INVARIANTS IN THE ENVELOPING ALGEBRA OF A SEMISIMPLE LIE ALGEBRA FOR THE ADJOINT ACTION OF A NILPOTENT LIE SUBALGEBRA

Philippe CALDERO

ABSTRACT. Let 𝔤 be a complex semisimple Lie algebra and let S(𝔤), resp. U(𝔤), be its symmetric, resp. enveloping algebra. It is shown in [1] that finite W-algebras can be realized, up to a central extension, as the algebra of invariants S(𝔤)^{𝔪}, resp. U(𝔤)^{𝔪}, for the adjoint action of a Lie subalgebra 𝔪 of 𝔤. For 𝔪 nilpotent, we use the Taylor lemma to give a description (generators and relations) for these algebras.

0. Introduction.

Let 𝔤 be a complex semisimple Lie algebra with enveloping algebra U(𝔤). We fix a Cartan subalgebra 𝔥 of 𝔤. Let 𝔤 = 𝔫 ⊕ 𝔥 ⊕ 𝔫^{−} be a triangular decomposition of 𝔤.

Recall, cf. [7], [Kostant, unpublished], (see also remark 2.1 ), that the center of U(𝔫) is a polynomial algebra and let z be the product of its natural generators. Then, {z^{n}} is a Ore set and let U(𝔤)_{z} be the localized algebra. In [1], [2], a class of algebras called finite W -algebras are shown to be, up to central extension, invariant algebras of type U(𝔤)_{z}^{𝔪}, where 𝔪 is a Lie subalgebra of 𝔤. The aim of this article is to describe these algebras when 𝔪 is nilpotent. We may without loss of generality restrict ourself to the case where 𝔪 is a Lie subalgebra of 𝔫. The paper is organized as follows.

Let A be an associative algebra and 𝔪 a Lie algebra of locally finite derivations on A. Then, 𝔪 is a nilpotent Lie algebra and we may fix a central sequence for 𝔪. This sequence permits us to apply the Taylor Lemma by induction, cf. Proposition 1.1. Therefore, we can give a description of A as an extension of the algebra of invariants A^{𝔪} and a projection ϕ from A onto A^{𝔪}.

Let R^{+} be the set of positive roots relative to the triangular decomposition of 𝔤. We follow [7] for the construction of a maximal set of strongly orthogonal roots and an ordering in R^{+}, cf. 1.3. For each ad 𝔥-stable Lie subalgebra 𝔪 of 𝔫, we may define, cf. 2.3, a central sequence which is compatible with this ordering.

In the second section, we study the adjoint action of 𝔪 on the (localized) symmetric algebra S(𝔤)_{z}. This algebra being commutative, the projector ϕ is an algebra projector. Let 𝔪 be an almost saturated Lie algebra, cf. 2.2, then, up to central extension, S(𝔤)_{z}^{𝔪} is isomorphic to the symmetric algebra of a quotient Lie algebra of 𝔤, cf. Proposition 2.2. Notice that this isomorphism is not compatible with the Poisson-Kirillov structure of these algebras. Nevertheless, in the case where 𝔪 is saturated, cf. 2.2, we can give generators and relations for S(𝔤)_{z}^{𝔪}, cf. Theorem 2.4 and Remark 2.4.

In the third section, we first use the Taylor Lemma to describe U(𝔤)_{z} as an extension of U(𝔤)_{z}^{𝔫}, (see [3] for a quantum analog of these results). T. Levasseur pointed out to me that this method was used in [8] and [9] in the framework of the adjoint action of a nilpotent Lie algebra on its enveloping algebra U. Recall that in this case, the invariant algebra is in the center and U is a central extension of a Weyl algebra.

We then obtain generators and relations for U(𝔤)_{z}^{𝔫} (this can be done in the same way for a saturated Lie algebra). U(𝔤)_{z}^{𝔫} has a polynomial base and the degree of the relations is lower than 2. Moreover, U(𝔤)_{z}^{𝔫} has a triangular decomposition with a Cartan subalgebra of dimension rk 𝔤. The algebra U(𝔤)_{z}^{𝔫} is similar to an enveloping algebra of a semisimple Lie algebra. As in [5], we may define Verma modules, O-category and Shapovalov determinant. For 𝔤 = sl_{3}, U(𝔤)_{z}^{𝔫} belongs to a class of algebra similar to U(sl_{2}) defined by Smith, [10].

