Lobachevskii Journal of Mathematics Vol. 4, 1999, 207–218 \begin{matrix}c \\ ◯ \end{matrix}D.P. Zhelobenko

D.P. Zhelobenko

ON FORMAL SERIES AND INFINITE PRODUCTS OVER LIE ALGEBRAS

(submitted by B.N.Shapukov)

Abstract. A brief survey of new methods for the study of nonstandard associative envelopes of Lie algebras is presented. Various extensions of the universal enveloping algebra U(𝔤) are considered, where 𝔤 is a symmetrizable Kac-Moody algebra. An elementary proof is given for describing the “extremal projector” over 𝔤 as an infinite product over U(𝔤). Certain applications to the theory of 𝔤-modules are discussed.

It is known that the theory of representations of Lie algebras can be completely reduced to the theory of U(𝔤)-modules, where U(𝔤) is the universal enveloping algebra of the Lie algebra 𝔤. The algebra U(𝔤) is an associative version of 𝔤, and any 𝔤-module V is naturally endowed with a U(𝔤)-module structure. However, for some aspects of the theory of U(𝔤)-modules, the use of certain nontrivial extensions of the algebra U(𝔤) is essential. For example, assume that the basic field is of characteristic 0. Then the algebra U(𝔤) has no zero divisors. This remarkable property has a negative aspect. Namely, the algebra U(𝔤) has no projection elements (idempotents), except 0 and 1. In contrast, for the case in which 𝔤 is reductive (dim 𝔤 < ∞), an extension F (𝔤) of U(𝔤) is defined in [3], with the property that F (𝔤) contains a large family (decomposition of unity) of projections. Namely, F (𝔤) consists of formal series (of special kind) over U(𝔤). This construction arises in [3,4] in the context of describing the so-called “extremal projector” p ∈ F (𝔤) projecting any F (𝔤)- module V onto its “extremal subspace” V^{e}, in the terminology of [3–10]. A similar construction is given in [2] for describing certain analogues of Laplace operators over U(𝔤). A general definition of F (𝔤) is proposed in [8, 9]. In [9], a generalization of F (𝔤) is considered for the class of Cartan type contragredient (associative) algebras. A study of formal series over U(𝔤) is closely related to the use of infinite products over U(𝔤). For example, in [6] a certain universal formula is proposed for describing the extremal projector p ∈ F (𝔤) as an infinite product generated by the Casimir element over U(𝔤). This formula holds for a wide class of symmetrizable Chevalley algebras (containing the family of Kac-Moody algebras). The aim of this paper is to give a brief survey of these methods. We consider, for simplicity, the case of Kac-Moody algebras (Section 1). In Section 2, we describe F (𝔤) (as well as similar algebras) as projective limits of quotient modules over U(𝔤). In Section 3, we give an elementary proof of the universal formula proposed in [6]. In Section 4, we discuss some problems of the theory of 𝔤-modules. A more detailed exposition on these questions may be found, for example, in [4, 6, 8]. See also [9] for a general scheme for Cartan type algebras.

The work is supported by the RFBR grant 98-01-00281.

1991 Mathematical Subject Classification. 17B35, 22E46, 17B37, 81R50.

Key words and phrases. Lie algebras, Kac-Moody algebras, enveloping algebras, quantum algebras, modules.

1. The algebra U(𝔤)

1.1. Let 𝔤 be a symmetrizable Kac-Moody algebra in the sense of [1], equipped with a Cartan subalgebra 𝔥 (dim 𝔥 < ∞) and Chevalley generators e_{i}, f_{i}(i ∈ I), where I is a finite set of indices. Correspondingly, U(𝔤) is the associative algebra over ℂ generated by the elements e_{i}, f_{i}(i ∈ I), h ∈ 𝔥, with the following three type of structure relations: (1) weight relations (over 𝔥), (2) commutation relations for e_{i}, f_{i}, namely

[e_{i}, f_{j}] = h_{i}δ_{ij},  (1.1)

where h_{i} ∈ 𝔥 and δ_{ij} is the Kronecker symbol, (3) the Serre relations for e_{i}, f_{i}(i ∈ I). The weight relations are of the form wht e_{i} = − wht f_{i} = α_{i} (i ∈ I), where the symbol wht x means the weight of x (with respect to ad𝔥), and α_{i} (i ∈ I) are the simple roots of the algebra 𝔤 (with respect to 𝔥). It is essential for us that the structure relations of U(𝔤) admit the ℤ-grading (resp. Q-grading) given by the rule: deg e_{i} = − deg f_{i} = 1, deg h = 0 (resp. wht e_{i} = − wht f_{i} = α_{i}, wht h = 0), for i ∈ I, h ∈ 𝔥, where Q = Σ_{i}ℤα_{i} (the weight lattice in 𝔥^{∗}). Consequently, the algebra U(𝔤) admits the corresponding gradings:

