}^{k} | k > m}.

(i’) If ξ_{1} − q^{−2k}ξ_{2} ≠ 0 for k = 1, 2, …, then im(α^{ξ}) = O(D \ GL_{q}).

(ii’) If ξ_{1} − q^{−2}ξ_{2}, …, ξ_{1} − q^{−2m}ξ_{2} are nonzero and ξ_{1} − q^{−2m−2}ξ_{2} = 0, then im(α^{ξ}) = O(D \ GL_{q})^{≤m}.

(iii’) If ξ_{1} − q^{−2k}ξ_{2} ≠ 0 for k = 1, 2, …, then ker(α^{ξ}) = ∑_{i=1}^{2} O(M_{q})(σ_{i} − σ_{i}(ξ)).

(iv’) If ξ_{1} − q^{−2}ξ_{2}, …, ξ_{1} − q^{−2m}ξ_{2} are nonzero and ξ_{1} − q^{−2m−2}ξ_{2} = 0, then ker(α^{ξ}) is the sum of ∑_{i=1}^{2} O(M_{q})(σ_{i}−σ_{i}(ξ)) and the O(GL_{q})-comodule generated by {x_{21}^{k} | k > m}.

As an immediate corollary we obtain the following.

Theorem 3.2 Assume that q is not a root of unity (or q = 1) and take a diagonal complex

matrix ξ which is not a scalar multiple of any integral power of [\begin{matrix}q^{2} & 0 \\ 0 & 1 \end{matrix}]. Then both β^{ξ} and

α^{ξ} map O(M_{q}) surjectively onto O(D \GL_{q}). In particular, im(β^{ξ}) = im(α^{ξ}) is a quantum homogeneous space. Moreover, the kernel of β^{ξ} is the right ideal of O(M_{q}) generated by the β-coinvariants τ_{1} − τ_{1}(ξ), τ_{2} − τ_{2}(ξ). The kernel of α^{ξ} is the left ideal of O(M_{q}) generated by σ_{1} − σ_{1}(ξ), σ_{2} − σ_{2}(ξ).

Remark 3.3 In some sense the map β^{ξ} (respectively α^{ξ}) is the best possible we can hope for. Suppose that φ : O(M_{q}) → O(D \ GL_{q}) is a map which is both a ℂ-algebra homomorphism and intertwines between the O(GL_{q})-corepresentations β and Δ. Then ker φ is a completely prime two-sided ideal in O(M_{q}), since O(D \ GL_{q}) is a domain. We shall see below that the trivial O(GL_{q})-corepresentation appears with multiplicity 1 in O(D \ GL_{q}). Therefore ker(φ) must contain τ_{1} − λ_{1} and τ_{2} − λ_{2} with some λ_{1}, λ_{2} ∈ ℂ. Elementary commutator computations (exploiting the fact that τ_{1} is non-central) show that φ(x_{12}) = φ(x_{21}) = 0, φ(x_{11}) ∈ ℂ, and φ(x_{22}) ∈ ℂ. In particular, im(φ) = ℂ for any φ compatible with both the algebra and comodule structures.

It will be convenient for the proof of Proposition 3.1 to pass from O(GL_{q}) to O(SL_{q}), the Hopf algebra quotient of O(GL_{q}) modulo the ideal generated by det_{q} − 1. We keep the symbol Δ to denote the comultiplication in O(SL_{q}). Denote by π the natural epimorphism π : O(GL_{q}) → O(SL_{q}), and a := π(x_{11}), b := π(x_{12}), c := π(x_{21}), d := π(x_{22}). The group K of diagonal matrices in SL(2, ℂ) is a quantum subgroup of O(SL_{q}) with surjective Hopf algebra homomorphism π_{K} : O(SL_{q}) → O(K) = ℂ[z^{±1}] given by π_{K}(a) = z, π_{K}(b) = 0 = π_{K}(c), π_{K}(d) = z^{−1}. There is a natural left coaction λ_{K} := (π_{K} ⊗ id) ∘ Δ and a right coaction ρ_{K} := (id ⊗ π_{K}) ∘ Δ of O(K) on O(SL_{q}). The subalgebra of λ_{K}-coinvariants is denoted by O(K \ SL_{q}).

We shall use the corepresentation theory of O(SL_{q}), the material below can be found for example in [6, 4.2]. We assume that q is not a root of unity (or q = 1). Then O(SL_{q}) is cosemisimple, so any O(SL_{q})-comodule decomposes as the direct sum of irreducible subcomodules. The degree n homogeneous component P_{n} (n ∈ ℕ_{0} := ℕ ∪ {0}) of the subalgebra ℂ⟨a, b⟩ of O(SL_{q}) is an n + 1-dimensional right O(SL_{q})-subcomodule. Denote the corresponding corepresentation by T_{n/2}. Then T_{l} : P_{2l} → P_{2l} ⊗ O(SL_{q}), l ∈ \frac{1}{2}ℕ_{0}, is a complete list of irreducible corepresentations of O(SL_{q}) up to isomorphism, see for example [6, 4.2.1]. We adopt the following notation: if T : V → V ⊗ O(SL_{q}) is a corepresentation of O(SL_{q}), then write V [n] := {v ∈ V | (id ⊗ π_{K}) ∘ T (v) = v ⊗ z^{n}}, where n ∈ ℤ. Since any O(K)-comodule decomposes as a direct sum of one-dimensional O(K)-subcomodules, the comodule V decomposes as ⨁_{n∈ℤ} V [n]. Assume that V is finite dimensional. By the character χ(T) of T we mean π_{K}(f), where f ∈ O(SL_{q}) is the sum of the diagonal matrix elements of T with repect to an arbitrarily chosen basis of V. In other words, χ(T) = ∑_{n∈ℤ} dim_{ℂ}(V [n])z^{n} ∈ O(K). (Note that our use of the word ‘character’ slightly differs from that of [6, 11.2.2], where f is called the character of T, and χ(T) = π_{K}(f) the character of the O(K)-corepresentation (id ⊗ π_{K}) ∘ T.) For the irreducible O(SL_{q})- corepresentations T_{l} we have that

dim(P_{2l}[n]) = \{ \begin{matrix}1, if n = 2l, 2l − 2, 2l − 4, …, −2l \\ 0, otherwise, \end{matrix}

so χ(T_{l}) = z^{l} + z^{l−2} + ⋯ + z^{−l}. In particular, the set \{χ(T_{l}) | l ∈ \frac{1}{2}ℕ_{0}\} is linearly independent, and the multiplicity of T_{j} as a direct summand of T equals the coefficient of χ(T_{j}) in χ(T), expressed as a linear combination of the elements χ(T_{l}).

The quantum quotient space O(S_{q,∞}^{2}) := O(K \SL_{q}) is a well studied object, belonging to the one-parameter family of quantum 2-spheres introduced in [11]. The facts we need about O(S_{q,∞}^{2}) can be found in [6, 4.5]. As a subalgebra of O(SL_{q}), the quantum 2-sphere O(S_{q,∞}^{2}) is generated by the elements

x_{−1} := (1 + q^{2})^{1/2}ac, x_{0} := 1 + (q + q^{−1})bc, x_{+1} := (1 + q^{2})^{1/2}db.

The right O(SL_{q})-corepresentation on O(S_{q,∞}^{2}) is isomorphic to T_{0} ⊕ T_{1} ⊕ T_{2} ⊕ · · ·. The linear span of the products of length at most n in the generators x_{−1}, x_{0}, x_{+1}, denoted by W^{n}, is a right O(SL_{q})-subcomodule. The corepresentation on W^{n} is isomorphic to ⨁_{l=0}^{n} T_{l}. Moreover, the simple subcomodule of W^{n} corresponding to the summand T_{n} is the subcomodule generated by x_{−1}^{n}, because W^{n}[2n] = ℂx_{−1}^{n} and W^{n}[d] = 0 if d > 2n. In particular, as an O(SL_{q})-comodule O(S_{q,∞}^{2}) is generated by the powers of x_{−1}.

The reason that we are able to switch from O(GL_{q}) to O(SL_{q}) is in the following lemma:

Lemma 3.4 The restriction of π maps O(D \ GL_{q}) isomorphically onto O(K \ SL_{q}).

Proof. The coaction λ_{D} defines a ℤ ⊕ ℤ-grading on O(GL_{q}): for (m, n) ∈ ℤ^{2} set

O(GL_{q})_{(m,n)} := {f ∈ O(GL_{q}) | λ_{D}(f) = t_{1}^{m}t_{2}^{n} ⊗ f},

then O(GL_{q}) = ⨁_{(m,n)∈ℤ^{2}} O(GL_{q})_{(m,n)}. By definition O(D \ GL_{q}) = O(GL_{q})_{(0,0)}. For the algebra generators of O(GL_{q}) we have x_{11}, x_{12} ∈ O(GL_{q})_{(1,0)}, x_{21}, x_{22} ∈ O(GL_{q})_{(0,1)}, and det_{q}^{−1} ∈ O(GL_{q})_{(−1,−1)}. Therefore O(D \ GL_{q}) contains x_{11}x_{21}/det_{q}, x_{12}x_{21}/det_{q}, x_{12}x_{22}/det_{q}. These elements are mapped via π onto a system of algebra generators of O(K \ SL_{q}).

Since det_{q} − 1 is not homogeneous with respect to this ℤ^{2}-grading and O(GL_{q}) is a domain, we have that ker(π) = (det_{q} − 1)O(GL_{q}) does not contain any homogeneous elements. In particular, the kernel of π is disjoint from O(D \ GL_{q}). □

The O(GL_{q})-coaction β is replaced by the O(SL_{q})-coaction

ψ := (id ⊗ π) ∘ β : O(M_{q}) → O(M_{q}) ⊗ O(SL_{q}),

and α is replaced by

φ := (id ⊗ π) ∘ α : O(M_{q}) → O(M_{q}) ⊗ O(SL_{q}).

For ξ ∈ M_{q}(ℂ) the maps ψ^{ξ}, φ^{ξ} : O(M_{q}) → O(SL_{q}) are defined in an obvious way: ψ^{ξ} = (ev_{ξ}⊗id)∘ψ = π∘β^{ξ} and φ^{ξ} = (ev_{ξ}⊗id)∘α = π∘α^{ξ}. Propositions 2.3 and 2.6 combined with Lemma 3.4 translate to the following. The map ψ^{ξ} intertwines between the O(SL_{q})- corepresentations ψ and Δ, whereas φ^{ξ} intertwines between the O(SL_{q})-corepresentations φ and Δ. We have that ker(ψ^{ξ}) = ker(β^{ξ}) and ker(φ^{ξ}) = ker(α^{ξ}). If ξ is diagonal then im(ψ^{ξ}) ⊆ O(K \ SL_{q}) with equality if and only if im(β^{ξ}) = O(D \ GL_{q}), and im(φ^{ξ}) ⊆ O(K \ SL_{q}) with equality if and only if im(α^{ξ}) = O(D \ GL_{q}). Moreover, im(β^{ξ}) = (O(D \ GL_{q}))^{n} if and only if im(ψ^{ξ}) = W^{n}, and im(α^{ξ}) = (O(D \ GL_{q}))^{n} if and only if im(φ^{ξ}) = W^{n}.

Proof of Proposition 3.1.

The kernel of ψ^{ξ} contains ∑_{i=1}^{2}(τ_{i}−τ_{i}(ξ))O(M_{q}) by Proposition 2.4, therefore ψ^{ξ} factors through the composition of the natural homomorphisms

O(M_{q}) ⟶^{μ} C ⟶^{η} B,

where C := O(M_{q})/(τ_{2} − τ_{2}(ξ))O(M_{q}), B := C/uC with u := μ(τ_{1} − τ_{1}(ξ)). Set y_{ij} := μ(x_{ij}). Note that τ_{2} − τ_{2}(ξ) = det_{q} − ξ_{1}ξ_{2} is a central element in O(M_{q}), so C is a quotient algebra of O(M_{q}). It is a domain by [5]. Since τ_{2} − τ_{2}(ξ) is a ψ-coinvariant, (τ_{2}−τ_{2}(ξ))O(M_{q}) is a subcomodule of O(M_{q}) by (4), hence C is a factorcomodule of O(M_{q}); the corresponding corepresentation is denoted by ψ_{C}. Similarly, B is a factorcomodule of C, with corepresentation ψ_{B}. Our next aim is to show that

ψ_{B} ≅ T_{0} ⊕ T_{1} ⊕ T_{2} ⊕ ⋯,  (7)

and moreover that, η(y_{21})^{r} generates the simple subcomodule of B corresponding to T_{r} for r ∈ ℕ_{0}.

Denote by O(M_{q})^{≤r} the linear span of monomials of degree ≤ r in the generators x_{ij}, and C^{r} := μ(O(M_{q})^{≤r}), B^{r} := η(C^{r}). Note that O(M_{q})^{≤r}, C^{r}, D^{r} are subcomodules. The symbols ψ_{C^{r}}, ψ_{B^{r}} stand for the obvious subcorepresentations of ψ_{C}, ψ_{B}.

For r ∈ ℕ_{0}, a vector space basis of C^{r} is

{y_{11}^{i+1}y_{12}^{j}y_{21}^{k}, y_{12}^{l}y_{21}^{m}y_{22}^{n} | i, j, k, l, m, n ∈ ℕ_{0}, i + j + k ≤ r − 1, l + m + n ≤ r} (8)

(by the same argument as the corresponding result for O(SL_{q}) is proved in [6, 4.1.5]). From (8) we see that the character of the corepresentation ψ_{C^{r}} is

χ(ψ_{C^{r}}) = ∑_{i+j+k≤r−1} z^{2(k−j)} + ∑_{l+m+n≤r} z^{2(m−l)}.  (9)

(This character does not depend on q or ξ.) In particular, C^{r}[n] = 0 if n > 2r, and C^{r}[2r] = ℂy_{21}^{r}, implying that the corepresentation ψ_{C^{r}} has a unique irreducible summand isomorphic to T_{r}, and y_{21}^{r} generates the corresponding simple subcomodule of C^{r}. It follows that η(y_{21}^{r}) is either zero, or it generates a simple subcomodule of B, on which the given corepresentation is isomorphic to T_{r}.

In order to show that η(y_{21}^{r}) is non-zero, we introduce a ℤ-grading deg on O(M_{q}) as follows: deg(x_{11}) := 1, deg(x_{22}) := −1, deg(x_{12}) := 0, deg(x_{21}) := 0. It is easy to check that deg extends to an algebra-grading of O(M_{q}). Moreover, ker(μ) is a homogeneous ideal, so C inherits the grading. Note that u is non-homogeneous, and deg(y_{21}^{r}) = 0. Since C is a domain, any non-zero multiple of u is non-homogeneous. Thus y_{21}^{r} is not contained in uC. In other words, η(y_{21}^{r}) is non-zero for any r ∈ ℕ_{0}.

Consequently, ψ_{B^{r}} contains a subcorepresentation isomorphic to T_{0} ⊕T_{1} ⊕⋯ T_{r}, on the subcomodule generated by 1, η(y_{21}), …, η(y_{21}^{r}). On the other hand, ker(η)∩C^{r} = uC ∩C^{r} clearly contains uC^{r−1}, so B^{r} is a factorcomodule of C^{r}/uC^{r−1}. Since u is a coinvariant and C is a domain, by (4) we get that the corepresentation on uC^{r−1} is isomorphic to ψ_{C^{r−1}}, hence the character of the corepresentation on C^{r}/uC^{r−1} is χ(ψ_{C^{r}}) − χ(ψ_{C^{r−1}}). Using (9) we get that

χ(ψ_{C^{r}}) − χ(ψ_{C^{r−1}}) = ∑_{i+j+k=r−1} z^{2(k−j)} + ∑_{l+m+n=r} z^{2(m−l)}

= ∑_{j+k≤r−1} z^{2(k−j)} + ∑_{l+m≤r} z^{2(m−l)}

= ∑_{s=0}^{r} χ(T_{s}).  (10)

So the corepresentation on C^{r}/uC^{r−1} is also isomorphic to ⨁_{s=0}^{r} T_{r}. It follows that C^{r} ∩ uC = uC^{r−1}, and ψ_{B^{r}} ≅ ⨁_{s=0}^{r} T_{r}. Moreover, the subcomodule corresponding to T_{s} is generated by η(y_{21}^{s}). This shows (7), since B = ⋃_{r=0}^{∞} B^{r}.

As an immediate consequence of (7) and the fact that ψ^{ξ} = ν ∘η ∘μ for some morphism ν : B → O(S_{q,∞}^{2}) of comodules, we obtain that the image of ψ^{ξ} is the subcomodule of O(S_{q,∞}^{2}) generated by {ν(η(y_{21}^{n})) | n ∈ ℕ_{0}} = {ψ^{ξ}(x_{21}^{n}) | n ∈ ℕ_{0}}. Moreover, the kernel of ψ^{ξ} is (η∘μ)^{−1}(ker(ν)). By (7) B is the direct sum of pairwise non-isomorphic simple subcomodules, and so any subcomodule is a direct sum of some of these simple subcomodules; in particular, ker(ν) is the O(SL_{q})-subcomodule of B generated by {η(y_{21}^{n}) | ν(η(y_{21}^{n})) = 0}. Therefore (η ∘ μ)^{−1}(ker(ν)) is the sum of ker(η ∘ μ) = ∑_{i=1,2}(τ_{i} − τ_{i}(ξ))O(M_{q}) and the subcomodule of O(M_{q}) generated by {x_{21}^{n} | ψ^{ξ}(x_{21}^{n}) = 0}. So (i), (ii), (iii), and (iv) follow from the fact that W^{n} ≤ O(S_{q,∞}^{2}) is generated by {x_{−1}^{r} | r = 0, 1, …, n} as an O(SL_{q})-comodule, and from the evaluation of ψ^{ξ}(x_{21}^{n}) given below.

Since ψ^{ξ} is not an algebra homomorphism, it is helpful to write it as the composition ψ^{ξ} = p ∘ Ψ, where Ψ is the map

Ψ := ((S ∘ π) ⊗ ev_{ξ} ⊗ π) ∘ Δ^{(2)} : O(M_{q}) → O(SL_{q}) ⊗ O(SL_{q}),

and p : O(SL_{q}) ⊗ O(SL_{q}) → O(SL_{q}) is the multiplication map in O(SL_{q}). Introduce a new multiplication ∗ on O(SL_{q}) ⊗ O(SL_{q}) given by (x_{1} ⊗ y_{1}) ∗ (x_{2} ⊗ y_{2}) = x_{2}x_{1} ⊗ y_{1}y_{2}. Observe that Ψ is an algebra homomorphism from O(M_{q}) to (O(SL_{q}) ⊗ O(SL_{q}), ∗). We have that

Ψ [\begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix}] = [\begin{matrix}ξ_{1}d ⊗ a − q^{−1}ξ_{2}b ⊗ c & ξ_{1}d ⊗ b − q^{−1}ξ_{2}b ⊗ d \\ −qξ_{1}c ⊗ a + ξ_{2}a ⊗ c & −qξ_{1}c ⊗ b + ξ_{2}a ⊗ d \end{matrix}] .  (11)

By induction on n we show that

ψ^{ξ}(x_{21}^{n}) = (−q)^{n}(ξ_{1} − ξ_{2})(ξ_{1} − q^{2}ξ_{2}) ⋯ (ξ_{1} − q^{2n−2}ξ_{2})c^{n}a^{n}.  (12)

Indeed, (12) holds for n = 1 by (11). Assume that (12) holds for n. Then

ψ^{ξ}(x_{21}^{n+1}) = p(Ψ(x_{21}^{n+1})) = p(Ψ(x_{21}^{n}) ∗ Ψ(x_{21}))
= p(Ψ(x_{21}^{n}) ∗ (−qξ_{1}c ⊗ a + ξ_{2}a ⊗ c))
= −qξ_{1}cp(Ψ(x_{21}^{n}))a + ξ_{2}ap(Ψ(x_{21}^{n}))c
= (−qξ_{1} + q^{2n+1}ξ_{2})cψ^{ξ}(x_{21}^{n})a,

and formula (12) easily follows for n + 1. Hence if ∏_{i=0}^{n−1}(ξ_{1} − q^{2i}ξ_{2}) ≠ 0, then ψ^{ξ}(x_{21}^{n}) is a non-zero scalar multiple of x_{−1}^{n}, and if ∏_{i=0}^{n−1}(ξ_{1} − q^{2i}ξ_{2}) = 0, then ψ^{ξ}(x_{21}^{n}) = 0.

The same arguments show (i’), (ii’), (iii’), (iv’), by using the fact that

φ^{ξ}(x_{21}^{n}) = (−q)^{n}(ξ_{1} − q^{−2}ξ_{2})(ξ_{1} − q^{−4}ξ_{2}) ⋯ (ξ_{1} − q^{−2n}ξ_{2})a^{n}c^{n}.
□

4 Co-orbit maps for non-diagonal 2 × 2 matrices

In this section we still restrict to the case N = 2. The kernel and the image of the co-orbit map can be described also when ξ is a non-diagonal ℂ-point of M_{q}.

Proposition 4.1 Let 0 ≠ q ∈ ℂ, and take ξ := [\begin{matrix}0 & ξ_{1} \\ 0 & 0 \end{matrix}], where ξ_{1} ≠ 0.

(i) The image of β^{ξ} is the subalgebra of O(GL_{q}) generated by x_{21}^{2}/det_{q}, x_{21}x_{22}/det_{q}, x_{22}^{2}/det_{q}. In particular, im(β^{ξ}) is a quantum homogeneous space.

(ii) The kernel of β^{ξ} is the right ideal of O(M_{q}) generated by τ_{1} and τ_{2}.

(i’) im(α^{ξ}) = ℂ⟨x_{21}^{2}/det_{q}, x_{21}x_{22}/det_{q}, x_{22}^{2}/det_{q}⟩.

(ii’) ker(α^{ξ}) = ∑_{i=1}^{2} O(M_{q})σ_{i}.

Again it is more convenient to switch from the O(GL_{q})-coactions to the corresponding O(SL_{q})-coactions.

Lemma 4.2 The restriction of π maps ℂ⟨x_{21}^{2}/det_{q}, x_{21}x_{22}/det_{q}, x_{22}^{2}/det_{q}⟩ isomorphically onto ℂ⟨c^{2}, cd, d^{2}⟩.

Proof. The subalgebras ℂ⟨x_{21}/det_{q}, x_{22}/det_{q}⟩ of O(GL_{q}) and ℂ⟨c, d⟩ of O(SL_{q}) are isomorphic to the coordinate ring of the quantum plane: the generators q-commute, and satisfy no more relations, as can be seen from standard basis arguments. The two algebras in our statement correspond to the subalgebra of the coordinate ring of the quantum plane generated by its quadratic elements, or in other words, the sum of the even degree homogeneous components. □

Proof of Proposition 4.1. As before, ψ = (id ⊗ π) ∘ β and ψ^{ξ} = (ev_{ξ} ⊗ id) ∘ ψ. The statements about β^{ξ} translate by Lemma 4.2 to the assertions ker(ψ^{ξ}) = ∑_{i=1}^{2} τ_{i}O(M_{q}) and im(ψ^{ξ}) = ℂ⟨c^{2}, cd, d^{2}⟩.

As in the proof of Proposition 3.1, decompose ψ^{ξ} as p ∘ Ψ, where Ψ = ((S ∘ π) ⊗ ev_{ξ} ⊗ π) ∘ Δ^{(2)} and p is the multiplication map in O(M_{q}). By definition we have

Ψ [\begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix}] = [\begin{matrix}ξ_{1}d ⊗ c & ξ_{1}d ⊗ d \\ −qξ_{1}c ⊗ c & −qξ_{1}c ⊗ d \end{matrix}] .  (13)

Since Ψ is an algebra homomorphism from O(M_{q}) to (O(SL_{q}) ⊗ O(SL_{q}), ∗), it is easy to see from the images of the generators that im(Ψ) is contained in ∑_{k=0}^{∞} ℂ⟨c, d⟩_{k} ⊗ ℂ⟨c, d⟩_{k}, where ℂ⟨c, d⟩_{k} denotes the degree k component of ℂ⟨c, d⟩. This space is mapped by p into W := ℂ⟨c^{2}, cd, d^{2}⟩, showing that im(β^{ξ}) ⊆ W. We have

ψ^{ξ}(x_{21}^{n}) = p ∘ Ψ(x_{21}^{n}) = p(Ψ(x_{21})^{n}) = (−1)^{n}q^{n}ξ^{n}c^{2n},

implying that im(ψ^{ξ}) contains c^{2n} for all n ∈ ℕ_{0}. Similarly, ψ^{ξ}(x_{21}^{i}x_{12}^{j}x_{11}^{k}) is a non-zero scalar multiple of (c^{2})^{i}(d^{2})^{j}(dc)^{k}, hence we obtain (i).

Set C := O(M_{q})/τ_{2}O(M_{q}) and B := C/μ(τ_{1})C, where μ : O(M_{q}) → C is the natural homomorphism. These are graded homomorphic images of O(M_{q}) (endowed with the usual grading). By Proposition 2.4 the linear map β^{ξ} : O(M_{q}) → W factors through B, denote by η : B → W the induced surjective linear map. We shall show that η is an isomorphism (this is clearly equivalent to the assertion ker(ψ^{ξ}) = ∑_{i=1}^{2} τ_{i}O(M_{q})). Define O(M_{q})^{≤r} as the degree ≤ r part of O(M_{q}), and write C^{r},B^{r}, W^{r} for the image of O(M_{q})^{≤r} in C, B, W. We know from [5] that C is a domain, and C^{r} has the basis (8). By homogeneity of μ(τ_{1}) we have C^{r} ∩ μ(τ_{1})C = μ(τ_{1})C^{r−1}. Hence dim(B^{r}) = dim(C^{r}) − dim(C^{r−1}) = (r + 1)^{2} (the latter equality follows for example from (10)). Since dim(W^{r}) = (r + 1)^{2} as well, the map η|_{B^{r}} : B^{r} → W^{r} is an isomorphism. This holds for all r ∈ ℕ_{0}, hence η : B → W is an isomorphism.

The same arguments work for α^{ξ}. □

A similar result holds when ξ = [\begin{matrix}0 & 0 \\ ξ_{1} & 0 \end{matrix}] is lower triangular.

Remark 4.3 ℂ⟨c^{2}, cd, d^{2}⟩ and O(S_{q,∞}^{2}) are isomorphic as right O(SL_{q})-comodules. However, they are not isomorphic quantum homogeneous spaces, since they are not isomorphic as algebras. Indeed, O(S_{q,∞}^{2}) is generated by x_{−1} and x_{+1} as a ℂ-algebra with unity. Assume that ℂ⟨c^{2}, cd, d^{2}⟩ is also generated by two elements. We may suppose that the generators are contained in the maximal ideal (c^{2}, cd, d^{2}). Then the images of the generators span the 3-dimensional ℂ-vector space (c^{2}, cd, d^{2})/(c^{2}, cd, d^{2})^{2}. This is a contradiction.

It is not difficult to check that O(SL_{q}) is a free left ℂ⟨c^{2}, cd, d^{2}⟩-module. Therefore by [13, Theorem 1] the right coideal subalgebra ℂ⟨c^{2}, cd, d^{2}⟩ of O(SL_{q}) can be realized as the space of coinvariants with respect to the left coaction of an appropriate quotient left O(SL_{q})-module coalgebra of O(SL_{q}) (called a left coisotropic quantum subgroup of O(SL_{q}) in [1]). This quotient left O(SL_{q})-module coalgebra can be viewed as the “stabilizer” of ξ.

5 The generic case

We return to quantum matrices of arbitrary size N, but assume that q is transcendental over the base field. It turns out that if the classical adjoint orbit of ξ ∈ M(N, ℂ) is of maximal dimension, then the kernel of the (quantized) co-orbit map coincides with the subset predicted by the classical theory, and is given in terms of coinvariants. For a diagonal ξ with pairwise different eigenvalues, the image of the co-orbit map also reflects the classical picture.

Let A be an arbitrary commutative ring and q a unit in A. We replace ℂ by A in the constructions of Sections 1 and 2, and define the A-bialgebra O_{A}(M_{q}), the A-Hopf algebras O_{A}(GL_{q}), O_{A}(D), the right coaction β_{A} : O_{A}(M_{q}) → O_{A}(M_{q}) ⊗ O_{A}(GL_{q}), the β_{A}-coinvariants τ_{1}, …, τ_{N} ∈ O_{A}(M_{q}). We say that an N × N matrix ξ is an Apoint of M_{q}, if the entries of ξ are elements of A satisfying the relations (2). Such a ξ determines an A-algebra homomorphism ev_{ξ} : O_{A}(M_{q}) → A, x_{ij} ↦ ξ_{ij}, and the co-orbit map β_{A}^{ξ} : O_{A}(M_{q}) → O_{A}(GL_{q}) is defined as in Definition 2.1. The A-algebra O_{A}(M_{q}) is graded, the generators x_{ij} having degree 1. Write O_{A}(M_{q})^{≤d} for the subspace spanned by the homogeneous components of degree ≤ d.

Let K be a subfield of ℂ, and let q be an indeterminate over K. We shall apply the above constructions in the case when A = K(q), the field of rational functions in q, or when A = K[q, q^{−1}], the ring of Laurent polynomials. When (A, q) = (K, 1), then O_{A}(M_{q}) becomes O_{K}(M), the coordinate ring of the space of N ×N matrices over K, and O_{A}(GL_{q}) becomes the coordinate ring O_{K}(GL) of the general linear group GL(N, K).

Denote by η : K[q, q^{−1}] → K the K-algebra surjection mapping q to 1. The symbol η will stand also for the induced surjections M(N, K[q, q^{−1}]) → M(N, K), O_{K[q,q^{−1}]}(GL_{q}) → O_{K}(GL).

Lemma 5.1 The elements f_{1}, …, f_{s} ∈ O_{K[q,q^{−1}]}(GL_{q}) are K(q)-linearly independent in O_{K(q)}(GL_{q}), if η(f_{1}), …, η(f_{s}) are K-linearly independent in O_{K}(GL).

Proof. Assume that a_{1}f_{1} + ⋯ + a_{s}f_{s} = 0 is a non-trivial relation with a_{i} ∈ K(q). Multiplying by an appropriate element of K[q] we may achieve that a_{i} ∈ K[q, q^{−1}] for all i. Since K[q, q^{−1}] is a unique factorization domain, cancelling an appropriate power of q − 1 we ensure that not all the a_{i} are contained in the ideal ⟨q − 1⟩. Then apply the map η to get a non-trivial K-linear relation ∑_{i=1}^{s} η(a_{i})η(f_{i}) = 0. □

We fix a K[q, q^{−1}]-point ξ of M_{q} such that the centralizer of η(ξ) in M(N, K) has dimension N; that is, the adjoint orbit of η(ξ) is of maximal dimension. For a fixed d ∈ ℕ_{0} set

X^{d} := O_{K(q)}(M_{q})^{≤d}, X_{0}^{d} := O_{K[q,q^{−1}]}(M_{q})^{≤d}, X_{1}^{d} := O_{K}(M)^{≤d},
b^{d} := β_{K(q)}^{ξ}|_{X^{d}}, b_{0}^{d} := β_{K[q,q^{−1}]}^{ξ}|_{X_{0}^{d}}, b_{1}^{d} := β_{K}^{η(ξ)}|_{X_{1}^{d}},

Y^{d} := im(b^{d}) ⊂ O_{K(q)}(GL_{q}), Y_{0}^{d} := im(b_{0}^{d}) ⊂ O_{K[q,q^{−1}]}(GL_{q}), Y_{1}^{d} := im(b_{1}^{d}) ⊂ O_{K}(GL).

Then we have a commutative diagram

X^{d} ⊃ X_{0}^{d} ⟶^{η} X_{1}^{d}
↓ b^{d}  ↓ b_{0}^{d}  ↓ b_{1}^{d}
Y^{d} ⊃ Y_{0}^{d} ⟶^{η} Y_{1}^{d}.

Denote by Z_{1}^{d} the kernel of b_{1}^{d}. It is the intersection of X_{1}^{d} and the ideal of O_{K}(M) generated by η(τ_{i} − τ_{i}(ξ)), i = 1, …, N, by the classical result [7, Theorem 10] (the base field is ℂ in this paper; since β^{η(ξ)} is obtained from β_{K}^{η(ξ)} by extending scalars, and since the τ_{i} are defined over the field of rational numbers, the cited results of [7] hold over any subfield K of ℂ). Moreover, by Lemma 5.2 below, Z_{1}^{d} has a K-basis of the form

Λ_{1} := ⋃_{i=1}^{N}{η(τ_{i} − τ_{i}(ξ))w | w ∈ Γ^{i}},

where Γ^{i} is an appropriate set of monomials in the variables x_{kl} of degree ≤ d − i, for i = 1, …, N.

Lemma 5.2 Let R be a finitely generated commutative polynomial algebra over K, endowed with the usual grading. For r ∈ R write r = r̄ + r̂, where r̄ is the highest degree homogeneous component of r. Let u_{1}, …, u_{N} be given elements of R, and assume that ū_{1}, …, ū_{N} is a regular sequence in R. Then for all f in the ideal generated by u_{1}, …, u_{N}, there exist elements f_{i} ∈ R, i = 1, …, N, such that f = ∑_{i=1}^{N} u_{i}f_{i}, and deg(u_{i}f_{i}) ≤ deg(f), i = 1, …, N.

Proof. Apply induction on N. The case N = 1 is trivial. Assume that the lemma is true for N − 1. Write f ∈ ∑_{i=1}^{N} u_{i}R as f = ∑_{i=1}^{N} u_{i}h_{i}, where d := max{deg(u_{i}h_{i}) | i = 1, …, N} is minimal. If d ≤ deg(f), then we are done. Suppose that d > deg(f). Without loss of generality we may assume that deg(u_{N}h_{N}) = d because any permutation of ū_{1}, …, ū_{N} is a regular sequence, see the corollary to [9, Theorem 16.3]. (We note that it is not essential for the proof to make this assumption, however, it simplifies the notation below.) Then

∑_{i:deg(u_{i}h_{i})=deg(u_{N} h_{N})} ū_{i}h̄_{i} = 0.

By assumption ū_{N} is not a zero-divisor modulo ∑_{i=1}^{N−1} ū_{i}R, hence h̄_{N} = ∑_{i=1}^{N−1} ū_{i}g_{i} with appropriate homogeneous elements g_{i}, where deg(ū_{i}g_{i}) = deg(h̄_{N}), i = 1, …, N − 1. Thus

u_{N}h_{N} = u_{N}(h̄_{N} + ĥ_{N})

= u_{N} ĥ_{N} + ∑_{i=1}^{N−1} u_{N} ū_{i}g_{i}

= u_{N} ĥ_{N} + ∑_{i=1}^{N−1} u_{N}(u_{i} − û_{i})g_{i}

= ∑_{i=1}^{N−1} u_{i}(u_{N}g_{i}) + u_{N}(ĥ_{N} − ∑_{i=1}^{N−1} ĥ_{i}g_{i}).

Obviously deg(u_{i}u_{N}g_{i}) ≤ d, deg(u_{N} û_{i}g_{i}) < d for i = 1, …, N − 1. Set b_{i} := h_{i} + u_{N}g_{i} for i = 1, …, N − 1, and b_{N} := ĥ_{N} − ∑_{i=1}^{N−1} û_{i}g_{i}. We have f = ∑_{i=1}^{N} u_{i}b_{i}, with deg(u_{i}b_{i}) ≤ d for i = 1, …, N − 1, and deg(u_{N}b_{N}) < deg(u_{N}h_{N}) = d. We claim that deg(∑_{i=1}^{N−1} u_{i}b_{i}) < d. Indeed, either deg(∑_{i=1}^{N−1} u_{i}b_{i}) ≤ deg(u_{N}b_{N}) < d, or deg(∑_{i=1}^{N−1} u_{i}b_{i}) > deg(u_{N}b_{N}) implying deg(∑_{i=1}^{N−1} u_{i}b_{i}) = deg(f) < d. By the induction hypothesis on N, there exist c_{1}, …, c_{N−1} ∈ R with ∑_{i=1}^{N−1} u_{i}b_{i} = ∑_{i=1}^{N−1} u_{i}c_{i}, and deg(u_{i}c_{i}) ≤ deg(∑_{i=1}^{N−1} u_{i}b_{i}) < d for i = 1, …, N − 1. Therefore

f = u_{1}c_{1} + ⋯ + u_{N−1}c_{N−1} + u_{N}b_{N},

where each summand on the right hand side has degree < d. This contradicts to our assumption on the minimality of d. □

Since η(τ_{1}), …, η(τ_{N}) is a regular sequence in O_{K}(M) by [7, Theorem 10], we may apply Lemma 5.2 to conclude the existence of the basis Λ_{1} in Z_{1}^{d}. Our next aim is to show that Λ_{1} can be lifted to a basis of the kernel b^{d}. There is a ℤ^{N}-grading on the spaces X^{d}, X_{1}^{d}, Y_{d}, Y_{1}^{d}, determined by a right coaction of O_{A}(D). This coaction is (id ⊗ π_{D}) ∘ β_{A} for X^{d}, X_{1}^{d}, and (id ⊗ π_{D}) ∘ Δ for Y^{d}, Y_{1}^{d}. (Recall that a coaction φ : V → V ⊗ O_{A}(D) yields the direct sum decomposition V = ⨁_{a∈ℤ^{N}}{v ∈ V | φ(v) = v ⊗ t_{1}^{a_{1}} ⋯ t_{N}^{a_{N}} }.) The maps b^{d}, b_{1}^{d}, η are compatible with the ℤ^{N}-grading, because b^{d}, b_{1}^{d} are homomorphisms of O_{A}(GL_{q})-comodules, hence they are homomorphisms of O_{A}(D)-comodules, whereas η maps a monomial in O_{K[q,q^{−1}]}(GL_{q}) to formally the same monomial in O_{K}(GL). Write χ(V) for the Hilbert series of a finite dimensional ℤ^{N}-graded vector space V, so χ(V) is an element of ℤ[t_{1}^{±1}, …, t_{N}^{±1}]. In other words, χ(V) is the character of the corresponding corepresentation of O_{A}(D) on V. Introduce a partial order ≥ on ℤ[t_{1}^{±1}, …, t_{n}^{±1}] by defining f ≥ g if all coefficients of f − g are non-negative. Obviously W ≤ V and χ(W) = χ(V) imply W = V.

Denote by Z^{d} the intersection of X^{d} and the right ideal of O_{K(q)}(GL_{q}) generated by τ_{i} − τ_{i}(ξ), i = 1, …, N. The same argument as in the proof of Proposition 2.4 shows that Z^{d} ⊆ ker(b^{d}).

Proposition 5.3 For all d ∈ ℕ_{0} we have the following:

(i) the kernel of b^{d} is Z^{d};

(ii) the Hilbert series χ(Y^{d}) equals χ(Y_{1}^{d}).

Proof. Choose a set Ω_{1} of monomials of degree ≤ d in O_{K}(M) such that Λ_{1} ∪ Ω_{1} is a K-basis of X_{1}^{d}. Set

Λ := {(τ_{i} − τ_{i}(ξ))ŵ | i = 1, …, N; w ∈ Γ^{i}},

where ŵ is a chosen monomial in X_{0}^{d} with η(ŵ) = w. The elements of Λ are multihomogeneous with respect to the ℤ^{N}-grading introduced on X^{d}. Similarly, lift each v ∈ Ω_{1} to a monomial v̂ ∈ X_{0}^{d} with η(v̂) = v. We obtain the set Ω := {v̂ | v ∈ Ω_{1}} of multihomogeneous elements in X^{d}.

As we noted above, Λ is a subset of ker(b^{d}), and it is K(q)-linearly independent by Lemma 5.1. Therefore

χ(ker(b^{d})) ≥ χ(Z^{d}) ≥ χ(Span_{K(q)}{Λ}) = χ(Span_{K}{Λ_{1}}) = χ(Z_{1}^{d}).  (14)

Since Z_{1}^{d} = ker(b_{1}^{d}), we have that Ω_{1} is mapped under b_{1}^{d} to a basis of Y_{1}^{d}. Again by Lemma 5.1 we have that b^{d}(Ω) is a K(q)-linearly independent subset of Y^{d}. Thus

χ(Y^{d}) ≥ χ(Span_{K(q)}{b^{d}(Ω)}) = χ(Span_{K}{b_{1}^{d}(Ω_{1})}) = χ(Y_{1}^{d}).  (15)

It is well known (see [6, 9.2.1 Proposition 6]) that there is a set of monomials in the variables x_{ij} which is a basis both for X^{d} and X_{1}^{d}. It follows that

χ(Y^{d}) + χ(ker(b^{d})) = χ(X^{d}) = χ(X_{1}^{d}) = χ(Y_{1}^{d}) + χ(Z_{1}^{d}).  (16)

Comparing (14), (15), (16) we obtain that all inequalities in (14) and (15) must be equalities. In particular, χ(Y^{d}) = χ(Y_{1}^{d}) in (15) and χ(ker(b^{d})) = χ(Span_{K(q)}{Λ}) in (14), implying ker(b^{d}) = Z^{d} = Span_{K(q)}{Λ}. □

As an immediate corollary we obtain the following.

Theorem 5.4 Let ξ be a K(q)-point of M_{q} such that the centralizer of η(ξ) in M(N, K) has dimension N. Then the kernel of β_{K(q)}^{ξ} coincides with the right ideal ∑_{i=1}^{N}(τ_{i} − τ_{i}(ξ))O_{K(q)}(M_{q}).

We determine the comodule structure of β_{K(q)}^{ξ} with the aid of corepresentation theory. The material we shall summarize below can be found in [6, 11.5] (the base field is ℂ in that book; however, the irreducible corepresentations of O(GL_{q}) are defined over the field of rational numbers, so the results mentioned below obviously hold over K and K(q)). In the sequel (A, q) will stand for either of (K(q), q) or (K, 1). Recall that O_{A}(GL_{q}) is cosemisimple. The irreducible corepresentations are indexed by the set P of dominant integral weights for GL(N). Denote by T_{q}(λ) the irreducible O_{A}(GL_{q})-corepresentation associated with λ ∈ P, write χ(T_{q}(λ)) for its character. Recall that by the character χ(T) of a finite dimensional O_{A}(GL_{q})-corepresentation T : V → V ⊗ O_{A}(GL_{q}) we mean the image under π_{D} of the sum of the diagonal matrix coefficients. In other words, the character χ(T) is the same as the Hilbert series χ(V) with respect to the ℤ^{N}-grading of V determined by the O_{A}(D)-corepresentation (id ⊗ π_{D}) ∘ T. Any O_{A}(GL_{q})-corepresentation decomposes as a direct sum of copies of the irreducible corepresentations T_{q}(λ). Given a finite dimensional corepresentation T, the multiplicity of T_{q}(λ) as a summand of T is the same as the (uniquely determined) coefficient of χ(T_{q}(λ)) in χ(T), expressed as a linear combination of the characters χ(T_{q}(μ)), μ ∈ P. We shall need also the fact that the character of the irreducible O_{K(q)}(GL_{q})-corepresentation T_{q}(λ) coincides with the character of the O_{K}(GL)-corepresentation T_{1}(λ) (which coincides with the character of the irreducible rational representation of GL(N, ℂ) associated with λ). Consequently, if we are given an O_{K(q)}(GL_{q})-corepresentation T : V → V ⊗ O_{K(q)}(GL_{q}) and an O_{K}(GL)- corepresentation U : W → W ⊗ O_{K}(GL) such that χ(V) = χ(W), then the multiplicity of T_{q}(λ) as a summand of T is the same as the multiplicity of T_{1}(λ) as a summand of U, for all λ ∈ P.

To simplify the notation we write T (λ) := T_{1}(λ). The O_{K}(GL)-corepresentation on im(β_{K}^{η(ξ)}) decomposes as

im(β_{K}^{η(ξ)}) ≅ ⨁_{λ∈P} m(λ)T (λ),

where m(λ) denotes the dimension of the zero weight space in the dual corepresentation T^{∗}(λ), by [7, Theorem 0.4 and formula (0.1.6)]. Although the result is stated in [7] in the language of representations, it can clearly be converted to the language of corepresentations. Indeed, observe that the character of the O_{K}(GL)-corepresentation on im(β_{K}^{η(ξ)}) coincides with the usual formal character of the natural right action of GL(N, K) on this space, and im(β_{K}^{η(ξ)}) is the coordinate ring of the closure of the adjoint orbit of η(ξ). So m(λ) is the dimension of the subspace of the underlying vector space of T^{∗}(λ) consisting of the vectors v with (id ⊗ π_{D}) ∘ T^{∗}(λ)(v) = v ⊗ 1. In particular, m(λ) is finite for all λ ∈ P, and does not depend on η(ξ) (still assuming that the centralizer of η(ξ) has dimension N). The multiplicity of T_{q}(λ) in the O_{K(q)}(GL_{q})-corepresentation on O_{K(q)}(GL_{q}) is the dimension of T_{q}(λ) (see [6, 11.5.4 Theorem 51 and 11.1.4 Proposition 8 (ii)]), hence the subcorepresentation on im(β_{K(q)}^{ξ}) decomposes as

im(β_{K(q)}^{ξ}) ≅ ⨁_{λ∈P} m_{q}^{ξ}(λ)T_{q}(λ)

with m_{q}^{ξ}(λ) finite for all λ ∈ P.

Proposition 5.5 Let ξ be a K(q)-point of M_{q} such that the centralizer of η(ξ) in M(N, K) has dimension N. Then for all λ ∈ P we have m_{q}^{ξ}(λ) = m(λ).

Proof. Denote by m^{d}(λ) the multiplicity of T (λ) as a summand in Y_{1}^{d}, and denote by m_{q}^{ξ,d}(λ) the multiplicity of T_{q}(λ) as a summand in Y^{d}. Obviously

m(λ) = sup{m^{d}(λ) | d ∈ ℕ_{0}}  (17)

and

m_{q}^{ξ}(λ) = sup{m_{q}^{ξ,d}(λ) | d ∈ ℕ_{0}},  (18)

because im(β_{K}^{ξ}) = ⋃_{d∈ℕ_{0}} Y_{1}^{d} and im(β_{K(q)}^{ξ}) = ⋃_{d∈ℕ_{0}} Y^{d}. On the other hand, χ(Y^{d}) = χ(Y_{1}^{d}) by Proposition 5.3, implying

m_{q}^{ξ,d}(λ) = m^{d}(λ) for all d ∈ ℕ_{0}, λ ∈ P.  (19)

So our statement follows from equations (17), (18), (19). □

If η(ξ) is diagonal with pairwise different eigenvalues, then the stabilizer of η(ξ) with respect to the classical adjoint action is the diagonal subgroup of GL(N, K), and the orbit of η(ξ) is closed in M(N, K). It follows then that the image of β_{K}^{η(ξ)} is O_{K}(D \ GL). In particular, the multiplicity of T (λ) as a direct summand of the O_{K}(GL)-corepresentation on O_{K}(D \ GL) is m(λ). We show that the same holds in the q-deformed case.

Lemma 5.6 The multiplicity of T_{q}(λ) as a direct summand of the O_{K(q)}(GL_{q})-corepresent-ation on O_{K(q)}(D \ GL_{q}) is m(λ) for all λ ∈ P.

Proof. In the sequel (A, q) will stand for either of (K(q), q) or (K, 1). First we describe a spanning set of O_{A}(D \ GL_{q}). For an N × N matrix a = (a_{ij}) with non-negative integer entries and a non-negative integer d set

x_{d}^{a} := det_{q}^{−d}x_{11}^{a_{11}}x_{12}^{a_{12}} ⋯ x_{NN}^{a_{NN}}

(the variables x_{ij} are ordered lexicographically). The elements of this form span O_{A}(GL_{q}), and

(π_{D} ⊗ id) ∘ Δ(x_{d}^{a}) = t_{d}^{a} ⊗ x_{d}^{a},

where

t_{d}^{a} := ∏_{i=1}^{N} t_{i}^{−d+∑_{j=1}^{N} a_{ij}}.

It follows that O_{A}(D \ GL_{q}) is spanned by the x_{d}^{a} with ∑_{j=1}^{N} a_{ij} = d for i = 1, …, N. Set

L_{A}^{d} := Span_{A}{x_{d}^{a} | ∑_{j=1}^{N} a_{ij} = d, i = 1, …, N}.

Observe that the given spanning set of L_{A}^{d} is actually a basis, because after multiplication by det_{q}^{d}, it becomes a set of linearly independent monomials of degree dN from the standard basis of the domain O_{A}(M_{q}). Clearly

O_{A}(D \ GL_{q}) = ⋃_{d∈ℕ_{0}} L_{A}^{d},

and L_{A}^{d} is a right O_{A}(GL_{q})-subcomodule of O_{A}(D \ GL_{q}) for all d.

Note that

(id ⊗ π_{D}) ∘ Δ(x_{d}^{a}) = x_{d}^{a} ⊗ t_{a,d} with t_{a,d} := ∏_{j=1}^{N} t_{j}^{−d+∑_{i=1}^{N} a_{ij}}.

It follows that

χ(L_{A}^{d}) = ∑_{a∈ω_{d}} t_{a,d},

where

ω_{d} := {a = (a_{ij}) ∈ M(N, ℕ_{0}) | ∑_{j=1}^{N} a_{ij} = d for i = 1, …, N}.

Observe that the above computation of χ(L_{A}^{d}) is the same when A = K(q) and when A = K. It follows that for all d ∈ ℕ_{0} the multiplicity of T_{q}(λ) in L_{K(q)}^{d} is the same as the multiplicity of T (λ) in L_{K}^{d}. Consequently, the multiplicity of T_{q}(λ) in O_{K(q)}(D \ GL_{q}) is the same as the multiplicity of T (λ) in O_{K}(D \ GL) = im(β_{K}^{η(ξ)}). □

Theorem 5.7 Let ξ be a diagonal K[q, q^{−1}]-point of M_{q} such that η(ξ) has pairwise different eigenvalues. Then the image of β_{K(q)}^{ξ} coincides with O_{K(q)}(D \ GL_{q}).

Proof. We have im(β_{K(q)}^{ξ}) ⊆ O_{K(q)}(D \ GL_{q}) by the same argument as in the proof of Proposition 2.6. The multiplicity of T_{q}(λ) is m(λ) both in O_{K(q)}(D \ GL_{q}) and in im(β_{K(q)}^{ξ}) by Proposition 5.5 and Lemma 5.6. This implies that im(β_{K(q)}^{ξ}) is the whole of O_{K(q)}(D \ GL_{q}). □

Finally we return to the coordinate ring of quantum matrices over ℂ. The results of this section have the following corollary in the framework of Section 2.

Corollary 5.8 Let ξ be a ℂ-point of M_{q} such that the centralizer of ξ in M(N, K) has dimension N. Assume that q ∈ ℂ^{∗} is transcendental over the subfield K of ℂ generated by the entries of ξ. Then the kernel of β^{ξ} coincides with the right ideal of O(M_{q}) generated by τ_{i} − τ_{i}(ξ), i = 1, …, N. If in addition ξ is diagonal, then the image of β^{ξ} is O(D \ GL_{q}).

Proof. Since q is transcendental over K, we may apply Theorems 5.4 and 5.7 for the kernel and the image of β_{K(q)}^{ξ}. The ℂ-linear map β^{ξ} is obtained from the K(q)-linear map β_{K(q)}^{ξ} by extending scalars to ℂ. Therefore a generating set of ker(β_{K(q)}^{ξ}) as a K(q)-vector space spans ker(β^{ξ}) over ℂ, and a generating set of im(β^{ξ}) as a K(q)-vector space spans im(β^{ξ}) over ℂ. □

It is straightforward to modify the statements of this section to obtain similar results for the coaction α.

References

[1] F. Bonechi, N. Ciccoli, R. Giachetti, E. Sorace, M. Tarlini, The coisotropic subgroup structure of SL_{q}(2, ℝ), Journal of Geometry and Physics 37 (2001), 190-200.

[2] T. Brzeziński, Quantum homogeneous spaces as quantum quotient spaces, J. Math. Phys. 37 (5) (1996), 2388–2399.

[3] M. S. Dijkhuizen and T. H. Koornwinder, Quantum homogeneous spaces, duality and quantum 2-spheres, Geom. Dedicata 52 (1994), 291–315.

[4] M. Domokos and T. H. Lenagan, Conjugation coinvariants of quantum matrices, Bull. London Math. Soc., to appear.

[5] D. A. Jordan, Normal elements of degree one in Ore extensions, Comm. Alg. 30 No. 2 (2002), 803–807.

[6] A. Klimyk and K. Schmüdgen, Quantum Groups and Their Representations, Springer-Verlag, Berlin, Heidelberg, New York, 1997.

[7] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.

[8] Yu. I. Manin, Quantum Groups and Non-commutative Geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988. vi+91 pp.

[9] H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

[10] B. Parshall and J. Wang, Quantum linear groups, Mem. Amer. Math. Soc 89 (1991), no.439.

[11] P. Podleś, Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.

[12] N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, (Russian), Algebra i Analiz 1 (1989), 178–206.

[13] M. Takeuchi, Relative Hopf modules - equivalences and freeness criteria, J. Alg. 60 (1979), 452–471.

M. Domokos: Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary E-mail: domokos@renyi.hu, domokos@maths.ed.ac.uk (Domokos is in Edinburgh until February 2004.)

R. Fioresi: Dipartimento di Matematica, Universita’ di Bologna Piazza Porta San Donato 5, 40126 Bologna, Italy E-mail: fioresi@dm.unibo.it

T. H. Lenagan: Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland E-mail: tom@maths.ed.ac.uk