I am in debt to F. Barbarin, E. Ragoucy, and P. Sorba for introducing me to finite W-algebras and their connection with centralizers. I also want to thank T. Levasseur and O. Mathieu, for precious remarks and discussions.

I. Notations and preliminaries.

1.1. Let A be a ℂ-algebra and 𝔫 be a finite dimensional Lie algebra of locally nilpotent derivations on A, dim 𝔫 = N. Then, 𝔫 is a nilpotent Lie algebra and admits a central sequence

𝔫 = 𝔫_{1} ⊃ 𝔫_{2} ⊃ … ⊃ 𝔫_{N+1} = {0},

of ideals such that dim 𝔫_{i}/𝔫_{i+1} = 1. For i, 1 ≤ i ≤ N, fix X_{i} ∈ 𝔫_{i}, X_{i} ∉ 𝔫_{i+1}. Then, X_{i} acts on the invariant subalgebra A^{𝔫_{i+1}} and let S be the set of i such that this action is non trivial. Suppose that for each i in S, there exists a_{i} ∈ A^{𝔫_{i+1}} such that X_{i}a_{i} = 1. By convention, we set a_{i} = 0 for i ∉ S. Now, for i ∈ S, let’s define the endomorphism of A^{𝔫_{i+1}} by

ϕ_{i} := ∑_{n≥0} \frac{(−1)^{n}}{n!} a_{i}^{n}X_{i}^{n}.

Let (I) = (n_{i}) in ℕ^{S}. Set X_{(I)} :=∏_{→} X_{i}^{n_{i}} (for the natural ordering), a_{(I)} :=∏_{←} a_{i}^{n_{i}}, (for

the reverse ordering), and (I)! = ∏ n_{i}!. The following proposition is a revisited version of the Taylor lemma : Proposition. We have :

(i) A = ⨁_{(I)∈ℕ^{S}} a_{(I)}A^{𝔫},

(ii) ϕ :=∏_{→} ϕ_{i} is a projector from A to A^{𝔫} whose kernel is ⨁_{(I)≠ø} a_{(I)}A^{𝔫}

(iii) Id_{A} = ∑_{(I)∈ℕ^{S}} \frac{a_{(I)}}{(I)!} ϕX_{(I)}

(iv) if A is commutative then ϕ is an algebra morphism and A^{𝔫} ≃ A/(a_{i}, i ∈ S). ♢

1.2. Let 𝔤 be a semisimple complex Lie algebra and U := U(𝔤) its enveloping algebra. Let n be the rank of 𝔤. We fix a Cartan subalgebra 𝔥 of 𝔤. Let R, resp. R^{+}, denote the set of all, resp. positive, roots with Δ = {α_{i}, 1 ≤ i ≤ n} a simple system corresponding to R^{+}. Let 𝔤_{α}, α ∈ R, be the root subspace, and for each α in R^{+}, fix non zero elements x_{α} and y_{α} in, respectively, 𝔤_{α} and 𝔤_{−α}. Define h^{α} the element of 𝔥 corresponding to the root α. Set

𝔫 = ⨁_{α∈R^{+}} 𝔤_{α}, 𝔫^{−} = ⨁_{α∈R^{+}} 𝔤_{−α}.

Let P denote the associated weight lattice generated by the fundamental weights ϖ_{i}, 1 ≤ i ≤ n, and P^{+} := ∑_{i} ℕϖ_{i} the semigroup of dominant weights.