U(𝔤) = ⨁_{n∈ℤ}U^{n}(𝔤) = ⨁_{μ∈Q} U_{μ}(𝔤),  (1.2)

where U^{n}(𝔤) (resp. U_{μ}(𝔤)) is the set of x ∈ U(𝔤) satisfying deg x = n (resp. wht x = μ). Recall that the set α_{i}(i ∈ I) is linearly independent. Hence U_{μ}(𝔤) ⊂ U^{d(μ)}(𝔤), where d(μ) = ∑_{i} μ_{i} for μ = ∑_{i} μ_{i}α_{i}. Recall that the relation wht x = μ means [h, x] = μ(h) x for any h ∈ 𝔥. The last condition may be written as follows:

hx = xh_{μ} for any h ∈ U(𝔥),  (1.3)

where h ↦ h_{μ} is an automorphism of U(𝔥) given by the rule h_{μ} = h+μ(h)1 for generators h ∈ 𝔥. It is convenient to use the canonical identification U(𝔥) = P (𝔥^{∗}), where P (𝔥^{∗}) means the polynomial algebra over 𝔥^{∗}. In these terms, the automorphism h ↦ h_{μ} is given by

h_{μ}(λ) = h(λ + μ),  (1.4)

for any λ ∈ 𝔥.

1.2. Let us describe the well-known contragredient structure of the algebra U(𝔤). First, we have 𝔤 = 𝔣 ⊕ 𝔥 ⊕ 𝔢, where 𝔢 (resp. 𝔥) is the subalgebra of 𝔤 generated by the elements e_{i} (resp. f_{i}). Hence we obtain, from the classical Poincaré-Birkhoff-Witt (PBW) theorem, the following free “triangular” decomposition:

U(𝔤) = U(𝔥)U(𝔣)U(𝔢).  (1.5)

Consequently, U(𝔤) is a free left U(𝔥)-module with a base A×B, where A (resp. B) is a basis of the vector space U(𝔣) (resp. U(𝔢)). Moreover, any permutation of the factors in (1.5) is admissible. In particular, U(𝔤) is a free right U(𝔥)- module with a base A × B. Setting A = {a_{i}}, B = {b_{i}} for i ∈ ℤ_{+} ,we obtain the following unique decomposition for the elements of U(𝔤):

p = Σ_{i,j}p_{ij}a_{i}b_{j}  (1.6)

with coefficients p_{ij} ∈ U(𝔥). We shall treat A (resp. B) as a ℤ-graded basis with homogeneous components A_{n} (resp. B_{n}). Note that A_{n} = A ∩ 𝔣_{n}, B_{n} = B ∩ 𝔢_{n}, where 𝔣_{n} = 𝔣_{1}^{n} (resp.𝔢_{n} = 𝔢_{1}^{n}) is the nth associative power of the linear hull 𝔣_{1} (resp. 𝔢_{1}) of the Chevalley generators f_{i} (resp. e_{i}). The coefficient p_{00} in (1.6) is called the free member of the element p. Similarly, one may choose A, B to be weighted (or Q-graded). For example, this condition is fulfilled for any PBW basis A (resp. B) consisting of ordered monomials of weight basic elements in 𝔣 (resp. 𝔢).

1.3. Let R(𝔥) = Fract U(𝔥) be the field of fractions of the (commutative) algebra U(𝔥). Setting

U′(𝔤) = R(𝔥) ⊗_{U(𝔥)} U(𝔤)  (1.7)

and using (1.3) for h ∈ R(𝔥), we equip U′(𝔤) with the structure of an algebra over ℂ (an extension of the algebra U(𝔤)). It is natural to consider U′(𝔤) as a set of elements (1.6) with coefficients p_{ij} ∈ R(𝔥). In particular, U′(𝔤) is a free left (resp. right) R(𝔥)-module. In terms of [8], the algebra U′(𝔤) is called the rational hull (with respect to U(𝔥)) of the algebra U(𝔤).

