Journal of Lie Theory Volume 13 (2003) 21–64 ◯^{C}2003 Heldermann Verlag

PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras

Benjamin Enriquez

Communicated by G. I. Olshanski

Abstract. We give proofs of the PBW and duality theorems for the quantum Kac-Moody algebras and quantum current algebras, relying on Lie bialgebra duality. We also show that the classical limit of the quantum current algebras associated with an untwisted affine Cartan matrix is the enveloping algebra of a quotient of the corresponding toroidal algebra; this quotient is trivial in all cases except the A_{1}^{(1)} case.

Introduction

The purpose of “quantum algebra” is to study deformations of various commutative (or cocommutative) algebras (or Hopf algebras). They are usually presented by generators and relations. Then a natural problem arises: to show that the deformed algebras have the same size as their undeformed counterparts. The first instance of such a result is the Poincaré-Birkhoff-Witt (PBW) theorem, which says that if 𝔤 is a Lie algebra over a field of characteristic zero, then the symmetrization map S^{•}(𝔤) → U(𝔤) is a linear isomorphism. In that case, U(𝔤) is viewed as a quantization of the Poisson algebra S^{•}(𝔤).

The problem of comparing the size of an algebra presented by generators and relations with that of the classical algebra is called with PBW problem. A useful tool for solving this problem is Bergman’s combinatorial “diamond” lemma ([4]). It was used in the quantum situation by several authors (notably Berger in [3] and Rosso in [26]). Another approach to the PBW problem involves constructing “enough” representations of the deformed algebra. For instance, Lusztig’s PBW result (Corollary 33.1.5 of [22]) relies on the study of integrable modules over quantized Kac-Moody algebras. A last, direct approach involves realizing the deformed algebra structure on some model space. The original PBW result can be proved in this way, by explicitly constructing a star-product on S^{•}(𝔤) ([5]). This is also the approach followed in the present paper in the case of quantum groups and quantum current algebras. In this situation, these algebras are realized as

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag (functional) shuffle algebras. We also use in an essential way the Hopf algebra structures.

1. Outline of results

1.1. Quantum Kac-Moody algebras.

Let A = (a_{ij})_{1≤i,j≤n} be a symmetrizable Cartan matrix. Let (d_{i})_{1≤i≤n} be the coprime positive integers such that the matrix (d_{i}a_{ij})_{1≤i,j≤n} is symmetric. Let r be the rank of A; we assume that the matrix (a_{ij})_{n−r+1≤i,j≤n} is nondegenerate.

Let 𝔤 be the Kac-Moody Lie algebra associated with A; let 𝔫_{+} be its positive pro-nilpotent subalgebra and (ē_{i})_{1≤i≤n} be the generators of 𝔫_{+} corresponding to the simple roots of 𝔤.

Let ℂ[[ℏ]] be the formal series ring in ℏ. Let U_{ℏ}𝔫_{+} be the quotient of the free algebra with n generators ℂ[[ℏ]]⟨e_{i}, i = 1, …, n⟩ by the two-sided ideal generated by the quantum Serre relations

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1 − a_{ij} \\ k \end{matrix} ]_{q^{d_{i}}} e_{i}^{k}e_{j}e_{i}^{1−a_{ij}−k} = 0,  (1)

where [ \begin{matrix}m \\ p \end{matrix} ]_{q} = \frac{[m]_{q}^{!}}{[p]_{q}^{!}[m−p]_{q}^{!}} , [k]_{q}^{!} = [1]_{q} ⋯ [k]_{q}, [k]_{q} = \frac{q^{k}−q^{−k}}{q−q^{−1}} , and q = e^{ℏ} ([9, 17]).

We will show:

Theorem 1.1. U_{ℏ}𝔫_{+} is a free ℂ[[ℏ]]-module, and the map e_{i} ↦ ē_{i} defines an algebra isomorphism of U_{ℏ}𝔫_{+}/ℏU_{ℏ}𝔫_{+} with U𝔫_{+}.

This Theorem may be derived from the Poincaré-Birkhoff-Witt (PBW) results of Lusztig’s book [22]; in the case 𝔤 = 𝔰𝔩_{n}, it can also be derived from those of Rosso ([26]), and in the cases when 𝔤 is semisimple or untwisted affine, from those of [21].

The proof presented here is based on the comparison of U_{ℏ}𝔫_{+} with a quantum shuffle algebra, Lie bialgebra duality and the Deodhar-Gabber-Kac theorem.

As a corollary of this proof, we show

Corollary 1.1. The map p_{ℏ} defined in Lemma 2.3 is an algebra isomorphism from U_{ℏ}𝔫_{+} to the subalgebra ⟨V ⟩ of the shuffle algebra Sh(V) defined in sect. 2..

This result was proved in [27]; it can also be derived from the results of [28]. Rosso’s proof uses the nondegeneracy of the pairing between opposite Borel quantum algebras ([22], Corollary 33.1.5; see also Theorem 1.2). Schauenburg shows that ⟨V ⟩ is isomorphic to the quotient of the free algebra generated by the e_{i} by the radical of a braided Hopf pairing. Together with [22], Corollary 33.1.5, this implies Corollary 1.1.

Define U_{ℏ}𝔫_{−} as the algebra with generators f_{i}, i = 1, …, n, and the same defining relations as U_{ℏ}𝔫_{+} (with e_{i} replaced by f_{i}). Define a grading on U_{ℏ}𝔫_{±} by (±ℕ)^{n} by deg(e_{i}) = ϵ_{i}, deg(f_{i}) = −ϵ_{i}, where ϵ_{i} is the ith basis vector of ℕ^{n}, and define the braided tensor products U_{ℏ}𝔫_{±} ¯⊗U_{ℏ}𝔫_{±} as the algebras isomorphic to U_{ℏ}𝔫_{±} ⊗_{ℂ[[ℏ]]} U_{ℏ}𝔫_{±} as ℂ[[ℏ]]-modules, with mutiplication rule

(x ⊗ y)(x′ ⊗ y′) = q^{−⟨deg(x′),deg(y)⟩}(xx′ ⊗ yy′);  (2)

we set ⟨ϵ_{i}, ϵ_{j}⟩ = d_{i}a_{ij}.

Then U_{ℏ}𝔫_{±} are endowed with braided Hopf algebra structures, defined by Δ_{+}(e_{i}) = e_{i} ⊗ 1 + 1 ⊗ e_{i}, and Δ_{−}(f_{i}) = f_{i} ⊗ 1 + 1 ⊗ f_{i}.

In [8], Drinfeld showed that there exists a unique pairing ⟨ , ⟩_{U_{ℏ}𝔫_{±}} of U_{ℏ}𝔫_{+} and U_{ℏ}𝔫_{−} with values in ℂ((ℏ)) = ℂ[[ℏ]][ℏ^{−1}], defined by

⟨e_{i}, f_{i′}⟩_{U_{ℏ}𝔫_{±}} = \frac{1}{ℏ}d_{i}^{−1}δ_{ii′},  (3)
⟨x, yy′⟩_{U_{ℏ}𝔫_{±}} = ∑⟨x^{(1)}, y⟩_{U_{ℏ}𝔫_{±}}⟨x^{(2)}, y′⟩_{U_{ℏ}𝔫_{±}},  (4)

and

⟨xx′, y⟩_{U_{ℏ}𝔫_{±}} = ∑⟨x, y^{(1)}⟩_{U_{ℏ}𝔫_{±}}⟨x′, y^{(2)}⟩_{U_{ℏ}𝔫_{±}}  (5)

for x, x′ in U_{ℏ}𝔫_{+} and y, y′ in U_{ℏ}𝔫_{−}, and Δ_{±}(z) = ∑ z^{(1)} ⊗ z^{(2)} (braided Hopf pairing axioms).

As a direct consequence of Corollary 1.1, we show:

Theorem 1.2. The pairing ⟨ , ⟩_{U_{ℏ}𝔫_{±}} between U_{ℏ}𝔫_{+} and U_{ℏ}𝔫_{−} is nondegenerate.

This result can be found in Lusztig’s book ([22], Corollary 33.1.5, Def. 3.1.1 and Proposition 3.2.4); it relies on the construction of dual PBW bases. Another argument using Lusztig’ results on integrable modules is in [29], and an argument using irreducible Verma modules is in [25].

We also show:

Proposition 1.1. For any α in ℕ^{n}, let U_{ℏ}𝔫_{±}[±α] be the part of U_{ℏ}𝔫_{±} of degree ±α, and let P [α] be the element of U_{ℏ}𝔫_{+}[α]⊗U_{ℏ}𝔫_{−}[−α] induced by ⟨ , ⟩_{U_{ℏ}𝔫_{±}}. Let Δ_{+} be the set of positive roots of 𝔤 (the ϵ_{i} are the simple roots). Let (ē_{α,i})_{α∈Δ_{+}} and (f̄_{α,i})_{α∈Δ_{+}} be dual Cartan-Weyl bases of 𝔫_{+} and 𝔫_{−}, and let e_{α,i}, f_{α,i} be lifts of the ē_{α,i}, f̄_{α,i} to U_{ℏ}𝔫_{±}. Then, if k is the integer such that α belongs to kΔ_{+}\(k−1)Δ_{+}, we have

P [α] = \frac{ℏ^{k}}{k!} _{α_{1},...,α_{k}∈Δ_{+}| ∑_{i} α_{i}=α;i_{j}}^{∑} e_{α_{1},i_{1}} ⋯ e_{α_{k},i_{k}} ⊗ f_{α_{1},i_{1}} ⋯ f_{α_{k},i_{k}} + o(ℏ^{k}).

The fact that P [α] has ℏ-adic valuation equal to k was stated by Drinfeld in [8].

•The case of a generic deformation parameter

It is easy to derive from the above results, PBW and nondegeneracy results in the case where the parameter q = e^{ℏ} is generic.

Corollary 1.2. Let q′ be an indeterminate, and let U_{q′}𝔫_{+} be the algebra over ℂ(q′) with generators e′_{i}, i = 1, …, r, and relations (1), with e_{i} and q = e^{ℏ} replaced by e′_{i} and q′. We have for any α in ℕ^{n}

dim_{ℂ(q′)} U_{q′}𝔫_{+}[α] = dim_{ℂ(q′)} U_{q′}𝔫_{+}[α].

Let (ē_{ν})_{ν∈I} be a basis of homogeneous elements of 𝔫_{+}. Let

ē_{ν} = ∑_{i_{j}∈{1,...,n}} C_{ν;i_{1},...,i_{k}} ē_{i_{1}} ⋯ ē_{i_{k}}

be expressions of the ē_{ν} in terms of the generators ē_{1}, …, ē_{n}.

Let C_{ν;i_{1},...,i_{k}}(q′) be rational functions of q′, such that

C_{ν;i_{1},...,i_{k}}(1) = C_{ν;i_{1},...,i_{k}},

and set e′_{ν} = ∑_{i_{j}∈{1,...,n}} C_{ν;i_{1},...,i_{k}}(q′)e_{i_{1}} ⋯ e_{i_{k}}. Then the family (∏_{ν} e_{ν}^{n_{ν}}), where the n_{ν} are in ℕ and vanish except for a finite number of them, forms a basis of U_{q′}𝔫_{+} (over ℂ(q′)).

Corollary 1.3. Let U_{q′}𝔫_{−} be the ℂ(q′)-algebra with generators f′_{i}, 1 ≤ i ≤ n, and relations (1), with e_{i} replaced by f′_{i}. Define the braided tensor squares U_{q′}𝔫_{±} ¯⊗U_{q′}𝔫_{±} using (2). We have a braided Hopf pairing ⟨ , ⟩_{ℂ(q′)} between U_{q′}𝔫_{+} and U_{q′}𝔫_{−}, defined by (3), (4) and (5). The pairing ⟨ , ⟩_{ℂ(q′)} is nondegenerate.

1.2. Quantum current and Feigin-Odesskii algebras.

Our next results deal with quantum current algebras. Assume that the Cartan matrix A is of finite type. Let L𝔫_{+} be the current Lie algebra 𝔫_{+}⊗ℂ[t, t^{−1}], endowed with the bracket [x ⊗ t^{n}, y ⊗ t^{m}] = [x, y] ⊗ t^{n+m}.

•Quantum affine algebras Let A be the quotient of the free algebra ℂ[[ℏ]]⟨e_{i}[k], i = 1, …, n, k ∈ ℤ⟩ by the two-sided ideal generated by the coefficients of monomials in the formal series identities

(q^{d_{i}a_{ij}}z − w)e_{i}(z)e_{j}(w) = (z − q^{d_{i}a_{ij}}w)e_{j}(w)e_{i}(z),  (6)

Sym_{z_{1},...,z_{1−a_{ij}}} ∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1 − a_{ij} \\ k \end{matrix} ]_{q^{d_{i}}} e_{i}(z_{1}) … e_{i}(z_{k})e_{j}(w)e_{i}(z_{k+1}) … e_{i}(z_{1−a_{ij}}) = 0,

(7)

where e_{i}(z) is the generating series e_{i}(z) = ∑_{k∈z} e_{i}[k]z^{−k}, and q = e^{ℏ}. Let U_{ℏ}L𝔫_{+} be the quotient A/(∩_{N>0}ℏ^{N}A). Define Ã as the quotient of the free algebra ℂ[[ℏ]]⟨e_{i}[k]_{A}, i = 1, …, nk ∈ ℤ⟩

by the two-sided ideal geberated by the coefficients of monimials of (6) and

∑_{k=0}^{1−a_{ij}}(−1)^{k} [ \begin{matrix}1 − a_{ij} \\ k \end{matrix} ]_{q^{d_{i}}} (e_{i}[0]_{A})^{k}e_{j}[l]_{A}(e_{i}[0]_{A})^{1−a_{ij}−k} = 0,  (8)

for any i, j = 1, …, n and l integer. Define Ũ_{ℏ}L𝔫_{+} as the quotient Ã/(∩_{N>0}ℏ^{N} Ã).

Theorem 1.3. 1) U_{ℏ}L𝔫_{+} is a free ℂ[[ℏ]]-module, and the map e_{i}[n] ↦ ē_{i} ⊗ t^{n} defines an algebra isomorphism from U_{ℏ}L𝔫_{+}/ℏU_{ℏ}L𝔫_{+} to UL𝔫_{+}.

2) Let U_{ℏ}L𝔫_{+}^{top} be the quotient of ℂ⟨e_{i}[k], i = 1, …, n, k ∈ ℤ⟩[[ℏ]] by the ℏ-adically closed two-sided ideal generated by the coefficients of monomials in relations (6) and (7). Then U_{ℏ}L𝔫_{+}^{top} is a topologically free ℂ[[ℏ]]-module; it is naturally the ℏ-adic completion of U_{ℏ}L𝔫_{+}, and the map e_{i}[n] ↦ ē_{i} ⊗ t^{n} defines an algebra isomorphism from U_{ℏ}L𝔫_{+}^{top}/ℏU_{ℏ}L𝔫_{+}^{top} to UL𝔫_{+}.

3) There is a unique algebra map from Ũ_{ℏ}L𝔫_{+} to U_{ℏ}L𝔫_{+}, sending each e_{i}[k]_{A} to e_{i}[k]; it is an algebra isomorphism.

4) Let Ũ_{ℏ}L𝔫_{+}^{top} be the quotient of ℂ⟨e_{i}[k]_{A}, i = 1, …, n, k ∈ ℤ⟩[[ℏ]] by the ℏ- adically closed two-sided ideal generated by the coefficients of monomials in relations (6) and (8). Then e_{i}[k]_{A} ↦ e_{i}[k] defines an algebra isomorphism between Ũ_{ℏ}L𝔫_{+}^{top} and U_{ℏ}L𝔫_{+}^{top}.

The statements 1) and 2) of this Theorem can be derived from the results of [2].

In [13, 14], Feigin and Odesskii defined the algebra FO, which may be viewed as a functional version of the shuffle algebra. FO is defined as

FO = ⊕_{k∈ℕ^{n}}FO_{k},  (9)

where if k = (k_{i})_{1≤i≤n}, we set

FO_{k} = \frac{1}{∏_{1≤β≤k_{j}^{1≤α≤k_{i}}^{i<j}}(t_{α}^{(i)} − t_{β}^{(j)})}ℂ[[ℏ]][(t_{j}^{(i)})^{±1}, i = 1, …, n, j = 1, …, k_{i}]^{𝔖_{k_{1}}×···×𝔖_{kn}},

where the product of symmetric groups acts by permutation of variables of each group of variables (t_{j}^{(i)})_{1≤j≤k_{i}}. FO_{k} therefore consists of rational functions in the t_{j}^{(i)}, symmetric in each group (t_{j}^{(i)})_{1≤j≤k_{i}}, regular except for poles when the variables go to 0 or infinity, and simple poles when variables of different “colors” collide. (9) defines a grading of FO by ℕ^{n}. The product on FO is also graded, and we have, for f in FO_{k} and g in FO_{l} (l = (l_{i})_{1≤i≤n}) ,

(f ∗ g)(t_{j}^{(i)})_{1≤i≤n,1≤j≤k_{i}+l_{i}} = Sym_{{t_{j}^{(1)}}} … Sym_{{t_{j}^{(n)}}}  (10)

( ∏_{1≤i≤N} ∏_{N+1≤j≤N+M} \frac{q^{⟨ϵ(i),ϵ(j)⟩}t_{i} − t_{j}}{t_{i} − t_{j}} f(t_{1}, …, t_{N})g(t_{N+1}, …, t_{N+M})),

where N = ∑_{i=1}^{n} k_{i} and M = ∑_{i=1}^{n} l_{i}; we set for any s,

t_{k_{1}+···+k_{s−1}+1} = t_{1}^{(s)}, …, t_{k_{1}+···+k_{s}} = t_{k_{s}}^{(s)}, t_{N+l_{1}+···+l_{s−1}+1} = t_{k_{s}+1}^{(s)}, …,
t_{N+l_{1}+···+l_{s}} = t_{k_{s}+l_{s}}^{(s)};

we also define ϵ(i) = ϵ_{k} if t_{i} = t_{l}^{(k)} for some l; as before, ⟨ϵ_{i}, ϵ_{j}⟩ = d_{i}a_{ij} for i, j = 1, …, n.

In the right side of (10), each symmetrization can be replaced by a sum over shuffles, since the argument in symmetric in each group of variables (t_{1}^{(s)}, …, t_{k_{s}}^{(s)}) and (t_{k_{s}+1}^{(s)}, …, t_{k_{s}+l_{s}}^{(s)}).

In Proposition 3.2, we define a topological braided Hopf structure on FO. In [11], we showed:

Proposition 1.2. There is a unique algebra morphism i_{ℏ} from U_{ℏ}L𝔫_{+} to FO, such that i_{ℏ}(e_{i}[n]) is the element (t_{1}^{(i)})^{n} of FO_{ϵ_{i}}.

Let us denote by LV the direct sum ⊕_{i=1}^{n}FO_{ϵ_{i}} and let ⟨LV ⟩ be the sub-ℂ[[ℏ]]-algebra of FO generated by LV . As a corollary of the proof of Theorem 1.3, we prove:

Corollary 1.4. i_{ℏ} is an algebra isomorphism between U_{ℏ}L𝔫_{+} and ⟨LV ⟩.

U_{ℏ}L𝔫_{+} is also endowed with a topological braided Hopf structure (the Drinfeld comultiplication); it is then easy to see that i_{ℏ} is compatible with both Hopf structures.

Define T (LV_{±}) as the free algebras ℂ[[ℏ]]⟨e_{i}[k]^{(T)}, 1 ≤ i ≤ n, k ∈ ℤ⟩ and ℂ[[ℏ]]⟨f_{i}[k]^{(T)}, 1 ≤ i ≤ n, k ∈ ℤ⟩. We have a pairing ⟨ , ⟩_{T (LV_{±})} between T (LV_{+}) and T (LV_{−}) defined by

⟨e_{i_{1}}[k_{1}]^{(T)} ⋯ e_{i_{p}}[k_{p}]^{(T)}, f_{j_{1}}[l_{1}]^{(T)} ⋯ f_{j_{p′}}[l_{p′}]^{(T)}⟩_{T (LV_{±})}

= δ_{pp′} \frac{1}{ℏ^{p}} ∑_{σ∈𝔖_{p}} res_{z_{1}=0} ⋯ res_{z_{p}=0}

( ∏_{s>t,σ^{−1}(s)<σ^{−1}(t)} \frac{q^{(ϵ_{is},ϵ_{i_{t}})}z_{s} − z_{t}}{z_{s} − q^{(ϵ_{is},ϵ_{i_{t}})}z_{t}} ∏_{s=1}^{p} \frac{1}{d_{j_{s}}} δ_{i_{σ(s),js}} ∏_{s=1}^{p} z_{i}^{k_{s}+l_{s}} ∏_{s=1}^{p} \frac{dz_{s}}{z_{s}}) (11)

where the ratios \frac{q^{(ϵ_{is} ,ϵ_{i_{t}})}z_{s}−z_{t}}{z_{s}−q^{(ϵ_{is} ,ϵ_{i_{t}})}z_{t}} are expanded for z_{t} ≪ z_{s}.

Let U_{ℏ}L𝔫_{−} be the quotient of T (LV_{−}) by the homomorphic image of the ideal defining U_{−ℏ}L𝔫_{+} by the map e_{i}[k]^{(T)} ↦ f_{i}[k]^{(T)}.

Proposition 1.3. (see [11]) This pairing induces a pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{±}} between U_{ℏ}L𝔫_{+} and U_{ℏ}L𝔫_{−}.

We then prove:

Theorem 1.4. The pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{±}} is nondegenerate.

•The form of the R-matrix Let us set A_{±} = U_{ℏ}L𝔫_{±}. Let a and b be two integers. Define I_{≥a}^{+} and I_{≤a}^{+}, as the right, resp. left ideals of A_{+} generated by the e_{i}[k], k ≥ a , resp. the e_{i}[k], k ≤ a. The ideals I_{≥a}^{+} and I_{≤a}^{+} are graded; for α in (±ℕ)^{n}, denote by I_{≥a}^{+}[α] and I_{≤a}^{+}[α] their component of degree α.

Proposition 1.4. For any α in ℕ^{n}, for any integers a and b, (I_{≤a}^{+}+I_{≥b}^{+})^{⊥}[−α] and [A_{+}/(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥}][α] are free finite-dimensional ℂ[[ℏ]]-modules. The pairing between A_{+} and A_{−} induces a nondegenerate pairing between them. Moreover, the intersection ∩_{a,b}(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥} is zero.

Denote by P_{a,b}[α] the corresponding element of

[A_{+}/(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥}][α] ⊗ (I_{≤a}^{+} + I_{≥b}^{+})^{⊥}[−α][ℏ^{−1}].

P_{a,b}[α] defines an element of lim_{←a,b}A_{+}/(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥} ⊗_{ℂ[[ℏ]]} A_{−}[ℏ^{−1}].

Let (ē_{β,i})_{β∈Δ_{+}} and (f̄_{β,i})_{β∈Δ_{+}} be dual Cartan-Weyl bases of 𝔫_{+} and 𝔫_{−}. Let e_{β,i}[p] and f_{β,i}[p] be lifts to U_{ℏ}L𝔫_{±} of ē_{β,i} ⊗ t^{p} and f̄_{β,i} ⊗ t^{p}. Then if α belongs to kΔ_{+} − (k − 1)Δ_{+}, P [α] has the form

P [α] = \frac{ℏ^{k}}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i} α_{i}=α;i_{j}}^{∑} ∑_{p_{1},...,p_{k}∈ℤ}

e_{α_{1},i_{1}}[p_{1}] … e_{α_{k},i_{k}}[p_{k}] ⊗ f_{α_{1},i_{1}}[−p_{1}] … f_{α_{k},i_{1}}[−p_{k}] + o(ℏ^{k})

(all but a finite number of elements of this sum belong to

(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥} ⊗_{ℂ[[ℏ]]} A_{−}[ℏ^{−1}].)

Let (h′_{i})_{i=1,...,n} be the basis of 𝔥, dual to (h_{i})_{i=1,...,n}. Set

K = exp(∑_{i=1}^{n} h_{i}[0] ⊗ h′_{i}[0] + ∑_{p>0} h_{i}[p] ⊗ h′_{i}[−p]).

Then the elements R[α] = KP [α] of lim_{←N}(U_{ℏ}L𝔟_{+}⊗U_{ℏ}L𝔟_{−})/I_{N}^{𝔟_{±},(2)}, where I_{N}^{𝔟_{±},(2)} is the ideal generated by the h_{i}[p] ⊗ 1, e_{i}[p] ⊗ 1, p ≥ N, and the 1 ⊗ f_{i}[p], p ≥ N, satisfy the R-matrix identity

∑_{γ∈ℕ^{n},β∈(±ℕ)^{n},β+γ=λ} R[γ]Δ(x)_{(β,α−β)} = ∑_{γ∈ℕ^{n},β∈(±ℕ)^{n},β+γ=λ} Δ′(x)_{(β,α−β)}R[γ],

for any λ ∈ ℤ^{r} and x in the double U_{ℏ}L𝔤 of U_{ℏ}L𝔟_{+} of degree α (the sums over the root lattice are obviously finite, and each product makes sense in lim_{←N}(U_{ℏ}L𝔤 ⊗ U_{ℏ}L𝔤)/I_{N}^{𝔤,(2)}, where I_{N}^{𝔤,(2)} is the left ideal generated by the x[p] ⊗ 1 and 1 ⊗ x[p], p ≥ N, x = e_{i}, h_{i}, f_{i}).

•Yangians Let us describe how the above results are modified in the case of Yangians. Let A^{rat} be the quotient of the free algebra ℂ[[ℏ]]⟨e_{i}[k]^{rat}, i = 1, …, n, k ∈ ℤ⟩ by the two-sided ideal generated by the coefficients of the relations

(z − w + ℏa_{ij})e_{i}(z)^{rat}e_{j}(w)^{rat} = (z − w − ℏa_{ij})e_{j}(w)^{rat}e_{i}(z)^{rat},  (12)
Sym_{z_{1},...,z_{1−a_{ij}}} ad(e_{i}(z_{1})^{rat}) ⋯ ad(e_{i}(z_{1−a_{ij}})^{rat})(e_{j}(w)^{rat}) = 0,  (13)

where we set e_{i}(z)^{rat} = ∑_{k∈ℤ} e_{i}[k]^{rat}z^{−k−1}, and let us set

U_{ℏ}^{rat}L𝔫_{+} = A^{rat}/ ∩_{N>0} ℏ^{N}A^{rat}.

Define also Ã^{rat} as the quotient of the free algebra ℂ[[ℏ]]⟨e_{i}[k]_{A^{rat}}, i = 1, …, n, k ∈ ℤ⟩ by the two-sided ideal generated by the coefficients of the relations (12) and

ad(e_{i}[0]_{A^{rat}})^{1−a_{ij}}e_{j}[l]_{A^{rat}} = 0,  (14)

for any i, j = 1, …, n and integer l.

Theorem 1.5. 1) U_{ℏ}L𝔫_{+} is a free ℂ[[ℏ]]-module. There is a unique algebra isomorphism from U_{ℏ}^{rat}L𝔫_{+}/ℏU_{ℏ}^{rat}L𝔫_{+} to UL𝔫_{+}, sending the class of e_{i}[k]^{rat} to ē_{i} ⊗ t^{k}.

2) There is a unique algebra morphism from Ũ_{ℏ}L𝔫_{+} to U_{ℏ}L𝔫_{+}, sending e_{i}[k]^{rat} to e_{i}[k]_{A^{rat}}; it is an isomorphism between these algebras.

3) Let U_{ℏ}L𝔫_{+}^{rat,top} and Ũ_{ℏ}L𝔫_{+}^{rat,top} be the quotients of

ℂ⟨e_{i}[k]^{rat}, i = 1, …, n, k ∈ ℤ⟩[[ℏ]] and ℂ⟨e_{i}[k]_{A^{rat}}, i = 1, …, n, k ∈ ℤ⟩[[ℏ]]

by the ℏ-adically closed two-sided ideals generated by the coefficients of monomials in relations (12) and (13), resp. (12) and (14). Then e_{i}[k]_{A^{rat}} ↦ e_{i}[k]^{rat} defines an algebra isomorphism between U_{ℏ}L𝔫_{+}^{rat,top} and Ũ_{ℏ}L𝔫_{+}^{rat,top}. U_{ℏ}L𝔫_{+}^{rat,top} is the ℏ-adic completion of U_{ℏ}L𝔫_{+}^{rat}; it is a topologically free ℂ[[ℏ]]-module.

Define FO^{rat} as the graded space FO, endowed with the product obtained from (10) by the replacement of each q^{λ}z − q^{μ}w by z − w + ℏ(λ − μ). FO^{rat} is an associative algebra and we have

Theorem 1.6. There is a unique algebra map i_{ℏ}^{rat} from U_{ℏ}L𝔫_{+}^{rat} to FO^{rat}, sending each e_{i}[k]^{rat} to e_{i} ⊗ t^{k}. i_{ℏ} is an isomorphism between U_{ℏ}L𝔫_{+}^{rat} and its subalgebra ⟨LV ⟩^{rat} generated by the degree one elements.

Define a pairing ⟨ , ⟩_{T (LV_{±}),rat} between T (LV_{+}) and T (LV_{−}) by the formula (11), where each q^{λ}z − q^{μ}w is replaced by z − w + ℏ(λ − μ). Let U_{ℏ}L𝔫_{−}^{rat} be the quotient T (LV_{−}) by the homomorphic image of the ideal defining U_{ℏ}L𝔫_{+}^{rat} by the map e_{i}[k]^{(T)} ↦ f_{i}[k]^{(T)}.

Theorem 1.7. ⟨ , ⟩_{T (LV_{±}),rat} induces a nondegenerate pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{±},rat} between U_{ℏ}L𝔫_{±}^{rat}. Define I_{a}^{±,rat} as the right, resp. left ideals of U_{ℏ}L𝔫_{±}^{rat} generated by the e_{i}[k]^{rat}, k ≥ a, resp. by the f_{i}[k]^{rat}, k ≥ a. Then for any α in ℕ^{n}, I_{a}^{+,rat}[α] ∩ (I_{b}^{−,rat}[−α])^{⊥} is a space of finite codimension in (I_{b}^{−,rat}[−α])^{⊥}. Let P [α]^{rat} be the corresponding element of

lim_{←a,b}(U_{ℏ}L𝔫_{+}^{rat}/I_{a}^{+,rat})[α] ⊗ (U_{ℏ}L𝔫_{−}^{rat}/I_{b}^{−,rat})[−α][ℏ^{−1}].

Let e_{β,i}[p]^{rat} be lifts to U_{ℏ}L𝔫_{±} of the ē_{β,i}[p]. Then if α belongs to kΔ_{+}−(k−1)Δ_{+}, P [α]^{rat} has the form

P [α]^{rat} = \frac{ℏ^{k}}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i} α_{i}=α;i_{j}}^{∑} ∑_{p_{1},...,p_{k}∈ℤ}

e_{α_{1},i_{1}}[p_{1}]^{rat} … e_{α_{k},i_{k}}[p_{k}]^{rat} ⊗ f_{α_{1},i_{1}}[−p_{1} − 1]^{rat} … f_{α_{k},i_{1}}[−p_{k} − 1]^{rat} + o(ℏ^{k}).

The proofs of the statements of this section are analogous to those of the quantum affine case and will be omitted.

1.3. Quantum current algebras of affine type (toroidal algebras) .

Assume that A is an arbitrary symmetrizable Cartan matrix. Define U_{ℏ}L𝔫_{+} and Ũ_{ℏ}L𝔫_{+} as in sect. 1..

Proposition 1.5. 1) Let F̃_{+} be the Lie algebra with generators x̃_{i}^{+}[k], 1 ≤ i ≤ n, k ∈ ℤ, and relations given by the coefficients of monomials in

(z − w)[x̃_{i}^{+}(z), x̃_{j}^{+}(w)] = 0, ad(x̃_{i}^{+}(z_{1})) ⋯ ad(x̃_{i}^{+}(z_{1−a_{ij}}))(x̃_{j}^{+}(w)) = 0,

where for x any x̃_{i}^{+}, x(z) is the generating series ∑_{k∈ℤ} x[k]z^{−k}. If we give degree ϵ_{i} to x̃_{i}^{+}[k], F̃_{+} is graded by set Δ_{+} of the roots of 𝔫_{+}.

Then U_{ℏ}L𝔫_{+}/ℏU_{ℏ}L𝔫_{+} and Ũ_{ℏ}L𝔫_{+}/ℏŨ_{ℏ}L𝔫_{+} are both isomorphic to the enveloping algebra U(F̃_{+}).

2) There is a unique Lie algebra morphism j_{+} : F̃_{+} → L𝔫_{+}, such that j_{+}(ẽ_{i}^{+}[k]) = ē_{i} ⊗ t^{k}. j_{+} is graded and surjective. The kernel of j_{+} is contained in ⊕_{α∈Δ_{+},α imaginary}F̃_{+}[α].

Let us assume now that A is untwisted affine. 𝔫_{+} is the isomorphic to a subalgebra of the loop algebra 𝔤̄[λ, λ^{−1}], with 𝔤̄ semisimple. Define 𝔱_{+} as the direct sum L𝔫_{+} ⊕ (⊕_{k>0,l∈ℤ}ℂK_{kδ}[l]), and endow it with the bracket such that the K_{kδ}[l] are central, and

[(x ⊗ t^{l}, 0), (y ⊗ t^{m}, 0)] = ([x ⊗ t^{l}, y ⊗ t^{m}], ⟨x̄, ȳ⟩𝔤̄(lk′′ − mk′)K_{(k′+k′′)δ}[l + m])

for x ↦ x̄ ⊗ λ^{k′}, y ↦ ȳ ⊗ λ^{k′′} by the inclusion 𝔫_{+} ⊂ 𝔤̄[λ, λ^{−1}], where ⟨ , ⟩𝔤̄ is an invariant scalar product on 𝔤̄.

Then 𝔱_{+} is a Lie subalgebra of the toroidal algebra 𝔱, which is the universal central extension of L𝔤 ([20, 23]). In what follows, we will set x[k]^{𝔱} = (x ⊗ t^{k}, 0).

Proposition 1.6. 1) When A is of affine Kac-Moody type, the kernel of j_{+} is equal to the center of F̃_{+}, so that F̃_{+} is a central extension of L𝔫_{+}.

2) We have a unique Lie algebra map j′ from 𝔱_{+} to F̃_{+} such that j′(ē_{i} ⊗ t^{n}) = ẽ_{i}[n]. This map is an isomorphism iff A is not of type A_{1}^{(1)}. If A is of type A_{1}^{(1)}, j′ is surjective, and its kernel is ⊕_{n∈ℤ}ℂK_{δ}[n].

In Remark 4.3, we discuss possible generalizations of Theorem 1.3 to the case of affine quantum current algebras, and the connection of Proposition 1.5 with the results of [15].

The basic idea of the constructions of the two first parts of this work is to compare the quantized algebras defined by generators and relations with quantum shuffles algebras. The idea to use shuffle algebras to provide examples of Hopf algebras dates back to Nichols ([24]). Later, Schauenburg ([28]) and Rosso ([27]) showed that the positive part U_{ℏ}𝔫_{+} of the Drinfeld-Jimbo quantized enveloping algebras are isomorphic to the subalgebra Sh(V) of quantum shuffle Hopf algebras generated in degree 1. Their results rely on Lusztig’s PBW or duality (nondegeneracy of Drinfeld’s pairing) results. A nonabelian generalization of Schauenburg’s result can be found in [1].

In sect. 2., we show that applying Drinfeld’s theory of Lie bialgebras to Sh(V) yields at the same time proofs of these results (PBW for U_{ℏ}𝔫_{+} and isomorphism of U_{ℏ}𝔫_{+} with Sh(V), and nondegeneracy of the pairing as a simple consequence), when the deformation parameter is formal or generic.

In sect. 3., we apply the same idea to quantum current algebras. These algebras, also know as “new realizations” algebras, depend on the datum of a Cartan matrix. In that situation, the proper replacement of shuffle algebras are the functional shuffle algebras introduced by Feigin and Odesskii ([13, 14]). We show that when the Cartan matrix is of finite type, the ideas of sect. 2. allow to complete the results of [11] on comparison of the quantum current algebras and the Feigin-Odesskii algebras. However, there are still some open problems in this direction, see Remark 3.16. We hope that the ideas of this section will help generalize the results of [12] from 𝔰𝔩_{2} to arbitrary semisimple Lie algebras. For this one should, in particular, find analogues of the quantum Serre relations for the algebras in genus ≥ 1.

In sect. 4., we consider the classical limit of the quantum current algebras in the case of an affine Kac-Moody Cartan matrix. We show that this classical limit is the enveloping algebra of a Lie algebra F̃_{+}, which is a central extension of the Lie algebra L𝔫_{+} of loops with values in the positive subalgebra 𝔫_{+} of the affine Kac-Moody algebra. F̃_{+} is graded by the roots of 𝔫_{+}, and its center is contained in the part of imaginary degrees. We show that in all affine untwisted cases, except the A_{1}^{(1)} case, F̃_{+} is isomorphic to a subalgebra 𝔱_{+} of the toroidal algebra 𝔱 introduced and studied in [23, 16, 30]. In the A_{1}^{(1)} case, we identify F̃_{+} with a quotient of 𝔱_{+}.

In the quantum case, the center of U_{ℏ}L𝔫_{+} seems closely connected with the central part of the affine elliptic algebras constructed in the recent work of Feigin and Odesskii ([15]). We hope that a better understanding of this center will enable to extend to toroidal algebras the results of sect. 3..

This paper grew from the notes of the DEA course I taught at univ. Paris 6 in february-april 1999. I would like to thank P. Schapira for giving me the opportunity to give this course and its participants, notably C. Grunspan, O. Schiffmann and V. Toledano Laredo, for their patience and attention.

I also would like to thank N. Andruskiewitsch and B. Feigin for valuable discussions. In particular, the idea that quantum shuffle algebras could be a tool to construct quantizations of Lie bialgebras is due to N. Andruskiewitsch; it is clear that this idea plays an important role in the present work. I also would like to thank N. Andruskiewitsch for his kind invitation to the univ. of Córdoba in August 1998, where these discussions took place.

2. Quantum Kac-Moody algebras (proofs of the results of sect. 1.)

2.1. PBW theorem and comparison with shuffle algebra (proofs of Theorem 1.1 and Corollary 1.1).

•Definition of Sh(V) and ⟨V ⟩ Let us set V = ⊕_{i=1}^{n}ℂv_{i}. Let ϵ_{i} be the ith basis vector of ℕ^{n}. Define the grading of V by ℕ^{n} by deg(v_{i}) = ϵ_{i}. Let Sh(V) be the quantum shuffle algebra constructed from V and the braiding V ⊗ V → V ⊗ V [[ℏ]], v_{i} ⊗ v_{j} ↦ q^{−d_{i}a_{ij}}v_{j} ⊗ v_{i}. That is, Sh(V) is isomorphic, as ℂ[[ℏ]]-module, to ⊕_{i≥0}V^{⊗i}[[ℏ]]. Denote the element z_{1} ⊗ ⋯ ⊗ z_{k} as [z_{1}| ⋯ |z_{k}]. The product is defined on Sh(V) as follows:

[z_{1}| ⋯ |z_{k}] [z_{n+1}| ⋯ |z_{k+l}]

= ∑_{σ∈Σ_{k,l}} q^{− ∑_{1≤i<j≤k+l,σ(i)>σ(j)}⟨deg(z_{i}),deg(z_{j})⟩}[z_{σ(1)}| ⋯ |z_{σ(k+l)}],

if the z_{i} are homogeneous elements of V , and where Σ_{k,l} is the subset of the symmetric group 𝔖_{k+l} consisting of shuffle permutations σ such that σ(i) < σ(j) if 1 ≤ i < j ≤ k or k + 1 ≤ i < j ≤ k + l; the bilinear form on ℕ^{n} is defined by form ⟨ϵ_{i}, ϵ_{j}⟩ = d_{i}a_{ij}.

Lemma 2.1. ⟨V ⟩ is the direct sum of its graded components, which are free ℂ[[ℏ]]-modules. It follows that ⟨V ⟩ is a free ℂ[[ℏ]]-module.

Proof. That ⟨V ⟩ is the direct sum of its graded components follows from its definition. These graded components are ℂ[[ℏ]]-submodules of finite-dimensional free ℂ[[ℏ]]-modules (the graded components of Sh(V)). Each graded component is therefore a finite-dimensional free module over ℂ[[ℏ]]. The Lemma follows.

•Crossed product algebras V and S Define linear endomorphisms h̃_{i}, i = 1, …, n and D̃_{j}, j = 1, …, n − r of V by the formulas

h̃_{i}(v_{j}) = a_{ij}v_{j}, D̃_{j}(v_{i}) = δ_{ij}v_{i}.

Extend the x̃, x ∈ {h_{i}, D_{j}} to linear endomorphisms of Sh(V) by the formulas

x̃([x_{1}| ⋯ |x_{n}]) = ∑_{k=1}^{n}[x_{1}| ⋯ |x̃(x_{k})| ⋯ |x_{n}].

It is clear that the x̃ define derivations of Sh(V). These derivations preserve ⟨V ⟩.

Define V and S as the crossed product algebras of ⟨V ⟩ and Sh(V) with the derivations h̃_{i}, D̃_{j}. More precisely, V and S are isomorphic, as ℂ[[ℏ]]-modules, to their tensor products ⟨V ⟩ ⊗_{ℂ[[ℏ]]} ℂ[h_{i}^{V}, D_{j}^{V}][[ℏ]] and Sh(V) ⊗_{ℂ[[ℏ]]} ℂ[h_{i}^{V}, D_{j}^{V}][[ℏ]] with ℏ-adically completed polynomial algebras in n + r variables. The products on V and S are then defined by the rules

(x ⊗ ∏_{s=1}^{2n−r}(X_{s}^{V})^{α_{s}})(y ⊗ ∏_{s=1}^{2n−r}(X_{s}^{V})^{β_{s}})

= ∑_{(i_{s})} ∏_{s=1}^{2n−r} ( \begin{matrix}α_{s} \\ i_{s} \end{matrix} ) (x ∏_{s=1}^{2n−r} X̃_{s}^{i_{s}}(y)) ⊗ ( ∏_{s=1}^{2n−r}(X_{s}^{V})^{α_{s}+β_{s}−i_{s}}) ,

where we set X_{s} = h_{s} for s = 1, …, n, and X_{s} = D_{s−n} for s = n + 1, …, 2n − r. In what follows, we will denote x ⊗ 1 and 1 ⊗ X_{s} simply by x and X_{s}, so that x ⊗ ∏_{s}(X_{s}^{V})^{α_{s}} will be x ∏_{s}(X_{s}^{V})^{α_{s}}.

S is then endowed with a Hopf ℂ[[ℏ]]-algebra structure (that is, all maps of Hopf algebra axioms are ℂ[[ℏ]]-module maps, and the tensor products are completed in the ℏ-adic topology), defined by

Δ_{V}(h^{V}) = h^{V} ⊗ 1 + 1 ⊗ h^{V} for h ∈ {h_{i}, D_{j}},

Δ_{V}([v_{i_{1}}| ⋯ |v_{i_{m}}]) = ∑_{k=0}^{m}[v_{i_{1}}| ⋯ |v_{i_{k}}] ⊗ exp(ℏ ∑_{j=1}^{k} d_{i_{j}}h_{i_{j}}^{V})[v_{i_{k+1}}| ⋯ |v_{i_{m}}]).

V is then a Hopf subalgebra of S.

Assign degrees 0 to the elements h_{i}^{V}, D_{j}^{V}, and ϵ_{i} to v_{i}. V is then the direct sum of its homogeneous components, which are free finite-dimensional modules over ℂ[h_{i}^{V}, D_{j}^{V}][[ℏ]]; the grading of V is compatible with its algebra structure.

•Hopf co-Poisson and Lie bialgebra structures Define V_{0} as V/ℏV.

Lemma 2.2. V_{0} is a cocommutative Hopf algebra.

Proof. Define Δ′_{V} as Δ_{V} composed with the permutation of factors. We have to show that for x in V, we have

(Δ_{V} − Δ′_{V})(x) ⊂ ℏ(V ⊗_{ℂ[[ℏ]]} V).  (15)

For x one of the h̄_{i}, D̄_{j}, (15) is clearly satisfied. On the other hand, if (15) is satisfied for x and y in V, then (Δ_{V} − Δ′_{V})(xy) is equal to (Δ_{V} − Δ′_{V})(x)Δ_{V}(y) + Δ′_{V}(x)(Δ_{V} − Δ′_{V})(y) and therefore belongs to ℏ(V ⊗_{ℂ[[ℏ]]} V). It follows that (15) holds for any x in V.

Lemma 2.3. 1) There exists a unique surjective Hopf algebra morphism p_{ℏ} from U_{ℏ}𝔟_{+} to V, such that p_{ℏ}(h_{i}) = h_{i}^{V} and p_{ℏ}(x_{i}^{+}) = [v_{i}].

2) The map D_{j} ↦ D̄_{j}, h_{i} ↦ h̄_{i}, x_{i}^{+} ↦ x̄_{i}^{+} extends to an isomorphism from U_{ℏ}𝔟_{+}/ℏU_{ℏ}𝔟_{+} to U𝔟_{+}.

3) p_{ℏ} induces a surjective cocommutative Hopf algebra morphism p from U_{ℏ}𝔟_{+} = U𝔟_{+} to V/ℏV = V_{0}.

Proof. That the quantum Serre relations are satisfied in Sh(V) by the [v_{i}] follows from [27], Lemma 14 (the proof relies on q-binomial coefficients identities, which are proved by induction); this proves the first part of the Lemma.

Let us show that U_{ℏ}𝔟_{+}/ℏU_{ℏ}𝔟_{+} is isomorphic U𝔟_{+}. U_{ℏ}𝔟_{+}/ℏU_{ℏ}𝔟_{+} is equal to the quotient of ℂ⟨h_{i}, D_{j}, x_{i}^{+}, i = 1, …, n, j = 1, …, 2n − r⟩[[ℏ]] by the sum of ℏℂ⟨h_{i}, D_{j}, x_{i}^{+}⟩[[ℏ]] and the closed ideal generated by the relations [h_{i}, e_{j}] = a_{ij}e_{j}, [D_{i}, e_{j}] = δ_{ij}e_{j} and the quantum Serre relations (1). This sum is the same as that of ℏℂ⟨h_{i}, D_{j}, x_{i}^{+}⟩[[ℏ]] and the closed ideal generated by [h_{i}, e_{j}] = a_{ij}e_{j}, [D_{i}, e_{j}] = δ_{ij}e_{j} and the classical Serre relations. The quotient of ℂ⟨h_{i}, D_{j}, x_{i}^{+}⟩[[ℏ]] by this last space is equal to U𝔟_{+}. This proves the second part of the Lemma.

The third part is immediate.

Proposition 2.1. Let 𝔩 be a Lie algebra and let J be a two-sided ideal of U𝔩 such that Δ_{U𝔩}(J) ⊂ U𝔩 ⊗ J + J ⊗ U𝔩. Then 𝔧 = J ∩ 𝔩 is an ideal of the Lie algebra 𝔩 and we have J = (U𝔩)𝔧 = 𝔧(U𝔩).

Proof. We first show:

Lemma 2.4. Let 𝔩 be a Lie algebra and let J be a left ideal of U𝔩 such that Δ_{U𝔩}(J) ⊂ J ⊗ U𝔩 + U𝔩 ⊗ J. Let 𝔧 be the intersection 𝔩 ∩ J. Then J is equal to (U𝔩)𝔧.

Proof of Lemma. Denote by (U𝔩)_{n} the subspace of U𝔩 spanned by the monomials in elements of 𝔩 of degree ≤ n. Let us set Δ̄_{U𝔩}(x) = Δ_{U𝔩}(x) − x ⊗ 1 − 1 ⊗ x. We have Δ̄_{U𝔩}((U Ł)_{n}) ⊂ ∑_{p,q>0,p+q=n}(U𝔩)_{p} ⊗ (U𝔩)_{q}. Denote by J_{n} the intersection J∩(U𝔩)_{n}. Then we have Δ̄_{U𝔩}(J_{n}) ⊂ ∑_{p,q>0,p+q=n} J_{p}⊗(U𝔩)_{q}+(U𝔩)_{p}⊗J_{q}.

Let us show by induction that J_{n} is contained in (U𝔩)_{n−1}𝔧. This is clear if n = 1; assume it is true at order n − 1. Let x be an element of J_{n}. Then Δ̄_{U𝔩}(x) is contained in ∑_{p,q>0,p+q=n}(U𝔩)_{p−1}𝔧 ⊗ (U𝔩)_{q} + (U𝔩)_{p} ⊗ (U𝔩)_{q−1}𝔧.

Let x̄ be the image of x in (U𝔩)_{n}/(U𝔩)_{n−1}. (U𝔩)_{n}/(U𝔩)_{n−1} is isomorphic to the nth symmetric power S^{n}𝔩. Let Δ_{S𝔩} be the coproduct of the symmetric algebra S𝔩, for which elements of degree 1 are primitive, and set Δ̄_{S𝔩}(x̄) = Δ_{S𝔩}(x̄)− x̄⊗1−1⊗x̄. Then Δ̄(x̄) is contained in ∑_{p,q>0,p+q=n}(S^{p−1}𝔩)𝔧⊗S^{q}𝔩+S^{p}𝔩⊗(S^{q−1}𝔩)𝔧. It follows that x̄ belongs to (S^{n−1}𝔩)𝔧. The difference of x with some element of (U𝔩)_{n−1}𝔧 therefore belongs to (U𝔩)_{n−1}, so that it belongs to J_{n−1} and by hypothesis to (U𝔩)_{n−2}𝔧. Therefore, x belongs to (U𝔩)_{n−1}𝔧. This proves the Lemma.

Let us prove Proposition 2.1. Lemma 2.4 implies that J = (U𝔩)𝔧 and its analogue for right Hopf ideals implies that J = 𝔧(U𝔩). Therefore, we have (U𝔩)𝔧 = 𝔧(U𝔩). Let us fix x in 𝔩 and j in 𝔧, then [x, j] belongs J; since it also belongs to 𝔩, [x, j] belongs to 𝔧. Therefore 𝔧 is an ideal of 𝔩.

Let J be the kernel of the cocommutative Hopf algebras morphism p defined in Lemma 2.3, 3). Let us set 𝔧 = J ∩ 𝔟_{+} and 𝔞 = 𝔟_{+}/𝔧. prop. 2.1 then implies

Lemma 2.5. The Lie algebra structure on 𝔟_{+} induces a Lie algebra structure on 𝔞. Moreover, V_{0} is isomorphic with U𝔞, and p can be identified with the quotient map U𝔟_{+} → U𝔞.

Define δ_{V_{0}} as \frac{Δ_{V}−Δ′_{V}}{ℏ} mod ℏ. δ_{V_{0}} is a linear map from V_{0} to the antisymmetric part of its tensor square ∧^{2}V_{0}. It obeys the rules

(Δ_{V_{0}} ⊗ id) ∘ δ_{V_{0}} = (δ_{V_{0}}^{2→23} + δ_{V_{0}}^{2→13}) ∘ Δ_{V_{0}},  (16)
Alt(δ_{V_{0}} ⊗ id) ∘ δ_{V_{0}}) = 0,
δ_{V_{0}}(xy) = δ_{V_{0}}(x)Δ_{V_{0}}(y) + Δ_{V_{0}}(x)δ_{V_{0}}(y) for x, y in V_{0},

if δ_{V_{0}}(y) = ∑_{i} y′_{i} ⊗ y′′_{i}, we set δ_{V_{0}}^{2→23}(x ⊗ y) = ∑_{i} x ⊗ y′_{i} ⊗ y′′_{i}, and δ_{V_{0}}^{2→13}(x ⊗ y) = ∑_{i} y′_{i} ⊗ x ⊗ y′′_{i}. These rules are the co-Leibnitz, co-Jacobi and Hopf compatibility conditions; they mean that (V_{0}, δ_{V_{0}}) is a Hopf co-Poisson algebra (see [8]).

Lemma 2.6. δ_{V_{0}} maps 𝔞 to ∧^{2}𝔞.

Proof. Let a be an element of 𝔞 and set δ_{V_{0}}(a) = ∑_{i} x_{i} ⊗ y_{i}, where (y_{i}) is a free family. Then (16) implies that Δ_{V_{0}}(x_{i}) ⊗ y_{i} = ∑_{i} x_{i} ⊗ 1 ⊗ y_{i} + 1 ⊗ x_{i} ⊗ y_{i}, so that each x_{i} is primitive and therefore belongs to 𝔞. So δ_{V_{0}}(a) belongs to 𝔞 ⊗ V_{0}. Since δ_{V_{0}}(a) is also antisymmetric, it belongs to ∧^{2}𝔞.

Call δ_{𝔞} the map from 𝔞 to ∧^{2}𝔞 defined as the restriction of δ_{V_{0}} to 𝔞. (𝔞, δ_{𝔞}) is then a Lie bialgebra, which means that δ_{𝔞} is a 1-cocyle of 𝔞 with values in the antisymmetric part of the tensor square of its adjoint representation, satisfying the co-Jacobi identity Alt(δ_{𝔞} ⊗ id)δ_{𝔞} = 0.

Remark 2.7. The Hopf co-Poisson algebra and Lie bialgebra axioms were introduced by Drinfeld in [8]. Drinfeld showed that the quantization of a cocommutative Hopf algebra lead to such structures. He also stated that there is an equivalence of categories between the category of Hopf co-Poisson algebras and that of Lie bialgebras. Lemma 2.6 can therefore be viewed as the proof of one part of this statement (from Hopf co-Poisson to Lie bialgebras). It is also not difficult to prove the other part (from Lie bialgebras to Hopf co-Poisson).

•Kac-Moody Lie algebras Let 𝔤 be the Kac-Moody Lie algebra associated with A. 𝔤 has generators x̄_{i}^{±}, h̄_{i}, i = 1, …, n and D̄_{j}, j = 1, …, n − r, and relations

[h, h′] = 0 if h, h′ ∈ {h̄_{i}, D̄_{j}},  (17)
[h̄_{i}, x̄_{i′}^{±}] = ±a_{ii′}x̄_{i′}^{±}, [D̄_{j}, x̄_{i}^{±}] = ±δ_{ij}x̄_{i}^{±},  (18)
ad(x̄_{i}^{±})^{1−a_{ij}}(x̄_{j}^{±}) = 0,  (19)
[x̄_{i}^{+}, x̄_{i′}^{−}] = δ_{ii′} h̄_{i}, for all i, i′ = 1, …, n, j = 1, …, n − r.

Let 𝔥 and 𝔫_{±} be the subalgebras of 𝔤 generated by {h̄_{i}, D̄_{j}} and {x̄_{i}^{±}}. Then 𝔤 has the Cartan decomposition 𝔤 = 𝔫_{+} ⊕ 𝔥 ⊕ 𝔫_{−}. Let us set 𝔟_{±} = 𝔥 ⊕ 𝔫_{±}. 𝔫_{±} and 𝔟_{±} are the Lie algebras with generators {x̄_{i}^{±}} and {h̄_{i}, D̄_{j}, x̄_{i}^{±}} and relations (19) for 𝔫_{±} and (17), (18) and (19) for 𝔟_{±}. 𝔤 is endowed with a nondegenerate bilinear form ⟨ , ⟩_{𝔤}, which is determined by ⟨h̄_{i}, h̄_{i′}⟩_{𝔤} = d_{i′}^{−1}a_{ii′}, ⟨x̄_{i}^{+}, x̄_{i′}^{−}⟩_{𝔤} = d_{i}^{−1}δ_{ii′}, ⟨h̄_{i}, D̄_{j}⟩_{𝔤} = d_{i}^{−1}δ_{ij}, and that its values for all other pairs of generators is zero (see [6, 19]).

•Hopf algebras U_{ℏ}𝔟_{±} Define U_{ℏ}𝔟_{±} as the algebras with generators h_{i}^{±}, D_{j}^{±} and x_{i}^{±}, and relations

[h_{i}^{±}, x_{i′}^{±}] = ±a_{ii′}x_{i′}^{±}, [D_{j}^{±}, x_{i}^{±}] = ±δ_{ij}x_{i}^{±},

and relations (1), with e_{i} replaced by x_{i}^{±}. It is easy to see that the maps e_{i} ↦ x_{i}^{+} and f_{i} ↦ x_{i}^{−} define algebra inclusions of U_{ℏ}𝔫_{±} in U_{ℏ}𝔟_{±}. We have Hopf algebra structures on U_{ℏ}𝔟_{+} and U_{ℏ}𝔟_{−}, defined by

Δ_{±}(h^{±}) = h^{±} ⊗ 1 + 1 ⊗ h^{±} for h ∈ {h̄_{i}, D̄_{j}}, Δ_{±}(x_{i}^{±}) = x_{i}^{±} ⊗ e^{±ℏd_{i}h_{i}^{±}} + 1 ⊗ x_{i}^{±}.

•Comparison lemmas Recall that V is the direct sum of its graded components. Its component of degree zero is ℂ[h_{i}^{V}, D_{j}^{V}][[ℏ]]. Let h_{i}^{𝔞}, D_{j}^{𝔞} be the images of h_{i}^{V}, D_{j}^{V} in V_{0}. We have an inclusion of ℂ[h_{i}^{𝔞}, D_{j}^{𝔞}] in V_{0}. It follows that the h_{i}^{𝔞} and D_{j}^{𝔞} are linearly independent and commute to each other.

On the other hand, as the elements h_{i}^{V} and D_{j}^{V} are primitive in V, the h_{i}^{𝔞} and D_{j}^{𝔞} are also primitive; it follows that they belong to 𝔞, and we have

p(h̄_{i}) = h_{i}^{𝔞}, p(D̄_{j}) = D_{j}^{𝔞}, δ_{𝔞}(h_{i}^{𝔞}) = δ_{𝔞}(D_{j}^{𝔞}) = 0.  (20)

The degree ϵ_{i} component of V is ℂ[h_{i}, D_{j}][[ℏ]] · [v_{i}], and the map x ↦ x[v_{i}] is a ℂ[[ℏ]]-module isomorphism from ℂ[h_{i}, D_{j}][[ℏ]] to this component. Therefore, [v_{i}] has a nonzero image in V/ℏV = V_{0}. Since we have

Δ_{V}([v_{i}]) = [v_{i}] ⊗ e^{ℏd_{i}h_{i}^{V}} + 1 ⊗ [v_{i}],

[v_{i}] mod ℏ is primitive in V_{0}. Therefore, [v_{i}] mod ℏ belongs to 𝔞; call v_{i}^{𝔞} this element of 𝔞. It is clear that

δ_{𝔞}(v_{i}^{𝔞}) = d_{i}v_{i}^{𝔞} ∧ h_{i}^{𝔞}.  (21)

On the other hand, p_{ℏ}(x̄_{i}^{+}) = [v_{i}] implies that

p(x̄_{i}^{+}) = v_{i}^{𝔞}.  (22)

Recall that we have a Lie bialgebra structure on 𝔟_{+}; it consists in a map δ_{𝔟_{+}} from 𝔟_{+} to ∧^{2}𝔟_{+}, which is uniquely determined by the conditions δ_{𝔟_{+}}(x̄_{i}^{+}) = d_{i}x̄_{i}^{+} ∧ h̄_{i}, δ_{𝔟_{+}}(h̄_{i}) = δ_{𝔟_{+}}(D̄_{j}) = 0 and that it satisfies the 1-cocycle identity.

Lemma 2.8. p is also a Lie bialgebra morphism from (𝔟_{+}, δ_{𝔟_{+}}) to (𝔞, δ_{𝔞}).

Proof. This means that

(∧^{2}p) ∘ δ_{𝔟_{+}}(x) = δ_{𝔞} ∘ p(x), for x in 𝔟_{+}.  (23)

For x equal to h̄_{i} and D̄_{j}, (23) follows from (20). It follows from (21) and (22) that

δ_{𝔞} ∘ p(x̄_{i}^{+}) = δ_{𝔞}(v_{i}^{𝔞}) = d_{i}v_{i}^{𝔞} ∧ h_{i}^{𝔞} = (∧^{2}p)(d_{i}x̄_{i}^{+} ∧ h̄_{i}) = ∧^{2}p(δ_{𝔞}(x̄_{i}^{+})),

so that (23) is also true for x = x̄_{i}^{+}.

Since p is a Lie algebra morphism, both sides of (23) are 1-cocycles of 𝔟_{+} with values in the antisymmetric part of the tensor square of (𝔞, ad_{𝔞}∘p). Therefore,

(23) holds on the subalgebra of 𝔟_{+} generated by the h̄_{i}, D̄_{j} and x̄_{i}^{+}, which is 𝔟_{+} itself.

Denote by 𝔥_{𝔞} the subspace of 𝔞 spanned by the h_{i}^{𝔞} and D_{j}^{𝔞}; it forms an abelian Lie subalgebra of 𝔞.

Since p_{ℏ}(h_{i}) = h_{i}^{V} and p_{ℏ}(D_{j}) = D_{j}^{V}, the restriction of p to the Cartan subalgebra 𝔥 of 𝔟_{+} is a Lie algebra isomorphism from 𝔥 to 𝔥_{𝔞}.

Define, for α in 𝔥^{∗}, the root subspace 𝔞[α] associated with α by

𝔞[α] = {x ∈ 𝔞|[p(h), x] = α(h)x, for all h in 𝔥},

and as usual

𝔟_{+}[α] = {x ∈ 𝔟_{+}|[h, x] = α(h)x, for all h in 𝔥}.

Lemma 2.9. 𝔞 is the direct sum of its root subspaces 𝔞[α], where α belongs to the set Δ_{+} ∪ {0} of roots of 𝔟_{+}. Each 𝔞[α] is finite dimensional. δ_{𝔞} is a graded map from 𝔞 to ∧^{2}𝔞, therefore the graded dual 𝔞^{∗} of 𝔞, defined as ⊕_{α}𝔞[α]^{∗}, has a Lie bialgebra structure.

Proof. For any α in 𝔥^{∗}, p maps 𝔟_{+}[α] to 𝔞[α]. It follows that 𝔞 is the sum of the root subspaces 𝔞[α], where α belongs to the set of roots of 𝔟_{+}. That this sum is direct is proved as in the case of 𝔟_{+}: let α_{i} be the root such that x̄_{i}^{+} belongs to 𝔟_{+}[α_{i}]. Then α_{1}, …, α_{n} form a basis of 𝔥^{∗} (they are the simple roots of 𝔤). Let (H̄_{1}, …, H̄_{n}) be the basis of 𝔥 dual to (α_{1}, …, α_{n}). Then for any family x_{α} of 𝔞[α] such that ∑_{α} x_{α} = 0, we have, by applying ad(p(H̄_{1}))^{k} to this equality, ∑_{(n_{i})∈ℕ^{n}} n_{1}^{k}x_{∑_{i} n_{i}α_{i}} = 0, which gives for any integer a_{1}, ∑_{(n_{i})∈ℕ^{n}|n_{1}=a_{1}} x_{∑_{i} n_{i}α_{i}} = 0; applying ad(p(H̄_{2}))^{k}, we get ∑_{(n_{i})∈ℕ^{n}|n_{1}=a_{1},n_{2}=a_{2}} x_{∑_{i} n_{i}α_{i}} = 0; finally each x_{α} vanishes.

That δ_{𝔞} is graded follows from the fact that its restriction to 𝔥_{𝔞} vanishes and from the cocycle identity.

Lemma 2.10. p is a Lie bialgebra isomorphism.

Proof. It follows from Lemma 2.8 that p^{∗} is an injective Lie bialgebra morphism from 𝔞^{∗} in the graded dual 𝔟_{+}^{∗} of 𝔟_{+}. Recall that 𝔟_{+}^{∗} is isomorphic, as a Lie algebra, to 𝔟_{−} = 𝔥 ⊕ 𝔫_{−}; this relies on the nondegeneracy of the invariant pairing between 𝔟_{+} and 𝔟_{−}, itself a consequence of [6] (in what follows, we will denote by 𝔥_{−} the Cartan subalgebra 𝔥 of 𝔟_{−}). We will show that the image of p^{∗} contains a generating family of 𝔟_{−}.

Let us denote by 𝔥_{𝔞^{∗}} the space of forms on 𝔞, which vanish on all the 𝔞[α], α ≠ 0. The duality beween 𝔟_{+} and 𝔟_{−} identifies 𝔥_{−} with the space of the forms on 𝔟_{+} which vanish on 𝔫_{+} = ⊕_{α≠0}𝔟_{+}[α]. We have p(𝔟_{+}[α]) ⊂ 𝔞[α] for any α, therefore

p^{∗}[(⊕_{α≠0}𝔞[α])^{⊥}] ⊂ (⊕_{α≠0}𝔟_{+}[α])^{⊥},

which means that p^{∗}(𝔥_{𝔞^{∗}}) ⊂ 𝔥_{−}. Since p^{∗} is injective and 𝔥_{𝔞^{∗}} and 𝔥_{−} have the same dimension, p^{∗} induces an isomorphism between 𝔥_{𝔞^{∗}} and 𝔥_{−}. It follows that the image of p^{∗} contains 𝔥_{−}.

Since x̄_{i}^{+} belongs to 𝔟_{+}[α_{i}], the element v_{i}^{𝔞} of 𝔞 defined before Lemma 2.8 belongs to 𝔞[α_{i}]. We have seen that v_{i}^{𝔞} is nonzero.

Let ξ_{i} be the element of 𝔞^{∗} which is 1 on v_{i}^{𝔞} and zero on each 𝔞[α], α ≠ α_{i}. For x in 𝔟_{+}[α], α ≠ α_{i}, ⟨p^{∗}(ξ_{i}), x⟩_{𝔟_{+}×𝔟_{−}} = ⟨ξ_{i}, p(x)⟩_{𝔞^{∗}×𝔞} = 0 because ξ_{i} vanishes on 𝔞[α]. It follows that p^{∗}(ξ_{i}) has weight −α_{i} in 𝔟_{−}. On the other hand, p^{∗}(ξ_{i}) is nonzero, because p^{∗} is injective, so it is a nonzero constant times x̄_{i}^{−}.

Since the image of p^{∗} contains 𝔥_{−} and the x̄_{i}^{−}, p^{∗} is an isomorphism.

Lemma 2.11. p_{ℏ} mod ℏ restricts to an isomorphism of ℕ^{n}-graded algebras from U𝔫_{+} to ⟨V ⟩/ℏ⟨V ⟩.

Proof. It follows from Lemma 2.10 that p_{ℏ} mod ℏ induces an isomorphism from U𝔟_{+} = U_{ℏ}𝔟_{+}/ℏU_{ℏ}𝔟_{+} to V_{0} = V/ℏV. Therefore it induces an isomorphism from U𝔫_{+} to its image in V_{0}. Since U𝔫_{+} coincides with the image of U_{ℏ}𝔫_{+} by the projection U_{ℏ}𝔟_{+} → U_{ℏ}𝔟_{+}/ℏU_{ℏ}𝔟_{+} = U𝔟_{+}, this image coincides with that of the composed map

U_{ℏ}𝔫_{+} → U_{ℏ}𝔟_{+} → V → V_{0}.  (24)

The image of the composed map U_{ℏ}𝔫_{+} → U_{ℏ}𝔟_{+} → V is equal to ⟨V ⟩. We have a ℂ[[ℏ]]-module isomorphism of V with ⟨V ⟩ ⊗_{ℂ[[ℏ]]} ℂ[h_{i}^{V}, D_{j}^{V}][[ℏ]], so that ℏV ∩ ⟨V ⟩ = ℏ⟨V ⟩. It follows that the image of ⟨V ⟩ by V → V_{0} is ⟨V ⟩/ℏ⟨V ⟩. Therefore the image of (24) is ⟨V ⟩/ℏ⟨V ⟩.

Proof of Theorem 1.1. Assign degree ϵ_{i} to the generator e_{i} of U_{ℏ}𝔫_{+}. Then U_{ℏ}𝔫_{+} is the direct sum of its homogeneous components (U_{ℏ}𝔫_{+})[α], α ∈ ℕ^{n}, which are finitely generated ℂ[[ℏ]]-modules. As a ℂ[[ℏ]]-module, (U_{ℏ}𝔫_{+})[α] is therefore isomorphic to the direct sum ⊕_{i=1}^{q_{α}}ℂ[[ℏ]]/(ℏ^{n_{i}^{(α)}}) ⊕ ℂ[[ℏ]]^{p_{α}} of its torsion part with a free module (see Lemma A1).

Lemma 2.3, 2) implies that U_{ℏ}𝔫_{+}[α]/ℏU_{ℏ}𝔫_{+}[α] = U𝔫_{+}[α] so

p_{α} + q_{α} = dimU𝔫_{+}[α].  (25)

On the other hand, ⟨V ⟩[α] is a free finite dimensional module over ℂ[[ℏ]], by Lemma 2.1 above, so it is isomorphic, as a ℂ[[ℏ]]-module, to ℂ[[ℏ]]^{p′_{α}}. p_{ℏ} restricts to a surjective ℂ[[ℏ]]-module morphism from U_{ℏ}𝔫_{+}[α] to ⟨V ⟩[α], therefore p_{ℏ} maps the torsion part of U_{ℏ}𝔫_{+}[α] to zero and

p_{α} ≥ p′_{α}.  (26)

Moreover,

p′_{α} = dimU𝔫_{+}[α]  (27)

by Lemma 2.11. It follows from (25), (26) and (27) that p_{α} = p′_{α} and q_{α} = 0.

This means that U_{ℏ}𝔫_{+} has no torsion, and is isomorphic to ⟨V ⟩. In view of Lemma 2.3, 2), this proves Theorem 1.1. This also proves Corollary 1.1.

2.2. Nondegeneracy of Hopf pairing (proof of Theorem 1.2).

Let (v_{i}^{∗}) be the basis of V^{∗} such that ⟨v_{i}^{∗}, v_{j}⟩ = d_{i}^{−1}δ_{ij}. Assign to v_{i}^{∗} the degree −ϵ_{i}. Let T (V^{∗}) be the tensor algebra ⊕_{i}(V^{∗})^{⊗i}[[ℏ]]. Define the braided tensor product strucutre on the tensor square of T (V^{∗}) according to (2). T (V^{∗}) is endowed with the braided Hopf structure defined by Δ_{T (V^{∗})}(v_{i}^{∗}) = v_{i}^{∗} ⊗ 1 + 1 ⊗ v_{i}^{∗}, for any i = 1, …, n. We have a surjective braided Hopf algebra morphism from T (V^{∗}) to U_{ℏ}𝔫_{−}, defined by v_{i}^{∗} ↦ f_{i}, for i = 1, …, n.

Then we have a braided Hopf pairing

⟨ , ⟩_{Sh(V)×T (V^{∗})} : Sh(V) × T (V^{∗}) → ℂ((ℏ)),

defined by the rules

⟨[v_{i_{1}}| ⋯ |v_{i_{k}}], ξ_{i′_{1}} ⋯ ξ_{i′_{k′}}⟩_{S×U_{ℏ}_{𝔟_{−}}} = \frac{1}{ℏ}δ_{kk′} ∏_{j=1}^{k}⟨v_{i_{j}}, ξ_{i′_{j}}⟩_{V ×V^{∗}}.  (28)

The ideal of T (V^{∗}) generated by the quantum Serre relations is in the radical of this pairing (see e.g. [22], chap. 1; this is a consequence of q-binomial identities).

It follows that ⟨ , ⟩_{Sh(V)×T (V^{∗})} induces a braided Hopf pairing

⟨ , ⟩_{Sh(V)×U_{ℏ}𝔫_{−}} : Sh(V) × U_{ℏ}𝔫_{−} → ℂ((ℏ)).

By Theorem 1.1, U_{ℏ}𝔫_{+} is a braided Hopf subalgebra of Sh(V). The restriction of ⟨ , ⟩_{Sh(V)×U_{ℏ}𝔫_{−}} to U_{ℏ}𝔫_{+} × U_{ℏ}𝔫_{−} therefore induces a braided Hopf pairing between U_{ℏ}𝔫_{+} and U_{ℏ}𝔫_{−}; since it coincides on generators with ⟨ , ⟩_{U_{ℏ}𝔫_{+}×U_{ℏ}𝔫_{−}}, it is equal to ⟨ , ⟩_{U_{ℏ}𝔫_{+}×U_{ℏ}𝔫_{−}}.

View V^{⊗k} as a subspace of Sh(V). Assign degree 1 to each element of V^{∗} in T (V^{∗}); then T (V^{∗}) is a graded algebra; we denote by T (V^{∗})^{(k)} is homogeneous component of degree k. The restriction of ⟨ , ⟩_{Sh(V)×T (V^{∗})} to V^{⊗k}×T (V^{∗})^{(k)} can be identified with the natural pairing of V^{⊗k} with (V^{∗})^{⊗k}, which is nondegenerate. Therefore the annihilator of T (V^{∗}) in Sh(V) for ⟨ , ⟩_{Sh(V)×T (V^{∗})} is zero. By Theorem 1.1, it follows that the annihilator of U_{ℏ}𝔫_{−} in U_{ℏ}𝔫_{+} for ⟨ , ⟩_{U_{ℏ}𝔫_{±}} is zero.

Since the pairing ⟨ , ⟩_{U_{ℏ}𝔫_{+}×U_{ℏ}𝔫_{−}} is graded and the graded components of U_{ℏ}𝔫_{+} and U_{ℏ}𝔫_{−} have the same dimensions (as ℂ[[ℏ]]-modules), the pairing

⟨ , ⟩_{U_{ℏ}𝔫_{+}×U_{ℏ}𝔫_{−}}

is nondegenerate.

2.3. The form of the R-matrix (proof of Proposition 1.1).

Let us endow U_{ℏ}𝔤 = U_{ℏ}𝔟_{+} ⊗ U_{ℏ}𝔫_{−} with the double algebra structure such that U_{ℏ}𝔟_{+} → U_{ℏ}𝔤, x_{+} ↦ x_{+} ⊗ 1, and U_{ℏ}𝔫_{−} → U_{ℏ}𝔤, x_{−} ↦ 1 ⊗ x_{−} are algebra morphisms and, if e_{i}^{𝔤} = e_{i} ⊗ 1, f_{i}^{𝔤} = 1 ⊗ f_{i}, h_{i}^{𝔤} = h_{i} ⊗ 1 and D_{j}^{𝔤} = D_{j} ⊗ 1,

[e_{i}^{𝔤}, f_{j}^{𝔤}] = δ_{ij} \frac{q^{d_{i}h_{i}^{𝔤}} − q^{−d_{i}h_{i}^{𝔤}}}{q^{d_{i}} − q^{−d_{i}}} ,

and

[h_{i}^{𝔤}, f_{j}^{𝔤}] = −a_{ij}f_{j}^{𝔤}, [D_{j}^{𝔤}, f_{j}^{𝔤}] = −δ_{ij}f_{j}^{𝔤}.

U_{ℏ}𝔤 is endowed with a topological Hopf algebra structure Δ : U_{ℏ}𝔤 ↦ U_{ℏ}𝔤 ˆ⊗U_{ℏ}𝔤 = lim_{←N}(U_{ℏ}𝔤 ⊗ U_{ℏ}𝔤)/ℏ^{N}(U_{ℏ}𝔤 ⊗ U_{ℏ}𝔤), extending Δ_{+} and Δ_{−} ([8]).

Let t_{0} be the element of 𝔥 ⊗ 𝔥 corresponding to the restriction of the invariant pairing of 𝔤 to 𝔥 and let R[α] be the element exp(ℏt_{0})P [α] of [lim_{←N}(U_{ℏ}𝔤 ⊗ U_{ℏ}𝔤)/ℏ^{N}(U_{ℏ}𝔤 ⊗ U_{ℏ}𝔤)][ℏ^{−1}]. Then we have the equalities

R[α − α_{i}](e_{i}^{𝔤} ⊗ q^{d_{i}h_{i}^{𝔤}}) + R[α](1 ⊗ e_{i}^{𝔤}) = (q^{d_{i}h_{i}^{𝔤}} ⊗ e_{i}^{𝔤})R[α] + (e_{i}^{𝔤} ⊗ 1)R[α − α_{i}], (29)

for any i = 1, …, n.

Lemma 2.12. For any nonzero α in ℕ^{n}, P [α] belongs to ℏU_{ℏ}𝔫_{+} ⊗ U_{ℏ}𝔫_{−}.

Proof. Let us show this by induction on the height of α (we say that the height of α = (α_{i})_{1≤i≤n} is ∑_{i=1}^{n} α_{i}). If α is a simple root α_{i}, P [α] = ℏe_{i}^{𝔤} ⊗ f_{i}^{𝔤}, so that the statement holds when deg(α) = 1.

Assume that we know that P [α] belongs to ℏU_{ℏ}𝔫_{+} ⊗ U_{ℏ}𝔫_{−} for any α of height < ν. Let α be of height ν. Let v be the ℏ-adic valuation of P [α], and assume that v ≤ 0. R[α] belongs to ℏ^{v}(U_{ℏ}𝔤 ˆ⊗U_{ℏ}𝔤), and since v ≤ 0 the equality (29) takes place in ℏ^{v}(U_{ℏ}𝔤 ˆ⊗U_{ℏ}𝔤). Let us set R_{α} = ℏ^{−v}R[α] mod ℏ; R_{α} is an element of U𝔟_{+} ⊗ U𝔟_{−}. Since ℏ^{−v}R[α − α_{i}] is zero mod ℏ, (29) implies that R_{α} commutes with each 1 ⊗ ē_{i}.

Lemma 1.5 of [19] says that if a belongs to 𝔫_{−} and commutes with each e_{i}, then a is zero. It follows that if x belongs to U𝔫_{−} and commutes with each ē_{i}, x is scalar; and if in addition x has nonzero degree, x is zero. Therefore R_{α} is zero.

It follows that v ≥ 1, which proves the induction.

It follows from [8] that the R[α] satisfy the quasi-triangular identities

(Δ ⊗ id)R[α] = ∑_{β,γ∈ℕ^{n},β+γ=α} R[β]^{(13)}R[γ]^{(23)},  (30)

(id ⊗ Δ)R[α] = ∑_{β,γ∈ℕ^{n},β+γ=α} R[β]^{(13)}R[γ]^{(12)}.  (31)

Let us set, for α ≠ 0, r[α] = R[α]/ℏ mod ℏ; r[α] belongs to U𝔟_{+} ⊗U𝔟_{−}. Dividing the equalities (30) and (31) by ℏ, we get (Δ_{U𝔟_{+}} ⊗ id)r[α] = r[α]^{(13)} + r[α]^{(23)} and (id ⊗ Δ_{U𝔟_{+}})r[α] = r[α]^{(12)} + r[α]^{(13)}. Therefore r[α] belongs to 𝔟_{+} ⊗ 𝔟_{−}.

Moreover, (29) implies the identity

δ(x)_{(β,α−β)} = [r[β − α], x ⊗ 1] + +[r[β], 1 ⊗ x]

for x in 𝔤[α], where we set (∑_{i} x_{i} ⊗ y_{i})_{(α,β)} = ∑_{i}(x_{i})_{(α)} ⊗ (y_{i})_{(β)} and we denote by x_{(α)} the degree α component of an element x of U𝔤.

It follows that r[α] is the element of 𝔫_{+}[α] ⊗ 𝔫_{−}[−α] corresponding to the invariant pairing of 𝔤.

Let us now prove by induction on k that if α belongs to kΔ_{+} − (k − 1)Δ_{+}, P [α] belongs to ℏ^{k}U_{ℏ}𝔫_{+} ⊗ U_{ℏ}𝔫_{−} and

P [α] = \frac{ℏ^{k}}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=α}^{∑} r[α_{1}] ⋯ r[α_{k}] + o(ℏ^{k}).

Assume that the statement is proved up to order k − 1 and let λ belong to kΔ_{+} − (k − 1)Δ_{+}. Then (30) and the induction hypothesis imply that

(Δ̃ ⊗ id)(P [λ]) = _{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} ∑_{l,l′>0,l+l′=k} \frac{ℏ^{k}}{l!l′!}  (32)
e_{α_{1},i_{1}} ⋯ e_{α_{l},i_{l}} ⊗ e_{α_{l+1},i_{l+1}} ⋯ e_{α_{k},i_{k}} ⊗ f_{α_{1},i_{1}} ⋯ f_{α_{k},i_{k}},

where Δ̃(x) = Δ(x) − x ⊗ 1 − 1 ⊗ x. Let σ be any permutation of {1, …, k}. For any α_{1}, …, α_{k} in Δ_{+}, such that ∑_{i=1}^{k} α_{i} = λ, we have

f_{α_{1},i_{1}} ⋯ f_{α_{k},i_{k}} = f_{α_{σ(1)},i_{σ(1)}} ⋯ f_{α_{σ(k)},i_{σ(k)}} + o(ℏ).

Indeed, the difference of both sides is a sum of products of the [f_{α_{s},i_{s}}, f_{α_{t},i_{t}}] with elements of 𝔫_{+}; but α_{s} + α_{t} does not belong to Δ_{+} by hypothesis on λ, so [f_{α_{s},i_{s}}, f_{α_{t},i_{t}}] = o(ℏ).

The right side of (32) can the be rewritten as

_{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} ∑_{l,l′>0,l+l′=k} \frac{ℏ^{k}}{l!l′!} \frac{1}{cardΣ_{l,l′}}

∑_{σ∈Σ_{l,l′}} e_{α_{σ(1)},i_{σ(1)}} ⋯ e_{α_{σ(l)},i_{σ(l)}} ⊗ e_{α_{σ(l+1)},i_{σ(l+1)}} ⋯ e_{α_{σ(k)},i_{σ(k)}} ⊗ f_{α_{1},i_{1}} ⋯ f_{α_{k},i_{k}},

where Σ_{l,l′} is the set of shuffle transformations of ((1, …, l), (l + 1, …, l + l′)). Therefore the right side of (32) is equal to

\frac{ℏ^{k}}{k!} Δ̃ (_{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} e_{α_{1};i_{1}} ⋯ e_{α_{k};i_{k}} ⊗ f_{α_{1};i_{1}} ⋯ f_{α_{k};i_{k}} ) + o(ℏ^{k}).

Let v be the ℏ-adic valuation of P [λ]. Assume that v < k. Set P̄[α] = ℏ^{−v}P [α] mod ℏ. Then if we call Δ_{0} the coproduct of U𝔫_{+}, and we set Δ̃_{0}(x) = Δ_{0}(x) − x ⊗ 1 − 1 ⊗ x, (32) gives (Δ̃_{0} ⊗ id)(P̄[α]) = 0, so that P̄[α] belongs to 𝔫_{+} ⊗ U𝔫_{−}; since P̄[α] also belongs to U𝔫_{+}[α] ⊗ U𝔫_{−}[−α] and α does not belong to Δ_{+}, P̄[α] is zero, contradiction. Therefore v ≥ k. Let us set P′[α] = ℏ^{−k}P [α] mod ℏ; we find that

(Δ̃_{0} ⊗ id) (P′[α] − \frac{1}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} ē_{α_{1};i_{1}} ⋯ ē_{α_{k};i_{k}} ⊗ f̄_{α_{1};i_{1}} ⋯ f̄_{α_{k};i_{k}} ) = 0,

so that P′[α] belongs to

\frac{1}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} ē_{α_{1};i_{1}} ⋯ ē_{α_{k};i_{k}} ⊗ f̄_{α_{1};i_{1}} ⋯ f̄_{α_{k};i_{k}} + 𝔫_{+} ⊗ U𝔫_{−};

as 𝔫_{+}[α] is zero, P′[α] is equal to

\frac{1}{k!} _{α_{1},...,α_{k}∈Δ_{+},∑_{i=1}^{k} α_{i}=λ;i_{j}}^{∑} ē_{α_{1};i_{1}} ⋯ ē_{α_{k};i_{k}} ⊗ f̄_{α_{1};i_{1}} ⋯ f̄_{α_{k};i_{k}},

which proves the induction.

2.4. The generic case (proof of Corollaries 1.2 and 1.3).

We have the equality U_{ℏ}𝔫_{+} ⊗_{ℂ[[ℏ]]} ℂ((ℏ)) = U_{q′}𝔫_{+} ⊗_{ℂ(q′)} ℂ((ℏ)), therefore the graded components of U_{q′}𝔫_{+} have the same dimension as those of U_{ℏ}𝔫_{+}. Corollaries 1.2 and 1.3) follow.

3. Quantum current algebras of finite type (proofs for Sections 1.)

3.1. PBW theorem and comparison with Feigin-Odesskii algebra (proofs of Theorem 1.3 and Corollary 1.4).

•Identification of algebras generated by the classical limits of quantum currents relations

Recall that A is now assumed of finite type. Define L𝔟_{+} as the Lie subalgebra (𝔥 ⊗ ℂ[t^{−1}]) ⊕ (𝔫_{+} ⊗ ℂ[t, t^{−1}]) of 𝔤 ⊗ ℂ[t, t^{−1}].

Proposition 3.1. Define U_{ℏ}L𝔟_{+} and Ũ_{ℏ}L𝔟_{+} as the algebra with generators h_{i}[k], i = 1, …, n, k ≤ 0 and x_{i}^{+}[k], i = 1, …, n, k ∈ ℤ, and relations

[h_{i}[k], h_{j}[l]] = 0, [h_{i}[k], x_{j}^{+}[l]] = \frac{q^{kd_{i}a_{ij}} − q^{−kd_{i}a_{ij}}}{2ℏkd_{i}} x_{j}^{+}[k + l],

and relations (6) and (7) among the x_{j}^{+}[k] (with e_{i} replaced by x_{i}^{+}), resp. (6) and (8). There are algebra isomorphisms from U_{ℏ}L𝔟_{+}/ℏU_{ℏ}L𝔟_{+} and Ũ_{ℏ}L𝔟_{+}/ℏŨ_{ℏ}L𝔟_{+} to UL𝔟_{+}, sending h_{i}[k] to h̄_{i} ⊗ t^{k} and x_{i}^{+}[k] to x̄_{i}^{+} ⊗ t^{k}.

Proof. U_{ℏ}L𝔟_{+}/ℏU_{ℏ}L𝔟_{+} is the algebra with generators h̄_{i}[k], ē_{i}[l], 1 ≤ i, j ≤ n, k ≤ 0, l ∈ ℤ and relations

[h̄_{i}[k], ē_{i}[l]] = a_{ij}ē_{i}[k + l],

and

(z − w)[ē_{i}(z), ē_{j}(w)] = 0,  (33)
Sym_{z_{1},...,z_{1−a_{ij}}} (ad(ē_{i}(z_{1})) ⋯ ad(ē_{i}(z_{1−a_{ij}}))(ē_{j}(w))) = 0,

where ē_{i}(z) = ∑_{k∈ℤ} ē_{i}[k]z^{−k}. It follows from (33) with i = j that we have [ē_{i}[n], ē_{j}[m]] = 0 for all n, m. Therefore, ad(ē_{i}(z_{1})) ⋯ ad(ē_{i}(z_{1−a_{ij}}))(ē_{j}(w)) is symmetric in the z_{i}, so that the last equation is equivalent to

ad(ē_{i}(z_{1})) ⋯ ad(ē_{i}(z_{1−a_{ij}}))(ē_{j}(w)) = 0.  (34)
On the other hand, Ũ_{ℏ}L𝔟_{+}/ℏŨ_{ℏ}L𝔟_{+} is the algebra with generators
h̄_{i}[k]′, ē_{i}[l]′, 1 ≤ i, j ≤ n, k ≤ 0, l ∈ ℤ

and relations

[h̄_{i}[k]′, ē_{i}[l]′] = a_{ij}ē_{i}[k + l]′,

and

(z − w)[ē_{i}(z)′, ē_{j}(w)′] = 0,  (35)

and

(ade_{i}[0]′)^{1−a_{ij}}e_{j}[k]′ = 0.  (36)

The algebras presented by the pairs of relations (33) and (34) on one hand, and (35) and (36) on the other, are isomorphic. Indeed, (33) and (35) are equivalent, and (34) implies (36); on the other hand, (33) implies that [e_{i}[0]′, e_{j}[k + l]′] = [e_{i}[k]′, e_{j}[l]′], so that [e_{i}[0]′, [e_{i}[0]′, e_{j}[k + k′ + l]′]] = [e_{i}[0]′, [e_{i}[k]′, e_{j}[k′ + l]′]] = [e_{i}[k]′, [e_{i}[0]′, e_{j}[k′ + l]′]], because the (33) implies that [e_{i}[0]′, e_{i}[k]′] = 0, therefore [e_{i}[0]′, [e_{i}[0]′, e_{j}[k + k′ + l]′]] = [e_{i}[k]′, [e_{i}[k′]′, e_{j}[l]′]]; one then proves by induction that (ade_{i}[0]′)^{p}(e_{j}[k+k_{1}+⋯+k_{p}]) = ade_{i}[k_{1}]′ ⋯ ade_{i}[k_{p}]′(e_{j}[k]). With p = 1−a_{ij}, this relation shows that the e_{i}[k]′ satisfy (34).

If follows that if F̃_{+} is the Lie algebra defined in Proposition 1.5, both quotient algebras U_{ℏ}L𝔟_{+}/ℏU_{ℏ}L𝔟_{+} and Ũ_{ℏ}L𝔟_{+}/ℏŨ_{ℏ}L𝔟_{+} are isomorphic to the crossed product of UF̃_{+} with the derivations h̃_{i}[k]′, defined by h̃_{i}[k]′(ē_{i}[l]) = a_{ij}ē_{i}[k + l].

It is clear that there is a unique Lie algebra morphism j_{+} from the Lie algebra F̃_{+} defined in Proposition 1.5 to 𝔫_{+} ⊗ ℂ[t, t^{−1}], sending ē_{i}[k] to x̄_{i}^{+} ⊗ t^{k}. Let us prove that it is an isomorphism.

For this, let us define F̃ as the Lie algebra with generators x̃_{i}^{±}[k], h̃_{i}^{±}[k], 1 ≤ i ≤ n, k ∈ ℤ, and relations given by the coefficients of the monomials in

(z − w)[x̃_{i}^{±}(z), x̃_{j}^{±}(w)] = 0, if x, y ∈ {x̃_{i}^{±}},
[h̃_{i}(z), h̃_{j}(w)] = 0,
[h̃_{i}(z), x̃_{j}^{±}(w)] = ±a_{ij}δ(z/w)x̃_{j}^{±}(w),
[x̃_{i}^{+}(z), x̃_{j}^{−}(w)] = δ_{ij}δ(z/w)h̃_{i}(z),
ad(x̃_{i}^{±}(z_{1})) ⋯ ad(x̃_{i}^{±}(z_{1−a_{ij}}))(x̃_{j}^{±}(w)) = 0,

where we set x̃(z) = ∑_{k∈ℤ} x[k]z^{−k} for x in {x̃_{i}^{±}, h̃_{i}}.

Lemma 3.1. (In this Lemma, 𝔤 may be an arbitrary Kac-Moody Lie algebra.) Let W be the Weyl group of 𝔤, and s_{i} be its elementary reflection associated to the root α_{i}. Then there is a unique action of W on F̃ such that

s_{i}(x̃_{i}^{±}[k]) = x̃_{i}^{∓}[k],
s_{i}(x̃_{j}^{±}[k]) = ad(x̃_{i}^{±}[0])^{−a_{ij}}(x̃_{j}^{∓}[k]), if j ≠ i,
s_{i}(h̃_{j}^{±}[k]) = h̃_{j}^{±}[k] − a_{ij}h̃_{i}^{±}[k].

Proof of Lemma. The proof follows the usual proof for Kac-Moody Lie algebras. For example, if j, k are different from i, we have

(z − w)[s_{i}(x̃_{j}^{±}(z)), s_{i}(x̃_{k}^{±}(w))]
= (z − w)[ad(x̃_{i}^{±}[0])^{−a_{ij}}(x̃_{j}^{±}(z)), ad(x̃_{i}^{±}[0])^{−a_{ik}}(x̃_{k}^{±}(w))]
= ad(x̃_{i}^{±}[0])^{−a_{ij}−a_{ik}} ((z − w)[x̃_{j}^{±}(z), x̃_{k}^{±}(w)])

because the Serre relations imply that ad(x̃_{i}^{±}[0])^{1−a_{il}}(x̃_{l}(u)) = 0 for j = k, l and u = z, w; therefore (z − w)[s_{i}(x̃_{j}^{±}(z)), s_{i}(x̃_{k}^{±}(w))] is zero.

Lemma 3.2. There is a unique Lie algebra isomorphism j from F̃ to 𝔤 ⊗ ℂ[t, t^{−1}], such that j(x̃[k]) = x̄ ⊗ t^{k}, for any x in {x_{i}^{±}, h_{i}} and k in ℤ.

Proof. Let F̃_{−} be the Lie algebra with generators x̄_{i}^{−}[k], 1 ≤ i ≤ n, k ∈ ℤ and relations (33) and (34), with x̄_{i}^{+}[k] replaced by x̄_{i}^{−}[k], and let H̃ be the abelian Lie algebra with generators h̄_{i}[k], 1 ≤ i ≤ n, k ∈ ℤ. There are unique Lie algebra morphisms from F̃_{±} and H̃ to F̃, sending the x̄_{i}^{±}[k] to x̃_{i}^{±}[k] and the h̄_{i}[k] to h̃_{i}[k]. These morphisms are injections, so that we will indentify F̃_{±} and H̃ with their images in F̃.

Moreover, let F_{±} be the free Lie algebras with generators x_{i}^{±}[k]^{F}, i = 1, …, n, k integer. Endow F_{±} ⊕ H̃ with the Lie algebra structure such that H̃ is abelian, F_{±} is a Lie subalgebra of F_{±} ⊕ H̃, and [h̃_{i}[k], x_{i}^{±}[l]^{F}] = ±a_{ij}x_{i}^{±}[k + l]^{F}

There are unique derivations Φ_{i,k}^{∓} from F_{±} to F_{±} ⊕ H̃ such that

Φ_{i,k}^{∓}(x_{i′}^{±}[l]) = δ_{ii′} h̃_{i}[k + l].

Let I_{±}^{F} be the ideals of F_{±} generated by relations (33) and (34); then computation shows that I_{±}^{F} are preserved by the Φ_{i,k}^{∓}. It follows that F̃ is the direct sum of its subspaces F̃_{±} and H̃.

The rules deg(x̃_{i}^{±}[k]) = (±ϵ_{i}, k) and deg(h̃_{i}[k]) = (0, k) define a Lie algebra grading of F̃ by ℤ^{n} × ℤ, because the relations of F̃ are homogeneous for this grading. Clearly, dimF̃_{±}[(±ϵ_{i}, k)] = 1 for any i and k, so that dimF̃[(±ϵ_{i}, k)] = 1.

Let α be any root on 𝔤. Then there is some simple root ±ϵ_{i} an element w of W such that α = w(±ϵ_{i}). Then F̃[(±ϵ_{i}, k)] = F̃[(α, k)] so that dimF̃[(α, k)] = 1.

It is clear that the map j defined in the statement of the Lemma defines a Lie algebra morphism. Define a grading by ℤ^{r} × ℤ on 𝔤 ⊗ ℂ[t, t^{−1}], by the rules deg(x ⊗ t^{k}) = (deg(x), k), for x a homogeneous (for the root grading) element of 𝔤. Then j is a graded map. Moreover, if α is in ±Δ_{+} and x is a nonzero element of 𝔤 od degree α, then x can be written as a ∑ λ_{i_{1},···,i_{p}}[x_{i_{1}}^{±}, […, x_{i_{p}}^{±}]]; then the image by j of ∑ λ_{i_{1},···,i_{p}}[x_{i_{1}}^{±}[0], […, x_{i_{p}}^{±}[k]]] is equal to x ⊗ t^{k}; therefore the map induced by j from F̃[(α, k)] to 𝔤 ⊗ ℂ[t, t^{−1}][(α, k)] is nonzero and therefore an isomorphism.

It follows that Kerj is equal to ∑_{α∈ℤ^{n}\(Δ_{+}∪{0}∪(−Δ_{+})),k∈ℤ} F̃[(α, k)]. Any element of F̃[(α, k)] is a linear combination of brackets [x_{i_{l}}^{±}[k_{l}], […, x_{i_{1}}^{±}[k_{1}]]], with ∑_{s=1}^{l} ±ϵ_{i_{s}} = α. Assume α is not a root of 𝔤 and let l′ be the smallest integer such that ∑_{s=1}^{l′+1} ±ϵ_{i_{s}} is not a root of 𝔤. Let us show that each [x_{i_{l′+1}}^{±}[k_{l′+1}], […, x_{i_{1}}^{±}[k_{1}]]] vanishes. It follows from the fact that j is an isomorphism when restricted to the parts of degree in Δ_{+} ∪ (−Δ_{+}) that we may write each [x_{i_{l′}}^{±}[k_{l′}], […, x_{i_{1}}^{±}[k_{1}]]] as a linear combination ∑_{s} λ_{s}[x_{i_{l′}}^{±}[k^{(s)}(i_{l′})] ⋯ x_{i_{1}}^{±}[k^{(s)}(i_{1})]], where for each s, i ↦ k^{(s)}(i) is a map from {1, …, n} to ℤ, such that k^{(s)}(i_{l′+1}) = k_{l′+1}. The defining relations for 𝔫_{±} hold among the x_{1}^{±}[k_{1}^{(s)}], …, x_{n}^{±}[k_{n}^{(s)}], therefore we have Lie algebra maps from 𝔫_{±} to F̃^{±} sending each x_{i}^{±} to x_{i}^{±}[k_{i}^{(s)}]. [x_{i_{l′+1}}^{±}[k^{(s)}(i_{l′+1})] ⋯ x_{i_{1}}^{±}[k^{(s)}(i_{1})]] is the image of zero by one of these maps, and is therefore zero. It follows that [x_{i_{l′}}^{±}[k_{l′}], […, x_{i_{1}}^{±}[k_{1}]]] vanishes, so that Kerj is zero.

End of the proof of the Proposition. The restriction of j to F̃_{+} coincides with the map j_{+} define before Lemma 3.1, therefore j_{+} induces an isomorphism between F̃_{+} and 𝔫_{+} ⊗ ℂ[t, t^{−1}].

•Crossed product algebras V^{L} and S^{L}

For k an integer ≤ 0 and 1 ≤ i ≤ n, define endomorphisms h̃_{i}[k] of FO by

(h̃_{i}[k]f)(t_{l}^{(i)}) = [∑_{j=1}^{n} ∑_{l=1}^{k_{j}} \frac{q^{kd_{i}a_{ij}} − q^{−kd_{i}a_{ij}}}{2ℏkd_{i}} (t_{l}^{(j)})^{k}] f(t_{l}^{(i)})  (37)

if f ∈ FO_{k}. The h̃_{i}[k] are derivations of FO. These derivations preserve LV , therefore they preserve ⟨LV ⟩.

Define V^{L} and S^{L} as the crossed product algebras of ⟨LV ⟩ and FO with the derivations x̃[k]: V^{L}, resp. S^{L} is equal to ⟨LV ⟩ ⊗ ℂ[h_{i}[k]^{V^{L}}, k ≤ 0], resp. FO ⊗ ℂ[h_{i}[k]^{V^{L}}, k ≤ 0]; both spaces are endowed with the products given by formula (28), where x now belongs to ⟨LV ⟩, resp. F O and the X_{s} are replaced by h_{i}[k], k ≤ 0. Define V̂^{L} and S^{L} as the partial ℏ-adic completions ℂ[h_{i}[k]^{V^{L}}, k < 0] ⊗ ⟨LV ⟩ ⊗_{ℂ[[ℏ]]} ℂ[h_{i}[0]^{V^{L}}][[ℏ]] and ℂ[h_{i}[k]^{V^{L}}, k < 0] ⊗ FO ⊗_{ℂ[[ℏ]]} ℂ[h_{i}[0]^{V^{L}}][[ℏ]].

Lemma 3.3. 1) The rules deg(h_{i}[k]) = 0, deg(t_{i}^{k}) = ϵ_{i} define gradings of ⟨LV ⟩, FO, V^{L}, S^{L}, V̂^{L} and Ŝ^{L} by ℕ^{n}, which are compatible with the inclusions. For X any of these algebras, we denote by X_{k} its homogeneous component of degree k. X is therefore the direct sum of the X_{k}.

2) For any k, ⟨LV ⟩_{k} is a free ℂ[[ℏ]]-modules with a countable basis.

3) For any k, V_{k}^{L} and S^{L} are free ℂ[[ℏ]][h_{i}[k]^{V^{L}}]-modules; and V̂_{k}^{L} and Ŝ^{L} are free ℂ[[ℏ]][h_{i}[k]^{V^{L}}, k < 0] ⊗_{ℂ[[ℏ]]} ℂ[h_{i}[0]^{V^{L}}][[ℏ]]-modules.

Proof. 1) is clear. ⟨LV ⟩_{k} is a ℂ[[ℏ]]-submodule of FO_{k}, and by Lemma A2, it is a free ℂ[[ℏ]]-module with a countable basis. This shows 2). 3) is a direct consequence of 2).

•Ideals and completions Define for N positive integer, I_{N} as the left ideal of ⟨LV ⟩ generated by the elements (t_{i}^{k}) of FO_{ϵ_{i}}, k ≥ N, i = 1, …, n. Define I_{N} and Î_{N} as the left ideals of V^{L} and V̂^{L} generated by the same family. For s ≥ 0, set I_{N}^{(s)} = ℏ^{−s}(I_{N} ∩ℏ^{s}⟨LV ⟩), and I_{N}^{(∞)} = ∪_{s≥0}I_{N}^{(s)}; define I_{N}^{(s)}, I_{N}^{(∞)} and Î_{N}^{(s)}, Î_{N}^{(∞)} in the same way.

For any integer a, define LV^{≥a} as the subspace of FO equal to the direct sum ⊕_{i=1}^{n}t_{i}^{a}ℂ[[ℏ]][t_{i}] and let ⟨LV^{≥a}⟩ be the subalgebra of FO generated by LV^{≥a}. Define I_{N}^{(0),≥a} as the left ideal of ⟨LV^{≥a}⟩ generated by the t_{i}^{k}, k ≥ N and I_{N}^{≥a} as the ideal of ⟨LV^{≥a}⟩ formed of the elements x such that for some k ≥ 0, ℏ^{k}x belongs to I_{N}^{(0),≥a}.

For any integer a and k in ℕ^{r}, define FO^{≥a} as the subspace of FO_{k} consisting of the rational functions

g(t_{α}^{(i)}) = \frac{1}{∏{n \atop i=1} ∏1≤α≤k_{i},1≤β≤k_{j}(t_{α}^{(i)}−t_{β}^{(j)})} f(t_{α}^{(i)}),

where the f(t_{α}^{(i)}) have degree ≥ a in each variable t_{α}^{(i)} and the total degree of g is ≥ (∑_{i} k_{i})a. Set FO^{≥a} = ⊕_{k∈ℕ^{n}}FO_{k}^{≥a}. Then FO^{≥a} is a subalgebra of F O.

Define I_{(N)}^{≥a} as the set of elements of FO^{(≥a)}, where f(t_{α}^{(i)}) has total degree N in the variables t_{α}^{(i)}, and let I_{N}^{≥a} be the direct sum ⊕_{k≥N}I_{N}^{≥a}.

Lemma 3.4. For (J_{N})_{N>0} a family of left ideals of some algebra A, say that (J_{N})_{N>0} has property (∗) if for any integer N > 0 and element a in A, there is an integer k(a, N) > 0 such that J_{N}a ⊂ J_{k(a,N)} for any N large enough, and k(N, a) tends to infinity with N, a being fixed. Then the inverse limit lim_{←N}A/J_{N} has an algebra structure.

Say that J_{N} has property (∗∗) if for any integer N > 0 and element a in A, there is are integer k′(a, N) and k′′(a, N) > 0 such that J_{N}a ⊂ J_{k′(a,N)} and aJ_{N} ⊂ J_{k′′(a,N)} for any N large enough, and k(N, a) tends to infinity with N, a being fixed. In that case also, the inverse limit lim_{←N}A/J_{N} has an algebra structure.

1) The family (I_{N}^{(∞)})_{N>0} of ideals of ⟨LV ⟩ has property (∗);

2) the family (I_{N}^{(∞)})_{N>0} of ideals of V^{L} has property (∗);
3) the family (Î_{N}^{(∞)})_{N>0} of ideals of V̂^{L} has property (∗);
4) the family (I_{N}^{≥a})_{N>0} of ideals of ⟨LV^{≥a}⟩ has property (∗);
5) the family (I_{N}^{≥a})_{N>0} of ideals of FO^{≥a} has property (∗∗).

Proof. Set for any a in A and N > 0, k′(a, N) = inf {k|J_{N}a ⊂ J_{k}}; then k′(N, a) tends to infinity with N, a being fixed and we have k′(N, a′) ≥ inf (k(N, a), p) if a′ belongs to a + J_{p}.

An element of lim_{←N}A/J_{N} is a family (a_{N})_{N>0}, a_{N} ∈ A/J_{N}, such that a_{N+1} + J_{N} = a_{N}. For a = (a_{N})_{N>0} and b = (b_{N})_{N>0} in lim_{←N}A/J_{N}, choose β_{N} in b_{N} and let N′(N, β_{N}) be the smallest integer N′ such that k′(N′, β_{N}) ≥ N; N′(N, β_{N}) is independent of the choice of β_{N}, we denote it N′(N, b). Choose then α_{N} in a_{N′(N,b)}; then α_{N}β_{N} + J_{N} is independent of the choice of α_{N} and β_{N}; one checks that α_{N+1}β_{N+1} + J_{N} = α_{N}β_{N} + J_{N}, so that (α_{N}β_{N} + J_{N})_{N>0} defines an element of lim_{←N}A/J_{N}. The product ab is defined to be this element. The construction is similar in the case of property (∗∗).

1) The equality

t_{i}^{k} ∗ t_{j}^{l} = q^{−nd_{i}a_{ij}}t_{i}^{k−n} ∗ t_{j}^{l+n} + q^{−d_{i}a_{ij}}t_{j}^{l} ∗ t_{i}^{k}

+ ∑_{n′=1}^{n−1}(q^{−(n′+1)d_{i}a_{ij}} − q^{−(n′−1)d_{i}a_{ij}})t_{j}^{l+n′} ∗ t_{i}^{k−n′} − q^{−(n−1)d_{i}a_{ij}}t_{j}^{l+n} ∗ t_{i}^{k−n},

where n = N − l and k ≥ 2N − l, implies that if k ≥ 2N − l, t_{i}^{k} ∗ t_{j}^{l} belongs to I_{N}. It follows that I_{2N−l} ∗ t_{j}^{l} ⊂ I_{N}. Set then k(t_{j}^{l}, N) = [\frac{1}{2}(N + l/2)] + 2; for a in ⟨LV ⟩, and any decomposition dec of a as a sum ∑_{(j_{i}),(l_{i})} λ_{(j_{i}),(l_{i})}t_{j_{1}}^{l_{1}} ∗⋯∗t_{j_{p}}^{l_{p}}, define k(a, N, dec) as the smallest of integers k(t_{j_{p}}^{l_{p}}, ⋯, k(t_{j_{1}}^{l_{1}}, N)); finally, define k(a, N) as the largest of all k(a, N, dec). The family (I_{N})_{N>0} has property (∗), with this function k(a, N). Then for any a in ⟨LV ⟩, (I_{N} ∩ ℏ^{s}V^{L})a ⊂ ℏ^{s}V^{L} ∩ I_{k(a,N)}, so that I_{N}^{(s)}a ⊂ I_{k(a,N)}^{(s)}. It follows that we have also I_{N}^{(∞)}a ⊂ I_{k(a,N)}^{(∞)}, so that the families (I_{N}^{(s)})_{N>0} and (I_{N}^{(∞)})_{N>0} have property (∗).

2) follows from the fact that I_{N}h_{j}[l]^{V^{L}} ⊂ I_{N+l}.

3) follows from the fact that for a any element of ℂ[h_{i}[0]^{V^{L}}][[ℏ]], we have Î_{N}^{(∞)}a ⊂ Î_{N}^{(∞)}.

4) is proved in the same way as 1) ans 2).

5) For k in ℕ^{n}, set |k| = ∑_{i=1}^{n} k_{i}. Then if f belongs to FO_{k}^{≥a}, we have f ∗I_{N}^{≥a} ⊂ I_{N+a|k|}^{≥a} and I_{N}^{≥a}∗f ⊂ I_{N+a|k|}^{≥a}. Therefore the family (I_{N}^{≥a})_{N} has property (∗∗).

It follows that the inverse limits

lim_{←N}⟨LV ⟩/I_{N}^{(∞)}, lim_{←N}V^{L}/I_{N}^{(∞)}, lim_{←N} V̂^{L}/Î_{N}^{(∞)},
lim_{←N}⟨LV^{≥a}⟩/I_{N}^{≥a} and lim_{←N}FO^{≥a}/I_{N}^{≥a}

have algebra structures. Moreover, as we have I_{N}^{(∞)}∩⟨LV ⟩ = I_{N}^{(∞)} and Î_{N}^{(∞)}∩V^{L} = I_{N}^{(∞)}, we have natural algebra inclusions

lim_{←N}⟨LV ⟩/I_{N}^{(∞)} ⊂ lim_{←N}V^{L}/I_{N}^{(∞)} ⊂ lim_{←N} V̂^{L}/Î_{N}^{(∞)}.

Moreover, there exists a function ϕ(k, N), tending to infinity with N, such that I_{N,k}^{≥a} ∩⟨LV^{≥a}⟩ ⊂ I_{ϕ(k,n)}. Indeed, if the k_{i} are ≥ a and t_{i_{1}}^{k_{1}} ∗⋯∗t_{i_{l}}^{k_{l}} belongs to I_{N,k}^{≥a} (l = |k|), then k_{1} + ⋯ + k_{l} ≥ N so that one of the k_{i} is ≥ N/l. The statement then follows from the proof of the above Lemma, 1). It follows that we have an algebra inclusion

lim_{←N}⟨LV^{≥a}⟩/I_{N}^{≥a} ⊂ lim_{←N}FO^{≥a}/I_{N}^{≥a}.

If a family (J_{N})_{N>0} of left ideals of the algebra A has property (∗), and B is any algebra, the family (J_{N} ⊗ B)_{N>0} also satisfies (∗); therefore the inverse limit lim_{←N}(A ⊗ B)/(J_{N} ⊗ B) has an algebra structure. It follows that we have algebra structures on lim_{←N}⟨LV ⟩ ⊗ A/I_{N}^{(∞)} ⊗ A, lim_{←N}V^{L} ⊗ A/I_{N}^{(∞)} ⊗ A and lim_{←N} V̂^{L} ⊗ A/Î_{N}^{(∞)} ⊗ A for any algebra A.

•Topological Hopf structures on V^{L} and S^{L} For i = 1, …, n and l ≥ 0, define K_{i}[−l] as the element of V^{L} (or S^{L})

K_{i}[−l] = e^{−ℏd_{i}h_{i}[0]^{V^{L}}}S_{l}(−2ℏd_{i}h_{i}[k]^{V^{L}}, k < 0),

where S_{l}(z_{−1}, z_{−2}, …) are the Schur polynomials in variables (z_{i})_{i<0}, which are determined by the relation exp(∑_{i<0} z_{i}t^{−i}) = ∑_{l≤0} S_{l}(z_{k})t^{−l}.

Proposition 3.2. There is unique graded algebra morphism Δ_{S^{L}} from S^{L} to lim_{←N}(S^{L} ⊗_{ℂ[[ℏ]]} Ŝ^{L})/(I_{N}^{S} ⊗_{ℂ[[ℏ]]} Ŝ^{L}), such that

Δ_{S^{L}}(h_{i}[k]) = h_{i}[k] ⊗ 1 + 1 ⊗ h_{i}[k]

for 1 ≤ i ≤ n and k ≤ 0, and its restriction to S_{k}^{L} is the direct sum of the maps Δ_{S^{L}}^{k′,k′′} : FO_{k}^{≥a} → lim_{←N}(I_{k′}^{≥a} ⊗_{ℂ[[ℏ]]} Ŝ_{k′′}^{L})/[FO_{k′,N}^{≥a} ⊗_{ℂ[[ℏ]]} Ŝ_{k′′}^{L}], where k = k′ + k′′, defined by

Δ_{S^{L}}^{k′,k′′}(P)

= ∑_{p_{1},...,p_{N′}≥0} (∏_{i=1}^{N′} u_{i}^{p_{i}}P′_{α}(u_{1}, …, u_{N′})) ⊗ (P′′_{α}(u_{N′+1}, …, u_{N}) ∏_{i=1}^{N′} K_{ϵ(i)}[−p_{i}]) ,

where N′ = ∑_{i=1}^{n} k′_{i}, N′′ = ∑_{i=1}^{n} k′′_{i}, N = N′ +N′′, the arguments of the functions in FO_{k′} and FO_{k′′} are respectively (t_{j}^{(i)})_{1≤i≤n,1≤j≤k′_{i}} and (t_{j}^{′(i)})_{1≤i≤n,1≤j≤k′′_{i}}; we set

(u_{1}, …, u_{k′_{1}}) = (t_{1}^{(1)}, …, t_{k′_{1}}^{(1)}), …, (u_{k′_{1}+...+k′_{n−1}+1}, …, u_{N′}) = (t_{1}^{(n)}, …, t_{k′_{n}}^{(n)}),
(u_{N′+1}, …, u_{N′+k′′_{1}}) = (t_{1}^{′(1)}, …, t_{k′′_{1}}^{′(1)}), …,
(u_{N′+k′′_{1}+...+k′′_{n−1}+1}, …, u_{N}) = (t_{1}^{′(n)}, …, t_{k′′_{n}}^{′(n)}),
t_{k′_{i}+j}^{(i)} = t_{j}^{′(i)}, j = 1, …, k′′_{i},
(t_{1}, …, t_{k_{1}}) = (t_{1}^{(1)}, …, t_{k_{1}}^{(1)}), …, (t_{k_{1}+...+k_{n−1}+1}, …, t_{N}) = (t_{1}^{(n)}, …, t_{k_{n}}^{(n)}),

and

∑_{α} P′_{α}(u_{1}, …, u_{N′})P′′_{α}(u_{N′+1}, …, u_{N})

= P (t_{1}, …, t_{N}) ∏_{1≤l≤N′,N′+1≤l′≤N} \frac{u_{l′} − u_{l}}{q^{⟨ϵ(l),ϵ(l′)⟩}u_{l′} − u_{l}} ,  (38)

for l in {1, …, N′}, resp. {N′ + 1, …, N}, ϵ(l) is the element of {1, …, n} such that u_{l} = t_{j}^{(ϵ(l))} , resp. u_{l} = t_{j}^{′(ϵ(l))} for some j; in (38), the ratios are expanded for u_{l} ≪ u_{l′}.

Proof. Define FO^{(2)} as the (ℕ^{r})^{2}-graded ℂ[[ℏ]]-module
FO^{(2)} = ⊕_{k,k′∈ℕ^{r}}FO_{k,k′}^{(2)},

where

FO_{k,k′}^{(2)} =

\frac{1}{∏_{1≤β≤k_{j}^{{i<j \atop 1≤α≤k_{i}}}}(t_{α}^{(i)} − t_{β}^{(j)}) ∏_{1≤β≤k′_{j}^{{i<j \atop 1≤α≤k′_{i}}}}(u_{α}^{(i)} − u_{β}^{(j)}) ∏_{1≤β≤k′_{j}^{{i,j \atop 1≤α≤k_{i}}}}(t_{α}^{(i)} − u_{β}^{(j)})} ·

·ℂ[[ℏ]][(t_{j}^{(i)})^{±1}, (u_{j′}^{(i)})^{±1}, j = 1, …, k_{i}, j′ = 1, …, k′_{i}]^{∏_{i=1}^{n} 𝔖_{k_{i}}×𝔖_{k′_{i}}},

where the groups 𝔖_{k_{i}} and 𝔖_{k′_{i}} act by permutation of the variables t_{j}^{(i)} and u_{j}^{(i)}. Define on FO^{(2)}, the graded composition map ∗ as follows: for f in FO_{k,k′}^{(2)} and g in FO_{l,l′}^{(2)},

(f ∗ g)(t_{1}, …, t_{P} , u_{1}, …, u_{P′}) = Sym_{t_{j}^{(1)}} ⋯ Sym_{t_{j}^{(n)}}Sym_{u_{j}^{(1)}} ⋯ Sym_{u_{j}^{(n)}}
(  ∏_{1≤i≤N,N+1≤j≤P} \frac{q^{⟨ϵ_{t}(i),ϵ_{t}(j)⟩}t_{i} − t_{j}}{t_{i} − t_{j}} ∏_{1≤i≤N′,N+1≤j≤P} \frac{q^{⟨ϵ_{u}(i),ϵ_{t}(j)⟩}u_{i} − t_{j}}{u_{i} − t_{j}}
∏_{1≤i≤N,N′+1≤j≤P′} \frac{q^{⟨ϵ_{t}(i),ϵ_{u}(j)⟩}t_{i} − u_{j}}{t_{i} − u_{j}} ∏_{1≤i≤N′,N′+1≤j≤P′} \frac{q^{⟨ϵ_{u}(i),ϵ_{u}(j)⟩}u_{i} − u_{j}}{u_{i} − u_{j}}
f(t_{1}, …, t_{N}, u_{1}, …, u_{M})g(t_{N+1}, …, t_{P} , u_{M+1}, …, u_{P})) ,

where N = ∑_{i} k_{i}, N′ = ∑_{i} k′_{i}, M = ∑_{i} l_{i}, M′ = ∑_{i} l′_{i}, P = N + M, P′ = N′ + M′, and (t_{1}, …, t_{k_{1}+k′_{1}}) = (t_{1}^{(1)}, …, t_{k_{1}+k′_{1}}^{(1)}), ..., (t_{k_{1}+···+k′_{n−1}+1}, …, t_{P}) = (t_{1}^{(n)}, …, t_{k_{n}+k′_{n}}^{(n)}), and (u_{1}, …, u_{l_{1}+l′_{1}}) = (u_{1}^{(1)}, …, u_{l_{1}+l′_{1}}^{(1)}), etc.,

(u_{l_{1}+···+l′_{n−1}+1}, …, u_{P′}) = (u_{1}^{(n)}, …, u_{l_{n}+l′_{n}}^{(n)}).

We set ϵ_{x}(α) = i if x_{α} = x_{j}^{(i)} for some j (x is t or u). It is easy to check that defines an algebra structure on FO^{(2)}.

Let Δ_{FO} be the linear map from FO to FO^{(2)}, which maps FO_{k} to ⊕_{k′+k′′=k}FO_{k′,k′′}^{(2)} as follows

Δ_{FO}(P)(t_{1}^{(1)}, …, t_{k′_{n}}^{(n)}, u_{1}^{(1)}, …, u_{k′′_{n}}^{(n)}) = P (t_{1}^{(1)}, …, t_{k′_{1}}^{(1)}, u_{1}^{(1)}, …, u_{k′′_{1}}^{(1)}, t_{1}^{(2)}, …, u_{k′′_{n}}^{(n)}).

Then it is immediate that Δ_{FO} defines an algebra morphism.

Consider the (ℕ^{n})^{2}-graded map

μ : FO^{(2)} → ∪_{a}lim_{←N}(FO^{≥a} ⊗_{ℂ[[ℏ]]} Ŝ^{L})/[FO_{N}^{≥a} ⊗_{ℂ[[ℏ]]} Ŝ^{L}]

defined by

μ(P) = ∑_{p_{1},...,p_{N′}≥0}(t_{1}^{p_{1}} ⋯ t_{N′}^{p_{N′}} P̄_{α}) ⊗ (P̄′_{α}K_{ϵ_{t}(1)}[−p_{1}] ⋯ K_{ϵ_{t}(N′)}[−p_{N′}])

if P (t_{1}, …, t_{N}, u_{1}, …, u_{N′}) belongs to FO_{k,k′}^{(2)} , and we set

P (t_{1}, …, u_{N′}) ∏_{1≤l≤N,1≤l′≤N′} \frac{t_{l} − u_{l′}}{q^{⟨ϵ_{t}(l),ϵ_{u}(l′)⟩}t_{l} − u_{l′}} = ∑_{α} P̄_{α}(t_{1}, …, t_{N})P̄′_{α}(u_{1}, …, u_{N′})

(the expansion is for u_{l′} ≪ t_{l}).

Let P, Q belong to FO_{k,l} and FO_{k′,l′}. μ(P)μ(Q) is equal to

∑_{α,β} (t_{1}^{p_{1}} ⋯ t_{N}^{p_{N}} P_{α}(t_{1}, ⋯, t_{N}) ⊗ P′_{α}(u_{1}, …, u_{P})K_{ϵ_{t}(1)}[−p_{1}] ⋯ K_{ϵ_{t}(N)}[−p_{N}]) ·(39)

· (t_{1}^{′p′_{1}} ⋯ t_{N′}^{′p′_{N′}}Q_{α}(t′_{1}, ⋯, t′_{N′}) ⊗ Q′_{α}(u′_{1}, …, u′_{P′})K_{ϵ_{t′}(1)}[−p′_{1}] ⋯ K_{ϵ_{t′}(1)}[−p′_{N′}]);

since

(∑_{i≥0} t^{i}K_{α}[−i])Q(u_{1}, …, u_{P}) = ∏_{i=1}^{P} \frac{u_{i} − q^{⟨ϵ_{u}(i),α⟩}t}{q^{⟨ϵ_{u}(i),α⟩}u_{i} − t}Q(u_{1}, …, u_{P})(∑_{i≥0} t^{i}K_{α}[−i]),

(39) is equal to

∑_{γ} ∑_{α,β} (f_{γ}(t_{1}, …, t_{N})t_{1}^{p_{1}} ⋯ t_{N}^{p_{N}} P_{α}(t_{1}, ⋯, t_{N}) ⊗ P′_{α}(u_{1}, …, u_{P})) ·  (40)

· (t_{1}^{′p′_{1}} ⋯ t_{N′}^{′p′_{N′}}Q_{α}(t′_{1}, ⋯, t′_{N′}) ⊗ g_{γ}(u′_{1}, …, u′_{P′})Q′_{α}(u′_{1}, …, u′_{P′})K_{ϵ_{t}(1)}[−p_{1}] ⋯

K_{ϵ_{t}(N)}[−p_{N}]K_{ϵ_{t′}(1)}[−p′_{1}] ⋯ K_{ϵ_{t′}(1)}[−p′_{N′}]),

with

∑_{γ} f_{γ}(t_{1}, …, t_{N})g_{γ}(u′_{1}, …, u′_{P′}) = ∏_{1≤i≤N,1≤j≤P′} \frac{u_{j} − q^{⟨ϵ_{u}(j),α⟩}t_{i}}{q^{⟨ϵ_{u}(j),α⟩}u_{j} − t_{i}} .

After some computation, one finds that (40) coincides with μ(P Q). Therefore μ is an algebra morphism.

It follows that the composition μ ∘ Δ_{FO} is an algebra morphism. This composition coincides with Δ_{S^{L}}, which is therefore an algebra morphism.

Remark 3.5. Δ_{S^{L}}^{k′,k′′} may also be expressed by the formula

Δ_{S^{L}}^{k′,k′′}(P)

= ∑_{p_{1},...,p_{N′}≥0} (∏_{i=1}^{N′} u_{i}^{p_{i}}P′′′_{α}(u_{1}, …, u_{N′})) ⊗ (∏_{i=1}^{N′} K_{ϵ(i)}[−p_{i}]P′′′′_{α}(u_{N′+1}, …, u_{N})) ,

where

∑_{α} P′′′_{α}(u_{1}, …, u_{N′})P′′′′_{α}(u_{N′+1}, …, u_{N})

= P (t_{1}, …, t_{N}) ∏_{1≤l≤N′,N′+1≤l′≤N} \frac{u_{l′} − u_{l}}{u_{l′} − q^{⟨ϵ(l),ϵ(l′)⟩}u_{l}} .

Corollary 3.1. There is a unique algebra morphism Δ_{V^{L}} from V^{L} to

lim_{←N}(V^{L} ⊗_{ℂ[[ℏ]]} V̂^{L})/(I_{N}^{(∞)} ⊗_{ℂ[[ℏ]]} V̂^{L}),

such that

Δ_{V^{L}}(h_{i}[k]^{V^{L}}) = h_{i}[k]^{V^{L}} ⊗ 1 + 1 ⊗ h_{i}[k]^{V^{L}}, 1 ≤ i ≤ n, k ≤ 0,

Δ_{V^{L}}(t_{i}^{k}) = ∑_{l≥0} t_{i}^{k+l} ⊗ K_{i}[−l] + 1 ⊗ t_{i}^{k} 1 ≤ i ≤ n, k ∈ ℤ.

Proof. For each i, Δ_{S^{L}} maps FO_{ϵ_{i}} to lim_{←N}(V^{L} ⊗_{ℂ[[ℏ]]} V̂^{L})/(I_{N}^{(∞)} ⊗_{ℂ[[ℏ]]} V̂^{L}), because I_{N,k}^{≥a} ∩ ⟨LV^{≥a}⟩ ⊂ I_{ϕ(k,n)}. It follows that Δ_{S^{L}} maps V^{L} to the same space. Call Δ_{V^{L}} the restriction of Δ_{S^{L}} to V^{L}. This restriction is clearly characterized by its values on h_{i}[k] and t_{i}^{k}.

•Construction of a Hopf algebra structure Recall that we showed in Lemma 3.3 that V^{L} is a free ℂ[[ℏ]]-module. Let us set V_{0}^{L} = V^{L}/ℏV^{L}.

Let \overline{I}_{N} be the image of the ideal I_{N}^{(∞)} of V^{L} by the projection from V^{L} to V_{0}^{L}. By Lemma 3.4, 2), and since the map from V^{L} to V_{0}^{L} is surjective, the ideals \overline{I}_{N} have property (∗), so that ∩_{N>0}\overline{I}_{N} is a two-sided ideal of V_{0}^{L}. Define W_{0} as the quotient algebra W_{0} = V_{0}^{L}/ ∩_{N>0} \overline{I}_{N}. We are going to define a Hopf algebra structure on W_{0}.

Since the V_{0}^{L} ⊗ \overline{I}_{N} have property (∗), lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}) has an algebra structure. Moreover, the projection

[lim_{←N}(V^{L} ⊗_{ℂ[[ℏ]]} V^{L})/(V^{L} ⊗_{ℂ[[ℏ]]} I_{N}^{(∞)})] ⊗_{ℂ[[ℏ]]} ℂ
→ lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}),

is an algebra isomorphism. Δ_{V^{L}} induces therefore an algebra morphism Δ_{V_{0}^{L}} from V_{0}^{L} to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}).

On the other hand, the \overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N} have property (∗), so that lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}) has an algebra structure. The composition of Δ_{V_{0}^{L}} with the projection

lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}) → lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N})

then yields an algebra morphism p′ ∘ Δ_{V_{L}^{0}} from V_{L}^{(0)} to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}).

We have for any k ≥ N, Δ_{V}(t_{i}^{k}) ∈ I_{N} ⊗ V̂ + V ⊗ Î_{N}, therefore Δ_{V^{L}}(I_{N}) ⊂ I_{N} ⊗_{ℂ[[ℏ]]} V̂ + V ⊗_{ℂ[[ℏ]]} Î_{N}, therefore Δ_{V_{0}^{L}}(\overline{I}_{N}) ⊂ \overline{I}_{N} ⊗ V_{0} + V_{0} ⊗ \overline{I}_{N}. It follows that p′ ∘ Δ_{V_{L}^{0}} maps the intersection ∩_{N>0}\overline{I}_{N} to the kernel of the projection lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}) → lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}).

We have then an algebra morphism Δ̃_{W_{0}} from W_{0} to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}).

In Proposition 1.3, we defined a surjective algebra morphism i_{ℏ} from U_{ℏ}L𝔟_{+} to V^{L}. It induces an algebra morphism i from UL𝔟_{+} to V_{0}^{L}, which is also surjective.

Let T be the free algebra generated by the h_{i}[k]^{(T)}, k ≤ 0, i = 1, …, n and e_{i}[k]^{(T)}, i = 1, …, n, k ∈ ℤ. We have a natural projection of T on UL𝔟_{+}, sending each x[k]^{(T)} to x ⊗ t^{k}; composing it with ι, we get a surjective algebra morphism π from T to V_{0}^{L}.

We have a unique algebra morphism Δ_{T} : T → T ⊗ T, such that

Δ_{T} (x_{i}[k]^{(T)}) = x_{i}[k]^{(T)} ⊗ 1 + 1 ⊗ x_{i}[k]^{(T)}.

Lemma 3.6. Let π_{V_{0}→W_{0}} be the natural projection from V_{0} to W_{0}. We have the identity

Δ̃_{W_{0}} ∘ π_{V_{0}→W_{0}} = (ν ∘ (π_{V_{0}→W_{0}} ⊗ π_{V_{0}→W_{0}})) ∘ Δ_{T} ,  (41)

where ν is the natural projection from (V_{0}^{L} ⊗ V_{0}^{L}) to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}).

Proof. The two sides are algebra morphisms from W_{0} to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}). The identity is satisfied on generators of W_{0}, therefore it is true.

Let J be the kernel of the projection π_{V_{0}→W_{0}} ∘ π from T to W_{0}. It follows from (41) that Δ_{T} (J) in contained in the kernel of ν ∘ (π ⊗ π), which is the preimage by π ⊗ π of Kerν; Kerν is equal to ∩_{N>0}(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}), which is (∩_{N>0}\overline{I}_{N}) ⊗ V_{0}^{L} + V_{0}^{L} ⊗ (∩_{N>0}\overline{I}_{N}). Therefore (π ⊗ π)^{−1}(Kerν) is J ⊗ T + T ⊗ J. We have shown that Δ_{T} (J) ⊂ J ⊗ T + T ⊗ J. We have shown:

Proposition 3.3. Δ_{T} induces a cocommutative Hopf algebra structure on T/J = W_{0}.

We will denote by Δ_{W_{0}} the coproduct induced by Δ_{T} on W_{0}.

•Compatibility of Δ_{W_{0}} with Δ_{V_{0}^{L}}

Recall that Δ_{V_{0}^{L}} is an algebra morphism from V_{0}^{L} to lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L}). Let us denote by Δ_{V_{0}^{L};N} the induced map from V_{0}^{L} to (V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L}). We have seen that for any integer p > 0, Δ_{V_{0}^{L};N}(\overline{I}_{p}) is contained in the image of \overline{I}_{p} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{p} by the projection map V_{0}^{L} ⊗ V_{0}^{L} → (V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L}). This image is

[(\overline{I}_{p} + \overline{I}_{N}) ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{p}]/(\overline{I}_{N} ⊗ V_{0}^{L}).

Therefore, Δ_{V_{0}^{L};N}(∩_{p>0}\overline{I}_{p}) is contained in the intersection of these spaces, which is

[\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ (∩_{p>0}\overline{I}_{p})]/(\overline{I}_{N} ⊗ V_{0}^{L}).
It follows that Δ_{V_{0}^{L};N} induces a linear map W_{0} → (W_{0} ⊗ W_{0})/(\overline{J}_{N} ⊗ W_{0}),

where \overline{J}_{N} is the image of \overline{I}_{N} by the projection map V_{0} → W_{0}.

It also induces an algebra morphism W_{0} → lim_{←N}(W_{0} ⊗ W_{0})/(\overline{J}_{N} ⊗ W_{0}). Then this algebra morphism factors through the coproduct map Δ_{W_{0}} de-

fined above. To check this, it is enough to check it on generators x[k] of W_{0}. It follows that Lemma 3.7. 1) i_{ℏ} induces a map i : UL𝔟_{+} → W_{0}, which is a surjective Hopf algebra morphism.

2) Let 𝔞_{L} be the Lie algebra of primitive elements of W_{0}. The restriction ι_{|L𝔟_{+}} of ι to L𝔟_{+} to 𝔞_{L} induces a surjective Lie algebra morphism.

Proof. Δ_{W_{0}} ∘ i and (i ⊗ i) ∘ Δ_{U𝔟_{+}} are both algebra morphisms from U𝔟_{+} to W_{0} ⊗ W_{0}. Their values on the x ⊗ t^{k} coincide, therefore they are equal. This shows 1).

2) follows directly from 1) and from Proposition 2.1.

•Construction of δ_{W_{0}} Define V̂^{L} ˆ⊗V̂^{L} as the tensor product

ℂ[X_{s}[0]^{V^{L}(1)}, X_{s}[0]^{V^{L}(2)}][[ℏ]] ⊗ ℂ[X_{s}[k]^{V^{L}(1)}, X_{s}[k]^{V^{L}(2)}, k > 0] ⊗ ⟨LV ⟩ ⊗_{ℂ[[ℏ]]} ⟨LV ⟩;

endow V̂^{L} ˆ⊗V̂^{L} with the unique ℏ-adically continuous algebra structure such that V^{L} ⊗_{ℂ[[ℏ]]} V^{L} → V^{L} ˆ⊗V^{L}, X_{s}[k]^{V^{L}(1)} ↦ X_{s}[k]^{V^{L}} ⊗ 1, X_{s}[k]^{V^{L}(2)} ↦ 1 ⊗ X_{s}[k]^{V^{L}}, 1⊗x⊗1 ↦ x⊗1, 1⊗1⊗x ↦ 1⊗x (where x is in ⟨LV ⟩) is an algebra morphism.

Define in the same way V̂^{L} ˆ⊗Î_{N}^{(∞)} as the tensor product

ℂ[X_{s}[k]^{V^{L}(1)}, X_{s}[k]^{V^{L}(2)}][[ℏ]] ⊗ ⟨LV ⟩ ⊗ I_{N}^{(∞)},

where the tensor products are over ℂ[[ℏ]]. Each V̂^{L} ˆ⊗Î_{N}^{(∞)} is then a left ideal of V̂^{L} ˆ⊗V^{L}.

Clearly, we have (V̂^{L} ˆ⊗V̂^{L})/ℏ(V̂^{L} ˆ⊗V̂^{L}) = V_{0}^{L} ⊗ V_{0}^{L}. Moreover,
[(V̂^{L} ˆ⊗V̂^{L})/(V̂^{L} ˆ⊗Î_{N}^{(∞)})]/ℏ[(V̂^{L} ˆ⊗V̂^{L})/(V̂^{L} ˆ⊗Î_{N}^{(∞)})] = (V_{0}^{L} ⊗ V_{0}^{L})/(V_{0}^{L} ⊗ \overline{I}_{N}).

Then Δ_{V^{L}} is an algebra morphism from V^{L} to lim_{←N}(V̂^{L} ˆ⊗V̂^{L})/(Î_{N}^{(∞)} ˆ⊗V̂^{L}). We again denote by Δ_{V^{L}} the composition of this map with the projection on lim_{←N}(V̂^{L} ˆ⊗V̂^{L})/(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)}). Define Δ′_{V^{L}} as Δ_{V}^{L} composed with the exchange of factors.

We have then

(Δ_{V^{L}} − Δ′_{V^{L}})(V^{L}) ⊂ lim_{←N}(ℏV̂^{L} ˆ⊗V̂^{L})/[(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)}) ∩ ℏV̂^{L} ˆ⊗V̂^{L}].

Since I_{N}^{(∞)} is divisible in V^{L}, we have

(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)}) ∩ ℏ(V̂^{L} ˆ⊗V̂^{L}) = ℏ(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)}),

so \frac{Δ_{V^{L}}−Δ′_{V^{L}}}{ℏ} is a linear map from V^{L} to

lim_{←N}(V̂^{L} ˆ⊗V̂^{L})/(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)}).

Define I_{N}^{(1)} as the image of Î_{N}^{(1)} in V_{0}^{L} by the projection V^{L} → V_{0}^{L}.

Let us set δ_{V_{0}^{L}} = \frac{Δ_{V^{L}}−Δ′_{V^{L}}}{ℏ} mod ℏ. Then δ_{V_{0}^{L}} is a linear map from V_{0}^{L} to

lim_{←N}(V_{0}^{L} ⊗ V_{0}^{L})/(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N}).

Moreover, \frac{Δ_{V^{L}}−Δ′_{V^{L}}}{ℏ} maps I_{N}^{(∞)} to the inverse limit

lim_{←M}(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)})/(Î_{M}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{M}^{(∞)}).

Therefore, δ_{V_{0}^{L}} maps \overline{I}_{N} to lim_{←M}(\overline{I}_{N} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{N})/(\overline{I}_{M} ⊗ V_{0}^{L} + V_{0}^{L} ⊗ \overline{I}_{M}). Therefore, δ_{V_{0}^{L}}(∩_{N} \overline{I}_{N}) is zero. It follows that δ_{V_{0}^{L}} induces a map δ_{W_{0}} from W_{0} to lim_{←N}(W_{0} ⊗ W_{0})/(\overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N}).

•Identities satisfied by δ_{W_{0}}

Lemma 3.8. δ_{W_{0}} satisfies

(Δ_{W_{0}} ⊗ id) ∘ δ_{W_{0}} = (δ_{W_{0}}^{2→23} + δ_{W_{0}}^{2→13}) ∘ Δ_{W_{0}},  (42)
Alt(δ_{W_{0}} ⊗ id) ∘ δ_{W_{0}}) = 0,  (43)
δ_{W_{0}}(xy) = δ_{W_{0}}(x)Δ_{W_{0}}(y) + Δ_{W_{0}}(x)δ_{W_{0}}(y) for x, y in W_{0},  (44)

where we use the notation of sect. 2.. The two first equalities are identities of maps from W_{0} to lim_{←N}W_{0}^{⊗3}/(\overline{J}_{N} ⊗ W_{0}^{⊗2} + W_{0} ⊗ \overline{J}_{N} ⊗ W_{0} + W_{0}^{⊗2} ⊗ \overline{J}_{N}).

Proof. Δ_{V^{L}} maps I_{N}^{(∞)} to

lim_{←M}(Î_{N}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{N}^{(∞)})/(Î_{M}^{(∞)} ˆ⊗V̂^{L} + V̂^{L} ˆ⊗Î_{M}^{(∞)}).

Therefore, (Δ_{V^{L}} ⊗id)⊗Δ_{V^{L}} and (id⊗Δ_{V^{L}})⊗Δ_{V^{L}} both define algebra morphisms from V^{L} to lim_{←N}(V̂^{L})^{ˆ⊗3}/[Î_{N}^{(∞)} ˆ⊗(V̂^{L})^{ˆ⊗2} + V̂^{L} ˆ⊗Î_{N}^{(∞)} ˆ⊗V̂^{L} + (V̂^{L})^{ˆ⊗2} ˆ⊗Î_{N}^{(∞)}]. These morphisms are the restrictions to V^{L} of (Δ_{S^{L}} ⊗ id) ⊗ Δ_{S^{L}} and (id ⊗ Δ_{S^{L}}) ⊗ Δ_{S^{L}}, which coincide, therefore they coincide.

The intersection ∩_{N>0}I_{N}^{(∞)} is a two-sided ideal of V^{L}. Define W as the quotient V^{L}/ ∩_{N>0} I_{N}^{(∞)}. Let J_{N} be the image of I_{N}^{(∞)} by the projection of V^{L} on W. Define Ŵ and Ĵ_{N} in the same way, replacing V^{L} and I_{N}^{(∞)} by V̂^{L} and Î_{N}^{(∞)}. Then Δ_{V^{L}} induces an algebra morphism

Δ_{W}: W → lim_{←N}(Ŵ ˆ⊗Ŵ)/(Ĵ_{N} ˆ⊗Ŵ + Ŵ ˆ⊗J_{N}).

Moreover, (Δ_{W} ⊗ id) ∘ Δ_{W} and (id ⊗ Δ_{W}) ∘ Δ_{W} define coinciding algebra morphisms from W to lim_{←N}(Ŵ^{ˆ⊗3})/[Ĵ_{N} ˆ⊗Ŵ^{ˆ⊗2} + Ŵ ˆ⊗J_{N} ˆ⊗Ŵ + Ŵ^{ˆ⊗2} ˆ⊗Ĵ_{N}]. Moreover, W is a free ℂ[[ℏ]]-module, and we have a topological Hopf algebra isomorphism of W/ℏW with W_{0}. The usual manipulations then imply the statements of the Lemma.

The identities of Lemma 3.8 are the topological versions of the co-Leibnitz, co-Jacobi and Hopf compatibility rules.

•Topological Lie bialgebra structure on 𝔞_{L} Define 𝔞_{L}^{(N)} as the intersection 𝔞_{L} ∩ \overline{J}_{N}.

Lemma 3.9. \overline{J}_{N} is the left ideal (U𝔞_{L})𝔞_{L}^{(N)} of W_{0} = U𝔞_{L}. Moreover, 𝔞_{L}^{(N)} is a Lie subalgebra of 𝔞_{L}.

Proof. J_{N} is a left ideal of W, therefore \overline{J}_{N} is a left ideal of W_{0}. Moreover, Δ_{W}(J_{N}) is contained in the inverse limit lim_{←M}(Ĵ_{N} ˆ⊗Ŵ + Ŵ ˆ⊗Ĵ_{N})/(Ĵ_{M} ˆ⊗Ŵ + Ŵ ˆ⊗Ĵ_{M}). It follows that Δ_{W_{0}}(\overline{J}_{N}) is contained in \overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N}.

The first statement of Lemma 3.9 now follows from Lemma 2.4. For x, y in 𝔞_{L}^{(N)}, [x, y] = xy − yx belongs to 𝔞_{L} and also to (U𝔞_{L})𝔞_{L}^{(N)}, so it belongs to 𝔞_{L}^{(N)}. Therefore 𝔞_{L}^{(N)} is a Lie subalgebra of 𝔞_{L}.

Lemma 3.10. 1) The restriction of δ_{W_{0}} to 𝔞_{L} defines a map δ_{𝔞_{L}} : 𝔞_{L} → lim_{←N}(𝔞_{L} ⊗ 𝔞_{L})/(𝔞_{L}^{(N)} ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(N)}).

2) For any element x of 𝔞_{L}, ad(x)(𝔞_{L}^{(N)}) in contained in 𝔞_{L}^{(N−k(x))}. The tensor square of the adjoint action therefore induces a 𝔞_{L}-module structure on lim_{←N}(𝔞_{L} ⊗ 𝔞_{L})/(𝔞_{L}^{(N)} ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(N)}). δ_{𝔞_{L}} is a 1-cocycle of 𝔞_{L} with values in this module.

3) We have δ_{𝔞_{L}}(𝔞_{L}^{(N)}) ⊂ lim_{←M}(𝔞_{L}^{(N)}⊗𝔞_{L}+𝔞_{L}⊗𝔞_{L}^{(N)})/(𝔞_{L}^{(M)}⊗𝔞_{L}+𝔞_{L}⊗𝔞_{L}^{(M)}). (δ_{𝔞_{L}} ∘ id) ∘ δ_{𝔞_{L}} therefore defines a map from 𝔞_{L} to lim_{←N}𝔞_{L}^{⊗3}/(𝔞_{L}^{(N)} ⊗ 𝔞_{L}^{⊗2} + 𝔞_{L} ⊗ 𝔞_{L}^{(N)} ⊗ 𝔞_{L} + 𝔞_{L}^{⊗2} ⊗ 𝔞_{L}^{(N)}). It satisfies the rule

Alt(δ_{𝔞_{L}} ⊗ id) ∘ δ_{𝔞_{L}} = 0.  (45)

Proof. Let us show 1). δ_{W_{0}} induces an map δ_{W_{0};N}: W_{0} → W_{0}^{⊗2}/(\overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N}). Let a belong to 𝔞_{L}. Let us write δ_{W_{0};N}(a) = ∑_{i} a_{i} ⊗ b_{i} mod \overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N}, with (a_{i})_{i} and (b_{i})_{i} finite families of W_{0} such that (b_{i} mod \overline{J}_{N})_{i} is a free family of W_{0}/\overline{J}_{N}. It follows from (42) that

∑_{i}(Δ_{W_{0}}(a_{i}) − a_{i} ⊗ 1 − 1 ⊗ a_{i}) ⊗ b_{i}

belongs to \overline{J}_{N} ⊗ W_{0}^{⊗2} + W_{0} ⊗ \overline{J}_{N} ⊗ W_{0} + W_{0}^{⊗2} ⊗ \overline{J}_{N}. Its image by the projection W_{0}^{⊗3} → [W_{0}^{⊗2}/(\overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N})] ⊗ [W_{0}/\overline{J}_{N}] its therefore zero. It follows that each a_{i} is such that Δ_{W_{0}}(a_{i})−a_{i} ⊗1−1⊗a_{i} belongs to \overline{J}_{N} ⊗W_{0} +W_{0} ⊗\overline{J}_{N}. Reasoning by induction on the degree of a_{i} (for the enveloping algebra filtration of W_{0}), we find that a_{i} belongs to 𝔞_{L}+\overline{J}_{N}. Therefore, δ_{W_{0};n}(a) belongs to the image of 𝔞_{L} ⊗ W_{0} in W_{0}^{⊗2}/(\overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ \overline{J}_{N}). Since δ_{W_{0};n}(a) is also antisymmetric, it belongs to the image of 𝔞_{L} ⊗ 𝔞_{L} in this space. This shows 1).

Let us show 2). For x en element of 𝔞_{L} and y an element of 𝔞_{L}^{(N)}, [x, y] = xy − yx belongs to \overline{J}_{N} + \overline{J}_{N−k(x)} = \overline{J}_{N−k(x)}; since it also belongs to 𝔞_{L}, [x, y] belongs to 𝔞_{L}^{(N)}. That δ_{𝔞_{L}} is a 1-cocycle then follows from (44).

Let us show 3). Δ_{W;M}(J_{N}) is contained in (J_{N} ⊗ W + W ⊗ J_{N})/(J_{M} ⊗ W + W ⊗ J_{M}). It follows that δ_{W_{0};M}(\overline{J}_{N}) is contained in (\overline{J}_{N} ⊗ W_{0} + W_{0} ⊗ øJ_{N})/(\overline{J}_{M} ⊗W_{0} +W_{0} ⊗øJ_{M}). Therefore, δ_{𝔞_{L}}(𝔞_{L}^{(N)}) is contained in lim_{←M}(𝔞_{L}^{(N)} ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(N)})/(𝔞_{L}^{(M)} ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(M)}). (45) in then a consequence of (43).

Define the restricted dual 𝔞_{L}^{⋆} of 𝔞_{L} as the subspace of 𝔞_{L}^{∗} composed of the forms ϕ on 𝔞_{L}, such for some N, ϕ vanishes on 𝔞_{L}^{(N)}.

Lemma 3.11. The dual map to δ_{𝔞_{L}} defines a Lie algebra structure on 𝔞_{L}^{⋆}.

Proof. Let ϕ, ψ belong to 𝔞_{L}^{⋆}. Let N be an integer such that ϕ, ψ vanish on 𝔞_{L}^{(N)}. For any integer M let δ̄_{𝔞_{L};M} be a lift to 𝔞_{L}^{⊗2} of the map δ_{𝔞_{L};M} from 𝔞_{L} to 𝔞_{L}^{⊗2}/(𝔞_{L}^{(M)}) ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(M)} induced by δ_{𝔞_{L}}.

Let x belong to 𝔞_{L}. Then M ≥ N, the number ⟨ϕ ⊗ ψ, δ̄_{𝔞_{L};M}(x)⟩ is independent of the lift δ̄_{𝔞_{L};M} and of M; it defines a linear form [ϕ, ψ] on 𝔞_{L}. The first statement of Lemma 3.10, 3), implies that [ϕ, ψ] actually belongs to 𝔞_{L}^{⋆}. It is clear that (ϕ, ψ) ↦ [ϕ, ψ] is linear and antisymmetric in ϕ and ψ. (45) implies that it satisfies the Jacobi identity.

•Topological Lie bialgebra structure on L𝔟_{+} Define for any integer N, (L𝔟_{+})^{(N)} as the Lie subalgebra of L𝔟_{+} generated by the x̄_{i}^{+} ⊗ t^{k}, k ≥ N, i = 1, …, n.

For any x in L𝔟_{+}, there exists an integer l(x) such that ad(x)((L𝔟_{+})^{(N)}) is contained in (L𝔟_{+})^{(N−l(x))}. It follows that lim_{←N}(L𝔟_{+})^{⊗2}/[(L𝔟_{+})^{(N)} ⊗ L𝔟_{+} + L𝔟_{+} ⊗ (L𝔟_{+})^{(N)}] and lim_{←N}(L𝔟_{+})^{⊗3}/[(L𝔟_{+})^{(N)} ⊗ L𝔟_{+}^{⊗2} + L𝔟_{+} ⊗ (L𝔟_{+})^{(N)} ⊗ L𝔟_{+} + L𝔟_{+}^{⊗2} ⊗ (L𝔟_{+})^{(N)}] have L𝔟_{+}-module structures.

Lemma 3.12. There is a unique map δ_{L𝔟_{+}} from L𝔟_{+} to

lim_{←N}(L𝔟_{+})^{⊗2}/[(L𝔟_{+})^{(N)} ⊗ L𝔟_{+} + L𝔟_{+} ⊗ (L𝔟_{+})^{(N)}],

such that δ_{L𝔟_{+}}(h̄_{i} ⊗ t^{k}) = 0 and

δ_{L𝔟_{+}}(x̄_{i}^{+} ⊗ t^{k}) = d_{i}Alt[\frac{1}{2}(h̄_{i} ⊗ 1) ⊗ (x̄_{i}^{+} ⊗ 1) + ∑_{l>0}(h̄_{i} ⊗ t^{−l}) ⊗ (x̄_{i}^{+} ⊗ t^{k+l})]

and δ_{L𝔟_{+}} is a 1-cocycle. Moreover, δ_{L𝔟_{+}} maps L𝔟_{+}^{(N)} to lim_{←M}(L𝔟_{+}^{(N)} ⊗ L𝔟_{+} + L𝔟_{+} ⊗ L𝔟_{+}^{(N)})/(L𝔟_{+}^{(M)} ⊗ L𝔟_{+} + L𝔟_{+} ⊗ L𝔟_{+}^{(M)}), and it satisfies the co-Jacobi identity Alt(δ_{L𝔟_{+}} ⊗ id) ∘ δ_{L𝔟_{+}} = 0.

Define the restricted dual (L𝔟_{+})^{⋆} to L𝔟_{+} as the subspace of (L𝔟_{+})^{∗} consisting of the forms on L𝔟_{+}, which vanish on some (L𝔟_{+})^{(N)}. The argument of Lemma 3.11 implies that δ_{L𝔟_{+}} induces a Lie algebra structure on (L𝔟_{+})^{⋆}.

Lemma 3.13. Define on 𝔤 ⊗ ℂ((t)), the pairing ⟨ , ⟩_{𝔤⊗ℂ((t))} as the tensor product of the invariant pairing on 𝔤 and ⟨f, g⟩ = res_{0}(fg \frac{dt}{t}). ⟨ , ⟩_{𝔤⊗ℂ((t))} an isomorphism of (L𝔟_{+})^{⋆} with the subalgebra L𝔟_{−} of L𝔤 defined as 𝔥 ⊗ ℂ[[t]] ⊕ 𝔫_{−} ⊗ ℂ((t)). This isomorphism is a Lie algebra antiisomorphism (that is, it is an isomorphism after we change the bracket of L𝔟_{−} into its opposite).

The map ι_{|L𝔟_{+}} defined in Lemma 3.7, 2), maps the generators of L𝔟_{+}^{(N)} to 𝔞_{L}^{(N)}; since 𝔞_{L}^{(N)} is a Lie subalgebra of 𝔞_{L} (Lemma 3.9), we have ι_{|L𝔟_{+}}(L𝔟_{+}^{(N)}) ⊂ 𝔞_{L}^{(N)}.

It follows that ι_{|L𝔟_{+}} induces a linear map ι^{⋆} from 𝔞_{L}^{⋆} to (L𝔟_{+})^{⋆} = L𝔟_{−}. Moreover, we have

δ_{𝔞_{L}} ∘ ι_{|L𝔟_{+}} = (ι_{L𝔟_{+}}^{⊗2}) ∘ δ_{L𝔟_{+}},  (46)

because both maps are 1-cocycles of L𝔟_{+} with values in lim_{←N}𝔞_{L}^{⊗2}/(𝔞_{L}^{(N)} ⊗ 𝔞_{L} + 𝔞_{L} ⊗ 𝔞_{L}^{(N)}), and coincide on the generators of L𝔟_{+}. Equation (46) then implies that ι^{⋆} : 𝔞_{L}^{⋆} → L𝔟_{−} is a Lie algebra morphism.

Let us set (L𝔟_{−})_{pol} = 𝔥 ⊗ ℂ[t^{−1}] ⊕ 𝔫_{−} ⊗ ℂ[t, t^{−1}]; (L𝔟_{−})_{pol} is the polynomial part of L𝔟_{−}.

Lemma 3.14. The image of ι^{⋆} contains (L𝔟_{−})_{pol}.

Proof. As we have seen, V^{L} is graded by ℕ^{n}. Each ideal I_{N}^{(∞)}is a graded ideal, so that W = V/ ∩_{N} I_{N}^{(∞)} is also graded by ℕ^{n}. Moreover, the degree 0 and ϵ_{i} components of I_{N}^{(∞)} are respectively 0 and ⊕_{k≥N}ℂ[[ℏ]][h_{i}[k]^{V^{L}}, k ≤ 0]t_{i}^{k}. Therefore, the components of ∩_{N}I_{N}^{(∞)} of degree 0 and ϵ_{i} are zero. The components of W of degrees 0 and ϵ_{i} are therefore respectively ℂ[[ℏ]][h_{i}[k], k ≤ 0] and ⊕_{l∈ℤ}ℂ[[ℏ]][h_{i}[k]^{V^{L}}, k ≤ 0]t_{i}^{l}.

W_{0} is also graded by ℕ^{n}, and its components of degrees 0 and ϵ_{i} are ℂ[h_{i}[k], k ≤ 0] and ⊕_{l∈ℤ}ℂ[h_{i}[k]^{V^{L}}, k ≤ 0]t_{i}^{l}.

The primitive part 𝔞_{L} of W_{0} is therefore also graded by ℕ^{n}, and the computation of Δ_{W_{0}} on W_{0}[0] and W_{0}[ϵ_{i}] shows that 𝔞_{L}[0] = ⊕_{1≤i≤n,k≤0}ℂh_{i}[k]^{V^{L}} and 𝔞_{L}[ϵ_{i}] = ⊕_{k∈ℤ}ℂt_{i}^{k}.

Define linear forms h_{i,k}^{∗} and e_{i,k}^{∗} on 𝔞_{L} by the rules that h_{i,k}^{∗} vanishes on ⊕_{α≠0}𝔞_{L}[α], and the restriction of h_{i,k}^{∗} to 𝔞_{L}[0] maps h_{j}[l]^{V^{L}} to δ_{ij}δ_{kl}; and e_{i,k}^{∗} vanishes on ⊕_{α≠ϵ_{i}}𝔞_{L}[α], and the restriction of e_{i,k}^{∗} to 𝔞_{L}[ϵ_{i}] maps t_{i}^{l} to δ_{kl}.

It follows from the computation of I_{N}^{(∞)}[0] and I_{N}^{(∞)}[ϵ_{i}] that the h_{i,k}^{∗} and e_{i,k}^{∗} vanish on all the \overline{J}_{N}, resp. on the \overline{J}_{N}, N ≥ k, adn therefore on all the 𝔞_{L}^{(N)}, resp. on the 𝔞_{L}^{(N)}, N ≥ k. It follows that the h_{i,k}^{∗} and e_{i,k}^{∗} actully belong to 𝔞_{L}^{⋆}.

Since the images of h̄_{i} ⊗ t^{k} and x̄_{i}^{+} ⊗ t^{k} by ι_{|L𝔟_{+}} are h_{i}[k]^{V^{L}} and t_{i}^{k}, the images of h_{i,k}^{∗} and e_{i,k}^{∗} by ι^{⋆} are the generators h̄_{i} ⊗ t^{k}, 1 ≤ i ≤ n, k ≥ 0 and x̄_{i}^{−} ⊗ t^{k}, 1 ≤ i ≤ n, k ∈ ℤ, of (L𝔟_{−})_{pol}.

The statement follows because ι^{⋆} is a Lie algebra morphism.

Lemma 3.14 implies that the kernel of ι_{|L𝔟_{+}} is contained in contains the annihilator of (L𝔟_{−})_{pol} in L𝔟_{+}. Since this annihilator is zero, ι_{|L𝔟_{+}} is injective. It follows that ι_{|L𝔟_{+}} is an isomorphism.

Therefore, ι : UL𝔟_{+} → W_{0} is also an isomorphism. Recall that ι was obtained from the surjective ℂ[[ℏ]]-modules morphism ι_{ℏ} = p ∘ i_{ℏ}, where p is the projection of V^{L} on W.

We now use:

Lemma 3.15. Let E and F be ℂ[[ℏ]]-modules, such that F is torsion-free and E is separated (i.e. ∩_{N>0}ℏ^{N}E = 0). Let π : E → F be a surjective morphism of ℂ[[ℏ]]-modules, such that the induced morphism π_{0} : E/ℏE → F/ℏF is an isomorphism of vector spaces. Then π is an isomorphism.

Proof. Let x belong to Kerp. π_{0}(x mod ℏ) is zero, therefore x belongs to ℏE. Set x = ℏx_{1}. ℏπ(x_{1}) is zero; since F is torsion-free, x_{1} belongs to Kerp. Therefore, Kerp ⊂ ℏKerp. It follows that Kerp ⊂ ∩_{N>0}ℏ^{N}E, so that Kerp = 0. It follows that π is an isomorphism.

Recall that U_{ℏ}L𝔫_{+} was defined as the quotient A/(∩_{N>0}ℏ^{N}A). It follows that U_{ℏ}L𝔫_{+} is separated. The above Lemma therefore shows that p ∘ i_{ℏ} is an isomorphism. Since p and i_{ℏ} are both surjective, they are both isomorphisms. Corollary 1.4 follows, together with ∩_{N>0}I_{N}^{(∞)} = 0 (from where also follows that ∩_{N>0}I_{N} = 0), and also, by Lemma 3.3, Theorem 1.3, 1).

It is then clear that the map U_{ℏ}L𝔫_{+} → U_{ℏ}L𝔫_{+}^{top} is injective and that U_{ℏ}L𝔫_{+}^{top} is the ℏ-adic completion of U_{ℏ}L𝔫_{+}. This proves Theorem 1.3, 2).

There is a unique algebra morphism ς from Ũ_{ℏ}L𝔫_{+} to U_{ℏ}L𝔫_{+}, which sends each e_{i}[k]_{A} to e_{i}[k]. As we have seen in Proposition 3.1, ς induces an isomorphism between Ũ_{ℏ}L𝔫_{+}/ℏŨ_{ℏ}L𝔫_{+} and U_{ℏ}L𝔫_{+}/ℏU_{ℏ}L𝔫_{+}. Moreover, Ũ_{ℏ}L𝔫_{+} is separated and by Theorem 1.3, 1), U_{ℏ}L𝔫_{+} is free. Lemma 3.15 then implies that ς is an isomorphism.

Remark 3.16. Let FO^{(0)} be the subspace of FO formed of the functions satisfying f(_{α}^{(i)}) = 0 when t_{1}^{(i)} = q_{d_{i}}^{2}t_{2}^{(i)} = ⋯ = q_{d_{i}}^{−2a_{ij}}t_{a_{ij}}^{(i)} = q_{d_{i}}^{−a_{ij}}t_{1}^{(j)} for any i, j. We showed in [11] that the image of U_{ℏ}L𝔫_{+} in FO is contained in FO^{(0)}[ℏ^{−1}]. It is natural to expect that this image is actually the subspace of FO^{(0)} consisting in the functions such that f(t_{1}, ⋯, t_{N}) = O(ℏ^{k}) whenever k out of the N variables t_{i} coincide.

3.2. Nondegeneracy of the pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{±}} (proof of Theorem 1.4).

Let us define T (LV) as the tensor algebra ⊕_{k≥0}(LV)^{⊗_{ℂ[[ℏ]]}k}, where LV = ⊕_{i=1}^{n}ℂ[[ℏ]][t_{i}, t_{i}^{−1}]. Denote in this algebra, the element t_{i}^{l} of LV as f_{i}[l]^{(T)}.

Define a pairing

⟨ , ⟩_{FO×T (LV)} : FO × T (LV) → ℂ((ℏ))

as follows: if P belongs to FO_{k},

⟨P, f_{i_{1}}[l_{1}]^{(T)} ⋯ f_{i_{N′}}[l_{N}]^{(T)}⟩_{FO×T (LV)}  (47)

= δ_{k,∑_{j=1}^{N} ϵ_{i_{j}}} res_{u_{N} =0} ⋯ res_{u_{1}=0}

(P (t_{1}, ⋯, t_{N}) ∏_{l<l′} \frac{u_{l′} − u_{l}}{q^{⟨ϵ_{i_{l′}} ,ϵ_{i_{l}}⟩}u_{l′} − u_{l}} u_{1}^{l_{1}} ⋯ u_{N}^{l_{N}} \frac{du_{1}}{u_{1}} ⋯ \frac{du_{N}}{u_{N}} ) ,

where we set as usual (t_{1}, …, t_{k_{1}}) = (t_{1}^{(1)}, …, t_{k_{1}}^{(1)}), etc.,

(t_{k_{1}+···+k_{n−1}+1}, …, t_{k_{1}+···+k_{n}}) = (t_{1}^{(n)}, …, t_{k_{n}}^{(n)}),

and u_{1} = t_{1}^{(i_{1})}, u_{2} = t_{1}^{(i_{2})} if i_{2} ≠ i_{1} and in general u_{s} = t_{ν_{s}+1}^{(i_{s})}, where ν_{s} is the number of indices t such that t < s and i_{t} = i_{s}.

Lemma 3.17. The pairing ⟨ , ⟩_{FO×T (LV)} verifies (T (LV))^{⊥} = 0.

Proof. Assume that the polynomial P of FO_{k} is such that (47) vanishes for any families of indices (i_{k}) and (l_{k}). Fix a family of indices (i_{k}) such that k = ∑_{j=1}^{N} ϵ_{i_{j}}. Since (47) vanishes for any family (l_{k}), the rational function P (t_{1}, ⋯, t_{N}) ∏_{l<l′} _{\frac{u_{l′}−u_{l}}{q^{⟨ϵ_{i_{l′}} ,ϵ_{i_{l}} ⟩}u_{l′}−u_{l}}} vanishes, therefore P is zero.

Let ⟨ , ⟩_{⟨LV ⟩×T (LV)} be the restriction of ⟨ , ⟩_{FO×T (LV)} to ⟨LV ⟩ × T (LV). Lemma 3.17 implies that T (LV)^{⊥} = 0 for this pairing. Using the isomorphism of Theorem 1.3 between ⟨LV ⟩ and U_{ℏ}L𝔫_{+}, we may view ⟨ , ⟩_{⟨LV ⟩×T (LV)} as a pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×T (LV)} between U_{ℏ}L𝔫_{+} and T (LV). So again T (LV)^{⊥} = 0 for ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×T (LV)}.

Let p be the quotient map from T (LV) to U_{ℏ}L𝔫_{+}. Composing

⟨ , ⟩_{U_{ℏ}L𝔫_{+}×T (LV)}

with p⊗id, we get a pairing ⟨ , ⟩_{T (LV)×T (LV)} between T (LV) and itself. It follows from (47) and (10) that ⟨ , ⟩_{T (LV)×T (LV)} is given by formula (11). Moreover, it follows from [11], Proposition 4.1 (relying on an identity of [18]) that ⟨ , ⟩_{T (LV)×T (LV)} induces a pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×U_{ℏ}L𝔫_{−}} between U_{ℏ}L𝔫_{+} and U_{ℏ}L𝔫_{−}.

Since ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×U_{ℏ}L𝔫_{−}} is induced by the pairing ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×T (LV)}, and the annihilator of T (LV)^{⊥} for this pairing is zero, we get that (U_{ℏ}L𝔫_{+})^{⊥} = 0 for ⟨ , ⟩_{U_{ℏ}L𝔫_{+}×U_{ℏ}L𝔫_{−}}. Exchanging the roles of U_{ℏ}L𝔫_{+} and U_{ℏ}L𝔫_{−}, we find that (U_{ℏ}L𝔫_{−})^{⊥} = 0. Theorem 1.4 follows.

Remark 3.18. This argument is completely similar to the proof of Theorem 1.2, the pairing between U_{ℏ}L𝔫_{+} and FO playing the role of the pairing beween U_{ℏ}𝔫_{+} and Sh(V).

3.3. The form of the R-matrix (proof of Proposition 1.4).

Let us define A_{+}^{a,b} as the subalgebra of U_{ℏ}L𝔫_{+} generated by the e_{i}[k], i = 1, …, n, a ≤ k ≤ b.

Lemma 3.19. A_{+}^{a,b} is a graded subalgebra of U_{ℏ}L𝔫_{+}. We have A_{+}^{a,b} + I_{≤a}^{+} + I_{≥b}^{+} = U_{ℏ}L𝔫_{+}. Moreover, the graded components of A_{+}^{a,b} are finite ℂ[[ℏ]]-modules.

Proof. Let us define A_{+}^{≤a} and A_{+}^{≥b} as the subalgebras of A_{+} generated by the e_{i}[k], k ≤ a (resp. k ≥ b). It follows from Theorem 1.3 that the product defines a surjective morphism from A_{+}^{≤a} ⊗ A_{+}^{a,b} ⊗ A_{+}^{≥b} to A_{+}. The Lemma follows.

Since I_{≥a}^{+}[α] + I_{≤b}^{+}[α] ⊂ (I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥⊥}, it follows from Lemma 3.19 that (I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥⊥} is a submodule of A_{+} with a complement of finite type. Moreover, this module is also divisible, so that A_{+}[α]/(I_{≥a}^{+}[α]+I_{≤b}^{+}[α])^{⊥⊥} is torsionfree. Since it is finitely generated, it follows that A_{+}[α]/(I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥⊥} is a free, finite-dimensional ℂ[[ℏ]]-module.

On the other hand, (I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥} is a submodule of

Hom_{ℂ[[ℏ]]}(A_{+}[α]/(I_{≥a}^{+}[α] + I_{≤b}^{+}[α]), ℂ[[ℏ]]),

and is therefore a ℂ[[ℏ]]-module of finite type. It is a submodule of A_{−}[−α], so it is torsion-free. It follows that (I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥} is also a free, finite-dimensional ℂ[[ℏ]]-module.

By construction, the pairing induced by ⟨ , ⟩_{U_{ℏ}L𝔫_{±}} between (I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥} and A_{+}[α]/(I_{≥a}^{+}[α] + I_{≤b}^{+}[α])^{⊥⊥} is nondegenerate.

The fact that P_{a,b}[α] defines an element of lim_{←a,b}A_{+}/(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥} ⊗_{ℂ[[ℏ]]} A_{−}[ℏ^{−1}] follows from the following fact: if F ⊂ G is an inclusion of finite dimensional vector spaces, and id_{F} and id_{G} are the identity elements of F ⊗ F^{∗} and G ⊗ G^{∗}, then their images in G ⊗ F^{∗} by the natural maps coincide.

Let x be any product of the f_{i}[k], with c ≤ k ≤ d. Then x is orthogonal to I_{≤−c}^{+} + I_{≥−d}^{+}. It follows that ∪_{a,b}(I_{≤a}^{+} + I_{≥b}^{+})^{⊥} = 0, therefore ∩_{a,b}(I_{≤a}^{+} + I_{≥b}^{+})^{⊥⊥} = 0.

The proof of Proposition 1.4 follows then the proof of Proposition 1.1.

Remark 3.20. We have in the 𝔰𝔩_{2} case

P = ∑_{r} ∑_{i_{1}<···<i_{r},n_{r}≥0} \frac{ℏ^{n_{1}+···+n_{r}}}{[n_{1}]_{q}^{!} ⋯ [n_{r}]_{q}^{!}} e_{i_{1}}^{n_{1}} ⋯ e_{i_{r}}^{n_{r}} ⊗ f_{−i_{1}}^{n_{1}} ⋯ f_{−i_{r}}^{n_{r}};

for r ≤ 2, this formula is shown in [7], App. B. It would be interesting to obtain analogous explicit formulas in Yangian or elliptic cases.

4. Toroidal algebras (proofs of Propsositions 1.5, 1.6)

4.1. Proof Proposition 1.5.

1) follows from the argument of the beginning of the proof of Proposition 3.1. The first statements of 2) are obvious.

The proof of Proposition 3.1 then implies that j_{+} induces a surjective Lie algebra morphism from F̃ to 𝔤 ⊗ ℂ[t, t^{−1}], which restricts to an isomorphim between ⊕_{α∈±Δ_{+},α real;k∈ℤ}F̃[(α, k)] and (⊕_{α∈±Δ_{+},α real}𝔤[α]) ⊗ ℂ[t, t^{−1}], which are the real roots part of both Lie algebras, and that F̃_{+}[(α, k)] = 0 if α does not belong to Δ_{+}.

It follows that Kerj_{+} is a graded subalgebra of F̃_{+}, contained in

⊕_{α∈Δ_{+},α imaginary,k∈ℤ}F̃_{+}[(α, k)].

4.2. Proof of Proposition 1.6.

Let us prove 1). Let us denote by Z(F̃_{+}) the center of F̃_{+}. Let us first prove that Kerj_{+} ⊂ Z(F̃_{+}). Let x belong to Kerj_{+}. We may assume that x is homogeneous of degree nδ. Then any nδ + α_{i}, which is a real root. [ẽ_{i}[k], x] is homogeneous of degree nδ + α_{i}, which is a real root. Since the restriction of j_{+} on the subspace of F̃ of degree nδ + α_{i} is injective, j_{+}([ẽ_{i}[k], x]) is nonzero unless [ẽ_{i}[k], x] is itself zero. But j_{+}([ẽ_{i}[k], x]) is equal to [j_{+}(ẽ_{i}[k]), j_{+}(x)], which is zero because j_{+}(x) = 0. Therefore, [ẽ_{i}[k], x] is equal to zero.

On the other hand, since j_{+} is surjective and the center of L𝔫_{+} is zero, Kerj_{+} = Z(F̃_{+}). This proves 1).

Let us prove 2). The argument used in the proof of 1) implies that Kerj_{+} is contained in the center Z(F̃) of F̃. In the same way, one proves that Kerj_{−} is contained in Z(F̃), therefore Kerj ⊂ Z(F̃). On the other hand, F̃ is perfect.

It follows that we have a surjective Lie algebra morphism j′ : 𝔱 → F̃, such that the composition 𝔱 → F̃ → L𝔤 is the natural projection of 𝔱 on L𝔤. Let j′_{+} be the restriction of j′ to 𝔱_{+}. For any i, k, we have j′_{+}(e_{i}[k]^{𝔱}) = ẽ_{i}[k]+k_{i,k}, with k_{i,k} in Ker(j). Let λ be any linear map from 𝔱_{+} to Ker(j), such that λ(e_{i}[k]^{𝔱}) = k_{i,k}. Set j̃′_{+} = j′_{+} − λ. Then j̃′_{+} is a Lie algebra map from 𝔱_{+} to F̃. Since j̃′_{+}(e_{i}[k]^{𝔱}) = ẽ_{i}[k] and the e_{i}[k]^{𝔱} generate 𝔱_{+}, the image of j̃′_{+} is contained in F̃_{+}.

Moreover, j̃′_{+} is graded, and it coincides with j′_{+} on the nonsimple roots subspace [𝔱_{+}, 𝔱_{+}] = ⊕_{α∈Δ_{+}\{ϵ_{i}}}𝔱_{+}[α] of 𝔱_{+}. It follows that the restrictions of j′ on [𝔱_{+}, 𝔱_{+}] and [𝔱_{−}, 𝔱_{−}] are graded.

Let us show that j̃′_{+} is surjective. Since the composition of j̃′_{+} with the projection j : F̃_{+} → L𝔫_{+} is the natural projection, it suffices to show that any element x of Kerj_{+} is contained in j̃′_{+}(𝔱_{+}). x belongs to the image of j′, so let us set x = j′(y), with y = y_{+} + y_{−} + y_{0}, y_{±} in [𝔱_{±}, 𝔱_{±}] and y_{0} in 𝔥_{𝔱} ⊕ ⊕_{i=1}^{n}𝔱_{+}[ϵ_{i}] ⊕ ⊕_{i=1}^{n}𝔱_{−}[−ϵ_{i}], where 𝔥_{𝔱} is the Cartan subalgebra of 𝔱 (defined as 𝔥[λ^{±1}] ⊕ Z_{0}, see Remark 4.3). Then j′(y_{±}) belong to [F̃_{±}, F̃_{±}] and j′(y_{0}) belongs to H̃ ⊕ ⊕_{i=1}^{n}F̃_{+}[ϵ_{i}] ⊕ ⊕_{i=1}^{n}F̃_{−}[−ϵ_{i}] ⊕ Ker(j). Moreover, the map from 𝔥_{𝔱} ⊕ ⊕_{i=1}^{n}𝔱_{+}[ϵ_{i}] ⊕ ⊕_{i=1}^{n}𝔱_{−}[−ϵ_{i}] to H̃ ⊕ ⊕_{i=1}^{n}F̃_{+}[ϵ_{i}] ⊕ ⊕_{i=1}^{n}F̃_{−}[−ϵ_{i}] induced by j′ is injective, therefore y_{0} = 0. It follows that y_{−} = 0 and x = j′(y_{+}) = j̃′_{+}(y_{+}), because j̃′_{+} coincides with j′ on [𝔱_{+}, 𝔱_{+}].

Lemma 4.1. 1) Assume that A is not of type A_{1}^{(1)}. There is a unique Lie algebra map j′′ from F̃_{+} to 𝔱_{+} such that j′′(ẽ_{i}[k]) = e_{i}[k]^{𝔱}, for any i = 0, …, n and k integer.

2) Assume that A is the Cartan matrix of type A_{1}^{(1)}. There is a unique Lie algebra map j′′ from F̃_{+} to 𝔱_{+}/ ⊕_{l∈ℤ} ℂK_{δ}[l] such that j′′(ẽ_{i}[k]) = e_{i}[k]^{𝔱}, for any i = 0, 1 and k integer.

Proof. One should just check that the defining relations of F̃_{+} are satisfied by the e_{i}[k]^{𝔱} (in the A_{1}^{(1)} case, by the images of e_{i}[k]^{𝔱} in 𝔱_{+}/ ⊕_{l∈ℤ} ℂK_{δ}[l]). This is the case when A is not of type A_{1}^{(1)}, because in that case we set e_{i} = x̄_{i} ⊗ λ^{δ_{i0}} and we always have ⟨x̄_{i}, x̄_{j}⟩𝔤̄ = 0 for i ≠ j.

If A is of type A_{1}^{(1)}, we have x_{0} = f̄⊗ λ, x_{1} = ē, therefore

[x_{0}[l], x_{1}[m]] = (−h̄λ ⊗ t^{l+m}, −mK_{δ}[l + m]),

so that [x_{0}[l + 1], x_{1}[m]] = [x_{0}[l], x_{1}[m + 1]] holds in 𝔱_{+}/ ⊕_{l∈ℤ} ℂK_{δ}[l].

Let us now prove Proposition 1.6, 2). The composition j̃′_{+}∘j′′ are Lie algebra maps from F̃_{+} → 𝔱_{+} → F̃_{+} (F̃_{+} → 𝔱_{+}/ ⊕_{k∈ℤ} ℂK_{δ}[k] → F̃_{+} in the A_{1}^{(1)} case), which map the generators ẽ_{i}[k] to themselves. Therefore, F̃_{+} can be viewed as a subalgebra of 𝔱_{+} (of 𝔱_{+}/ ⊕_{k∈ℤ} ℂK_{δ}[k] in the A_{1}^{(1)} case). This subalgebra contains the elements e_{i}[k]^{𝔱} of 𝔱_{+} (resp. of 𝔱_{+}/ ⊕_{k∈ℤ} ℂK_{δ}[k]). Since the Lie subalgebra of 𝔱_{+} generated by the e_{i}[k]^{𝔱} is 𝔱_{+} itself, the image of F̃_{+} is equal to 𝔱_{+} (resp. to 𝔱_{+}/ ⊕_{k∈ℤ} ℂK_{δ}[k]). This proves Proposition 1.6, 2).

Remark 4.2. Proposition 1.6, 1), can also be obtained using the presentation given in [23] of 𝔱. In this paper, one shows that 𝔱 is isomorphic to the algebra Ḟ with generators ė_{i}^{±}[k], ḣ_{i}[k] and ċ and relations ad(ė_{i}^{±}[0])^{1−a_{ij}}(ė_{j}^{±}[k]) = 0, [ė_{i}^{±}[k], ė_{i}^{±}[l]] = 0, [ḣ_{i}[k], ė_{j}^{±}[l]] = ±a_{ij}ė_{j}^{±}[k + l], [ė_{i}^{+}[k], ė_{j}^{−}[l]] = δ_{ij}ḣ_{i}[k + l] + iδ_{i+j,0}⟨e_{i}, f_{i}⟩𝔤̄c, [ḣ_{i}[k], ḣ_{j}[k]] = kδ_{k+l,0}⟨h_{i}, h_{j}⟩𝔤̄c, c central. It is then clear that there is a Lie algebra map from 𝔱 to F̃. On the other hand, the system of relations [ė_{i}^{+}[k], ė_{i}^{+}[l]] = ad(ė_{i}^{±}[0])^{1−a_{ij}}(ė_{j}^{±}[k]) = 0 is not a presentation of 𝔱_{+}, because the ideal generated by these relations is not preserved by the analogues of the Φ_{i,k}^{±} of the proof of Lemma 3.2.

Remark 4.3. Toroidal Manin triples. It is easy to define an extension of the Lie algebra 𝔱 with an invariant scalar product. Recall first ([23, 20]) that if 𝔤 is the central extension of the Lie algebra 𝔤̄[λ, λ^{−1}], 𝔱 is the universal central extension of 𝔤̄[λ^{±1}, μ^{±1}]. We have therefore

𝔱 = 𝔤̄[λ^{±1}, μ^{±1}] ⊕ Z(𝔱).

Z(𝔱) is isomorphic to Ω_{𝔸}^{1}/d𝔸, where 𝔸 = ℂ[λ^{±1}, μ^{±1}]. We have

Z(𝔱) = ⊕_{k,l∈ℤ}K_{kδ}[l] ⊕ ℂc,

with K_{kδ}[l] = the class of \frac{1}{k}λ^{k}μ^{l−1}dμ if k ≠ 0 , K_{0}[l] = the class of μ^{l} \frac{dλ}{λ}, c = the class of \frac{dμ}{μ}.

Define for k, l in ℤ, D̃_{kδ}[l] as the derivations of 𝔤̄[λ^{±1}, μ^{±1}] equal to λ^{k}μ^{l}(lλ∂_{λ} − kμ∂_{μ}) if k ≠ 0 and to μ^{l}λ∂_{λ} if k = 0, and d̃ as the derivation μ∂_{μ}.

Endow ℂ^{×2} with the coordinates (λ, μ) and consider on this space the Poisson structure defined by {λ, μ} = λμ. Let Ham(ℂ^{×2}) be the Lie algebra of Hamiltonian vector fields on ℂ^{×2} generated by the functions λ^{k}μ^{l}, k, l ∈ ℤ^{2}, log λ and log μ. For any function f on ℂ^{×2}, denote by V_{f} the corresponding Hamiltonian vector field. Then Ham(ℂ^{×2}) is a Lie algebra, and the map V_{λ^{k}μ^{l}} ↦ D̃_{kδ}[l], for (k, l) ≠ (0, 0), V_{1} ↦ 0, V_{log λ} ↦ D̃_{0}[0], V_{log μ} ↦ d̃, defines a Lie algebra map from Ham(ℂ^{×2}) to Der(𝔤̄[λ^{±1}, μ^{±1}]).

The formula V_{f}(∑_{i} a_{i}db_{i}) = ∑_{i}{f, a_{i}}db_{i} + a_{i}d{f, b_{i}} defines an action of Ham(ℂ^{×2}) on Ω_{𝔸}^{1}/d𝔸, that is on Z(𝔱). Define D̄_{kδ}[l] and d̄ as the following endomorphisms of 𝔱: D̄_{kδ}[l](x, 0) = (D̃_{kδ}[l](x), 0), d̄(x, 0) = (d̃(x), 0), and D̄_{kδ}[l](0, ω) = (0, V_{λ^{k}μ^{l}}(ω)) for (k, l) ≠ (0, 0), D̄_{0}[0](0, ω) = (0, V_{log λ}(ω)), d̄(0, ω) = (0, V_{log μ}(ω)). These endomorphisms again define derivations of 𝔱, and we have now a Lie algebra map from Ham(ℂ^{×2}) to Der(𝔱). Let 𝔱̃ be the corresponding crossed product Lie algebra of 𝔱 with Ham(ℂ^{×2}). We denote by D_{kδ}[l] and d the elements of t̃ implementing the extensions of the derivations D̄_{kδ}[l] and d̄ to 𝔱.

Define for a, b integers, x[a, b] as the element (x ⊗ λ^{a}μ^{b}) of 𝔤̄[λ^{±1}, μ^{±1}]. Define the bilinear form ⟨ , ⟩_{𝔱} by

⟨x[a, b], x′[a′, b′]⟩_{𝔱} = ⟨x, x′⟩𝔤̄δ_{a+a′,0}δ_{b+b′,0},

⟨D_{kδ}[l], K_{k′δ}[l′]⟩_{𝔱} = δ_{k+k′,0}δ_{l+l′,0}, ⟨d, c⟩_{𝔱} = 1,

and all other pairings of elements x[a, b], K_{kδ}[l], D_{k′δ}[l′], c and d are zero. Then ⟨ , ⟩_{𝔱} is an invariant nondegenerate bilinear form on 𝔱̃. Let us define D as the image of Ham(ℂ^{×2}) in 𝔱̃. Let us set D_{>}, D_{<} and

D_{0} as its subspaces ⊕_{k>0,l∈ℤ}ℂD_{kδ}[l], ⊕_{k<0,l∈ℤ}ℂD_{kδ}[l] and ⊕_{l∈ℤ}ℂD_{0}[l] ⊕ ℂd. We have then D = D_{>} ⊕ D_{0} ⊕ D_{<}. In the same way, define Z_{>}, Z_{<} and Z_{0} as the subspaces ⊕_{k>0,l∈ℤ}ℂK_{kδ}[l], ⊕_{k<0,l∈ℤ}ℂK_{kδ}[l] and ⊕_{l∈ℤ}ℂK_{0}[l] ⊕ ℂc of Z(𝔱). We have then Z(𝔱) = Z_{>} ⊕ Z_{<} ⊕ Z_{0}.

Recall we defined 𝔥_{𝔱} as the subalgebra 𝔥̄[λ^{±1}] ⊕ Z_{0} of 𝔱. 𝔥̃_{𝔱} = 𝔥_{𝔱} ⊕ D_{0} is then a Lie subalgebra of 𝔱̃. In the spirit of the new realizations, we split 𝔥̃_{𝔱} in two parts.

Let us set 𝔥_{+} = 𝔥̄[λ] ⊕ Z_{0}, 𝔥_{−} = 𝔥̄[λ^{−1}] ⊕ D_{0}; then 𝔥_{+} + 𝔥_{−} = 𝔥̃_{𝔱}, and 𝔥_{+} ∩ 𝔥_{−} is 𝔥̄.

Define L𝔫_{+} and L𝔫_{−} as the linear spans of the x[a, b], a ∈ ℤ, b > 0 (b ≥ 0 if x ∈ 𝔫̄_{+}), resp. of the x[a, b], a ∈ ℤ, b < 0 (b ≤ 0 if x ∈ 𝔫̄_{−}). L𝔫_{±} are Lie subalgebras of 𝔤̄[λ^{±1}, μ^{±1}]. L𝔫_{±} ⊕ Z_{±} and L̃𝔫_{±} = L𝔫_{±} ⊕ D_{±} ⊕ Z_{±} are also Lie subalgebras of 𝔱̃. Set 𝔱̃_{±} = L̃𝔫_{±} ⊕ 𝔥_{±}.

Endow 𝔱̃ × 𝔥̄ with the scalar product ⟨ , ⟩_{_{𝔱}×𝔥̄} defined by

⟨(x, h), (x′, h′)⟩_{_{𝔱}×𝔥̄} = ⟨x, x′⟩_{𝔱} − ⟨h, h′⟩𝔥̄.

Let p_{±} be the natural projection of 𝔱̃_{±} on 𝔥̄. Identify 𝔱̃_{±} as the Lie subalgebras of {(x, ±p_{±}(x)), x ∈ 𝔱̃_{±}} of 𝔱̃ × 𝔥̄[λ, λ^{−1}]. 𝔱̃_{±} are supplementary isotropic subspaces of 𝔱̃× 𝔥̄[λ, λ^{−1}] and define therefore a Manin triple. This Manin triple is a central and cocentral extension (by Z(𝔱) and D) of the Manin triple

(𝔤̄[λ^{±1}, μ^{±1}] × 𝔥̄, L𝔫_{+} ⊕ 𝔥̄[λ], L𝔫_{−} ⊕ 𝔥̄[λ^{−1}])

which is a part of the new realizations Manin triple (𝔤[μ^{±1}] × 𝔥, L𝔟_{+}, L𝔟_{−}). One may also consider “intermediate” Manin triples, for example

({𝔤̄[λ^{±1}, μ^{±1}] ⊕ Z_{>} ⊕ D_{<}} × 𝔥̄, L𝔫_{+} ⊕ 𝔥̄[λ] ⊕ Z_{>}, L𝔫_{−} ⊕ 𝔥̄[λ^{−1}] ⊕ D_{<}) .

It is a natural problem to quantize the corresponding Lie bialgebra structures on 𝔱̃_{±}. For this, one can think of the following program:

1) to compute the centers of U_{ℏ}L𝔫_{+} and (following [15]) the center of FO. By duality, these central elements should provide derivations of U_{ℏ}L𝔫_{−} (and FO) of imaginary degree. Compute these derivations and relations between them. One could expect that the algebra generated by the derivations is some difference analogue of the Lie algebra Ham(ℂ^{×2}).

2) it should then be easy, following Theorem 1.3, to prove that the analogue of i_{ℏ} is an isomorphism, and to derive from there the quantization of the Lie bialgebra L𝔟_{+}.

We hope to return to these questions elsewhere.

A Appendix: Lemmas on ℂ[[ℏ]]-modules

Lemma A1. Let E be a finitely generated ℂ[[ℏ]]-module. Let E_{tors} = {x ∈ E|ℏ^{k}x = 0 for some k > 0} be the torsion part of E. Then E_{tors} is isomorphic to a direct sum ⊕_{i=1}^{p}ℂ[[ℏ]]/(ℏ^{n_{i}}), where n_{i} are positive integers, and E is isomorphic to the direct sum of E_{tors} and a free module ℂ[[ℏ]]^{p′}.

Proof. As E is finitely generated, we have a surjective ℂ[[ℏ]]-modules morphism ℂ[[ℏ]]^{N} → E. Let K be the kernel of this morphism. Then E is isomorphic to ℂ[[ℏ]]^{N}/K.

Let us determine the form of K. Let us set K̄_{i} = K ∩ ℏ^{i}ℂ[[ℏ]]^{N}. Then we have ℏK̄_{i} ⊂ K̄_{i+1}. Let us set E_{0} = ℂ^{N}, and F_{i} = ℏ^{−i}K̄_{i} mod ℏ. Then we have F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ E_{0}. Let p the integer such that F_{k} = F_{p} for k ≥ p. We can then find a basis (v_{i})_{1≤i≤N} of E_{0} such that (v_{1}, …, v_{dimF_{0}}) is a basis of F_{0}, (v_{1}, …, v_{dimF_{1}}) is a basis of F_{1}, etc., (v_{1}, …, v_{dimF_{p}}) is a basis of F_{p}. Then K is the submodule ⊕_{i}(⊕_{k=dimF_{i−1}+1}^{dimF_{i}}ℏ^{i}ℂ[[ℏ]]v_{k}) of E_{0}[[ℏ]].

It follows that the quotient E_{0}[[ℏ]]/K is isomorphic to a direct sum

⊕_{i=1}^{p}ℂ[[ℏ]]/(ℏ^{n_{i}}) ⊕ ℂ[[ℏ]]^{p′}.

The statement of the Lemma follows.

Corollary A1. Any ℂ[[ℏ]]-submodule of a finite-dimensional free ℂ[[ℏ]]-mo-dule is free.

Proof. This follows from the fact that such a submodule has no torsion and from the above Lemma.

We have also

Lemma A2. Let E be a free ℂ[[ℏ]]-module with countable basis (v_{i})_{i≥0}. Any countably generated ℂ[[ℏ]]-submodule of E is free and has a countable basis.

Proof. We repeat the reasoning of the proof of Lemma A1. Let (w_{i})_{i≥0} be a countable family of E and let F be the sub-ℂ[[ℏ]]-module of E generated by the w_{i}.

Set F̄_{i} = F ∩ ℏ^{i}E and F_{i} = ℏ^{−i}F mod ℏ. Generating families and bases for the F_{i} can be constructed inductively as follows.

A generating family for F_{0} is (w_{i} mod ℏ)_{i≥0}. We can then construct by induction a partition of ℕ in subsets (i_{k}) and (j_{k}) such that (w_{i_{k}} mod ℏ)_{k≥0} is a basis of Span(w_{i} mod ℏ)_{i≥0}.

Let λ_{kk′} be the scalars such that w_{j_{k}} − ∑_{k′} λ_{kk′}w_{i_{k′}} belongs to ℏE. Set w_{k}^{(1)} = ℏ^{−1}[w_{j_{k}} − ∑_{k′} λ_{kk′}w_{i_{k′}}]. Then a generating family of F_{1} is (w_{i_{k}}, w_{k}^{(1)} mod ℏ). We then construct by induction a partition of ℕ in subsets (i_{k}^{(1)}) and (j_{k}^{(1)}) such that (w_{i_{k}}, w_{i_{k}^{(1)}}^{(1)} mod ℏ) is a basis of F_{1}.

It is clear how to continue this procedure. Then (w_{i_{k}}, w_{i_{k}^{(1)}}^{(1)}, w_{i_{k}^{(2)}}^{(2)} …) forms a basis of F.

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B. Enriquez IRMA and Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg, France enriquez@math.u-strasbg.fr

Received March 2, 2001 and in final form June 10, 2002