Let W be Weyl group associated with R. We denote by ( , ) the W -invariant form on P, as well as its corresponding form on 𝔥. As usual, we denote by ht(α) for the height of α and αˇ = 2\frac{α}{(α,α)} for all α in R. We have (α_{j}, ϖ_{i}) = δ_{ij} \frac{(α_{i},α_{i})}{2}. Set d_{i} = (α_{i}, ϖ_{i}). For each Lie algebra 𝔪 , let S(𝔪), resp. U(𝔪), denote the symmetric, resp. enveloping, algebra of 𝔪. S(𝔪) is endowed with the so-called Poisson-Kirillov structure defined through the gradation functor, [6, 2.8.7]. Let { , } be the Poisson-Kirillov bracket. As in [4, 4.2], let N_{α,β}, α, β ∈ R be the constant structures of 𝔤. In particular, N_{α,−α} = 1, α ∈ R^{+}, N_{−α,−β} = −N_{α,β}, α, β ∈ R, and N_{α,β} = 0 if α + β is not a root.

1.3. We now follow the construction of [7]. Let’s decompose R into simple components R_{i}, i ∈ ℕ with β_{i} the highest root of R_{i}. Then, the set of roots of R_{i} which are orthogonal to β_{i} is again a root system and let R_{ij}, j ∈ ℕ, be its simple components. This (finite) process defines, inductively, a subset K of ℕ ∪ ℕ^{2} ∪ …, and for each K in K a simple root system R_{K} with highest root β_{K}. Then, M := {β_{K}, K ∈ K} is a maximal set of strongly orthogonal roots. K admits a partial ordering through L ≤ K if L = (K, l_{1}, …, l_{t}), l_{i} ∈ ℕ. For K in K, we define the following sets

R_{K}^{+} = R^{+} ∩ R_{K}, Γ_{K} = {δ ∈ R_{K}, (δ, β_{K}) > 0}, Δ_{K} = Δ ∩ Γ_{K}, Γ_{K}^{o} = Γ_{K}\{β_{K}}.

The following lemma results from [7] : Lemma. We have :

(i) R^{+} is a disjoint union of the Γ_{K}, K ∈ K, (ii) let K ∈ K and α ∈ Γ_{K}^{o}, then (β_{K}, α) = \frac{1}{2}(β_{K}, β_{K}), and \overline{α} := β_{K} − α ∈ Γ_{K}^{o}, (iii) Let α ∈ Γ_{K}, δ ∈ R^{+}. If α + δ is a root, then it belongs to Γ_{L}, L ≥ K. If β_{K} + δ is

a root, then it belongs to Γ_{L}, L > K, (iv) Let α ∈ Γ_{K}, δ ∈ Γ_{L}^{o}, K < L, if δ − α is a root, then it belongs to Γ_{L}^{o}. ♢

For α in R^{+}, let K(α) denote the element of K such that α ∈ Δ_{K(α)}, cf. (i). Set K(i) = K(α_{i}).

II. Invariants in the symmetric algebra.

Let 𝔪 be an 𝔥-stable Lie subalgebra of 𝔫. Then, 𝔪 acts on S(𝔤) and U(𝔤) by the adjoint action. In order to describe the decomposition given by Proposition 1.1 (i), we first need to study the algebra S(𝔫)^{𝔫}. 2.1. Fix a lexicographic (total) ordering ≺ in R^{+} such that α ≺ β ⇒ (K(α), ht(α)) < (K(β), ht(β)). Set

𝔫_{⪰α} = ⊕_{δ⪰α}𝔫_{δ}, 𝔫_{≻α} = ⊕_{δ≻α}𝔫_{δ}.

By Lemma 1.3, 𝔫_{⪰α}, α ∈ R^{+} is a central sequence for 𝔫. Remark that, by Lemma 1.3 (iii).

(2.1.1)  ad y_{α}(x_{β_{K}} + S(𝔫_{≻β_{K}})) = N_{β_{K},−α}x\overline{α} + S(𝔫_{≻β_{K}}), K = K(α).

Now, let’s define inductively in an appropriate localization of S(𝔫)

(2.1.2)  z_{K} = (∏_{→} ϕ_{δ})y_{β_{K}} , for δ ∈ ⋃_{L>K} Γ_{L}^{o}, z_{>K} = ∏_{L>K} z_{L}, K ∈ K,

a_{α} = \frac{2}{(β_{K}, β_{K})}ad y_{α}(z_{K(α)})z_{K(α)}^{−1}, α ∈ R^{+}\M, K = K(α), z_{>K} = ∏_{L>K} z_{L}, K ∈ K.

As in 1.1, we can construct projectors ϕ_{α}, for α ∈ R^{+}\M and ϕ :=∏_{→} ϕ_{α}. This gives :

Proposition. Each z_{K}, K ∈ K is central in the localized Poisson-Kirillov-algebra S(𝔫)_{z_{>K}}. Let z be their product, then ϕ is an (algebra) projector from S(𝔫)_{z} onto S(𝔫)_{z}^{𝔫}. Proof. Notice first that if z_{K(α)} is ad𝔫-invariant, then, by Lemma 1.3 (ii) ad x_{α}(a_{α}) = 1. Moreover, a_{α} is 𝔫_{≻α}-invariant.

Now, by Proposition 1.1 (iv) and (2.1.1), we have by (inverse) induction on K

(2.1.3)  S(𝔫)_{z_{>K}}^{𝔫_{≻β_{K}}} ≃ S(𝔫)_{z_{>K}} /(a_{α}, α ∈ ⋃_{L>K} Γ_{L}^{o}) ≃ S(𝔫)_{z_{>K}} /(x\overline{α}, α ∈ ⋃_{L>K} Γ_{L}^{o}).

Indeed, suppose this is true for L > K. Then, by Lemma 1.3 (iii), the image of y_{β_{K}} is ad𝔫_{≻β_{K}} invariant in the third algebra of (2.1.3). As well, the adjoint action of y_{β_{K}} is trivial on this algebra. This proves the proposition. ♢

Remark. This implies that the localized algebra S(𝔫)_{z}^{𝔫} is isomorphic to the algebra of the group generated by M. [7, 4.12] is then a direct consequence of this fact.

2.2. For β ∈ M, set β = β_{K} and

(2.2.1)  a^{β} = \frac{1}{(β, β)}ad y_{β}(z_{K})z_{K}^{−1}.

By construction, ad x_{β}(a^{β}) = 1. Fix an 𝔥-stable Lie subalgebra 𝔪 of 𝔫 and let Ω(𝔪) be its set of weights. Let z := z(𝔪) be the product of the z_{K} such that K = K(α) for α in 𝔪. We now study the adjoint action of 𝔪 on S(𝔤)_{z}. We start with some definitions.

Definition. The Lie subalgebra 𝔪 above is called saturated if α ∈ Ω(𝔪) ⇒ Γ_{K(α)} ⊂ Ω(𝔪). It is called almost saturated, if it verifies the following property : let δ ∈ Γ_{L}, α ∈ Γ_{K} ∩ Ω(𝔪), L > K, with δ − α ∈ R^{+}, then \overline{δ − α} ∈ Ω(𝔪).

Note that 𝔫 is a saturated Lie algebra. A saturated Lie subalgebra of 𝔫 is clearly almost saturated. If 𝔪 is the nilradical of a parabolic subalgebra of 𝔤, then it is an almost saturated Lie subalgebra (but generally not saturated).

For α in R^{+}, with β = β_{K(α)}, set ε_{α} = N_{−α,β}. By [4, Theorem 4.2.1], we have ε_{α} = ±1 and ε\overline{α} = −ε_{α}. Set n_{α} = \frac{2}{(β,β)}ε_{α}.

Let’s start with the lemma : Lemma. Let α ∈ R^{+}\M, K = K(α) = K(β), and β ∈ M. We have :

(i) z_{K} = x_{β_{K}} + S(𝔫_{≻β_{K}})_{z_{>K}}

(ii) a_{α} = n_{α}x\overline{α}z_{K}^{−1} + S(𝔫_{≻β_{K}})_{z_{>K}} , a^{β} = −\frac{1}{(β, β)}h^{β}z_{K}^{−1} + S(𝔫_{≻β})_{z_{>K}} ,

(iii) S(𝔫_{≻β})_{z_{>K}} is an ad𝔫_{−α}-module, (iv) the algebra S(𝔫_{≻β})_{z_{>K}} is generated by a_{δ}, δ ∈ ⋃_{L>K} Γ_{L}^{o}, and z_{L}, L > K

(v) up to a multiplicative scalar,

ad x_{δ}(a_{α}) = \{ \begin{matrix}a_{α−δ} & if α − δ ∈ Γ_{K}, \\ 1 & if δ = α \\ 0 & if not. \end{matrix}

and ad x_{α}(a^{β}) = a\overline{α}

Proof. (i) follows from the definition of z_{K}. (ii), and (v) are consequences of (2.1.1), (2.1.2), (2.2.1). (iv) can be deduced, by induction, from (ii). (iii) is consequence of Lemma 1.3 (v). ♢

This leads to the following proposition. Proposition. Let 𝔪 be an almost saturated Lie subalgebra of 𝔫, then

S(𝔤)_{z}^{𝔪} ≃ S(𝔤)_{z}/(x\overline{α}, h^{β}; α ∈ Ω(𝔪)\M, β ∈ Ω(𝔪) ∩ M).

Proof. Let α ∈ Ω(𝔪)\M, K = K(α). ¿From (i) and (ii) of the previous lemma and Lemma 1.3 (iv), we have, up to a scalar a_{α} = x\overline{α} + I_{K}, where I_{K}, is the ideal in S(𝔫_{≻β_{K}})_{z_{>K}} generated by the x_{δ−α}, δ ∈ ⋃_{L>K} Γ_{L}^{o}. As 𝔪 is almost saturated, I_{K} is generated by elements x\overline{μ}, μ ∈ Ω(𝔪).

Now, by Proposition 1.1 (iv), S(𝔤)_{z}^{𝔪} ≃ S(𝔤)_{z}/(a_{α}, a^{β}; α, β ∈ Ω(𝔪)). This proves the proposition. ♢

Remark. This proposition shows that, up to a central extension, the invariant algebra S(𝔤)_{z}^{𝔪} is the algebra of regular functions on a (localized) subspace of 𝔤. In [1], x\overline{α}’s are the constraint conditions and the h^{β}’s are the symmetry fixing.

2.3. Fix a saturated Lie subalgebra 𝔪 and let m = dim 𝔪. ¿From Proposition 1.1 (i), we have the (ordered) decomposition S(𝔤)_{z} = ⨁_{(I)∈ℕ^{m}} a_{(I)}S(𝔤)_{z}^{𝔪}, where a_{(I)} is an ordered product of a_{α}’s and a^{β}’s. Before calculating the commutation relations inside S(𝔤)_{z}^{𝔪}, we first study the Poisson commutations in S(𝔤)_{z} involving the a_{α}’s and a^{β}’s. This is given by the following proposition.

Proposition. Let α, δ in Γ\M, K = K(α), and β, β′ in M. Then : (i)

\{a_{α}, a_{δ}\} = \{ \begin{matrix}n_{α}z_{K}^{−1} & if δ = \overline{α} \\ 0 & otherwise. \end{matrix} ,

(ii)

{a^{β}, a^{β′}} = 0,

\{a^{β}, a_{α}\} = \{ \begin{matrix}\frac{1}{2}a_{α}z_{K}^{−1} & if K(α) = K(β) \\ 0 & otherwise. \end{matrix}

(iii) \{a_{α}, −\}, resp. \{a^{β} + \frac{z_{K}^{−1}}{(β,β)}h^{β}), −\}, is zero on S(𝔤)_{z}^{𝔪}.

Proof. Suppose that K(α) is not smaller than K(δ). Then, up to a scalar, {a_{α}, a_{δ}} = \{a_{α}, x\overline{δ}\} + \{a_{α}, S(𝔫_{>K(δ)})_{z_{>K(δ)}}\} = 0, cf. Lemma 2.2 (ii), (v).

Now, by Lemma 1.3 (ii) if K = K(α) = K(δ), α − δ belongs to Γ_{K} ∪ {0} iff α = δ. Thus, by Lemma 2.2 (v), if \{a_{α}, x\overline{δ}\} is non zero, then α = \overline{δ}. This gives the ”otherwise” part of (i). Note that \{a_{α}, a\overline{α}\} = n_{α}z_{K}^{−1} by Lemma 2.2 (ii), (v). Thus, (i) holds. (ii) is proved in the same manner. As 𝔪 is saturated, S(𝔤)_{z}^{𝔪} PK-commutes with S(𝔫_{>K})_{z_{>K}}. This gives (iii). ♢

2.4. We can now give generators and relations for S(𝔤)_{z}^{𝔫}. Let ϕ denotes the (algebra) projector on S(𝔤)_{z}^{𝔫}.

Set (2.4.1)

w_{α} := ϕ(y_{α}), α ∈ R^{+}, w_{0}^{δ} := ϕ(h^{δ}), δ ∈ P, w_{−β} := ϕ(x_{β}) = z_{K(β)}, β ∈ M.

Fix a set Λ of independent weights generating the orthogonal of M in P. Then, by Proposition 1.1, we have the following natural generators of S(𝔤)_{z}^{𝔫} : w_{α}, α ∈ R^{+} ∪ −M, w_{0}^{δ}, δ ∈ Λ. Moreover, by Proposition 1.1 (i), (iv), they generate a polynomial base for the space S(𝔤)_{z}^{𝔫}. Now, we calculate generators of S(𝔤)_{z}^{𝔫} by using the Taylor formula ( Proposition 1.1 (iii)) at order 1.

Theorem. The Poisson bracket on S(𝔤)_{z}^{𝔫} is determined by

1) the w_{−β}, β ∈ M are central elements.

2)  {w_{0}^{δ}, w_{0}^{δ′}} = 0,  {w_{0}^{δ}, w_{α}} = −(δ, α)w_{α}, δ ∈ Λ, α ∈ R^{+},

3)  \{w_{α}, w_{α′}\} = N_{α′,α}w_{α+α′} + ∑_{K∈K, ν∈Γ_{K}∪\{0\}} \frac{M_{α,α′}^{ν}}{w_{−β_{K}}} w_{α−ν}w_{α′−ν̄},

with w_{α−ν} = w_{0}^{α}, if ν = α and w_{α′−ν̄} = w_{0}^{α′}, if ν̄ = α′. The constants being given by

M_{α,α′}^{ν} = −n_{ν}N_{ν,−α}N_{ν̄,−α′}, if ν ∈ Γ_{K}^{0},

M_{α,α′}^{β} = −\frac{N_{β_{K},−α}(β_{K}, α′)}{(β_{K}, β_{K})}, M_{α,α′}^{0} = \frac{N_{β_{K},−α′}(β_{K}, α)}{(β_{K}, β_{K})}.

Proof. We sketch the proof of the last formula. Let I denote the kernel of ϕ. We have, by Proposition 1.1 (iii) :

y_{α} = w_{α} + ∑ a_{ν}N_{ν,−α}w_{α−ν} + ∑ a^{β}N_{β,−α}w_{α−β} [I^{2}].

This formula, and the corresponding one for α′, give by Proposition 2.3 :

N_{α′,α}y_{α+α′} = \{y_{α}, y_{α′}\} = \{w_{α}, w_{α′}\} − ∑_{K∈K, ν∈Γ_{K}∪\{0\}} \frac{M_{α,α′}^{ν}}{w_{−β}} w_{α−ν}w_{α′−ν̄} [I].

Acting by ϕ on both sides yields the desired formula. ♢

Remark. These formulas can be easily generalized for a saturated Lie subalgebra of 𝔫. Recall that if 𝔪 is not saturated, Proposition 2.3 (iii) may not hold.

III. Nilpotent invariants in the enveloping algebra.

With the help of the Taylor Lemma, cf. Proposition 1.1, we can transpose Proposition 2.3 and 2.4 inside the enveloping algebra of 𝔤. 3.1. As in the previous section, Proposition 1.1, (2.1.2), (2.2.1), (2.4.1) enable us to define inside some localization of U(𝔤) the analog of ϕ, z_{K}, z_{>K}, K ∈ K, z, a_{α}, α ∈ R^{+}\M, a^{β}, β ∈ M, w_{α} α ∈ R^{+}, w_{0}^{δ}, δ ∈ P, w_{−β}, β ∈ M. Replacing the PKbracket by the usual bracket, the formulas of Proposition 2.3 remain exact in U(𝔤)_{z} (with the same proof).

Now, in order to formulate an analog of Proposition 2.4, we need to introduce some more notation. 3.2. ℕ^{R^{+}} ordered as in 2.1. Definition. Let I be the set of I in ℕ^{R^{+}} such that I_{α} = 0 except for α ∈ R^{+}\M and α ≺ \overline{α}. Let I ∈ I, define \overline{I} in ℕ^{R^{+}} by \overline{I}_{α} = I\overline{α}, α ∈ R^{+}\M, \overline{I}_{β} = 0, β ∈ M. For I ∈ I, B ∈ ℕ[M], let I^{B} ∈ ℕ^{R^{+}} be such that I_{α}^{B} = I_{α}, α ∈ R^{+}\M, I_{β} = B_{β}, β ∈ M. Define \overline{I}^{β} in the same way. For I ∈ ℕ^{R^{+}}, set

| I |= ∑_{α∈R^{+}} I_{α}, | I |_{K}= ∑_{α∈Γ_{K}^{o}} I_{α}, I! = ∏_{α∈R^{+}} I_{α}!, α_{I} = ∑ I_{α}α,

x_{I} =∏_{→} x_{α}^{I_{α}},  n_{I} = ∏ n_{α}^{I_{α}},  z_{K(I)} = ∏ z_{K(α)}^{I_{α}}.

Let a_{I} be as in 1.1. For α ∈ R^{+}, let N_{I,−α} be the constant such ad x_{I}(y_{α}) = N_{I,−α}y_{α−α_{I}}.

In order to understand the commutations inside U(𝔤)_{z}^{𝔫}, we need to compute the commutators [a_{I}w_{α}, a_{I′}w_{α′}] modulo Ker ϕ, cf Proposition 1.1 (ii). Lemma. Let I ∈ I, B ∈ ℕ[M], and let w_{λ} and w_{λ′} be elements in U(𝔤)_{z}^{𝔫} of weights λ and λ′ in P. Then :

ϕ[a_{I}w_{λ}, a\overline{I}w_{λ′}] = (−1)^{|I|+1}I!n_{I}z_{K(I)}^{−1}w_{λ′}w_{λ},

ϕ[a_{I^{B}}w_{λ}, a\overline{I}w_{λ′}] = (−1)^{|B|} ∏_{β∈M}(\frac{| I |_{K(β)}}{2} + \frac{(β, λ′)}{(β, β)})^{B_{β}}. z_{K(B)}^{−1}ϕ[a_{I}w_{λ}, a\overline{I}w_{λ′}],

ϕ[w_{λ}, a_{B}w_{λ′}] = (−1)^{|B|} ∏_{β∈M}(\frac{(β, λ′)}{(β, β)})^{B_{β}}. z_{K(B)}^{−1}w_{λ}w_{λ′}.

We can now state the analog of Theorem 2.4 for U(𝔤)^{𝔫}. Theorem. The algebra U(𝔤)_{z} has the following decomposition :

U(𝔤)_{z} = ⨁_{I∈ℕ^{R^{+}}} a_{I}U(𝔤)^{𝔫}.

Moreover, the algebra U(𝔤)_{z}^{𝔫} is generated by the w_{α}’s, α ∈ R^{+}, the w_{0}^{δ}’s, δ ∈ Λ and the central elements w_{−β}^{±1}, β ∈ M. The relations are :

{w_{0}^{δ}, w_{0}^{δ′}} = 0, {w_{0}^{δ}, w_{α}} = −(δ, α)w_{α}, δ ∈ Λ, α ∈ R^{+},

\{w_{α}, w_{α′}\} = N_{α′,α}w_{α+α′} − ∑_{I,I′} \frac{N_{I,−α}N_{I′,−α′}}{I!I′!} ϕ[a_{I}w_{α−α_{I}} , a_{I′}w_{α′−α_{I′}}],

where the non zero images of the commutators are given by the previous lemma. Proof. The first assertion follows from Proposition 1.1 (i). Now, let’s prove that the algebra U(𝔤)_{z}^{𝔫} is generated by the given elements. Let u, v ∈ U(𝔤)_{z}. Applying Proposition 1.1 (iii) to the equality Id(uv) = Id(u).Id(v), we obtain that ϕ(uv) is a linear combination of ϕ(ad x_{I}(u))ϕ(ad x_{J}(v)). The adjoint action of 𝔫 being nilpotent on U(𝔤)_{z}, we obtain the assertion. As in Theorem 2.4, the proofs of the commutations formulas are straightforward. ♢

Remark. In the above proof, we showed that ϕ(uv) is a combination of ϕ(ad x_{I}(u))ϕ(ad x_{J}(v)). This may be precised by a formula provided by Proposition 1.1 (iii) and the previous Lemma.

3.3. Set k = ℂ[w_{−β}^{±1}, β ∈ M] and W^{0} = R[w_{0}^{δ}, w_{β}, δ ∈ Λ, β ∈ M]. By Theorem 3.2, W^{0} is a commutative subalgebra of U(𝔤)_{z}^{𝔫} (this is an analog of the Cartan subalgebra but the w_{β} are not semisimple). Let W^{+}, resp. W^{−}, the k-space with polynomial base generated by the w_{α}’s, α ∈ R^{+}\M, α ≺ \overline{α}, resp. α ⪰ \overline{α}. We have the following decomposition W := U(𝔤)_{z}^{𝔫} = W^{+} ⊗_{k} W^{0} ⊗_{k} W^{−}. For λ in k^{M} × k^{Λ}, we can define the

Verma module M(λ) := W/ ∑_{α,β,δ} W w_{α} + W (w_{β} − λ_{β}) + W (w_{0}^{δ} − λ_{δ}), where w_{α} ∈ W^{+}.

The algebra U(𝔤)_{z}^{𝔫} is similar to an enveloping algebra of a semisimple Lie algebra. T. Levasseur pointed out to me that for 𝔤 = sl_{3}, cf. [1], [2], [5], it belongs to a class of algebra similar to U(sl_{2}) defined by Smith, [10].

REFERENCES

1. F. BARBARIN, E. RAGOUCY, P. SORBA. Non-polynomial realizations of Walgebras, to appear in Int. J. Mod. Phys. A.

2. J. de BOER and T. TJIN, Quantization and representation theory of finite W algebras, Comm. Math. Phys., 158, (1993), 485.

3. P. CALDERO. On the Gelfand-Kirillov conjecture for quantum algebras, in preparation.

4. R.W. CARTER. Simple groups of Lie type, Interscience, John Wiley, London 1972.

5. K. DE VOS, P. VAN DRIEL. The Kazhdan-Lusztig conjecture for finite W -algebras, preprint.

6. J. DIXMIER. Algèbres enveloppantes, Cahiers scientifiques n^{∘} 37, Gauthiers-Villars, Paris, 1974.

7. A. JOSEPH. A preparation theorem for the prime sprectrum of a semisimple Lie algebra, J. Algebra, 48, (1977), 241-289.

8. A. JOSEPH. A generalization of the Gelfand-Kirillov conjecture, Amer. J. Math., 99, (1977), 1151-1165.

9. Y. NOUAZÉ, P. GABRIEL. Idéaux premier de l’algèbre enveloppante d’une algèbre de Lie nilpotente, J. Algebra, 6, 77-99, (1967).

10. S.P. SMITH. A class of algebra similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc., 322, (1990), n^{0} 1, 285-314.

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

10