1.4. The algebra U(𝔤) (resp. U′(𝔤)) admits the involutive antiautomorphism (the Chevalley involution) x ↦ x′ defined on the generators by the rule e′_{i} = f_{i}, f′_{i} = e_{i}, h′ = h (i ∈ I, h ∈ 𝔥). In particular, we set h′ = h (h ∈ R(𝔥)) in the case U′(𝔤).

2. The algebra F (𝔤)

2.1. Starting from the decreasing set of left ideals J_{n} = U(𝔤)𝔢_{n}, we define Π_{n} = U(𝔤)/J_{n+1}. The projective limit

Π(𝔤) = lim_{n} Π_{n}  (2.1)

is defined to be the set of all regular sequences p = (p_{n}), for p_{n} ∈ Π_{n}. The regularity means that p_{n} = π_{nm}p_{m}(n ≤ m), where π_{nm} : Π_{m} → Π_{n} is the canonical projection. The set Π(𝔤) is naturally equipped with a structure of left 𝔤-module. Moreover, Π(𝔤) may be naturally endowed with a structure of right 𝔤-module (Section 2.4). Hence Π(𝔤) admits a structure of 𝔤-bimodule. Viewing Π_{n} as a subset in Π_{m} for n < m, we can write the elements p ∈ Π(𝔤) as formal series

p = Σ_{n}δp_{n},  (2.2)

where δp_{n} = p_{n} − p_{n−1} (p_{−1} = 0). To be exact, let us note that U(𝔢) = E_{n} ⊕ U(𝔢) 𝔢_{n+1}, where E_{n} is the sum of the components 𝔢_{k}(0 ≤ k ≤ n). Using the triangular decomposition (1.5), we find

Π_{n} ≃ U(𝔥)U(𝔣)E_{n},  (2.3)

for any n ∈ ℤ_{+}. Hence

Π_{n} = ⨁_{k=0}^{n} δΠ_{k},  (2.4)

where δΠ_{k} corresponds to the component 𝔢_{k} ⊂ E_{n}. Respectively, the series (2.2) may be interpreted as a formal sum of members δp_{n} ∈ δΠ_{n} (n ∈ ℤ_{+}). It is natural also to use the grading 𝔣_{n}(n ∈ ℤ_{+}) of U(𝔣). The element p ∈ Π(𝔤) becomes a double series of the form

p = Σ_{m,n}δp_{mn},  (2.5)

with members δp_{mn} ∈ δΠ_{mn}, where δΠ_{mn} corresponds to the component 𝔣_{m}𝔢_{n} (in U(𝔤)). Choosing any ℤ-graded basis A×B (Section 1.2), we may represent (2.5) as a formal series of the form (1.6).

2.2. Definition. Note that any ideal J_{n} is ℤ-graded with the components J_{n}^{k} = J_{n} ∩ U^{k}(𝔤). Correspondingly, Π_{n} is ℤ-graded with the components Π_{n}^{k} = U^{k}(𝔤)/J_{n+1}^{k}. Setting Φ^{k}(𝔤) = lim_{n} Π_{n}^{k} (as a subset of Π(𝔤)), we define Φ(𝔤) to be the direct sum of the subspaces Φ^{k}(𝔤):

Φ(𝔤) = ⨁_{k} Φ^{k}(𝔤).  (2.6)

Similarly, starting from the Q-gradings J_{nμ} = J_{n} ∩ U_{μ}(𝔤), Π_{nμ} = U_{μ}(𝔤)/J_{n+1,μ}, and setting Ψ_{μ}(𝔤) = lim Π_{nμ}, we define

Ψ(𝔤) = ⨁_{μ} Ψ_{μ}(𝔤).  (2.7)

Note that Ψ_{μ}(𝔤) ⊂ Φ^{d(μ)}(𝔤), hence Ψ(𝔤) ⊂ Φ(𝔤). As a result, we have

U(𝔤) ⊂ Ψ(𝔤) ⊂ Φ(𝔤),  (2.8)

where U(𝔤) is identified with a set of finite series in Π(𝔤). Moreover, this construction may be applied to the rational hull U′(𝔤) of U(𝔤) (Section 1.3). It is enough to replace J_{n}, Π_{n}, … by their rational hulls J′_{n}, Π′_{n}, . . .. As a result, we obtain the following analog of the chain (2.8):

U′(𝔤) ⊂ Ψ′(𝔤) ⊂ Φ′(𝔤).  (2.9)

In particular, we are interested in the case F (𝔤) = Ψ′(𝔤), in the notation of [3–8]. The corresponding Q-gradation is written as follows:

F (𝔤) = ⨁_{μ} F_{μ}(𝔤).  (2.10)

2.3. Theorem. Any of the spaces (2.8),(2.9) is an algebra, with respect to the multiplication of formal series (1.6) (or (2.2),(2.5)). Moreover, the algebras Φ(𝔤), Φ′(𝔤) (resp.Ψ(𝔤), Ψ′(𝔤)) are ℤ-graded (resp. Q-graded). This result can be proved quite easily. Starting from Φ(𝔤) (or Φ′(𝔤)), we reduce the question to the study of the following products:

A_{k}B_{l}A_{m}B_{n},  (2.11)

for fixed s = l−k, t = n−m. A contribution of (2.11) into A_{p}B_{q} is restricted by the rule k ≤ p, n ≤ q, hence l ≤ p+s, m ≤ q+t. We find that this contribution is finite and hence well defined. As a result, we obtain Φ^{s}(𝔤)Φ^{t}(𝔤) ⊂ Φ^{s+t}(𝔤). Passing to Ψ(𝔤), note that for any f, g ∈ Ψ(𝔤) we have fg ∈ Φ(𝔤). Using the weight rule (for homogeneous components of f, g), we obtain fg ∈ Φ(𝔤). Moreover, we have Φ_{μ}(𝔤)Φ_{ν}(𝔤) ⊂ Φ_{μ+ν}(𝔤). Note also that the Chevalley involution of U(𝔤) (Section 1.4) has a unique extension to any of the algebras (2.8), (2.9).

2.4. Remark. The language of projective limits is convenient to obtain an invariant description of F (𝔤), independently of the choice of the bases A, B. Another version of such description is given in [3] (see also Section 2.7).

Applying the arguments of Section 2.3 to the case Π(𝔤), one can easily obtain that Π(𝔤) also has a structure of an algebra, with respect to multiplication of formal series (2.2). However, this is not true for a set ∑(𝔤) of the formal series (1.6).

For example, let 𝔤 = 𝔰𝔩_{2}, with canonical basis e, f, h and relations [e, f] = h, [h, e] = 2e, [h, f] = −2f. Note that e^{n}f^{n} ≡ n!h(h − 1) ⋯ (h − n + 1) mod J_{1}. Then the product of elements x = Σ_{n}e^{n}, y = Σ_{n}f^{n} is not defined in ∑(𝔤) (also in ∑′(𝔤)).

2.5. Definition. A module V over U(𝔤) (or U′(𝔤)) is called locally nilpotent over 𝔢 if 𝔢_{n}x = 0 (n ≥ n_{0}(x)) for any x ∈ V. The category LN(𝔢) of such modules is closed with respect to the tensor operations (⊕, ⊗) and structure operations (passing to submodules and quotient modules). It is clear that the elements p ∈ Π(𝔤) (resp. p ∈ Π′(𝔤)) may be considered as operators for any module of category LN(𝔢) over U(𝔤) (resp. U′(𝔤)). Namely, we set px = p_{n}x(n ≥ n_{0}(x)) for any x ∈ V. In particular, the module V admits a structure of Φ(𝔤)-module (resp. Φ′(𝔤)-module). A standard example of U(𝔤)- modules of category LN(𝔢) is the highest weight (or extremal) U(𝔤)-module. An example for U′(𝔤) is given below.

2.6. Example. The universal Verma module V (𝔤) = U(𝔤)/J_{1} (resp. M(𝔤) = U′(𝔤)/J′_{1}) is a module of category LN(𝔢). Note that each of modules V = V (𝔤), M(𝔤) is generated by the single vector 1 (the image of 1 ∈ U(𝔤)). It is clear also that V admits a Q-grading V_{λ}(λ ∈ Q) with respect to the adjoint action of the algebra 𝔥. This action is well defined on V, since 𝔥 normalizes 𝔢. Moreover, V admits the ℤ_{+}-grading V_{n} generated (over U(𝔥) or R(𝔥)) by the components 𝔣_{n}1 (n ∈ ℤ_{+}).

2.7. Remark. Let us consider V = M(𝔤) as a right R(𝔥)-module. The weight condition wht a = μ for a ∈ EndV is defined by aV_{λ} ⊂ V_{λ+μ} for any λ ∈ Q. We set

E(𝔤) = ⨁_{μ} E_{μ}(𝔤),  (2.12)

where E_{μ}(𝔤) is the largest subspace in End_{R(𝔥)}V corresponding to the weight μ ∈ Q. It is clear that E(𝔤) is a Q-graded algebra with respect to the grading (2.12). In [4,8], it is shown that the action of F (𝔤) on M(𝔤) defines an isomorphism of graded algebras F (𝔤) ≃ E(𝔤). A similar description may be given for the algebra Φ(𝔤) (see [9]). In particular, M(𝔤) is an exact F (𝔤)-module.

3. Extremal projectors

3.1. Definition. Let z be a generalized Casimir element introduced in [1] as an operator on “bounded” 𝔤-modules. As is noted in [6], we may consider z as an element of F_{0}(𝔤):

z = z_{0} + z_{+},  (3.1)

where z_{+} ∈ F (𝔤)𝔢_{1} and z_{0} ∈ U(𝔥) is defined as the following quadratic element:

z_{0}(λ) = (λ, λ + 2ρ),  (3.2)

for λ ∈ 𝔥^{∗}. Here (.,.) is the canonical bilinear form in 𝔥^{∗} [1] and ρ ∈ 𝔥^{∗} is defined by the rule ρ(h_{i}) = 1 for any i ∈ I. Moreover, the element z ∈ F (𝔤) is central and z′ = z (with respect to the Chevalley involution of F (𝔤)).

Setting z_{ε}(λ) = z_{0}(λ + ε), we define the corresponding affine function a_{ε} = z_{ε} − z_{0}, i.e.,

a_{ε}(λ) = 2(λ + ρ, ε) + (ε, ε),  (3.3)

for ε ∈ Q_{+} := Σ_{i}ℤ_{+}α_{i}.

3.2. Theorem. There exists exactly one solution of the following system of equations on F (𝔤):

𝔢p = p𝔣 = 0, p_{00} = 1.  (3.4)

Moreover, we have p^{2} = p = p′. The solution of (3.4) may be given as an infinite product

p = ∏_{ε≠0}(1 − a_{ε}^{−1}z_{+}),  (3.5)

defined as a regular sequence p = (p_{n}) ∈ F_{0}(𝔤), where p_{n} = π_{n}p̄_{n}, p̄_{n} being a sequence of partial products in (3.5).

The result is proved in [6], by using the technique of induced modules. We present below an elementary proof (Sections 3.4 – 3.8). Another proof (Section 3.9) is based on the isomorphism F (𝔤) ≃ E(𝔤).

3.3. Corollary. The element (3.5) projects any F (𝔤)-module V onto its “extremal subspace” V^{e} = ker𝔢 parallel to im𝔣. In particular, we have:

V = ker𝔢 ⊕ im𝔣.  (3.6)

Here im𝔣 means the linear hull of subspaces imf (f ∈ 𝔣). Actually this result follows immediately from (3.4). See [8], for example. In the terminology of [8], an element p satisfying (3.4) is called an extremal projector of the algebra F (𝔤).

3.4. We set J_{n}^{⋆} = F (𝔤)𝔢_{n} (a series of left ideals of the algebra F (𝔤)), and set

X_{n} = ⨁_{k=0}^{n−1} 𝔣_{k}𝔢_{k}, Y_{n} = ˆ ⨁_{k≥n}𝔣_{k}𝔢_{k}.  (3.7)

Note that F_{0}(𝔤) is the direct sum of the subspaces R(𝔥)X_{n}, R(𝔥)Y_{n}. Below we will need the folowing relation:

ˆ⨁_{k≥n} 𝔣_{k}Y_{1}𝔢_{k} ⊂ Y_{k+1}.  (3.8)

Actually, (3.8) may be easily verified.

3.5. Lemma. Let π_{ε} = a_{ε} − z_{+} (ε ∈ Q_{+}), and let

π_{n} = ∏_{|ε|≤n} π_{ε},  (3.9)

where | ε | = d(ε) for ε ∈ Q_{+}. Then we have π_{n} ∈ J_{n+1}^{⋆} for any n ∈ ℤ_{+} such that lim_{n}π_{n} = 0, i.e.

π := ∏_{ε} π_{ε} = 0.  (3.10)

Proof. Setting Z = R(𝔥)[z], we define Z_{n} = ZY_{n}. Expose that π_{n} ∈ Z_{n+1}, hence π_{n} ∈ J_{n+1}^{⋆}. Actually, we have π_{0} = −z_{+} ∈ Y_{1}. In general, we have π_{n} = θ_{n}π_{n−1}(n ≥ 1), where θ_{n} is the product of the factors π_{ε} with | ε |= n. Note that π_{ε} ∈ Z, hence the multiplication on θ_{n} preserves Z_{n}. Moreover, using (1.3) and the fact that the element z is central, we find π_{ε}x = xπ_{ε+α} for any x ∈ U_{α}(𝔤). In particular, we have

π_{ε}U_{−ε}(𝔣) ⊂ U_{−ε}(𝔣)π_{0} ⊂ U_{−ε}(𝔣)Y_{1}.  (3.11)

Representing 𝔣_{n} as a sum of weight components U_{−ε}(𝔣) with | ε |= n, we may apply (3.11) to the corresponding factors π_{ε} of θ_{n}. Moreover, using (3.8), we obtain θ_{n}Z_{n} ⊂ Z_{n+1}. Applying this rule inductively to the relation π_{n−1} ∈ Z_{n}, we obtain π_{n} ∈ Z_{n+1} for any n ∈ ℤ_{+}.

3.6. Lemma. Setting p_{ε} = 1 − a_{ε}^{−1}z_{+}, we define

p̄_{n} = ∏_{1≤|ε|≤n} p_{ε},  (3.12)

for n ∈ ℤ_{+}. Then we have 𝔢_{1}p̄_{n} ∈ J_{n+1}^{⋆}, p̄_{n}𝔣_{1} ∈ J_{n}^{⋆} for any n ∈ ℤ_{+}. Proof. Note that p̄_{n} = λ_{n}π̄_{n}, where λ_{n} ∈ R(𝔥) and π̄_{n} is the product of π_{ε} with 1 ≤| ε | ≤ n. Using (1.3) for x = e_{i} (i ∈ I), we obtain

e_{i}π̄_{n} = π_{n}^{∗}e_{i},  (3.13)

where π_{n}^{∗} is the product of π_{δ} for δ = ε − α_{i}(1 ≤| ε |≤ n). It is clear that this product contains any factor π_{δ} for | δ |= n − 1, hence π_{n}^{∗} is a multiple of π_{n−1}. Using Lemma 3.5, we obtain e_{i}π̄_{n} ∈ J_{n+1}^{⋆}. Similarly, we have

π̄_{n}f_{i} = f_{i}π_{n}^{∗},  (3.14)

and the preceding arguments show that π̄_{n}f_{i} ∈ J_{n}^{⋆}.

3.7. Proof of Theorem 3.2. According to Lemma 3.6, we obtain p̄_{n+1} ∈ (1 + J_{1}^{⋆})p̄_{n} ≡ p̄_{n} mod J_{n+1}^{⋆}, hence the element p = (p_{n}) ∈ F_{0}(𝔤) with p_{n} = π_{n}p̄_{n} is well defined. Replacing p by the elements of 𝔢_{1}p, p𝔣_{1} and using Lemma 3.6, we obtain (3.4). The conditon p_{00} = 1 is immediate from (3.5). Moreover, we have p − 1 ∈ J_{1}^{∗}, (p − 1)p = 0, hence p^{2} = p. Finally, using the rule z′_{+} = z_{+}, we obtain p′ = p. Assuming that q ∈ F (𝔤) satisfies (3.4), we also obtain (p − 1)q = 0, hence q = pq. On the other hand, we have q − 1 ∈ 𝔣F (𝔤), hence p(q − 1) = 0 and p = pq. As a result, we obtain q = p (the uniqueness of solution of (3.4)).

3.8. Remark. It is easy to see that any central element z ∈ F (𝔤) satisfies a “characteristic equation” of the form (3.10). Moreover, assuming z_{ε} ≠ z_{0} for ε ≠ 0, we obtain for z an analog of decomposition (3.5).

3.9. Remark. Another proof of Theorem 3.2 may be given by direct calculation of (3.5) as an operator on M(𝔤). Consider the ℤ_{+}-grading M_{n} = R(𝔥)𝔣_{n}1 of M(𝔤), and note that M_{0} = R(𝔥)1, 𝔢M_{0} = 0. Repeating the argument of Section 3.6, we find p̄_{n}𝔣_{k} ⊂ J_{n−k+1}^{⋆}(n ≥ k ≠ 0), hence p̄_{n}M_{k} = 0. Note also that p̄_{n}1 = 1. Therefore

(p̄_{n} − p_{0})M_{k} = 0 for n ≥ k,  (3.15)

where p_{0} is the projection onto M_{0} parallel to the sum of M_{k}(k ≠ 0). As a result, the sequence p̄_{n} ∈ E(𝔤) is stable on any vector x ∈ M(𝔤), and the product (3.5) is equal to the element p_{0} ∈ E(𝔤). Finally, using the isomorphism E_{0}(𝔤) ≃ F_{0}(𝔤), we obtain p ∈ F_{0}(𝔤). It is clear that p = p_{0} satisfies (3.4). Repeating the concluding argument of Section 3.7, we obtain a new proof of Theorem 3.2.

3.10. Corollary. The projector p ∈ F_{0}(𝔤) may be considered as a locally rational series over 𝔥^{∗}, with denominators as products of affine functions a_{ε} (0 ≠ ε ∈ Q_{+}). In [6], an essentially finer result is obtained. Namely, there exists a representation p = lim_{F} p_{F}, for certain subproducts of (3.5), such that any denumerator of p_{F} contains a_{ε} only for ε = nα (0 ≠ n ∈ ℤ_{+}, α ∈ Δ_{+}). In other words, the singularities of p are concentrated on the intersection of hyperplanes a_{ε} = 0, where ε = nα (0 ≠ n ∈ ℤ_{+}, α ∈ Δ_{+}).

3.11. Example. For the algebra 𝔤 = 𝔰𝔩_{2} (Section 2.4), we have z_{+} = fe, z_{0} = h(h + 2)/4. The projector p ∈ F (𝔤) is given by the following relation:

p = ∑_{n=0}^{∞} \frac{(−1)^{n}}{n!} φ_{n}^{−1}f^{n}e^{n} = ∏_{n=1}^{∞} (1 − \frac{fe}{n(h + n + 1)}) ,  (3.16)

where φ_{n} = Π_{j+1}^{n}(h + j + 1).

3.12. Remark. For dim 𝔤 < ∞, there exists a remarkable representation of p as a finite product of factors p_{α}(α ∈ Δ_{+}) similar to (3.16), with respect to “normal orderings” in Δ_{+}. An exposition (including the history of the question) may be found in [8]. Certain analogs of these results are known for infinite dimensional Lie algebras and their quantum envelopes. For example, similar results are announced in [11] for affine Kac-Moody algebras. On the other hand, Theorem 3.2 holds for a wide category of symmetrizable Chevalley algebras [6,8].

4. The category LN(𝔢)

4.1. Let us demonstrate certain applications of Theorem 3.2 to the theory of 𝔤-modules of category LN(𝔢) (Section 2.5). Let V be a module of category LN(𝔢). Note that V ≠ 0 implies V^{e} ≠ 0, where V^{e} = ker e (extremal subspace of the module V). First, we consider the case of U′(𝔤)-modules.

4.2. Theorem. The category LN(𝔢) over U′(𝔤) is semisimple, with exactly one simple object, namely M(𝔤).

Proof. It is clear (Section 3.8) that M(𝔤)^{e} = M_{0}. Hence any submodule V ≠ 0 contains an element h · 1 with 0 ≠ h ∈ R(𝔥). Consequently, 1 ∈ V and V = M(𝔤). Therefore, M(𝔤) is a simple U′(𝔤)-module. Moreover, for any simple module V of category LN(𝔢) we have V = U′(𝔤)v_{0} for any 0 ≠ v_{0} ∈ V^{e}. Hence V ≃ M(𝔤)/N, where N is a quotient (with respect to J′_{1}) of the annulator of v_{0} in U′(𝔤). Since the module M(𝔤) is simple, we obtain N = 0, i.e. V ≃ M(𝔤). Now, let V be a module of category LN(𝔢). Setting W = U′(𝔤)V^{e}, it is easy to verify that W is a direct sum of simple submodules V_{ω} = U′(𝔤)ω, where ω runs through a fixed basis of the vector space V^{e}. Hence W is a multiple of M(𝔤). If V ≠ W, there exists x ∉ W such that 𝔢x ⊂ W. Note that p ≡ 1 mod F (𝔤)𝔢, hence px ≡ x mod W, and we obtain px ∉ W.

Contradiction with px ∈ V^{e}. As a result, we obtain V = W, and V is a multiple of M(𝔤).

4.3. Passing to the theory of U(𝔤)-modules, we restrict our attention to 𝔥graded 𝔤-modules of the form

V = ⨁_{μ} V_{μ},  (4.1)

where V_{μ} is the weight subspace of V corresponding to a weight μ ∈ 𝔥^{∗}. Let O(𝔢) be a subcategory of LN(𝔢) consisting of modules (4.1). It is known that O(𝔢) contains a series of Verma modules

M(λ) = V (𝔤)/J(λ),  (4.2)

where λ ∈ 𝔥^{∗} and J(λ) is the submodule of V (𝔤) generated by the elements h − λ(h)1(h ∈ 𝔥). Moreover, any simple object of category O(𝔢) is of the form

V (λ) = M(λ)/N(λ),  (4.3)

where N(λ) is the largest submodule of M(λ) that does not contain the cyclic vector 1_{λ} = 1 + J(λ). It is clear also that 𝔢1_{λ} = 0 and h1_{λ} = λ(h)1_{λ} for any h ∈ 𝔥 (i.e. 1_{λ} is an extremal vector of E(λ)).

4.4. It is convenient to write the elements p ∈ F (𝔤) in the form

p = Σ_{i}ω_{i}p_{i},  (4.4)

where ω_{i} denotes basic elements more general than in Section 1.2 and p_{i} ∈ R(𝔥). The action of p on V_{μ} is reduced to the corresponding specialization

p(μ) = Σ_{i}ω_{i}p_{i}(μ),  (4.5)

at the point μ ∈ 𝔥^{∗}. The operator (4.4) is well defined on V_{μ} if the coefficients p_{i} have no singularities (for some choice of basis ω_{i}) at the point μ.

4.5. A vector x ∈ V is called primitive [1] if there exists a submodule N ⊂ V such that v ∉ N, 𝔢v ⊂ N. The subspace V′ of primitive vectors of V inherits the grading (4.1):

V′ = ⨁_{μ} V′_{μ},  (4.6)

where V′_{μ} = V_{μ} ∩ V′. A vector x ∈ V′ is called singular if 𝔢x ≠ 0 or 𝔢x = 0, x ∈ 𝔣V. A module V of category O(𝔢) is called regular if the extremal projector p ∈ F (𝔤) is well defined on any singular vector v ∈ V. In particular, a module V is regular if the projector p ∈ F (𝔤) is well defined on V′. Let O_{reg}(𝔢) be a subcategory of O(𝔢) consisting of regular 𝔤-modules.

4.6. Theorem. Category O_{reg}(𝔢) coincides with the largest semisimple subcategory in O(𝔢). A proof is quite similar to that of Theorem 4.2. For example, let V = U(𝔤)v_{0} be the cyclic module generated by v_{0} ∈ V_{λ}^{e}. Using (1.5), we find V = ℂv_{0} ⊕ 𝔣V (the directness of the sum follows from usual weight considerations). Using p𝔣 = 0, we find V^{e} = ℂv_{0}, hence V is simple, V ≃ V (λ). Now, the argument of Section 4.2 allows to show that any regular module V is semisimple. Conversely, any semisimple module V is free on singular vectors, hence V is regular. Note also that the knowledge of singularities of the projector p (Section 3.9) allows one to give an effective criterion of semisimplicity in the category O(𝔢).

4.7. Example. A module V of category O(𝔢) is called dominant if any weight μ of V′ is dominant, i.e. (μ, α_{i}) ≥ 0 for any i ∈ I. Using (3.3) we find that μ is regular (with respect to p). Hence the module V is semisimple. In other words, the subcategory O_{+}(𝔢) in O(𝔢) consisting of dominant modules is semisimple. Note that this fact is known for the subcategory O_{int} of “integrable” 𝔤-modules. In particular, for dim 𝔤 < ∞, we obtain a new proof of the classical Weyl theorem on the semisimplicity of finite dimensional 𝔤-modules.

4.8. Other applications of extremal projectors may be found in [8,10]. In [8], the projector p is used to study “hypersymmetries” of extremal equations 𝔢x = 0 (x ∈ V), for a wide class of 𝔤-modules V. In particular, a structure theory of corresponding hypersymmetry algebras is proposed. As an example, a theory of Mickelsson algebras is developed for reductive pairs of Lie algebras (see also [4,5]). In [10], a general construction of extremal projectors is given for a category of Cartan type associative algebras. For example, this construction is useful in the study of quantum Serre algebras, in context of the theory of Drinfeld-Jimbo algebras [7,9].

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Independent University of Moscow, Kedrova 13 - 2 - 73, 117036 Moscow, