Primitive ideals of C_{q}[SL(3)]

Timothy J. Hodges^{∗} University of Cincinnati Cincinnati, OH 45221-0025, U.S.A.

Thierry Levasseur Université de Bretagne Occidentale 29287 Brest Cedex, France

Abstract

The primitive ideals of the Hopf algebra C_{q}[SL(3)] are classified. In particular it is shown that the orbits in Prim C_{q}[SL(3)] under the action of the representation group H ≅ C^{∗}×C^{∗} are parameterized naturally by W × W where W is the associated Weyl group. It is shown that there is a natural oneto-one correspondence between primitive ideals of C_{q}[SL(3)] and symplectic leaves of the associated Poisson algebraic group SL(3, C).

Introduction

The primitive spectrum of a noncommutative affine algebra is the natural generalization of the variety associated to a commutative affine algebra. When the noncommutative algebra A is a deformation of a commutative algebra B, one expects to find a close correspondence between the primitive ideals of A and the symplectic leaves of the associated Poisson structure on the variety Max(B). For instance if g is a solvable complex Lie algebra, then the primitive ideals of the enveloping algebra U(g) correspond to the coadjoint orbits in g^{∗}, which are the symplectic leaves for the Kostant-Kirillov Poisson structure on g^{∗}.

A similar close correspondence seems likely to occur for quantum groups and related algebras. Let G be a semi-simple complex Lie group and let C_{q}[G] be the associated quantum group as defined in [16]. There is a standard Poisson Lie group structure on G associated to C_{q}[G]. The primitive ideals of C_{q}[G] are expected to correspond bijectively to the symplectic leaves of G. This correspondence may be verified for SL(2) by direct calculation. In this paper we study the primitive ideals of C_{q}[SL(n)] and prove that the primitive ideals of C_{q}[SL(3)] correspond exactly to the symplectic leaves of SL(3).

When q is real, q ≠ 1, C_{q}[G] together with a natural involution * can be viewed as a deformation of C[K], the algebra of functions on a maximal compact subgroup K of G. In a series of articles in [18, 19, 20] Soibelman, and Vaksman showed that the unitary representations of C_{q}[K] correspond to the symplectic leaves of K.

Fix a maximal torus H in G. Then G has a natural H-invariant Poisson structure [4]. A description of the symplectic leaves of G may be deduced from the work of Semenov-Tian-Shansky and Lu and Weinstein [11, 17]; an outline of this description is given in Appendix A. Let W be the Weyl group of G. The symplectic leaves fall into H-orbits parameterized by W × W. Let D = G × G, identify G with the diagonal subgroup of D and let G_{r} be the dual group. Denote by p the natural projection G → D/G_{r}. The symplectic leaves of G are precisely the connected components of the inverse images of the left G_{r}-orbits in D/G_{r}. Set Γ = ker p and Ḡ = p(G). Then Γ is a finite subgroup of H and Ḡ = G/Γ is an open subset of D/G_{r}. For each w ∈ W × W, let C_{w} be the image of the corresponding Bruhat cell of D in D/G_{r}. Denote by Cẇ a fixed G_{r}-orbit in C_{w}. Then Cẇ ≅ C^{l} × (C^{∗})^{s} and C_{w} is the union of the H-translates of Cẇ. Each symplectic leaf of G is then a finite cover of hCẇ ∩ Ḡ for some w ∈ W × W and some h ∈ H.

∗Partially supported by a grant from the N.S.A.

Key words and phrases: Quantum group, primitive ideal, Poisson Lie group, symplectic leaves.

AMS Mathematics Subject Classification (1980): 17B37, 16W30, 16S80, 16S30, 58F06, 81R50.

In section two we prove some preliminary results about the primitive spectrum of C_{q}[SL(n)]. The group H occurs again in the quantum case as the character group and Prim C_{q}[SL(n)] therefore decomposes into the union of the H-orbits. Following ideas of Soibelman [18, 19], we define for each w ∈ W ×W a locally closed H-invariant subset Prim_{w} of Prim C_{q}[SL(n)]. It may be shown that Prim_{w} is nonempty for all w and that Prim C_{q}[SL(n)] = ⨆_{w} Prim_{w}. We conjecture that each Prim_{w} is a single H-orbit and that the elements of Prim_{w} are in bijection with the leaves of type w. This conjecture is proved in sections three and four for C_{q}[SL(3)]. The truth of the conjecture for C_{q}[SL(2)] was proved earlier by S.P Smith and the first author. This result is outlined in Appendix B.

In order to describe the symplectic leaves of G one passes first to Ḡ. Similarly, in order to describe the primitive ideals of C_{q}[G], we first study the invariant subalgebra C_{q}[Ḡ] = C_{q}[G]^{Γ}. The quantum analog of C_{w} ∩ Ḡ is a certain localization of a homomorphic image of C_{q}[G] denoted by B_{w}. The key result in section three is the decomposition of B_{w} as the tensor product Bẇ ⊗ C[H_{w}] where Bẇ is a quantum analog of Cẇ ∩ Ḡ and C[H_{w}] is the algebra of functions on the torus H_{w} = H/Stab _{H}Cẇ.

Acknowledgements. The authors would like to thank S.P. Smith for many interesting conversations concerning these questions. The first author would also like to thank H.J. Koelink for bringing his work to his attention. Much of this work was done while the first author was visiting the Université de Bretagne Occidentale.

1 Preliminaries

1.1 In this section we introduce the basic definitions and notation that we shall be using. We denote by g the Lie algebra sl(n, C) and by G the Lie group SL(n, C). We follow the standard notation in Bourbaki for the roots, weights, Weyl group etc. associated to g. Other notation is listed at the end of the paper.

1.2 Let q ∈ C^{∗}. We shall assume throughout this paper that q is not a root of unity. We denote by Q the set {q^{n} | n ∈ Z}. Let [a_{ij}] be the Cartan matrix associated to g. Recall that the quantum universal enveloping algebra associated to g is defined to be the algebra U_{q}(g) generated by K_{i}^{±1}, X_{i}^{±}, 1 ≤ i ≤ n−1 with relations

K_{i}^{−1}K_{i} = K_{i}K_{i}^{−1} = 1, K_{i}X_{j}^{±} = q^{±a_{ij}}X_{j}^{±}K_{i}

[X_{i}^{+}, X_{j}^{−}] = δ_{ij} \frac{K_{i}^{2} − K_{i}^{−2}}{q^{2} − q^{−2}} , K_{i}K_{j} = K_{j}K_{i}

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1 − a_{ij} \\ k \end{matrix} ] (X_{i}^{±})^{k}X_{j}^{±}(X_{i}^{±})^{1−a_{ij}−k} = 0, if i ≠ j

where [ \begin{matrix}m \\ k \end{matrix} ] = ∏_{j=1}^{k} \frac{q^{2(m−j+1)} − q^{−2(m−j+1)}}{q^{2j} − q^{−2j}}

(see, for example, [12]). The algebra U_{q}(g) is a Hopf algebra. The comultiplication Δ is defined by

Δ(X_{i}^{±}) = X_{i}^{±} ⊗ K_{i}^{−1} + K_{i} ⊗ X_{i}^{±}, Δ(K_{i}) = K_{i} ⊗ K_{i}

and the counit and antipode by

ϵ(X_{i}^{±}) = 0, ϵ(K_{i}) = 1, S(K_{i}) = K_{i}^{−1}, S(X_{i}^{±}) = −q^{∓2}X_{i}^{±}.

There is also a C-linear antiautomorphism a ↦ a^{∗} given by (X_{i}^{±})^{∗} = X_{i}^{∓}, (K_{i})^{∗} = K_{i}. It is easily verified that Δ(a^{∗}) = Δ(a)^{∗} (where (a ⊗ b)^{∗} = a^{∗} ⊗ b^{∗}) and S(S(a)^{∗}) = a^{∗}.

1.3 Set U^{o} = C[K_{i}^{±1} | 1 ≤ i ≤ n − 1]. Let M be a U^{o}-module. If χ is a character of U^{o} define the χweight space of M by M_{χ} = {x ∈ M | ux = χ(u)x, ∀u ∈ U^{o}}. Set Ω(M) = {χ | M_{χ} ≠ 0}. Let P be the set of weights of g and let {α_{1}, …, .α_{n−1}} be a fixed set of positive roots. Each weight λ ∈ P induces a character of U^{o} via λ(K_{i}) = q^{(λ,α_{i})}, 1 ≤ i ≤ n − 1. We denote by M_{λ} the associated weight space.

Define C to be the category of finite dimensional U_{q}(g) modules such that M = ⨁_{μ∈P} M_{μ}. Since C is closed under finite direct sums, tensor products and passage to the dual module, we may define the restricted dual of U_{q}(g) with respect to C. This is the associated quantum group C_{q}[G]. Thus

C_{q}[G] = {f ∈ U_{q}(g)^{∗} | Kerf ⊇ AnnM for some M ∈ C}

The algebra C_{q}[G] then has a natural Hopf algebra structure induced in the usual way from that on U_{q}(g). There is also an anti-automorphism on C_{q}[G] induced from that on U_{q}(g) by ℓ^{∗}(u) = ℓ(S(u)^{∗}) for all ℓ ∈ C_{q}[G] and all u ∈ U_{q}(g).

Let π : U_{q}(g) ⟶ End(M) ≅ M_{m}(C), π(a) = [π_{ij}(a)], be an m-dimensional representation of U_{q}(g) where M is an object of C. The elements π_{ij} ∈ U_{q}(g)^{∗} are called the matrix elements or matrix coefficients of the representation π. It is clear that these π_{ij} belong to C_{q}[G] and that the set of all such π_{ij} for all possible M in C, spans C_{q}[G] as a vector space. Recall the following useful formulas:

Δπ_{ij} = ∑_{k} π_{ik} ⊗ π_{kj}, π_{ij}π_{kl} = (π_{ij} ⊗ π_{kl}) ∘ Δ, S(π_{ij}) = π_{ij} ∘ S, ϵ(π_{ij}) = π_{ij}(1)

1.4 The category C is in some sense a deformation of the category of finite dimensional modules over the Lie algebra g [12]. Denote by P_{+} the set of dominant weights of g. For each dominant weight Λ ∈ P_{+} there is a simple module L(Λ) in C and an element v_{Λ} ∈ L(Λ) such that

1. L(Λ) = U_{q}(g)v_{Λ} = ⨁_{μ∈P,μ≤Λ} L(Λ)_{μ};

2. L(Λ)_{Λ} = Cv_{Λ}, X_{i}^{+}v_{Λ} = 0, 1 ≤ i ≤ n − 1; (v_{Λ} is called the highest weight vector of L(Λ))

3. the set of weights Ω(Λ) = Ω(L(Λ)) and the multiplicities are the same as for the corresponding simple g-module.

Any M ∈ C decomposes as M = ⨁_{Λ∈P_{+}} L(Λ)^{m_{Λ}}. The representation ring of C is generated by the classes of the simple modules L(ϖ_{i}) corresponding to the fundamental dominant weights ϖ_{i}, 1 ≤ i ≤ n−1. Moreover each L(Λ) occurs as a subquotient of a suitable power of the standard representation L(ϖ_{1}). On the other hand the dual of L(ϖ_{1}) is isomorphic to L(ϖ_{n−1}) which is isomorphic to the (n − 1)-th quantum exterior power of L(ϖ_{1}). Hence if the matrix coefficients with respect to the natural basis e_{1}, …, e_{n} of L(ϖ_{1}) are denoted by X_{ij} then the matrix coefficients corresponding to L(ϖ_{n−1}) are the quantum minors defined by:

D_{ij} = ∑_{σ∈S_{n−1}}(−q^{2})^{ℓ(σ)}X_{1,σ(1)} … X_{i−1,σ(i−1)}X_{i+1,σ(i+1)} … X_{n,σ(n)}

where S_{n−1} denotes the symmetric group acting in the usual way as bijections from {1, …, i − 1, i + 1, …, n} to {1, …, j − 1, j + 1, …, n}. .

From these and related facts one deduces the following well-known description of the Hopf algebra C_{q}[G].

Theorem 1.4.1 (a) The algebra C_{q}[G] is generated by the X_{ij}, 1 ≤ i, j ≤ n, with relations:

X_{iℓ}X_{jℓ} = q^{2}X_{jℓ}X_{iℓ}, ∀ℓ, ∀i < j, X_{ℓi}X_{ℓj} = q^{2}X_{ℓj}X_{ℓi}, ∀ℓ, ∀i < j,
X_{ℓi}X_{mj} = X_{mj}X_{ℓi}, ∀ℓ < m, ∀i > j,
X_{ℓi}X_{mj} − X_{mj}X_{ℓi} = (q^{2} − q^{−2})X_{ℓj}X_{mi}, ∀ℓ < m, ∀i < j,

Det_{q} = ∑_{σ∈S_{n}}(−q^{2})^{ℓ(σ)}X_{σ(1),1} … X_{σ(n),n} = 1.

(b) The Hopf algebra structure is given by

Δ(X_{ij}) = ∑_{k} X_{ik} ⊗ X_{kj},  S(X_{ij}) = (−q^{2})^{i−j}D_{ji},  ϵ(X_{ij}) = δ_{ij}.

(c)The involution * is given by (X_{ij})^{∗} = (−q^{2})^{j−i}D_{ij}.

(d) Furthermore δ_{ij} = ∑_{k}(−q^{2})^{k−j}X_{ik}D_{jk} = ∑_{k}(−q^{2})^{i−k}D_{ki}X_{kj} = ∑_{k}(−q^{2})^{j−k}D_{jk}X_{ik} = ∑_{k}(−q^{2})^{k−i}X_{kj}D_{ki}

The reader is referred to [16] and [14] for further details concerning this algebra.

1.5 The generators described in the above section are not well suited to the study of the primitive ideals. A more natural set of generators is the following. This notation was first introduced by Soibelman in [18].

Recall that L(ϖ_{k}) ≅ ⋀^{k} L(ϖ_{1}) (the k-th quantum exterior power of L(ϖ_{1})) and that Ω(ϖ_{k}) = W ϖ_{k} where W denotes the Weyl group. Recall that W may be naturally identified with the symmetric group S_{n} by letting the reflection with respect to the simple root α_{i} correspond to the transposition (i, i + 1). Let \underline{i} = \{i_{1}, …, i_{k}\} be a subset of \{1, …, n − 1\} such that i_{1} < … < i_{k}. Define e\underline{i} = e_{i_{1}} ∧ … ∧ e_{i_{k}}. Then the weight spaces of L(ϖ_{k}) are exactly the Ce\underline{i}. For any w ∈ W define e_{wϖ_{k}} to be the element e\underline{i} of L(ϖ_{k}) where \underline{i} is the ordered multi-index associated to \{w(1), …, w(k)\}. It is easily verified that e_{wϖ_{k}} ∈ L(ϖ_{k})_{wϖ_{k}}. Define e_{−wϖ_{k}}^{∗} ∈ L(ϖ_{k})^{∗} to be the dual basis element coresponding to e_{wϖ_{k}} and denote by ⟨−, −⟩ the natural pairing between a vector space and its dual.

Definition For each k = 1, …, n − 1 and each w ∈ W we define elements c_{k,w}^{±} ∈ C_{q}[G] by: ∀u ∈ U_{q}(g), c_{k,w}^{+}(u) = ⟨e_{−wϖ_{k}}^{∗}, ue_{ϖ_{k}}⟩, c_{k,w}^{−}(u) = ⟨e_{−ww_{0}ϖ_{n−k}}^{∗}, ue_{w_{0}ϖ_{n−k}}⟩.

Thus c_{k,w}^{+} (respectively c_{k,w}^{−}) is a matrix coefficient of L(ϖ_{k}) (respectively L(ϖ_{n−k})). In particular we have that c_{1,(1,i)}^{+} = X_{i1}, c_{n−1,(i,n)}^{+} = D_{in}, c_{1,(1,i)}^{−} = D_{i1}, c_{n−1,(i,n)}^{−} = X_{in}. The general element c_{k,w}^{±} can be interpreted as a general quantum minor as defined in [14]. In the notation of that article,

c_{j,w}^{+} = ξ_{{1,...,j}}^{w{1,...,j}}, c_{j,w}^{−} = ξ_{{j+1,...,n}}^{w{j+1,...,n}}

where w{1, …, k} = {w(1), …, w(k)} etc.

One of the key properties of these matrix elements is that they generate C_{q}[G]. In fact a slightly stronger statement is true. Let A_{+} be the subalgebra of C_{q}[G] generated by the elements of the form c_{i,w}^{+} and let A_{−} be the algebra generated by the elements of the form c_{i,w}^{−}.

Theorem 1.5.1 The linear map A_{−} ⊗A_{+} ⟶ C_{q}[G] given by a⊗b ↦ ab is an epimorphism of C-vector spaces.

Proof. This result is Theorem 3.1 of [19]. It suffices to check that the definition of A_{±} given there is in fact the same as the one given above. □

1.6 On occasion we will need a notation for a coordinate function coming from an arbitrary representation in C. Our notation again follows Soibelman [19].

Let Λ ∈ P_{+}. Recall that L(Λ) = ⨁_{λ∈Ω(Λ)} L(Λ)_{λ}, L(Λ)^{∗} ≅ L(−w_{0}Λ) and L(Λ)_{−μ}^{∗} = [L(Λ)_{μ}]^{∗} Each module L(Λ) carries a non degenerate bilinear contravariant form (− | −)_{Λ} such that (av | w)_{Λ} = (v | a^{∗}w)_{Λ} for all a ∈ U_{q}(g) and v, w ∈ L(Λ). Such a form is unique up to a scalar multiple [7]. Choose an orthonormal basis {v_{μ}^{(j)} | μ ∈ Ω(Λ), 1 ≤ j ≤ dim L(Λ)_{μ}} of L(Λ) with respect to (− | −)_{Λ}. Let {ℓ_{−λ}^{(i)}} be the dual basis in L(Λ)^{∗}. Then each ℓ_{−λ}^{(i)} identifies with (v_{λ}^{(i)} | −)_{Λ} and ℓ_{−λ}^{(i)} ∈ L(Λ)_{−λ}^{∗}. Hence ⟨ℓ_{−λ}^{(i)}, v_{μ}^{(j)}⟩ = (v_{λ}^{(i)} | v_{μ}^{(j)})_{Λ} = δ_{λμ}δ_{ij}. We define elements c_{−λ,i,μ,j}^{Λ} of C_{q}[G] by:

∀u ∈ U_{q}(g), c_{−λ,i,μ,j}^{Λ}(u) = ⟨ℓ_{−λ}^{(i)}, uv_{μ}^{(j)}⟩ = (v_{λ}^{(i)} | uv_{μ}^{(j)})_{Λ}.

For convenience we use the following abreviations:

c_{−λ,i,μ,j}^{Λ} = \{ \begin{matrix}c_{−λ,μ,j}^{Λ} & if dim L(Λ)_{λ} = 1 \\ c_{−λ,i,μ}^{Λ} & if dim L(Λ)_{μ} = 1 \\ c_{−λ,μ}^{Λ} & if dim L(Λ)_{λ} = dim L(Λ)_{μ} = 1 \end{matrix}

The first two parts of the following lemma are taken from [19]. The third part is a consequence of the general formula in section 1.3.

Lemma 1.6.1 (a) S(c_{−λ,i,μ,j}^{Λ}) ∈ Cc_{μ,j,−λ,i}^{−w_{0}Λ}

(b) (c_{−λ,i,μ,j}^{Λ})^{∗} ∈ Cc_{λ,i,−μ,j}^{−w_{0}Λ}

(c) Δ(c_{−λ,i,μ,j}^{Λ}) = ∑_{ν,k} c_{−λ,i,ν,k}^{Λ} ⊗ c_{−ν,k,μ,j}^{Λ}

Notice that c_{k,w}^{+} ∈ Cc_{−wϖ_{k},ϖ_{k}}^{ϖ_{k}} and c_{k,w}^{−} ∈ Cc_{wϖ_{k},−ϖ_{k}}^{ϖ_{n−k}} = C(c_{k,w}^{+})^{∗}.

1.7 Let R^{±} denote the set of positive and negative roots respectively. Denote by b^{±} = h ⊕ n^{±} the Borel subalgebras associated to R^{±}. We denote by U_{q}(b^{±}) the Hopf subalgebras of U_{q}(g) generated by {K_{i}, X_{i}^{±} | 1 ≤ i ≤ n − 1} respectively (we call them the Borel subalgebras).

As in [19] we define the following ideals of C_{q}[G] which play a fundamental role in what follows:

I^{+}(w, Λ) = ⟨c_{−μ,i,Λ}^{Λ} | 1 ≤ i ≤ dim L(Λ)_{μ}, v_{μ}^{(i)} ∉ U_{q}(b^{+})v_{wΛ}⟩
I^{−}(w, Λ) = ⟨c_{μ,i,−Λ}^{−w_{0}Λ} | 1 ≤ i ≤ dim L(−w_{0}Λ)_{−μ}, v_{−μ}^{(i)} ∉ U_{q}(b^{−})v_{−wΛ}⟩

Notice that in the definition of I^{−}(w, Λ) the v_{−μ}^{(i)}’s belong to L(−w_{0}Λ)_{−μ}. Notice also that the condition v_{μ}^{(i)} ∉ U_{q}(b^{+})v_{wΛ} can be expressed in the form c_{−μ,i,wΛ}^{Λ}(u) = 0 for all u ∈ U_{q}(b^{+}).

Define τ to be the involutive automorphism τ = ∗ ∘ S. For any U_{q}(g)-module M we denote by M^{τ} the twisted module where the action of an element u ∈ U_{q}(g) on an element v ∈ M is given by u · v = τ(u)v. Then it is easily verified that L(−w_{0}Λ) ≅ L(Λ)^{τ}. This isomorphism takes v_{−μ}^{(i)} ∈ L(−w_{0}Λ) onto v_{μ}^{(i)} ∈ L(Λ)_{μ}. Since τ(U_{q}(b^{+})) = U_{q}(b^{−}) we obtain that

v_{μ}^{(i)} ∉ U_{q}(b^{+})v_{wΛ} ⟺ v_{−μ}^{(i)} ∉ U_{q}(b^{−})v_{−wΛ}.

Therefore Lemma 1.6.1 shows that I^{+}(w, Λ)^{∗} = I^{−}(w, Λ).

1.8 We shall need some elementary facts about the Bruhat ordering on W. We take the reverse of the usual Bruhat ordering introduced in [3]. Thus e ≤ w ≤ w_{0} for all w ∈ W. For each fundamental weight ϖ_{i} we denote the stabiliser of ϖ_{i} in W by W_{i} = Stab(ϖ_{i}). Denote by Ŵ_{i} a fixed transversal of W_{i} in W.

Definition Fix i ∈ {1, …, n − 1}. Let y, w ∈ W. we say that y ≤_{i} w if and only if yϖ_{i} ≥ wϖ_{i}.

It is clear that ≤_{i} is right W_{i}-invariant and that the induced ordering on W/W_{i} is a partial ordering. In order to keep the notation consistent, we shall sometimes use the notation =_{i} for equivalence modulo W_{i}. The proof of the following proposition is similar to standard arguments concerning the Bruhat ordering (for instance [3, §7.7]).

Proposition 1.8.1 Let i ∈ {1, …, n − 1} and let y, w ∈ W.

1. The following are equivalent: (a) y ≤_{i} w; (b) v_{yϖ_{i}} ∈ U_{q}(b^{+})v_{wϖ_{i}}; (c) v_{wϖ_{i}} ∈ U_{q}(b^{−})v_{yϖ_{i}}.

2. y ≤_{i} w ⇔ yw_{0} ≥_{n−i} ww_{0}.

3. y ≤ w ⇔ y ≤_{i} w for all i.

Example If we identify W as above with the symmetric group S_{n}, then the subgroup W_{1} = Stab_{W} (ϖ_{1}) identifies with the group S_{n−1} = Sym {2, …, n} and we may take Ŵ_{1} to be {e = (1, 1), (1, 2), …, (1, n)}. The ordering ≤_{1} is then given by

W_{1} <_{1} (1, 2)W_{1} <_{1} ⋯ <_{1} (1, n)W_{1}.

Similarly W_{n−1} = Sym {1, …, n−1} and we may take the transversal Ŵ_{n−1} to be {(n, 1), (n, 2), …, (n, n) = e}. The ordering ≤_{n−1} is then given by

W_{n−1} <_{n−1} (n, n − 1)W_{n−1} <_{n−1} ⋯ <_{n−1} (n, 1)W_{n−1}.

In the case we shall be most interested in (when n = 3) these are of course the only two cases.

2 The Algebras A_{w}, B_{w} and C_{w}.

2.1 In order to study C_{q}[G] in more detail we introduce algebras A_{w}, B_{w} and C_{w} defined for each w ∈ W × W. The motivation for the definitions of these algebras comes from the structure of the symplectic leaves in G. Recall the notation of Appendix A. There are natural maps G ⟶ Ḡ ↪ D/G_{r} and a symplectic leaf of G is a connected component of the inverse image of a left G_{r}-orbit of D/G_{r}. The Bruhat cells C_{w} of D/G_{r} are disjoint unions of isomorphic leaves of “type w”. Just as in this geometric case it is natural to study the symplectic leaves by type, so in the study of C_{q}[G] it is natural to classify primitive ideals by type. The algebras C_{w}, B_{w} and A_{w} correspond to the cell C_{w} and its inverse image in Ḡ and G respectively.

2.2 Setting Λ = ϖ_{i} in 1.7 we obtain the ideals I^{±}(w, ϖ_{i}). From Lemma 1.6.2 and Proposition 1.8.1 it follows that

I^{±}(w, ϖ_{i}) = ⟨c_{i,y}^{±} | y ≰_{i} w⟩.

Henceforth, the principal objects of interest will be the ideals defined for each w = (w_{+}, w_{−}) ∈ W × W by:

I_{w} = ∑_{i=1}^{n−1}(I^{+}(w_{+}, ϖ_{i}) + I^{−}(w_{−}, ϖ_{i})) = ⟨c_{i,y}^{ε} | 1 ≤ i ≤ n − 1, y ≰_{i} w_{ε}⟩.

and the sets, defined also for each w = (w_{+}, w_{−}) ∈ W × W by

E_{w} = {c_{i,w_{+}}^{+}, c_{i,w_{−}}^{−} | i = 1, ..., n − 1} .

We shall also occasionally use the following notation. For y ∈ W we define I^{±}(y) = ∑_{i=1}^{n−1} I^{±}(y, ϖ_{i}) and E^{±}(y) = {c_{i,y}^{±} | i = 1, ..., n}. For w = (w_{+}, w_{−}), we define I_{w}^{±} = I^{±}(w_{±}), and E_{w}^{±} = E^{±}(w_{±}).

Theorem 2.2.1 Let w ∈ W. The image of c_{i,w}^{ε} is normal in C_{q}[G]/I^{ε}(w, ϖ_{i}). In fact we have that

c_{−λ,i,μ,j}^{Λ}c_{i,w}^{ε} = γc_{i,w}^{ε}c_{−λ,i,μ,j}^{Λ} (mod I^{ε}(w, ϖ_{i})) for some γ ∈ Q.

Proof. Recall that C_{q}[G] = C [c_{−λ,i,μ,j}^{Λ} | Λ ∈ P_{+}] and that c_{i,w}^{+} is a scalar multiple of c_{−wϖ_{i},ϖ_{i}}^{ϖ_{i}}. The

ideal J_{0}(wϖ_{i}, ϖ_{i}) defined in [19] is precisely the ideal I^{+}(w, ϖ_{i}) defined above. The result for c_{i,w}^{+} then follows from [19, Prop. 3.2]. Applying the involution ∗ yields the result for c_{i,w}^{−}. □

Corollary 2.2.2 For any w = (w_{+}, w_{−}) ∈ W × W, the elements of E_{w} (respectively E_{w}^{ε}) are normal in C_{q}[G]/I_{w} (respectively C_{q}[G]/I_{w}^{ε}).

Now let w = (w_{+}, w_{−}) ∈ W × W. Denote by E_{w} the multiplicatively closed set generated by the images of the elements of E_{w} in C_{q}[G]/I_{w}. Since E_{w} consists of normal elements we may localize with respect to this set. Denote the localized algebra

A_{w} = (C_{q}[G]/I_{w})_{E_{w}} .

It is not immediately clear that A_{w} ≠ 0 since it could happen that E_{w} ∩ I_{w} ≠ ∅. In the next few subsections we shall prove the following result:

Theorem 2.2.3 For all w ∈ W × W, A_{w} ≠ 0.

The idea of the proof is to construct a non-zero A_{w}-module by tensoring together certain “fundamental” C_{q}[G]-modules. This technique was used by Soibelman in [19, §5]; the idea is apparently due to Drinfeld. It is a quantum analog of the proof that p^{−1}(C_{w}) ≠ ∅ given in Appendix A.

Definition. A non-zero C_{q}[G]-module is said to be of type w ∈ W × W if (i) I_{w}M = 0 and (ii) ∀c ∈ E_{w}, M = cM (i.e., M is E_{w}-divisible).

It is a standard fact that a module of type w has a natural structure as an A_{w}-module. Thus the theorem will be a consequence of the existence of a nontrivial module of type w for all w ∈ W × W.

2.3 For each i ∈ {1, ..., n − 1}, denote by U_{q}(sl_{i}(2)), the Hopf subalgebra generated by {X_{i}^{+}, X_{i}^{−}, K_{i}^{±1}}; denote by U_{q}(b_{i}^{ε}) the subalgebra generated by {X_{i}^{ε}, K_{i}^{±1}}. Consider the following commutative diagram of inclusions:

\begin{matrix}U_{q}(b_{i}^{ε}) & ⟶ & U_{q}(sl_{i}(2)) \\ ↓ & ↘^{φ_{ε,i}} & ↓ φ_{i} \\ U_{q}(b^{ε}) & φ_{ε}^{⟶} & U_{q}(g) \end{matrix}

Since U_{q}(b^{ε}) is a Hopf subalgebra, the subspace U_{q}(b^{ε})^{⊥} = {f ∈ C_{q}[G] | f(U_{q}(b^{ε})) = 0} is an ideal of C[G]. Define C_{q}[B^{ε}] = C_{q}[G]/U_{q}(b^{ε})^{⊥} and define similarly C_{q}[B_{i}^{ε}] and C_{q}[SL_{i}(2)]. Then we have a commutative diagram of surjections,

\begin{aligned}C_{q}[B_{i}^{ε}] & ⟵ & C_{q}[SL_{i}(2)] \\ ↑ & ↖^{φ_{ε,i}^{∗}} & ↑ φ_{i}^{∗} \\ C_{q}[B^{ε}] & φ_{ε}^{∗}^{⟵} & C_{q}[G] \end{aligned}

It is easily verified that the canonical isomorphism, U_{q}(sl_{i}(2)) ≅ U_{q}(sl(2)) induces an isomorphism C_{q}[SL(2)] ≅ C_{q}[SL_{i}(2)] such that the kernel of C_{q}[SL_{i}(2)] → C_{q}[B_{i}^{+}] is I_{(e,s)} (and likewise Ker (C_{q}[SL_{i}(2)] → C_{q}[B_{i}^{−}]) = I_{(s,e)}). From the Theorem in Appendix B, we know that there exist C_{q}[SL(2)] modules M^{+} and M^{−} of type (s, e) and (e, s) respectively. Define M_{i}^{±} to be the modules M^{±} considered as C_{q}[G] modules via the map C_{q}[G] ⟶ C_{q}[SL_{i}(2)] ˜ ⟶C_{q}[SL(2)]. Then in particular we have that Ann M_{i}^{±} ⊇ ker φ_{i,∓}^{∗}.

Proposition 2.3.1 The modules M_{i}^{+} and M_{i}^{−} are of type (s_{i}, e) and (e, s_{i}) respectively.

Proof. We give the proof for M_{i}^{+}. We first need to show that I_{(s_{i},e)} ⊂ Ann M_{i}^{+}; it is enough to show that I^{+}(s_{i}, ϖ) + I^{−}(e, ϖ) ⊂ Ann M_{i}^{+} for each fundamental weight ϖ. Notice that I^{−}(e, ϖ) = ⟨c_{μ,−ϖ}^{−w_{0}ϖ} | v_{−μ} ∉ U_{q}(b^{−})v_{−ϖ}⟩ = ⟨c_{μ,−ϖ}^{−w_{0}ϖ} | φ_{−}^{∗}(c_{μ,−ϖ}^{−w_{0}ϖ}) = 0⟩ ⊂ Ker φ_{−}^{∗} ⊂ Ann M_{i}^{+}. On the other hand, I^{+}(s_{i}, ϖ) = ⟨c_{−λ,ϖ}^{ϖ} | v_{λ} ∉ U_{q}(b^{+})v_{s_{i}ϖ}⟩. Suppose that I^{+}(s_{i}, ϖ) ⊄ Ker φ_{i}^{∗}. Then exists a λ such that v_{λ} ∉ U_{q}(b^{+})v_{s_{i}ϖ} and v_{λ} ∈ U_{q}(sl_{i}(2))v_{ϖ}. Since X_{i}^{+}v_{ϖ} = 0 and X_{i}^{+}v_{s_{i}ϖ} ∈ Cv_{ϖ}, we obtain v_{λ} ∈ U_{q}(sl_{i}(2))v_{ϖ} = Cv_{ϖ} + Cv_{s_{i}ϖ} ⊆ U_{q}(b^{+})v_{s_{i}ϖ}, a contradiction.

It remains to show that M_{i}^{+} is E_{(s_{i},e)}-divisible. Recall that elements of E_{(s_{i},e)} are of the form c_{j} = c_{−s_{i}ϖ_{j},ϖ_{j}}^{ϖ_{j}} or c′_{j} = c_{ϖ_{j},−ϖ_{j}}^{−w_{0}ϖ_{j}}. We first compute φ_{i}^{∗}(c_{j}) acting on M^{+} via the identification C_{q}[SL_{i}(2)] = C_{q}[SL(2)]. The U_{q}(sl_{i}(2)) module generated by v_{ϖ_{j}} is either trivial (when (ϖ_{j}, α_{i}) = 0) or is the fundamental representation with highest weight vector v_{ϖ_{i}} (when (ϖ_{j}, α_{i}) = 1). It follows that φ_{i}^{∗}(c_{j}) = (c_{−sρ,ρ}^{ρ})^{(ϖ_{j},α_{i})} for which M_{i}^{+} is divisible by definition. A similar reasoning gives that φ_{i}^{∗}(c′_{j}) = (c_{−sρ,sρ}^{ρ})^{(ϖ_{j},α_{i})} which again acts divisibly by definition on M^{+}. □

2.4 We now show that modules of type w = (w_{+}, w_{−}) can be constructed by forming the tensor product of modules of the form M_{i}^{±} using the reduced decomposition of w_{+} and w_{−} . The fundamental result is the following.

Theorem 2.4.1 Let M be a C_{q}[G]-module of type (w_{+}, w_{−}). If s_{i}w_{+} > w_{+} (respectively s_{i}w_{−} > w_{−}) then M_{i}^{+} ⊗ M (respectively M ⊗ M_{i}^{−}) is a C_{q}[G]-module of type (s_{i}w_{+}, w_{−}) (respectively of type (w_{+}, s_{i}w_{−})).

Proof. We prove the assertion in the case s_{i}w_{+} > w_{+}.

(i) I^{−}(w_{−}, ϖ) ⊆ Ann(M_{i}^{+} ⊗ M) for all fundamental representations ϖ.

We denote c_{λ,μ}^{−w_{0}ϖ} by c_{λ,μ}. A standard generator for I^{−}(w_{−}, ϖ) is then of the form c_{λ,−ϖ} where v_{−λ} ∉ U_{q}(b^{−})v_{−w_{−}ϖ}. The action of c_{λ,−ϖ} is given by the comultiplication Δ(c_{λ,−ϖ}) = ∑_{μ∈Ω(−w_{0}ϖ)} c_{λ,μ} ⊗ c_{−μ,−ϖ} Suppose that the action is nontrivial. Then there exists a μ such that both factors c_{λ,μ} and c_{−μ,−ϖ} act non-trivially on M_{i}^{+} and M respectively. Since M is of type (w_{+}, w_{−}) this implies that v_{μ} ∈ U_{q}(b^{−})v_{−w_{−}ϖ}. Since Ann(M_{i}^{+}) ⊇ Ker(φ_{−}^{∗}), we must have that φ_{−}^{∗}(c_{λ,μ}) ≠ 0; thus v_{−λ} ∈ U_{q}(b^{−})v_{μ} ⊆ U_{q}(b^{−})v_{−w_{−}ϖ}, a contradiction.

(ii) I^{+}(s_{i}w_{+}, ϖ) ⊆ Ann(M_{i}^{+} ⊗ M) for all fundamental weights ϖ.

For these calculations we abbreviate c_{λ,μ}^{ϖ} by c_{λ,μ}. Then a standard generator of I^{+}(s_{i}w_{+}, ϖ) is c_{−λ,ϖ} where v_{λ} ∉ U_{q}(b^{+})v_{s_{i}w_{+}ϖ}. The action on M_{i} ⊗ M is given by: Δ(c_{−λ,ϖ}) = ∑_{μ∈Ω(ϖ)} c_{−λ,μ} ⊗ c_{−μ,ϖ}. Suppose that there exists a μ such that both c_{−λ,μ} and c_{−μ,ϖ} act non-trivially on M_{i}^{+} and M respectively. Then by definition and Proposition 1.8.1, v_{μ} ∈ U_{q}(b^{+})v_{w_{+}ϖ} ⊆ U_{q}(b^{+})v_{s_{i}w_{+}ϖ}. On the other hand, since Ann(M_{i}^{+}) ⊇ Ker(φ_{i}^{∗}) we must have v_{λ} ∈ U_{q}(sl_{i}(2))v_{μ}. Since s_{i}w_{+} > w_{+}, X_{i}^{−}v_{s_{i}w_{+}ϖ} = 0. Since moreover [X_{k}^{+}, X_{i}^{−}] = δ_{ik}(q^{2} − q^{−2})^{−1}(K_{i}^{2} − K_{i}^{−2}), it follows easily that

U_{q}(sl_{i}(2))U_{q}(b^{+})v_{s_{i}w_{+}ϖ} ⊆ U_{q}(b^{+})v_{s_{i}w_{+}ϖ}

which implies that v_{λ} ∈ U_{q}(b^{+})v_{s_{i}w_{+}ϖ}, a contradiction.

(iii) M_{i}^{+} ⊗ M is E_{(s_{i}w_{+},w_{−})}-divisible.

Let ϖ be a fundamental representation. We continue with the notation of part (ii). The action of c_{s_{i}w_{+}ϖ,ϖ} is given by:

Δ(c_{−s_{i}w_{+}ϖ,ϖ}) = ∑_{μ∈Ω(ϖ)} c_{−s_{i}w_{+}ϖ,μ} ⊗ c_{−μ,ϖ}.

Suppose μ is such that the corresponding summand is non-trivial. Then we have that (a) v_{μ} ∈ U_{q}(b^{+})v_{w_{+}ϖ} and (b) v_{s_{i}w_{+}ϖ} ∈ U_{q}(b_{i}^{−})v_{μ}. Consider the U_{q}(sl_{i}(2))-submodule of L(ϖ) containing v_{w_{+}ϖ}. Since s_{i}w_{+} > w_{+} it has highest weight w_{+}ϖ and lowest weight s_{i}w_{+}ϖ. If (w_{+}ϖ, α_{i}) = 0, the representation is trivial; otherwise (w_{+}ϖ, α_{i}) = 1 and the representation is the fundamental representation. From (b) we obtain μ = s_{i}w_{+}ϖ + pα_{i} = w_{+}ϖ + (p − (w_{+}ϖ, α_{i}))α_{i} where p is an integer between 0 and (w_{+}ϖ, α_{i}). From (a) we deduce that p = (w_{+}ϖ, α_{i}) and so μ = w_{+}ϖ. Thus for any m′ ∈ M_{i}^{+} and m ∈ M,

c_{−s_{i}w_{+}ϖ,ϖ} m′ ⊗ m = c_{−s_{i}w_{+}ϖ,w_{+}ϖ} m′ ⊗ c_{−w_{+}ϖ,ϖ}m.

By hypothesis M is c_{−w_{+}ϖ,ϖ}-divisible. On the other hand, φ_{i}^{∗}(c_{−s_{i}w_{+}ϖ,w_{+}ϖ}) = (c_{−sρ,ρ}^{ρ})^{(w_{+}ϖ,α_{i})} and M_{i}^{+} is divisible with respect to this element. Hence M_{i}^{+} ⊗ M is c_{−s_{i}w_{+}ϖ,ϖ}^{ϖ}-divisible. The proof for elements of the form c_{w_{−}ϖ,−ϖ}^{−w_{0}ϖ} is similar. □

Corollary 2.4.2 Let w_{+} = s_{i_{1}}...s_{i_{k}}, w_{−} = s_{j_{1}}...s_{j_{m}} be reduced expressions for w_{+} and w_{−} in w. Then

M_{i_{1}}^{+} ⊗ ... ⊗ M_{i_{k}}^{+} ⊗ M_{j_{m}}^{−} ⊗ .. ⊗ M_{j_{1}}^{−}

is a module of type (w_{+}, w_{−}).

This completes the proof of Theorem 2.2.3. These results generalize slightly [19, Propositions 5.1, 5.2].

2.5 Let R(C_{q}[G]) denote the set of one-dimensional representations of C_{q}[G]. Since C_{q}[G] is a Hopf algebra, R(C_{q}[G]) has a natural group structure. Let X = (X_{ij}) be the matrix of coordinate functions as described in 1.4. Since the X_{ij} generate C_{q}[G], there is a natural map from R(C_{q}[G]) to M_{n}(C) given by χ ↦ (χ(X_{ij})) = χ(X). It is easily verified that this is an isomorphism of R(C_{q}[G]) onto the set of invertible diagonal matrices. Since R(C_{q}[G]) is naturally isomorphic to this complex torus we shall denote it by H.

For any Hopf algebra A, there is a natural action of R(A) as automorphisms of A given by r_{χ}(a) = ∑ a_{(1)}χ(a_{(2)}) for all χ ∈ R(A) and a ∈ A. In the case A = C_{q}[G] the action of H on C_{q}[G] is therefore algebraic and given by r_{χ}(X) = Xχ(X).

Denote by Γ the subgroup of H corresponding to matrices with entries equal to ±1. Denote by γ_{i} for i = 1, ..., n − 1, the element with −1 in the (i, i) and (i + 1, i + 1) position and 1’s elsewhere. Obviously Γ is generated by the γ_{i}. Using the description of c_{i,w}^{ε} as a quantum minor given in 1.5 it is easily verified that the action of γ_{i} on the elements c_{i,w}^{ε} is given by

γ_{i}(c_{j,w}^{ε}) = \{ \begin{matrix}c_{j,w}^{ε} & if j ≠ i \\ −c_{j,w}^{ε} & if j = i. \end{matrix}

Definition. We denote by B = C_{q}[Ḡ] = C_{q}[G]^{Γ} the algebra of elements of C_{q}[G] invariant under the action of Γ.

Definition. Let w = (w_{+}, w_{−}) ∈ W × W. Recall that A_{w} = (A/I_{w})_{E_{w}}. Since γ(I_{w}) ⊆ I_{w} and γ(E_{w}) ⊆ E_{w} for all γ ∈ Γ, there is a natural induced action of Γ on A_{w}. We define B_{w} = A_{w}^{Γ}.

Notice that B_{w} = (B/(I_{w} ∩ B))_{(E_{w}∩B)}. In order to simplify the notation we continue to denote by c_{i,y}^{ε} the image of c_{i,y}^{ε} in A_{w}.

It is fairly easy to see that A_{w} has a natural structure as a crossed product of the dual group Γ̂ over B_{w}. Denote by Γ̂ the dual group of Γ and denote by γ̂_{i} the element of Γ̂ such that γ̂_{i}(γ_{j}) = (−1)^{δ_{ij}}. Define a map ϕ : Γ̂ → A_{w} by ϕ(γ̂_{i_{1}} … γ̂_{i_{t}}) = c_{i_{1},w_{+}}^{+} … c_{i_{t},w_{+}}^{+} if i_{1} < ⋯ < i_{t}. Then A_{w} is a crossed product of Γ̂ over B_{w} via ϕ in the sense of [13, 1.5.8].

2.6 Fix w = (w_{+}, w_{−}) ∈ W × W.

Definition. Let y ∈ W. In A_{w} set z_{i,y}^{ε} = c_{i,y}^{ε}(c_{i,w_{ε}}^{ε})^{−1} and t_{i} = c_{i,w_{−}}^{−}(c_{i,w_{+}}^{+})^{−1}.

Clearly these elements belong to B_{w}. We define C_{w} to be the subalgebra of B_{w} generated by the set

{z_{i,y}^{ε} | ε = ±, i = 1, ..., n − 1, y ∈ W } ∪ {t_{i}^{±1} | i = 1, ..., n − 1} .

Clearly z_{i,y}^{ε} = 0 for y >_{i} w_{ε} and z_{i,w_{ε}}^{ε} = 1. Thus

C_{w} = C [t_{i}^{±1}, z_{i,y}^{ε} | ε = ±, y <_{i} w_{ε}, i = 1, ..., n − 1] .

We now show that B_{w} is the localization of C_{w} with respect to an appropriate normal element. Recall [14, §2] that the relation Det_{q} = 1 may be written, for each i = 1, ..., n − 1, as 1 = ∑_{y∈Ŵ_{i}} α_{i,y}c_{i,y}^{+}c_{i,y}^{−} where α_{i,y} ∈ Q and Ŵ_{i} is a transversal of W_{i} in W. Using Theorem 2.2.1 and the description of the c_{i,w}^{ε} as quantum minors given in 1.5, we obtain that C_{w} contains the elements

d_{i} = (c_{i,w_{+}}^{+})^{−1}(c_{i,w_{−}}^{−})^{−1} = ∑_{y∈Ŵ_{i}} β_{i,y}z_{i,y}^{+}z_{i,y}^{−}

where β_{i,y} ∈ Q. Define d to be d_{1}d_{2}...d_{n−1}.

Theorem 2.6.1 The element d is a normal element of C_{w} and B_{w} = C_{w}[d^{−1}].

Proof. It follows easily from Theorem 2.2.1 that dA_{w} = A_{w}d. Since each z_{i,y}^{ε} is an eigenvector for conjugation by d, it is clear that dC_{w}d^{−1} = C_{w}. Thus d is a normal element of C_{w}. It follows from Theorem 1.5.1 that A_{w} is spanned by elements of the form vd^{t} where v is a word in the c_{i,y}^{ε} and t is a non-negative integer. Such words are clearly eigenvectors for the action of Γ. Hence B_{w} is spanned by the words with eigenvalue 1; that is, words for which the number of occurrences in v of elements of the form c_{i,y}^{ε}, for a fixed i is even, say 2m_{i}. For such words it follows from the normality of the elements c_{i,w_{ε}}^{ε} (Theorem 2.2.1) that if t > m_{i} for all i, then vd^{t} ∈ C_{w}. Hence for all b ∈ B_{w}, there exists a positive integer m such that bd^{m} ∈ C_{w}. □

2.7 We shall also be interested in the subalgebras of elements invariant under the action of the whole group H. There is a natural induced algebraic action of H on A_{w} and B_{w} Let λ ∈ C^{∗} and let h = λe_{ii} + λ^{−1}e_{i+1,i+1}. Then it follows from the description of the c_{i,y}^{ε} as quantum minors that

h(c_{j,y}^{±}) = \{ \begin{matrix}λ^{±1}c_{j,y}^{±} & if j = i \\ c_{j,y}^{±} & if j ≠ i. \end{matrix}

It is thus clear that the elements z_{i,y}^{ε} are H-invariant.

Theorem 2.7.1 (i) C_{w}^{H} = C[z_{i,y}^{ε} | ε = ±, 1 ≤ i ≤ n − 1, y ∈ W].

(ii) A_{w}^{H} = B_{w}^{H} = C_{w}^{H}[d^{−1}].

(iii) The monomials t_{1}^{r_{1}}...t_{n−1}^{r_{n−1}} for (r_{1}, ..., r_{n−1}) ∈ Z^{n−1} form a basis for C_{w} as a left or right C_{w}^{H}- module and a basis for B_{w} as a left or right B_{w}^{H}-module.

Proof. Denote by S the subalgebra of C_{w} generated by the z_{i,y}^{ε}. Clearly S ⊆ C_{w}^{H}. On the other hand, since C_{w} is generated over S by the t_{i} which are invertible elements normalising S (t_{i}S = St_{i}), it follows that the given monomials span C_{w} as a left or right S-module. It is also clear that if h = λe_{ii} + λ^{−1}e_{i+1,i+1}, then h(t_{i}) = λ^{−2}t_{i} and h(t_{j}) = t_{j} for j ≠ 1. Thus each distinct monomial corresponds to a different character of H. Hence the monomials must be linearly independent over C_{w}^{H}. Thus C_{w} = ⨁_{r∈Z^{n−1}} St^{r} where t^{r} = t_{1}^{r_{1}}...t_{n−1}^{r_{n−1}} if r = (r_{1}, ..., r_{n−1}). This proves (i) and the first part of (iii). The remaining assertions then follow easily. □

2.8 We are now in a position to formulate more precisely the conjectures made in the introduction concerning Prim C_{q}[G]. Although we only consider here the case when G = SL(n), similar conjectures may be made in the general case. The reader is referred to Appendix A for a description of the symplectic leaves of G.Denote by A the algebra C_{q}[G].

Definition For each w ∈ W × W, define Spec_{w}A = {P ∈ Spec A | P ⊇ I_{w} and P ∩ E_{w} = ∅} and Spec_{w}B = {P ∈ Spec B | P ⊇ I_{w} ∩ B_{w} and P ∩ E_{w} = ∅}. Elements of Spec_{w}A and Spec_{w}B are said to be of type w. Set Prim_{w}A = Spec_{w} A ∩ Prim A and Prim_{w}B = Spec_{w} B ∩ Prim B.

The action of H on A described above induces an action of H on Prim A for which the locally closed subsets Prim_{w}A are invariant for all w ∈ W × W. Since the action of H is algebraic, Stab_{H}P is a closed subgroup of H and H/Stab_{H}P is a torus for all P ∈ Prim A.

Conjecture 1: Prim A = ⨆_{w∈W ×W} Prim_{w}A and Prim_{w}A is a non-empty H-orbit for all w ∈ W ×W. If Pẇ is a primitive ideal of type w, then H/Stab_{H} Pẇ is a torus of rank rk G − s(w). Hence there is a bijection β : Prim A → Symp G such that β(Prim_{w}A) = Symp_{w}G.

Conjecture 2: One may define a bijection β : Prim A → Symp G as in Conjecture 1 such that β is order reversing and GKdim A/P = dim β(P) for all P ∈ Prim A.

Both conjectures are known to be true in the case G = SL(2, C) (see Appendix B). Conjecture 1 is proved in section 4 in the case when G = SL(3, C).

3 The Adjoint Action

3.1 Henceforth we restrict our attention to the case G = SL(3). We shall denote the algebra C_{q}[SL(3)] by A. In order to study the ideals of A we look at the ideals of C_{w} and B_{w} invariant under the adjoint action. At the same time we study in detail the structure of the algebra C_{w}^{H}, showing that it is an iterated Ore extension in the sense of [13]. We shall therefore be interested in bases consisting of standard monomials as defined below.

Definition Let Y = {y_{1}, y_{2}, …, y_{t}} be an indexed set of elements. The standard monomials in Y are defined to be the elements y^{r} = y_{1}^{r_{1}} … y_{t}^{r_{t}} where r = (r_{1}, …, r_{t}) ∈ N^{t}.

3.2 We shall show that for each w, there exists a certain set of z_{i,y}^{ε} such that for a suitably chosen ordering, the standard monomials in these z’s form a basis for C_{w}^{H}. Clearly we should exclude from such a set all the z_{i,y}^{ε} for which y ≮_{i} w_{ε}. The Plucker relations imply that certain other relations are redundant.

Definition Fix w ∈ W × W. Define

Z = {z_{i,y}^{ε} | ε = ±, y <_{i} w_{ε}, i = 1, 2} − {z_{2,w_{ε}w_{0}}^{ε} | ε = ±}.

Define I to be the corresponding index set; that is,

I = {(ε, y, i) | ε = ±, y <_{i} w_{ε}, i = 1, 2} − {(ε, 2, w_{ε}w_{0})}.

Theorem 3.2.1 C_{w}^{H} = C[Z]

Proof. From Theorem 2.7.1 and the remarks at the beginning of 2.6, it suffices to show that if w_{ε}w_{0} < w_{ε}, then z_{2,w_{ε}w_{0}}^{ε} ∈ C[Z]. The Plucker relations given in Theorem 1.4.1 (d) imply that in A_{w}

∑_{y∈Ŵ_{1}, y≤_{1}w_{+}} α_{y}c_{1,y}^{+}c_{2,yw_{0}}^{+} = 0, for some α_{y} ∈ Q.

Multiplying by (c_{1,w_{+}}^{+})^{−1}(c_{2,w_{+}}^{+})^{−1} and using the fact that z_{1,w_{+}}^{+} = 1, we obtain:

z_{2,w_{+}w_{0}}^{+} = ∑_{y∈Ŵ_{1}, y<_{1}w_{+}} γ_{y}z_{1,y}^{+}z_{2,yw_{0}}^{+}, for some γ_{y} ∈ Q.

Now for y <_{1} w_{+}, z_{2,yw_{0}}^{+} is either 0 or an element of Z. Hence z_{2,w_{+}w_{0}}^{+} ∈ C[Z], as required. A similar argument works for z_{2,w_{−}w_{0}}^{−}.

Remark It is important to notice that if w_{+} <_{2} w_{+}w_{0}, then the above relation collapses to 0 = 0. Nontrivial relations for z_{2,w_{+}w_{0}}^{+} only occur when w_{+} or w_{−} belongs to {(1, 3), (1, 3, 2), (1, 2, 3)}.

3.3 The ordered indexing on the set Z will be induced from the following ordering on the set S = {(ε, i, y) | ε = ±, i = 1, 2 and y ∈ Ŵ_{i}}.

Definition Define a total ordering on the set S by:

(ε′, i′, y′) < (ε, i, y) iff \{ \begin{matrix}i′ < i; & or \\ i′ = i and y′ >_{i} y; & or \\ i′ = i and y′ =_{i} y and ε′ = +, ε = −. \end{matrix}

Since Ŵ_{i} is totally ordered by ≤_{i}, it easy to see that this defines a total ordering on S.

The required commutation relations on the z_{i,y}^{ε} follow from the following commutation relations in C_{q}[SL(3)].

Proposition 3.3.1 Suppose that (ε′, i′, y′) < (ε, i, y). Then there exists an α ∈ C^{∗} such that

c_{i′,y′}^{ε′}c_{i,y}^{ε} = αc_{i,y}^{ε}c_{i′,y′}^{ε′} + ∑_{j} β_{j}a_{j}a′_{j}

where β_{j} ∈ C, a_{j} ∈ {c_{i,u}^{ε} | (ε, i, u) < (ε, i, y)} and a′_{j} ∈ {c_{i′,u}^{ε′} | (ε′, i′, u) < (ε, i, y)}

Proof. The result may be deduced from the commutation relations given in [8, 2.1,2.2,2.13-2.16] using the equations in Section 1.5. Alternatively, one may use the more general formula [19, §3.8] which follows from the form of the universal R-matrix for U_{q}(sl(3, C)). □

3.4 We define R(ε, i, y) = C[z_{j,u}^{η} | (η, j, u) < (ε, i, y)].

Proposition 3.4.1 The algebra R(ε, i, y) is spanned by the standard monomials in {z_{i′,y′}^{ε′} ∈ Z | (ε′, i′, y′) < (ε, i, y)} In particular, the algebra C_{w}^{H} is spanned by the standard monomials in the elements of Z.

Proof. It follows from the proof of Theorem 3.2.1 that z_{2,w_{ε}w_{0}}^{ε} ∈ R(ε, 2, w_{ε}w_{0}). On the other hand Proposition 3.3.1 implies that R(ε, i, y)[z_{i,y}^{ε}] is spanned as a left R(ε, i, y) module by the powers of z_{i,y}^{ε}. The result then follows by induction. □

3.5 In order to show that the standard monomials from Proposition 3.4.1 form a basis for C_{w}^{H}, we consider the adjoint action of C_{q}[SL(3)]. Let us recall the basic definitions and properties for the adjoint action of a Hopf algebra on a bimodule.

Let (R, Δ, ϵ, S) be a complex Hopf algebra and let M be an R-bimodule. The adjoint action of R on M is defined by: (ad h)x = h_{(1)}xS(h_{(2)}) forall h ∈ R and x ∈ M where we are using the Sweedler notation together with the obvious summation convention. We set M^{ad} = {x ∈ M | (ad h)x = ϵ(h)x, ∀h ∈ R}. It is easily seen that M^{ad} = {x ∈ M | hx = xh, ∀h ∈ R}.

The map ad: R ⟶ End_{C}M is a homomorphism of algebras and in this way M becomes a left Rmodule via ad. Suppose now that M also has the structure of a C-algebra compatible with its bimodule structure; i.e.

∀x, y ∈ M, ∀h ∈ R, h(xy) = (hx)y and (xy)h = x(yh)

Then under the adjoint action, M has the structure of a R-module algebra in the sense that (ad h)(xy) = (ad h_{(1)})(x)(ad h_{(2)})(y).

3.6 These generalities apply to the Hopf algebra C_{q}[SL(3)] and any bimodule M. Recall that C_{q}[SL(3)] = C[X_{ij} | 1 ≤ i, j ≤ 3] where the X_{ij} are the coordinate functions for the standard 3-dimensional representation of U_{q}(sl(3, C)). Since Δ(X_{ij}) = ∑_{k} X_{ik} ⊗ X_{kj}, the adjoint action of X_{ij} is given by (ad X_{ij})m = ∑_{k} X_{ik}mS(X_{kj}) for all m ∈ M. Denote ad X_{ij} by ad_{ij}, and define the adjoint matrix of m to be [ad m] = [ad_{ij} m]_{1≤i,j≤n}. Denote by X the matrix of coordinate functions (X_{ij}) ∈ M_{n}(A) and by S(X) the matrix (S(X_{ij})). It follows easily from the coalgebra structure of A that S(X) = X^{−1}.

Proposition 3.6.1 Let ϕ : C_{q}[SL(3)] → B be a C-algebra map. Then for any b ∈ B, [ad b] = ϕ(X)bϕ(S(X)). The map [ad −] : B → M_{n}(B) is an algebra map. In particular, [ad bc] = [ad b][ad c] for all b, c ∈ B.

Proof. The formula for [ad b] is clear. For simplicity, drop the ϕ and consider M_{n}(A) as acting on M_{n}(B) via ϕ. Then [ad bc] = XbcS(X) = XbIcS(X) = XbS(X)XcS(X) = [ad b][ad c]. □

3.7 In this section we study the adjoint action of A on the subalgebra generated by the elements t_{1}^{±1}, t_{2}^{±1} defined in section 2.6. To simplify the notation a little, set

a = w_{−}(1), b = w_{+}(1), c = w_{+}(3), d = w_{−}(3).

In this notation, t_{1} = D_{a1}X_{b1}^{−1}, t_{2} = X_{d3}D_{c3}^{−1} and t_{1}t_{2} = q^{2(δ_{a,c}−δ_{b,d})}t_{2}t_{1}. Recall that by Theorem 2.7.1, the elements t_{1}^{n}t_{2}^{m} for n, m ∈ Z form a basis for the subalgebra C[t_{1}^{±1}, t_{2}^{±1}]. Denote by F_{i}(α) the diagonal scalar matrix with the scalar α in the (i, i)−th position and 1’s elsewhere on the diagonal.

Lemma 3.7.1 With the above notation we have that

[ad t_{1}] = F_{b}(q^{2})F_{a}(q^{−2})t_{1} and [ad t_{2}] = F_{c}(q^{−2})F_{d}(q^{2})t_{2}

Proof. It is easily verifed that X_{b1}X = F_{b}(q^{2})XF_{1}(q^{−2})X_{b1} (mod I_{w}) and similarly that D_{a1}S(X) = F_{1}(q^{2})S(X)F_{a}(q^{−2})D_{a1} (mod I_{w}). Combining these two identities gives the formula for [ad t_{1}]. The proof of the second equality is similar. □

Proposition 3.7.2 The algebra C[t_{1}^{±1}, t_{2}^{±1}]^{ad} is a subalgebra of the centre of A_{w} equal to:

(i) C[t_{1}^{±1}, t_{2}^{±1}] if w_{+} = w_{−};

(ii) C[t_{i}^{±1}] if w_{+} = w_{−}(W_{i}) and w_{+} ≠ w_{−};

(iii) C if w_{+} ≠ w_{−}(W_{i}) for i = 1 and 2 but w_{−} ≠ w_{+}w_{0};

(iv) C[(t_{1}t_{2}^{−1})^{±1}] if w_{−} = w_{+}w_{0}.

Proof. It is clear that C[t_{1}^{±1}, t_{2}^{±1}]^{ad} has as a basis the set of all monomials t_{1}^{n}t_{2}^{m} which are ad A-invariant. Now

[ad t_{1}^{n}t_{2}^{m}] = F_{b}(q^{2n})F_{c}(q^{−2m})F_{a}(q^{−2n})F_{d}(q^{2m})t_{1}^{n}t_{2}^{m}.

The result then follows easily. □

Notice that the dimension of C[t_{1}^{±1}, t_{2}^{±1}]^{ad} is therefore 2 − s(w), where s(w) is the length of a shortest expression for w_{+}w_{−}^{−1} as a product of reflections.

3.8 The adjoint action of A on C_{w}^{H} is a little more complicated. As usual let w = (w_{+}, w_{−}) ∈ W × W. As before, set a = w_{−}(1), b = w_{+}(1), c = w_{+}(3), d = w_{−}(3) and set p = q^{2} − q^{−2}.

Lemma 3.8.1 Let y be an arbitrary element of W and set r = y(1) and s = y(3). The adjoint action on z_{i,y}^{ε} is given by

[ad z_{1,y}^{+}] = F_{r}(q^{−2})F_{b}(q^{2})z_{1,y}^{+} − ∑_{i=r+1}^{b} pz_{1,(1,i)}^{+}e_{ir}

[ad z_{1,y}^{−}] = F_{r}(q^{−2})F_{a}(q^{2})z_{1,y}^{−} + q^{2(r+1−a)} ∑_{i=r+1}^{a}(−1)^{r−i−1}pz_{1,(1,i)}^{−}e_{ri}

[ad z_{2,y}^{+}] = F_{c}(q^{−2})F_{s}(q^{2})z_{2,y}^{+} − q^{2(s−c−1)} ∑_{i=c}^{s−1}(−1)^{s−i−1}pz_{2,(i,3)}^{+}e_{si}

[ad z_{2,y}^{−}] = F_{d}(q^{−2})F_{s}(q^{2})z_{2,y}^{−} + ∑_{i=d}^{s−1} pz_{2,(i,3)}^{−}e_{is}

Proof. Recall that z_{1,y}^{+} = X_{r1}X_{a1}^{−1}. One verifies first that for r ≤ j ≤ a

XX_{j1} = X_{j1}F_{j}(q^{−2})XF_{1}(q^{2}) − ( ∑_{i=j+1}^{a} pX_{i1}e_{ij}) XF_{1}(q^{2}) (mod I_{w})

Hence XX_{a1}^{−1} = X_{a1}^{−1}F_{a}(q^{2})XF_{1}(q^{−2}). Putting these two formulas together yields the desired result. A similar calculation proves the other three formulas. □

3.9 Thus for each z_{i,y}^{ε} the matrix [ad z_{i,y}^{ε}] is of the form D + N where D is diagonal and N is a strictly upper or lower triangular matrix with all its non-zero entries in a single row or column. Furthermore the nonzero entry in N that is furthest from the diagonal is a scalar. Since this entry is of particular importance we define ϕ to be the function that associates to z_{i,y}^{ε} this position. That is, for a fixed w we define

ϕ(z_{1,y}^{+}) = (w_{+}(1), y(1)), ϕ(z_{2,y}^{+}) = (y(3), w_{+}(3))
ϕ(z_{1,y}^{−}) = (y(1), w_{−}(1)), ϕ(z_{2,y}^{−}) = (w_{−}(3), y(3))

This map is not injective on the set of all z_{i,y}^{ε} since for instance when w_{+} = (13), ϕ(z_{2,e}^{+}) = (3, 1) = ϕ(z_{1,e}^{+}). However when ϕ is restricted to Z we do have injectivity.

Lemma 3.9.1 The map ϕ restricted to Z is injective.

Proof. Clearly ϕ(z_{i,y}^{+}) ⊂ {(k, l) | k > l} and ϕ(z_{i,y}^{−}) ⊂ {(k, l) | k < l} so we may consider the two cases separately. Suppose that ϕ(z_{1,y}^{+}) = ϕ(z_{2,y′}^{+}) where y <_{1} w_{+} and y′ <_{2} w_{+}. This means that (w_{+}(1), y(1)) = (y′(3), w_{+}(3)). Hence y = w_{+}w_{0}(W_{1}) and y′ = w_{+}w_{0}(W_{2}). Since z_{2,w_{+}w_{0}}^{+} ∉ Z, the result follows. The other case is similar. □

Proposition 3.9.2 Let y ∈ W and suppose that ϕ(z_{i,y}^{ε}) = (k, l). Set [ad z_{i,y}^{ε}] = [a_{ij}]. Then [ad (z_{i,y}^{ε})^{n}] = [ad z_{i,y}^{ε}]^{n} = [a_{ij}(n)] where

(i) a_{ii}(n) = a_{ii}^{n} ∈ C^{∗}(z_{i,y}^{ε})^{n}

(ii) a_{ij} = 0 implies a_{ij}(n) = 0.

(iii) a_{kl}(n) = ad_{kl} (z_{i,y}^{ε})^{n} ∈ C^{∗}(z_{i,y}^{ε})^{n−1}.

Proof. Write [a_{ij}] = D + N where D is diagonal and N is strictly upper or lower triangular. Then because of the particular form of N, we have that ND^{i}N = 0 for any i. Hence [a_{ij}(n)] = (D + N)^{n} = D^{n} + ∑_{s=0}^{n−1} D^{s}ND^{n−s−1}. The first two assertions are then clear, as is the fact that

a_{kl}(n) = ∑_{s=0}^{n−1}(q^{2})^{±(n−2s−1)}a_{kl}(z_{i,y}^{ε})^{n−1}.

Since q is not a root of unity, the coefficient on the right hand side is non-zero. □

The lemma states that if (k, l) = ϕ(z_{i,y}^{ε}), then ad_{kl} behaves rather like a partial differential operator with respect to z_{i,y}^{ε}. However, on an arbitrary standard monomial it is important to apply these operators in the correct order. This necessitates defining a new ordering on the standard monomials.

Let I = {(ε, y, i) | ε = ±, y <_{i} w_{ε}, i = 1, 2} − {(±, 2, w_{ε}w_{0})} be the index set corresponding to the set Z and let K = ϕ(I) (where ϕ is the obvious induced map on I). For each w let ≺ be a total ordering on the set {(i, j) | i, j = 1, 2, 3, i ≠ j} satisfying

(1, i) ≻ (1, i′) ≻ (2, 3) ≻ (3, j) ≻ (3, j′) ≻ (2, 1)

and i and i′ are chosen so that if (1, i) and (1, i′) are both in ϕ(I) then the ordering ≺ reverses the ordering induced by ϕ. We denote by ≺ the induced ordering on the subset K. The ordering induced by ≺ on I via ϕ^{−1} will also be denoted by ≺. The ordering ≺ on I extends naturally to a lexicographic ordering on N^{I} which will again be denoted by ≺.

Theorem 3.9.3 Let m ∈ N^{I} and let ϕ(m) be its image in N^{K}. Let M^{m} be a standard monomial in the z_{i,y}^{ε} with respect to the order defined in 3.3 and let X^{ϕ(m)} be the standard monomial in the X_{ij} with respect to the ordering on K defined above. Then

(i) ad X^{ϕ(m)} M^{m} ∈ C^{∗}; (ii) ad X^{ϕ(m)} M^{n} = 0 for all n ≺ m.

Proof. Define Supp(m) = {η ∈ I | m_{η} ≠ 0}. For (i, j) ∈ K, define f_{ij} to be the element of N^{I} such that (f_{ij})_{ξ} = δ_{ξ,φ^{−1}(i,j)}. It suffices to prove that for any monomial M^{m} and any (i, j) ⪰ Max(ϕ(Supp(m)),

(ad X_{ij})M^{m} = \{ \begin{matrix}cM^{m−f_{ij}} & if (i, j) = Max(ϕ(Supp(m)); \\ 0 & if (i, j) ≻ Max(ϕ(Supp(m)); \end{matrix}

for some c ∈ C^{∗}. Suppose that M^{m} = Z_{1}^{m_{1}} … Z_{t}^{m_{t}} where Z_{i} ∈ Z. Then ad_{ij}M^{m} is the (i, j)-entry of ad (Z_{1}^{m_{1}}) … ad (Z_{t}^{m_{t}}). The form of these matrices was computed in Proposition 3.9.1. A lengthy but routine calculation shows in all cases that if k is such that (i, j) = ϕ(Z_{k}) = Max(ϕ(Supp(m)), then

(ad X_{ij})M^{m} = ad_{ii}Z_{1}^{m_{1}} … ad_{ii}Z_{k−1}^{m_{k−1}}ad_{ij}Z_{k}^{m_{k}}ad_{jj}Z_{k+1}^{m_{k+1}} … ad_{jj}Z_{t}^{m_{t}},

and that if (i, j) ≻ Max(ϕ(Supp(m)), then (ad X_{ij})M^{m} = 0. Hence the result above follows from Proposition 3.9.1. □

3.10 We now come to the most important results of the section. For each character ν ∈ R(A) let us denote by C_{w}^{ν} the ν-isotypic part of C_{w} under the adjoint action. Denote by Soc C_{w} the socle of C_{w} under this action.

Theorem 3.10.1 1. The algebras C_{w} and C_{w}^{H} are iterated Ore extensions. Hence C_{w}, C_{w}^{H} and B_{w} are all domains.

2. Soc C_{w} = ⨁_{ν∈R(A)} C_{w}^{ν} = C[t_{1}^{±1}, t_{2}^{±1}]. Hence C_{w}^{ad} = C[t_{1}^{±1}, t_{2}^{±1}]^{ad} is as described in section 3.7.2.

3. If ν ∈ R(A) is such that C_{w}^{ν} ≠ 0, then there exists an invertible element u_{ν} such that C_{w}^{ν} = u_{ν}C_{w}^{ad}.

Proof. Theorem 3.9.3 implies that the standard monomials in the elements of Z form a basis for C_{w}^{H}. The fact that C_{w}^{H} is an iterated Ore extension is an induction based on [2, 1.3] using Proposition 3.4.1. Theorem 2.7.1 implies that C_{w} is an Ore extension of C_{w}^{H}. Thus C_{w} and C_{w}^{H} are both domains. Since B_{w} is a localization of C_{w} (Theorem 2.6.1), it too is a domain.

Now let f ∈ C_{w}. We may write f in the form f = ∑_{n≤m} α_{n}M^{n} where M^{n} is the monomial described in 3.9, α_{n} ∈ C[t_{1}^{±1}, t_{2}^{±1}] for all n and α_{m} ≠ 0. By Theorem 3.9.3, there exists an a ∈ A such that a is a product of elements of the form X_{ij} and such that

(ad a)M_{n} = \{ \begin{matrix}0 & if n < m \\ 1 & if n = m \end{matrix}

Now α_{n} = ∑_{χ∈R(A)} α_{n,χ}, where α_{n,χ} ∈ C[t_{1}^{±1}, t_{2}^{±1}]_{χ}. Moreover,

(ad a)α_{n,χ}M_{n} = (ad a_{(1)})α_{n,χ}(ad a_{(2)})M_{n} = χ(a_{(1)})α_{n,χ}ad a_{(2)}M_{n} = α_{n,χ}ad (r_{χ}(a))M_{n}.

But r_{χ}(a) = λ_{χ}a for some non-zero scalar λ_{χ}. Thus

(ad a)f = ∑_{χ} λ_{χ}α_{m,χ} ∈ C[t_{1}^{±1}, t_{2}^{±1}]\{0}.

Since C[t_{1}^{±1}, t_{2}^{±1}] is a semi-simple ad-A module, this proves the second assertion. The third statement then follows easily from 3.7. □

Remark We can identify C_{w}^{ad} with C[H_{w}], the algebra of functions on the torus H_{w} = H/Stab _{H}Cẇ (see Theorem A.3.1).

4 Primitive Spectrum of C_{q}[G]

4.1 We begin with a result showing that the study of Spec A and Spec B may be reduced to the study of Spec A_{w} and Spec B_{w}, w ∈ W × W respectively.

Proposition 4.1.1 Let P ∈ Spec A (resp. Spec B). Then there exists a unique w ∈ W × W such that P ⊃ I_{w} (resp. P ⊃ I_{w} ∩ B) and P ∩ E_{w} = ∅.

Proof. Let P ∈ Spec A. Define the elements w_{+}^{(i)} ∈ Ŵ_{i}, i = 1, 2 to be the smallest elements of Ŵ_{i} such that c_{i,y}^{+} ∈ P for all y > w_{(i)}^{+}. We want to show that there exists a w_{+} ∈ W ×W such that w_{+} = w_{+}^{(i)}(W_{i})

for i = 1, 2. It is easily verified that this will occur if and only if w_{+}^{(1)}w_{0} ≠ w_{+}^{(2)}(W_{2}). Suppose that w_{+}^{(1)}w_{0} = w_{+}^{(2)}(W_{2}). Recall the Plucker relation

∑_{y∈Ŵ_{1}}(−q^{2})^{y(1)−1}c_{1,y}^{+}c_{2,yw_{0}}^{+} = 0

Now for y >_{1} w_{+}^{(1)}, c_{1,y}^{+} ∈ P by definition. On the other hand, if y <_{1} w_{+}^{(1)}, then yw_{0} >_{2} w_{+}^{(1)}w_{0} = w_{+}^{(2)}(W_{2}) by Proposition 1.8. Hence c_{2,yw_{0}}^{+} ∈ P. The remaining term, which is a scalar multiple of c_{1,w_{+}^{(1)}}^{+}c_{2,w_{+}^{(2)}}^{+}, must therefore lie in P also. However neither c_{1,w_{+}^{(1)}}^{+} nor c_{2,w_{+}^{(2)}}^{+} lie in P by hypothesis.

Moreover c_{1,w_{+}^{(1)}}^{+} is normal modulo P by Lemma 2.1. This contradicts the fact that P is prime.

A similar argument produces an analogous element w_{−}. Thus there exists an element w = (w_{+}, w_{−}) such that c_{i,y}^{±} ∈ P for all y > w_{±} and c_{i,w_{±}}^{±} ∉ P for i = 1, 2. In other words, P ⊃ I_{w} and P ∩ E_{w} = ∅. It is clear that such an element must be unique.

Now let P ∈ SpecB. By [13, 10.2.10], there exists a Q ∈ SpecA such that P is minimal over Q ∩ B. By the first part of the proof there exists a w such that Q ⊃ I_{w} and Q ∩ E_{w} = ∅. Hence it is clear that P ⊃ I_{w} ∩ B. Suppose that c ∈ P ∩ E_{w}.From the minimality of P over Q ∩ B and the fact that c is normal modulo I_{w} it follows easily that c ∈ Q, a contradiction. □

Corollary 4.1.2 Identify Spec A with {P ∈ Spec A | P ⊃ I_{w}, P ∩E_{w} = ∅}. Then Spec A = ⨆_{w∈W ×W} Spec A_{w} where ⨆ denotes the disjoint union. Similarly Spec B = ⨆_{w∈W ×W} Spec B_{w}.

The analogous result concerning the primitive spectrum is also true. However, this is a subtler question and the proof requires the characterization of the primitive ideals as the locally closed elements of SpecA.

4.2 We now return to the study of B_{w} and C_{w}. Define the algebra Cẇ by:

Cẇ = \{ \begin{matrix}C_{w}^{H}, & if w_{+} = w_{−}; \\ C_{w}^{H}[t_{j}^{±1}], & if w_{+} = w_{−}(W_{i}) and w_{+} ≠ w_{−}(W_{j}); \\ C_{w}^{H}[t_{1}^{±1}, t_{2}^{±1}], & if w_{+} ≠ w_{−}(W_{i}) for i = 1, 2 but w_{−} ≠ w_{+}w_{0}; \\ C_{w}^{H}[t_{1}^{±1}], & if w_{−} = w_{+}w_{0}. \end{matrix}

and define Bẇ to be Cẇ[d^{−1}]. Then it is clear from 2.7.1 and 3.7.2 that C_{w} ≅ Cẇ⊗C_{w}^{ad} and B_{w} ≅ Bẇ⊗C_{w}^{ad}. Moreover both Cẇ and Bẇ are integral domains by 3.10.1. We now show that Bẇ is simple. It will then follow that all prime ideals of B_{w} are induced from C_{w}^{ad}.

Lemma 4.2.1 Let I be an ideal of B_{w} (respectively C_{w}). Then I is an adA-submodule if and only if I = (I ∩ C_{w}^{ad})B_{w} (respectively I = (I ∩ C_{w}^{ad})C_{w}).

Proof. Since B_{w} is a localization of C_{w} and C_{w} is ad-invariant, it is enough to prove the result for C_{w}. Let I be an ideal of C_{w} and suppose that I strictly contains (I ∩ C_{w}^{ad})C_{w}. Choose f ∈ I\(I ∩ C_{w}^{ad})C_{w} and write f (as in the proof of 3.10.1) as f = ∑_{n≤m} α_{n}M^{n} where α_{n} ∈ Soc C_{w} for all n and α_{m} ≠ 0. Assume that m is minimal for such elements. The argument used in the proof of 3.10.1 implies that I contains ∑_{χ} λ_{χ}α_{m,χ} for some nonzero scalars λ_{χ} . Since I is adA-invariant it therefore contains each λ_{χ}α_{m,χ}. But λ_{χ}α_{m,χ} ∈ (C_{w})_{χ} = u_{χ}C_{w}^{ad} for some unit u_{χ}. Thus λ_{χ}α_{m,χ} ∈ (I ∩ C_{w}^{ad})C_{w} and so α_{m}M_{m} ∈ (I ∩ C_{w}^{ad})C_{w}, contradicting the minimality of m. □

Theorem 4.2.2 B_{w} ≅ Bẇ ⊗ C_{w}^{ad} where Bẇ is a simple algebra. The center of B_{w} is C_{w}^{ad} and all ideals of B_{w} are generated by their intersection with the center. Thus Spec B_{w} ≅ Spec C_{w}^{ad} and Prim B_{w} ≅ Prim C_{w}^{ad}. All primitive ideals of B_{w} are maximal and all prime ideals are completely prime. If P ∈ Prim B_{w} then GKdim B_{w}/P = l(w) + s(w).

Proof. Let P_{e} be the ideal of B_{w} generated by elements of the form t−1 where t ∈ {t_{1}^{n}t_{2}^{m} | n, m ∈ Z}∩C_{w}^{ad}. Then clearly Bẇ ≅ B_{w}/P_{e}. Hence P_{e} is a completely prime ideal of B_{w}. From the lemma we have that P_{e} is a maximal adA-invariant ideal of B_{w}. Since A_{w} is a finite normalizing extension of B_{w}, it follows from ‘Lying over’ and ‘Going up’ [13, 10.2], that P_{e} is in fact a maximal ideal of B_{w}. Hence Bẇ is simple. Because Bẇ satisfies the nullstellensatz [13, 9.1], it follows that Bẇ is central simple and the assertion concerning the spectrum is a consequence of [3, 4.5.1]. By the nullstellensatz again, the primitive ideals are generated by the maximal ideals of C_{w}^{ad}. Since the quotient of B_{w} by such an ideal will always be isomorphic to Bẇ, all the primitive ideals are completely prime. Since every prime ideal is an intersection of primitives it follows easily that all the prime ideals are completely prime. The assertion concerning the Gelfand-Kirillov dimension follows from the description of Bẇ as a localization of an Ore extension and a slight generalization of [13, 8.2.10]. □

4.3 We may now use Corollary 4.1.2 to deduce some global results about the primitive spectrum of B. We shall say that a Noetherian C-algebra R satisfies the Dixmier-Moeglin condition if the following conditions are equivalent for a prime ideal P: (a) P is primitive; (b) P is rational (the center of the ring of fractions of R/P is C); (c) P is locally closed in Spec R. Recall that the action of H by right translation on B induces a natural action of H on Prim B.

Theorem 4.3.1 In the notation of section 2.8, we have that

Prim B = ⨆_{w∈W ×W} Prim_{w}B.

Moreover Prim_{w}B is a nonempty H-orbit for each w ∈ W × W. If Qẇ is a primitive ideal of type w, then H/Stab_{H}Qẇ is a torus of rank 2 − s(w). All primitive ideals of B are completely prime. B satisfies the Dixmier-Moeglin condition.

Proof. Let P be a primitive ideal of B of type w. Then by the nullstellensatz [13, 9.1] and [3, 4.1.6] P B_{w} is maximal. On the other hand if P is a prime ideal of B of type w and P B_{w} is maximal, then any prime ideal strictly containing P intersects the set E_{w} of regular elements nontrivially. Hence the set P is locally closed in Spec B and again by the nullstellensatz [13, 9.1.8], P must be primitive. The fact that all prime ideals of B are completely prime follows immediately from 4.1.2 and 4.2.2 by standard facts about localization. □

Remark Notice that these results imply that for any primitive ideal P of B there exists an Ore set E_{w} and a normal element d such that (B/P)_{E_{w}} ≅ Cẇ[d^{−1}] and Cẇ is an iterated Ore extension. This should be compared with the structure of primitive factors of the enveloping algebra of a solvable Lie algebra [13, §14.8].

4.4 We now deduce the main theorem. Recall that A = C_{q}[G].

Theorem 4.4.1 In the notation of section 2.8, we have that

Prim A = ⨆_{w∈W ×W} Prim_{w}A.

Moreover Prim_{w}A is a nonempty H-orbit for each w ∈ W × W. The map P ↦ P A_{w} is an isomorphism between Prim_{w}A and Prim A_{w}. If Pẇ is a primitive ideal of type w, then H/Stab_{H}Pẇ is a torus of rank rk G − s(w). GKdim A/Pẇ = l(w) + s(w). A satisfies the Dixmier-Moeglin condition.

Proof. Let Pẇ be a primitive ideal of A of type w. It follows from sections 4.2 and 4.3 that PẇA_{w} is a primitive ideal of A_{w} and that Pẇ ∩ B is a primitive ideal of B of type w. Furthermore the prime ideals of A lying over a given primitive ideal of B form a Γ-orbit and are all primitive. The fact that the Dixmier-Moeglin condition passes from B to A follows from [9]. □

4.5 As noted in the proof of Theorem 4.4.1, it follows from the description of the primitive ideals of B_{w} that if P ∈ Prim A_{w}, then P ∩ B_{w} is a primitive ideal of B_{w} and that the primitive ideals lying over a fixed primitive ideal of B_{w} form a nontrivial Γ-orbit. Using a detailed analysis of the structure of A_{w} as a crossed product of Γ̂ over B_{w}, one can calculate the exact number of primitives of A_{w} sitting over a given primitive of B_{w}.

Proposition 4.5.1 Let P ∈ Prim A_{w}. Then P ∩ B_{w} is a maximal ideal of B_{w}. Conversely for all maximal ideals Q of B_{w} the number of primitive ideals P of A_{w} such that P ∩ B_{w} = Q is:

\{ \begin{matrix}4 & if w = (e, e); \\ 2 & if w_{+} = w_{−} = e(W_{i}) and w_{+} or w_{−} ≠ e(W_{j}) \\ 1 & otherwise. \end{matrix}

All primitive ideals of A_{w} are maximal and completely prime.

In particular this last result implies that all prime ideals of C_{q}[G] are completely prime. Goodearl and Letzter [6] have recently proved that all prime ideals of C_{q}[SL(n)] are completely prime.

Remark The authors have recently generalized the results of this section, proving Conjecture 1 of 2.8 for C_{q}[SL(n)].

LIST OF NOTATION

1.2 Q, U_{q}(g) 3.2 Z, I 1.3 C_{q}[G] 3.3 S 1.4 X_{ij} 3.4 R(ε, i, y) 1.5 c_{i,w}^{ε} 3.5 R^{ad} 1.6 c_{−λ,i,μ,j}^{Λ} 3.6 [ad m] 1.7 U_{q}(b^{±}), I^{±}(w, Λ) 3.7 F_{i}(α) 1.9 W_{i}, Ŵ_{i}, ≤_{i}, =_{i} 3.9 ϕ 2.2 I_{w}, E_{w}, E_{w}, A_{w} 3.10 C_{w}^{ad}, H_{w} 2.5 H, Γ, B, B_{w} 4.2 Cẇ, Bẇ 2.6 z_{i,y}^{ε}, t_{i}, C_{w}, d_{i}, d 2.7 C_{w}^{H} 2.8 Prim_{w}, Symp_{w}

A Symplectic Leaves in a Semi-simple Poisson Lie Group

A.1 Let G be a connected complex semisimple Lie group with Lie algebra g. Let h be a Cartan subalgebra of g, let R be the associated root system and R^{+} a choice of positive roots. Denote by κ(−, −) the Killing form on g. Let n^{±} = ⊕_{α∈R^{±}}g_{α} and let b^{±} = h ⊕ n^{±}. Let d = g × g. The Iwasawa decomposition of d (as defined in [3, 1.13.14]) is then d = g ⊕ a ⊕ u^{+} where g is identified with the diagonal subalgebra of d, a = {(x, −x) | x ∈ h} and u^{+} = {(x, y) | x ∈ n^{+}, y ∈ n^{−}}. Define the bilinear form ⟨−, −⟩ on d by:

⟨(x_{1}, y_{1}), (x_{2}, y_{2})⟩ = \frac{1}{2}(κ(x_{1}, x_{2}) − κ(y_{1}, y_{2}))

Denote a⊕u^{+} by g_{r}. Then (g, g_{r}, d) is a Manin triple in the sense of [4]. There is then a Poisson Lie group structure on G associated to this triple [4]. The corresponding Poisson tensor is the tensor π defined by π(g) = l_{g^{∗}}R − r_{g^{∗}}R where R = \frac{1}{2} ∑_{α>0} E_{α} ∧ E_{−α} ∈ g ∧ g and l_{g^{∗}} and r_{g^{∗}} are the differentials of left and right translation respectively. The associated local double Lie group is then (G, G_{r}, D) where D = G×G; G is identified with the diagonal subgroup {(x, x) | x ∈ G}; G_{r} = AU^{+} where A = {(x, x^{−1}) | x ∈ H} and U^{+} = {(x, y) ∈ N^{+}, y ∈ N^{−}} and H, N^{±} and B^{±} are the closed connected subgroups of G associated to h, n^{±} and b^{±} respectively.

Consider the map p : G → D/G_{r}. Define Γ to be G ∩ G_{r} = ker p. It is easily seen that Γ = {(h, h) ∈ H | h^{2} = 1}. Hence Γ is a finite subgroup of D isomorphic to Z_{2}^{rkG}. Define Ḡ to be G/Γ ≅ GG_{r}/G_{r}. Since GG_{r} is open in D, it follows that Ḡ is an open subset of D/G_{r}. Since π is H-invariant (and therefore Γ-invariant), it induces a Poisson tensor on Ḡ.

Recall that a symplectic leaf of a Poisson variety is defined to be a maximal connected symplectic subvariety. We denote by Symp G the set of symplectic leaves of G. There is a natural partial order on Symp G given by inclusions of closures.

Theorem A.1.1 1) The symplectic leaves of Ḡ are of the form Ḡ ∩ G_{r}xG_{r}/G_{r} for some x ∈ G.

2) The symplectic leaves of G are the connected components of the inverse images of the symplectic leaves of Ḡ.

Proof. Since p : G → Ḡ is étale, we have that for all x ∈ G, T_{x}G ≅ T_{p(x)}Ḡ ≅ T_{p(x)}D/G_{r}. We recall some results from [11]. The left action of G_{r} on D/G_{r} induces a map σ from the Lie algebra g_{r} to the Lie algebra of vector fields on D/G_{r}. For α ∈ g_{r} we denote by σ_{x}(α) the corresponding element of T_{x}G. The bilinear form ⟨−, −⟩ identifies g_{r} with g^{∗}. Therefore, each α ∈ g_{r} induces a right invariant 1-form α_{r} on G. Define the right dressing vector field on G by ⟨ρ_{x}(α), ξ⟩ = π_{x}(α_{r}(x), ξ) for all ξ ∈ T_{x}^{∗}G. By [11, 3.13], ρ_{x}(α) = −σ_{x}(α) for all α ∈ g_{r} and x ∈ G. Hence

rk π_{x} = dim σ_{x}(g_{r}) = dim G_{r}xG_{r}/G_{r}, ∀x ∈ G.

It is easily seen that G_{r}xG_{r}/G_{r} ∩ Ḡ is a Poisson subvariety of Ḡ; hence it is a symplectic subvariety by the above equality. The theorem then follows easily. □

A.2 Denote by Q = T U^{+} = HG_{r} the positive Borel subgroup of D. Recall the Bruhat decomposition D = ⨆_{w∈W ×W} QwQ = ⨆_{w∈W ×W} QwG_{r}. For each w ∈ W × W we fix a representative ẇ of w in the normaliser of T and we set: Cẇ = G_{r}ẇG_{r}/G_{r}, C_{w} = QwG_{r}/G_{r} = ⋃_{h∈H} hCẇ. Hence D/G_{r} = ⨆_{w∈W ×W} C_{w}. Set Bẇ = Cẇ ∩ Ḡ, B_{w} = C_{w} ∩ Ḡ, A_{w} = p^{−1}(B_{w}). Fix a connected component Aẇ of p^{−1}(Bẇ). Notice that QwG_{r} ∩ G ≠ ∅ for all w ∈ W × W. This can be proved as follows by induction on l(w) (the length of w). Assume that s is a simple reflection; so s = (s_{α}, e) or (e, s_{α}) for some α ∈ R^{+}. If s = (s_{α}, e) we have that QsQ ∩ G = (B^{+}s_{α}B^{+}, B^{−}) ∩ G ≠ ∅ since B^{+}s_{α}B^{+} ∩ B^{−} ≠ ∅; similarly for s = (e, s_{α}). In the general case, set w = sw′ where s is a simple reflection and l(w) = l(w′) + 1. Then by induction QwQ∩G ⊃ (QsQ∩G)(Qw′Q∩G) ≠ ∅. Therefore B_{w} = C_{w} ∩Ḡ ≠ ∅ and since C_{w} = ⋃_{h∈H} hCẇ, we have that hCẇ ∩ Ḡ ≠ ∅ for all h ∈ H. These observations together with the theorem of section one give the following description of the symplectic leaves.

Theorem A.2.1 1) Each symplectic leaf of Ḡ is of the form hBẇ for some h ∈ H and w ∈ W × W. 2) Each symplectic leaf of G is of the form hAẇ for some h ∈ H and some w ∈ W × W.

Let w = (w_{+}, w_{−}) ∈ W × W. Define A′_{w} = w(A) ∩ A = {a ∈ A | aẇG_{r} = ẇG_{r}}. Set A_{w} = A/A′_{w}. Then A_{w} is a torus of rank s(w) = dim A − dim A′_{w} = codim_{h} ker(w_{+}w_{−}^{−1} − I). When G = SL(n, C) we have that s(w) = min{m | w_{+}w_{−}^{−1} = r_{1} … r_{m} where r_{i} is a transposition for all i }.

Define U_{w}^{±} = w(U^{±}) ∩ U^{+} and recall that we have an isomorphism of varieties U^{+} ≅ U_{w}^{−} × U_{w}^{+}, and that U_{w}^{−} ≅ C^{l(w)}. Thus we have that Cẇ = AU^{+}ẇG_{r}/G_{r} = AU_{w}^{−}ẇG_{r}/G_{r}. Using a standard argument one verifies that the multiplication A_{w} ×U_{w}^{−} → Cẇ is an isomorphism. Thus we have proved the following proposition.

Proposition A.2.2 Cẇ ≅ A_{w} × U_{w}^{−} where A_{w} is a torus of rank s(w) and U_{w}^{−} ≅ C^{l(w)}. Hence dim Cẇ = l(w) + s(w).

A.3 Let w ∈ W × W. Set H′_{w} = {h ∈ H | hG_{r}ẇG_{r} = G_{r}ẇG_{r}}. Then H′_{w} is a closed subgroup of H and H_{w} = H/H′_{w} is a torus of rank rk G − s(w). We have that C_{w} = HCẇ and the same argument as in the previous subsection shows that the multiplication map H_{w} × Cẇ → C_{w} is an isomorphism.

The group G_{r} acts by left translation on C_{w} and therefore on the product H_{w} × Cẇ. It is easily seen that the algebra of G_{r}-invariant functions on C_{w} is C[H_{w}]. This proves the first part of the theorem below. The second part is a consequence of the description given above.

Theorem A.3.1 1) The G_{r}-orbits in C_{w} are the fibres of the natural projection C_{w} → G_{r}\\C_{w} ≅ H_{w}. 2) The symplectic leaves of type w in Ḡ are the fibres of the induced projection B_{w} → H_{w}.

We now summarize the results about the set Symp G of symplectic leaves in G. Denote by Symp_{w}G the set of symplectic leaves of type w ∈ W × W.

Theorem A.3.2 1) Symp G = ⨆_{w∈W ×W} Symp_{w}G.

2) For each w ∈ W × W, Symp_{w}G is a nonempty H-orbit. If Aẇ is a fixed symplectic leaf of type w, then H/Stab _{H}Aẇ is a torus of rank rk G − s(w).

3) The dimension of a leaf of type w is l(w) + s(w).

B The case G = SL(2, C)

B.1 In this appendix we outline the classification of primitive ideals of C_{q}[SL(2)] and of symplectic leaves of SL(2, C). The proofs of the two theorems below are straightforward calculations. In the notation of the section 1.4, C_{q}[SL(2)] is generated by the elements a = X_{11}, b = X_{12}, c = X_{21}, and d = X_{22} subject to the relations ab = q^{2}ba, ac = q^{2}ca bd = q^{2}db, bc = cb, ad − da = (q^{2} − q^{−2})bc, and ad − q^{2}bc = 1. The Weyl group in this case is just W = {e, s} where s^{2} = e. The ideals I_{w} for w ∈ W × W are given by I_{(e,e)} = (b, c), I_{(s,e)} = (b), I_{(e,s)} = (c) and I_{(s,s)} = (0).

Theorem B.1.1 The following is a complete list by type of the primitive ideals of C_{q}[SL(2)]:

\begin{matrix}(e, e) : & P_{(e,e),λ} = (b, c, a − λ, d − λ^{−1}),  λ ∈ C^{∗} \\ (s, e) : & P_{(s,e)} = I_{(s,e)} = (b) \\ (e, s) : & P_{(e,s)} = I_{(e,s)} = (c) \\ (s, s) : & P_{(s,s),λ} = (b − λc),  λ ∈ C^{∗} \end{matrix}

All prime ideals of C_{q}[SL(2)] are completely prime.

Remark Let M^{+} and M^{−} be modules with annihilators P_{(s,e)} and P_{(e,s)} respectively. Then M^{+} and M_{−} are modules of type (s, e) and (e, s) respectively. The existence of such modules is used in section 2.3.

B.2 We now describe explicitly the symplectic leaves of SL(2, C). We continue to denote the coordinate functions of the standard representation of SL(2, C) by a, b, c and d as above. The standard Poisson bracket is then given by: {a, b} = −ab, {a, c} = −ac, {b, d} = −bd, {b, c} = 0 and {a, d} = −2bc.

Theorem B.2.1 The following is a complete list by type of the symplectic leaves of SL(2, C).

\begin{matrix}(e, e) : & \{[ \begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix} ]\} , λ ∈ C^{∗} \\ (s, e) : & \{[ \begin{matrix}α & 0 \\ γ & α^{−1} \end{matrix} ] | α, γ ∈ C^{∗}\} \\ (e, s) : & \{[ \begin{matrix}α & β \\ 0 & α^{−1} \end{matrix} ] | α, β ∈ C^{∗}\} \\ (s, s) : & \{[ \begin{matrix}α & λγ \\ γ & δ \end{matrix} ] | γ ∈ C^{∗}, αδ − λγ^{2} = 1\} , λ ∈ C^{∗}. \end{matrix}

Combining these two theorems yields a positive answer to all the conjectures given in section four. Corollary B.2.2 There is an order preseving bijection β : Prim C_{q}[SL(2)] → Symp SL(2, C). Furthermore, if L = β(P), then dim L = GKdim C_{q}[SL(2)]/P.

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