ON THE GELFAND-KIRILLOV CONJECTURE FOR QUANTUM ALGEBRAS

Philippe CALDERO

RESUME. Soit q un nombre complexe non racine de l’unité et 𝔤 une ℂ-algèbre de Lie semi-simple. On note U_{q}(𝔤) l’algèbre enveloppante quantifiée de Drinfeld et Jimbo, U_{q}(𝔫^{−})⊗U^{0}⊗U_{q}(𝔫) sa décomposition triangulaire, et ℂ_{q}[G] le groupe quantique associé. On décrit explicitement Fract U_{q}(𝔫) et Fract ℂ_{q}[G] comme corps de Weyl quantique. Nous utilisons pour cela un analogue quantique du lemme de Taylor.

ABSTRACT. Let q be a complex not a root of unity and 𝔤 be a semi-simple Lie ℂ-algebra. Let U_{q}(𝔤) be the quantized enveloping algebra of Drinfeld and Jimbo, U_{q}(𝔫^{−})⊗U^{0}⊗U_{q}(𝔫) be its triangular decomposition, and ℂ_{q}[G] the associated quantum group. We describe explicitly Fract U_{q}(𝔫) and Fract ℂ_{q}[G] as a quantum Weyl field. We use for this a quantum analogue of the Taylor lemma.

0. Introduction.

Let q be a non zero complex number which is not a root of unity. In this article, a ℂ-algebra defined by generators X_{i}, 1 ≤ i ≤ m, and relations X_{i}X_{j} = q^{a_{i,j}}X_{j}X_{i}, 1 ≤ i < j ≤ m, a_{i,j} ∈ ℤ, will be called ”the algebra of regular functions on an affine quantum space”. Its skew field of fractions will be called the quantum Weyl field. The X_{i}, 1 ≤ i ≤ m, will be called system of q-commuting generators (SQCG).

Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n. Let R be the root system associated to the choice of a Cartan subalgebra 𝔥. We denote Δ = {α_{i}} the set of simple roots of R, P the lattice of associated weights generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} ℕϖ_{i} the lattice of dominant weights. Let G be the simply connected group associated to 𝔤 and U_{q}(𝔤) the Drinfeld and Jimbo’s quantized enveloping algebra. We define as in the classical case its ”nilpotent” subalgebra U_{q}(𝔫) and the quantum algebra of regular functions on the group ℂ_{q}[G]. A theorem of J. Alev and F. Dumas, c.f. [1], asserts that Fract U_{q}(𝔫) is a quantum Weyl field when 𝔤 is of type A_{n}. In [15], A. Joseph proves that this property is verified for all semi-simple Lie algebras 𝔤 when q is generic. We prove in this article that Fract U_{q}(𝔫) and Fract ℂ_{q}[G] are quantum Weyl fields when 𝔤 is semi-simple and when q is not a root of one, see [9] for the case where q is a root of one. The method we used provides a system of q-commuting generators.

Inspired by [12, Th. 3.2], we essentially used the quantum analogue of the Taylor Lemma. This lemma asserts that if 1) δ is a locally nilpotent σ-derivation, cf. 1.1, on a ℂ-algebra A and 2) there exists an element a such that δ(a) = 1, then a is (right) transcendant on the invariant algebra A^{δ} and A ≃ A^{δ}[a]. Our results are proved as follows :

As a first step, we give, cf. [Proposition 2.1], a multi-parametered version of the Taylor lemma for the locally nilpotent action (as a bialgebra) of the Borel subalgebra U_{q}(𝔟) on an algebra A. The difficulty encountered in the quantum case is the following : the generators E_{β}, (β being a positive root) of the Poincaré-Birkoff-Witt base of U_{q}(𝔫) do not act as σ-derivations on A. To get round this problem, we can, from a reduced decomposition of the longest element w_{0} in the Weyl group, define a total order on the set of these generators and obtain a decreasing sequence of subalgebras U_{q}(𝔫_{β}) of U_{q}(𝔫), cf. [10, Lemma 1.7]. With the help of a result of S.Z. Levendorskii and Y.S Soibelman, cf. [17, 2.4.1], we obtain that E_{β} acts as a σ-derivation on the subalgebra of U_{q}(𝔫_{β^{<}})-invariants of A, β^{<} being the root preceding β. So, we can inductively apply the Taylor Lemma and prove Proposition 2.1. As a second step, we apply Proposition 2.1, see also [Assertion 2.2], to the (right) regular action of U_{q}(𝔟) on ℂ_{q}[G]. Recall, cf. 1.4, that ℂ_{q}[G] is generated as a space by the coefficients c_{μ,ν}^{λ} of the simple finite dimensional U_{q}(𝔤)-modules L_{q}(λ), λ ∈ P^{+}. Let w_{0} = s_{i_{1}} … s_{i_{N}} be a reduced decomposition of w_{0} into a product of elementary reflections. Let β = β_{l} := s_{i_{1}} … s_{i_{l−1}}(α_{i_{l}}) and y_{l} = s_{i_{1}} … s_{i_{l}}. By using the Lusztig automorphisms and the Weyl character formula, we prove that c_{β} := c_{y_{l−1}ϖ_{i_{l}} ,ϖ_{i_{l}}}^{ϖ_{i_{l}}}, is U_{q}(𝔫_{β^{<}})-invariant. Moreover, with the help of the R-matrix, we prove, cf. Proposition 2.3, that the c_{β} q-commute, i.e. commute up to a power of q. By the quantized Taylor lemma and the Drinfeld duality, we obtain the claimed theorem for Fract U_{q}(𝔫). We may specify the description of Fract U_{q}(𝔫) as in [1, Th. 2.15], cf. Theorem 3.2. We give similar results for the quantum algebras S_{w}^{+} of regular functions on a Schubert variety, cf. [14, 10.3.1 (3)]. On this subject, we remark that the elements c_{β} belong to the Lakshmibai-Reshetikhin base of standard momomials, [16]. After localization, they generate a polynomial base.

As a third step, we show that our method works for ℂ_{q}[G]. If ρ is the sum of fundamental weights, then the elements d_{β} = c_{y_{l−1}ρ,−y_{l}ρ}^{ρ}, d′_{β} = c_{y_{l−1}ρ,−y_{l−1}ρ}^{ρ} and c_{w_{0}ϖ_{i},ϖ_{i}}^{ϖ_{i}} generate the quantum Weyl field Fract ℂ_{q}[G]. This theorem is a consequence of the Taylor Lemma for the regular action of U_{q}(𝔟)⊗U_{q}(𝔟)^{opp} on ℂ_{q}[G]. Note that this result was proved by A.N. Panov for G = SL_{n} and generic q, [21].

In the classical case, the Gelfand-Kirillov conjecture asks if the enveloping algebra of 𝔤 is a Weyl field. In [12], A. Joseph gives a generalization of the Gelfand-Kirillov conjecture, replacing the enveloping algebra of 𝔤 by an algebra on which 𝔫 acts by derivations. The title of our article must be understood in the sense of this generalization. At the present time, we do not know if Fract U_{q}(𝔤) is a quantum Weyl field. As for the classical case, this assertion may be shown when 𝔤 has type A_{n}, see [19].

We are in debt to T. Levasseur for his remarks and observations.

1. Preliminaries and notations.

1.1. Let 𝔤 be a semi-simple Lie ℂ-algebra of rank n. We fix a Cartan sub-algebra 𝔥 of 𝔤. Let 𝔤 = 𝔫^{−} + 𝔥 + 𝔫 be the triangular decomposition and {α_{i}}_{i} be a base of the root system Δ resulting from this decomposition. We note 𝔟 = 𝔫+𝔥 and 𝔟^{−} = 𝔫^{−} +𝔥 the two opposed Borel sub-algebras. Let P be the weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} ℕϖ_{i} the monoid of integral dominant weights. We note ρ the sum of fundamental weights. Let W be the Weyl group, generated by the reflections corresponding to the simple roots s_{α_{i}}. Let w_{0} be the longest element of W. We note ( , ) the W -invariant form on P. We have (α_{j}, ϖ_{i}) = δ_{ij} \frac{(α_{i},α_{i})}{2}.

1.2. Let q be a non zero complex number not a root of unity and U_{q}(𝔤) be the simply connected quantized enveloping algebra, defined as in [14, 3.2.9]. Let U_{q}(𝔫), resp. U_{q}(𝔫^{−}), be the subalgebra generated by the canonical generators E_{α_{i}}, resp. F_{α_{i}}, of positive, resp. negative, weights. For all λ in P, let τ(λ) the corresponding element in the algebra U^{0} of the torus of U_{q}(𝔤). We have the triangular decomposition U_{q}(𝔤) = U_{q}(𝔫^{−})⊗U^{0} ⊗U_{q}(𝔫). We set

(1.2.1)  U_{q}(𝔟) = U_{q}(𝔫) ⊗ U^{0},  U_{q}(𝔟^{−}) = U_{q}(𝔫^{−}) ⊗ U^{0}.

U_{q}(𝔤) is endowed with a structure of Hopf algebra with comultiplication Δ, and antipode S.

We fix the following notations, where t is a complex not root of one, n a non negative integer and α a positive root : [n]_{t} = \frac{1−q^{t}}{1−q}, [n]_{t}! = [n]_{t}[n − 1]_{t} … [1]_{t}, q_{α} = q^{\frac{(α,α)}{2}} .

1.3. For w in W, let T_{w} be the Lusztig automorphism, [18], associated to w. We fix a decomposition of the longest element of the Weyl group w_{0} = s_{i_{1}} … s_{i_{N}}, where N = dim 𝔫. This decomposition settles an order, noted <, into the set Δ^{+} of positive roots : β_{N} = s_{i_{1}} … s_{i_{N−1}}(α_{i_{N}}), …, β_{2} = s_{i_{1}}(α_{i_{2}}), β_{1} = α_{i_{1}}. Then, we introduce the following elements in U_{q}(𝔫) : E_{β_{s}} = T_{i_{1}} … T_{i_{s−1}}(E_{i_{s}}). We define in the same way F_{β_{s}} = T_{i_{1}} … T_{i_{s−1}}(F_{i_{s}}).

We know, cf. [18], that these elements generate a Poincaré-Birkoff-Witt base of U_{q}(𝔫). We have, by [22], see also [10, Lemma 1.7] :

Proposition. Let U_{q}(𝔫_{β}) be the space generated by the ordered products ∏_{→} E_{α}^{k_{α}}, α ∈

Δ^{+}, α ≤ β, k_{α} ∈ ℕ. Then, U_{q}(𝔫_{β}) is a subalgebra of U_{q}(𝔫). Moreover, if μ < β, we have E_{μ}E_{β} − q^{−(μ,β)}E_{β}E_{μ} ∈ ∑_{α<β} U_{q}(𝔫_{β})E_{α}. ♢

1.4. The dual U_{q}(𝔤)^{∗} is endowed with a structure of left, resp. right, U_{q}(𝔤)-module by u.c(a) = c(au), resp. c.u(a) = c(ua), u, a ∈ U_{q}(𝔤), c ∈ U_{q}(𝔤)^{∗}. In the same way, if M is a left U_{q}(𝔤)-module, we endow the dual M^{∗} with a structure of right U_{q}(𝔤)-module by ξu(v) = ξ(uv), u ∈ U_{q}(𝔤), ξ ∈ M^{∗}, v ∈ M.

For all λ in P^{+}, let L_{q}(λ) be the simple U_{q}(𝔤)-module with highest weight λ. We know that L_{q}(λ) verifies the Weyl character formula ; for all w in W we note v_{wλ} extremal vector of weight wλ. For all integral dominant weight λ, we fix a weight base (v_{μ}), μ ∈ Ω(L_{q}(λ)), of L_{q}(λ). We note (v_{μ}^{∗}) its dual base. ¿From [14, 10.2], we have the assertion

Assertion. Let λ be a integral dominant weight and w an element of the Weyl group. Fix a space M and an isomorphism ϕ : M → L_{q}(λ)^{∗}. We can endow M with the structure of right U_{q}(𝔤)-module by : v^{∗}.u = ϕ^{−1}(ϕ(v^{∗})T_{w}(u)), v^{∗} ∈ M. Then, the U_{q}(𝔤)-module M is isomorphic to L_{q}(λ)^{∗} and ϕ^{−1}(v_{wλ}^{∗}), resp. ϕ^{−1}(v_{ww_{0}λ}^{∗}), is its highest weight, resp. lowest weight, vector. ♢

For all ξ in L_{q}(λ)^{∗} and v in L_{q}(λ), let c_{ξ,v}^{λ} in U_{q}(𝔤)^{∗} given by c_{ξ,v}^{λ}(u) = ξ(uv), u ∈ U_{q}(𝔤). Then, we have u.c_{ξ,v}^{λ} = c_{ξ,uv}^{λ} and c_{ξ,v}^{λ}.u = c_{ξu,v}^{λ}. If ξ, resp. v, has weight ν, resp μ, we set (if no confusion occurs), c_{ν,μ}^{λ} = c_{ξ,v}^{λ}. For all integral dominant weight λ, let C(λ), resp. C^{+}(λ), be the space generated by the c_{ξ,v}^{λ}, resp. c_{ξ,λ}^{λ}, ξ ∈ L_{q}(λ)^{∗}, v ∈ L_{q}(λ). We note R = ℂ_{q}[G] = ⨁_{λ∈P^{+}} C(λ), R^{+} = ⨁_{λ∈P^{+}} C^{+}(λ). R^{+} and R are subalgebras of the Hopf dual of U_{q}(𝔤).

For w in W, we define the quantized algebra S_{w}^{+} of regular functions on the Schubert variety, (see [14,15] for details) : S_{w}^{+} is the inductive limit of (c_{wλ,λ}^{λ})^{−1}V_{w}^{+}(λ)^{∗}, for λ in P^{+}, where V_{w}^{+}(λ)^{∗} is the dual of the Demazure module V_{w}(λ), naturally identified as a quotient of C^{+}(λ).

1.5. We know that U_{q}(𝔤) is an almost cocommutative Hopf algebra, cf. [11]. Let R = R_{(1)} ⊗ R_{(2)}, be the R-matrix of U_{q}(𝔤). It verifies RΔ = Δ^{t}R, where t is the twist. ¿From this property, follows easily :

(1.5.1)  (c.R_{(1)})(c′.R_{(2)}) = (R_{(2)}.c′)(R_{(1)}.c), c, c′ ∈ ℂ_{q}[G].

We recall the expression of the R-matrix as an ordered product [17, 3.3] :

(1.5.2)  R = ( ∏_{α∈Δ^{+}} exp_{q_{α}^{−2}}((1 − q_{α}^{−2})E_{α} ⊗ F_{α}))τ(γ) ⊗ τ(γ),

where γ ∈ P, exp_{t}(x) = ∑_{n≥0} \frac{x^{n}}{[n]_{t}!}.

2. A quantum Taylor Lemma.

2.1. We have the following lemma, whose proof is analogue to [20, 1.1], [8, Proposition 1.1] :

Lemma. Let A be an ℂ-algebra, σ a ℂ-automorphism of A, δ a σ-derivation of A, i.e. δ(ab) = δ(a)b + σ(a)δ(b), a, b ∈ A. Let A^{δ} be the algebra of δ-invariants in A. Suppose that 1) δ is locally nilpotent, 2) σδσ^{−1} = Qδ, Q ∈ ℂ^{∗}, Q not root of one, 3) there exists a in A such that δ(a) ∈ ℂ^{∗}. Then, A = A^{δ}[a] and a is (right) transcendantal on A^{δ}, i.e. A = ⨁_{p≥0} A^{δ}a^{p}.

Proof. By 2), A^{δ} is σ-stable. Moreover, we have : δ^{p}(a^{p}) = [p]_{Q^{−1}}!δ(a)^{p}. This implies the direct sum in the claimed equality. Let u in A, with degree p, i.e. p is the greatest integer such that u_{0} := δ^{p}(u) ≠ 0. Clearly, the element u_{0} is in A^{δ}. We prove the result by induction on p by considering u − \frac{1}{[p]_{Q^{−1}} !δ(a)}σ^{−p}(u_{0})a^{p}, of degree ≤ p − 1. ♢

Let A be an ℂ-algebra such that U_{q}(𝔟) acts (as a bialgebra) on A, i.e. A is a U_{q}(𝔟)- module and a(uv) = a_{(1)}ua_{(2)}v, u, v ∈ A, a ∈ U_{q}(𝔟), Δ(a) = a_{(1)} ⊗ a_{(2)}. Suppose that this action is locally finite. We set A^{0} = A and we note A^{l}, 1 ≤ l ≤ N, the algebra of U_{q}(𝔫_{β_{l}})-invariants in A. This proposition follows from the lemma.

Proposition. Let A be an algebra defined as above. Suppose that for all β in Δ^{+}, there exists a_{β} in A such that E_{α}.a_{β} = δ_{αβ}, α ≤ β, where δ_{αβ} is the Kroenecker symbol. Then, for all l, 1 ≤ l ≤ N, we have

A = ⨁_{(k_{1},...,k_{l})∈ℕ^{l}} A^{l}a_{β_{l}}^{k_{l}} … a_{β_{1}}^{k_{1}}.

Proof. We note ϕ : U_{q}(𝔟) → End(A), the natural morphism for this action. By Proposition 1.3 and [17, 2.4.1], δ := ϕ(E_{β_{l}}) is a ϕ(τ(β_{l}))-derivation on A^{l−1}. The conditions of the previous lemma are satisfied because 1) δ is locally nilpotent on A^{l−1},

2) τ(β_{l}) and E_{β_{l}} q-commute, 3) a := a_{β_{l}} is in A^{l−1} and verifies δ(a) = 1 by the definition. The proposition is obtained by induction on l using the previous lemma. ♢

2.2. We shall see that if A is one of the algebras considered in the introduction, then the elements a_{β} of Proposition 2.1 exist in some localization of A, and not in the algebra A. For the classical case, cf. [12, Th. 2.6], it is enough to localize by a set S generated by 𝔫-invariant elements in A. We can then apply the Taylor lemma to A_{S}. In the quantum case, the Taylor Lemma needs some refinements. We slightly modify Lemma 2.1 to get

Assertion. Let C be a noetherian domain on ℂ, σ a ℂ-automorphism of C and δ be a σ-derivation on C which verifies 1) and 2) of Lemma 2.1. Suppose s (non zero) and s′ q-commuting in C and such that δ(s′) = s ∈ C^{δ}. Let a = s′s^{−1} in Fract C and M = ⋃_{p≥0} Cs^{−p}. Then δ acts on M and (as spaces) C ⊂ M = ⨁_{p≥0} M^{δ}a^{p}. Proof. The element s in δ-invariant, so δ extends as a locally nilpotent derivation on M. Clearly, C is a subset of M and δ(a) = 1. The direct sum is proved as in Lemma 2.1. As s and s′ q-commute, we have ⨁_{p≥0} M^{δ}a^{p} ⊂ M. The reverse inclusion is an easy induction as in the proof of Lemma 2.1. ♢

We now give a condition on A which implies that Fract A is isomorphic to a quantum Weyl field.

Definition. Let A be a noetherian domain. We say that A verifies the property (P) if the following hypotheses are verified

(i) U_{q}(𝔟) acts (as a Hopf algebra) on the ℂ-algebra A and this action is locally finite. Let B = A^{N} be the subalgebra of U_{q}(𝔫)-invariant elements in A.

(ii) B is generated by elements c^{i}, 1 ≤ i ≤ m, and B = ℂ[c^{m}] … [c^{2}][c^{1}] is an algebra

of functions on a quantum affine space with SQCG {c^{1}, …, c^{m}}. (iii) there exists non zero elements c_{β}, β ∈ Δ^{+}, in A which q-commute, q-commute with

c^{i}, 1 ≤ i ≤ m, and verify : E_{α}.c_{β} = δ_{αβ}c_{β}^{>}, α ≤ β, where c_{β}^{>} is either c_{γ}, γ > β, E_{β}-invariant, or c^{j}, 1 ≤ j ≤ m.

Proposition. Let A be a noetherian domain. If A verifies the property (P), then Fract A is isomorphic to a quantum Weyl field. To be precise, if B is the algebra of U_{q}(𝔫)-invariant elements in A, then Fract A is isomorphic to the skew field of fractions of B[c_{β_{N}}] … [c_{β_{2}}][c_{β_{1}}]. Proof. Suppose that A verifies the property (P). For all β = β_{i} in Δ^{+}, note S_{i} the multiplicative set generated by s_{i} := c_{β}^{>}, cf. (iii). We can define the following elements : a_{β} = (c_{β}^{>})^{−1}c_{β} ∈ Fract A. In the context of the previous Assertion, we set (improperly) C_{S} = ⋃_{p≥0} Cs^{−p}, where S is the multiplicative set generated by s. Recall that A is a domain. The conditions of the assertion are easily verified from the property (P). We have : (∗) A = A^{0} ⊂ A_{S_{1}}^{1}[a_{β_{1}}] ⊂ A_{S_{2}}^{2}[a_{β_{2}}][s_{1}^{−1}, a_{β_{1}}] ⊂ … ⊂ A_{S_{N}}^{N} [a_{β_{N}}][s_{N−1}^{−1}, a_{β_{N−1}}, …, s_{1}^{−1}, a_{β_{1}}].

Moreover, from (iii), c_{β_{N}}^{>} ∈ B, so c_{β_{N}} ∈ B[a_{β_{N}}]. Inductively, we can prove that B[a_{β_{N}}] … [a_{β_{1}}] contains all the s_{i}. This and (*) imply that Fract A is isomorphic to the skew field of fractions of B[a_{β_{N}}] … [a_{β_{1}}]. The property (P) (iii) asserts that the a_{α} are in the skew field of fractions of B[c_{β_{N}}] … [c_{β_{2}}][c_{β_{1}}]. By the Taylor Lemma, these extensions are (right) transcendantal. Our proposition follows. ♢

2.3. Fix 1 ≤ l ≤ N and β = β_{l}. The reduced decomposition of w_{0} being fixed as in 1.3, we define the elements y_{l} of the Weyl group : y_{0} = Id, y_{l} = s_{i_{1}}s_{i_{2}} … s_{i_{l}}, l > 0. Then, we introduce in R^{+} : c_{β} = c_{y_{l−1}ϖ_{i_{l}} ,ϖ_{i_{l}}}^{ϖ_{i_{l}}} , c_{β}^{>} = c_{β}.E_{β}.

Lemma. Let I_{l} = {p, l < p ≤ N | i_{p} = i_{l}}. If I_{l} is empty, c_{β_{l}}^{>} = c_{w_{0}ϖ_{i_{l}} ,ϖ_{i_{l}}}^{ϖ_{i_{l}}}. If not, let l′ be the minimal element in I_{l}, then c_{β_{l}}^{>} = c_{β_{l′}} (up to a multiplicative scalar). Proof. Fix l. Set j = i_{l}. We show the second assertion of the lemma, the first is similar. We have y_{l′−1}(ϖ_{j}) = y_{l−1}s_{i_{l}} … s_{i_{l′−1}}(ϖ_{j}) = y_{l−1}s_{i_{l}}(ϖ_{j}) = y_{l−1}(ϖ_{j} − α_{j}) = y_{l−1}(ϖ_{j})−β. From Assertion 1.4, with λ = ϖ_{j} and w = y_{l−1}, it is enough to prove that v_{ϖ_{i_{l}}}^{∗} .E_{ϖ_{i_{l}}} = v_{s_{i_{l}} ϖ_{i_{l}}}. This is clear by the Weyl character formula and we can conclude the lemma. ♢

Proposition. Let S be the multiplicative set generated by the c_{α}^{>}, α ∈ Δ^{+}. S is a Ore set in R^{+}. Let a_{β} = (c_{β}^{>})^{−1}c_{β} ∈ S^{−1}R^{+}. We have a_{α′}a_{α} = q^{(α,α′)}a_{α}a_{α′}, α, α′ ∈ Δ^{+}, α′ < α. Proof. As in the previous proof, we fix β = β_{l} in Δ^{+}, 1 ≤ l ≤ N, and i_{l} = j. From Assertion 1.4, the extremal vector v_{y_{l−1}ϖ_{j}}^{∗} in L_{q}(ϖ_{j})^{∗} is annihilated by the right action of E_{α}, α > β, and F_{α}, α ≤ β. So, this holds for c_{β}. From (1.5.1) and (1.5.2), we can deduce that the c_{β}, β ∈ Δ^{+}, q-commute. By Lemma 2.3 and by [14, Corollary 9.1.4], this is also true for the c_{β}^{>}, β ∈ Δ^{+}. The first assertion of the proposition follows from [loc. cit., Lemma A.2.9], and [loc. cit., Lemma 9.1.10]. We used [loc. cit., Proposition 9.1.5] to calculate the exponent of q in the formula. ♢

Remark. The fact that S is a Ore set is not essential for the next section. Indeed, the elements a_{β} defined above exist a least in Fract R^{+}, because R^{+} is a noetherian domain, cf. [14,9.1.11].

3. Applications.

3.1. In this section, we give a list of quantum algebras which verify the desired property.

Theorem. The skew field of fractions of the algebra R^{+}, resp. S_{w}^{+}, resp. U_{q}(𝔫), resp. U_{q}(𝔟), is isomorphic to a quantum Weyl field of dimension N + n, resp. l(w) + n, resp. N, resp. N + n.

Proof. By [14, Chap. 7, Chap. 9], all the algebras in the claim are noetherian domains. Let’s verify the assertions of the property (P).

For R^{+}, the action of (i) is the right regular action, which is locally finite. The algebra of U_{q}(𝔫)-invariant elements in R^{+} is generated by the c_{w_{0}ϖ_{i},ϖ_{i}}^{ϖ_{i}}, 1 ≤ i ≤ n which q-commute by [14, 9.1.4]. Then, (iii) is given by Lemma 2.3 and Proposition 2.3.

The assertion for S_{w}^{+} is similar. We may consider, without loss of generality, the case where w = s_{i_{N−l(w)+1}} … s_{i_{N}}. Set β := β_{N−l(w)+1}. We prove the theorem with the help of (i) the right action of U_{q}(𝔫), (ii) the U_{q}(𝔫)-invariant elements c_{wϖ_{i},ϖ_{i}}^{ϖ_{i}}, 1 ≤ i ≤ n, (iii) the q-commuting elements c_{α}, α ≤ β. Indeed, (i) and (iii) are clear, and (ii) follows from the fact that (V_{w}^{+}(λ)^{∗})^{U_{q}(𝔫)} is generated by c_{wλ,λ}^{λ}.

Let φ be the restriction homomorphism from U_{q}(𝔤)^{∗} to U_{q}(𝔟^{−})^{∗} and J^{−} = Kerφ ∩ R. We know, cf. [14, 9.1.10, 9.2.11], that φ restricts to an embedding from R^{+} to R/J^{−}, and this map is surjective up to localization. It is also well known, cf. [3], that there exists an algebra antihomomorphism R/J^{−} ≃ U_{q}(𝔟). The theorem being true for R^{+}, it is true for U_{q}(𝔟). Let’s consider now the previous isomorphism extent to the skew field of fractions. We remark, cf. [4, Lemme 3.4], [5, I Prop. 4.2], that the image of a_{β} is in Fract U_{q}(𝔫). Thus, we can conclude by (1.2.1). ♢

As in [15, Cor. 6], we have the following corollary.

Corollary. Let P minimal primitive ideal of U_{q}(𝔤). Then, Fract U_{q}(𝔤)/P is isomorphic to a quantum Weyl field of dimension 2N. ♢

3.2. In this section section, we make more explicit the system of q-commuting generators (SQCG) of Fract U_{q}(𝔫) for the classical simple Lie algebras 𝔤. Recall that these generators are, from the proof of Theorem 3.1, the images of the elements c_{β}, β ∈ R^{+}, by the Drinfeld (anti)homomorphism, followed by the natural projection on U_{q}(𝔫).

We know, [6], that there is a natural embedding from L_{q}(λ)^{∗} to U_{q}(𝔫) which maps the lowest weight vector of L_{q}(λ)^{∗} to 1. It maps the highest weight vector to an element e_{λ} of the q-center of U_{q}(𝔫), cf. [7]. An element of U_{q}(𝔫) will be called almost maximal, and be noted e_{λ}^{i}, if it is the image of a vector v_{−s_{i}w_{0}λ} ∈ L_{q}(λ)^{∗}, 1 ≤ i ≤ n, λ ∈ P^{+}. Remark that those elements can be explicitly computed with the help of [6, Lemme 3.3].

With the standard notations of [2, Planches I à IV], we recall the canonical embeddings of Dynkin diagrams :

A_{1} ⊂ … ⊂ A_{n}, A_{1} ⊂ B_{2} ⊂ … ⊂ B_{n}, A_{1} ⊂ C_{2} ⊂ … ⊂ C_{n}, A_{3} ⊂ D_{4} ⊂ … ⊂ D_{n}.

If X is a Dynkin label for a classical simple Lie algebra, we note X^{−} the previous label for the embedding sequence above. We can inductively define the reduced decomposition of the longest element of the Weyl group w_{0}(X) for the Lie algebra of type X by :

w_{0}(X) = w_{0}(X^{−}). \{ \begin{matrix}s_{n} … s_{2}s_{1}, & if X = A_{n} \\ s_{1} … s_{n−1}s_{n}s_{n−1} … s_{1} & if X = B_{n} or C_{n} \\ s_{1} … s_{n−2}s_{n}s_{n−1} … s_{1} & if X = D_{n} \end{matrix}

This decomposition of w_{0} permits to obtain inductively our SQCG :

Theorem. The system of q-commuting generators corresponding to the simple classical Lie algebra 𝔤 of type X is inductively given by

SQCG(X) = SQCG(X^{−}) ⋃ ∪_{i=1}^{n}e_{ϖ_{i}} ⋃ \{ \begin{matrix}ø, & if X = A_{n} \\ ∪_{i=1}^{n−1}e_{ϖ_{i}}^{i} & if X = B_{n} or C_{n} \\ ∪_{i=1}^{n−2}e_{ϖ_{i}}^{i} & if X = D_{n} \end{matrix}

♢

Remark. This theorem is a generalization of [1, Th. 2.15] for the classical Lie algebras. Note that, except for e_{ϖ_{n}} if 𝔤 has type B_{n}, C_{n}, D_{n} and e_{ϖ_{n−1}}, if 𝔤 has type D_{n}, all those elements can be obtained as quantum determinants of a ”basic” matrix. This can be obtained as in [6], see also [14, 7.5.5], by considering exterior powers of L_{q}(ϖ_{1}). 3.3. Now, we give the proof of a similar theorem for the algebra R = ℂ_{q}[G]. For all i, j, 1 ≤ i, j ≤ N, we note R^{i,j} the subalgebra of elements in R which are invariant for the right action of U_{q}(𝔫_{β_{j}}) and for the left action of U_{q}(𝔫_{β_{i}}) (recall that these actions commute ). Set B = R^{N,N}.

Fix β = β_{l}. We define the following elements in C(ρ) :

d_{β} = c_{y_{l−1}ρ,−y_{l}ρ}, d′_{β} = c_{y_{l−1}ρ,−y_{l−1}ρ}.

By Assertion 1.4, we prove that d_{β} is invariant for the left action of E_{α}, α ≤ β, and of F_{α}, α > β. Moreover, d_{β} is invariant for the right action of E_{α}, α < β, and of F_{α}, α ≥ β. In the same way, d′_{β} is invariant for the left and right action of E_{α}, α < β, and of F_{α}, α ≥ β. Hence, by (1.5.1) and (1.5.2), the elements d_{α} and d′_{α}, α ∈ Δ^{+}, q-commute. We have clearly d_{β} ∈ R^{l,l−1} and d_{β} ∈ R^{l−1,l−1}. Moreover, up to a non zero multiplicative scalar, E_{β}.d′_{β} = d_{β} and d_{β}.E_{β} = d′_{β_{l+1}}, l ≠ N, d_{β}.E_{β} = c_{w_{0}ρ,ρ} ∈ B, l = N. ℂ_{q}[G] being noetherian [14, 9.2.2], we can easily modify the proof of Proposition 2.2 to obtain :

Theorem. Fract ℂ_{q}[G] is isomorphic to a quantum Weyl field of dimension 2N + n. To be more precise, Fract ℂ_{q}[G] is isomorphic to the skew Weyl field of fractions of B[d_{β_{N}}][d′_{β_{N}}] … [d_{β_{2}}][d′_{β_{2}}][d_{β_{1}}][d′_{β_{1}}]. ♢ The generators of B, the d_{β} and d′_{β}, can be easily expressed as a product of elements of C(ϖ_{i}), 1 ≤ i ≤ n. Consider the algebra ℂ_{q}[SL_{2}] generated by the elements of the

quantum matrix : ( \begin{matrix}a & c \\ b & d \end{matrix} ). Then, the system of q-commuting generators provided by

the theorem is {b, a, c}.

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Mathematics Subject Classification : Quantum groups, quantum weyl fields, Rmatrix.

Institut Girard Desargues, UPRES-A-5028 Université Claude Bernard Lyon I, Bat 101 69622 Villeurbanne Cedex, France e-mail : caldero@desargues.univ-lyon1.fr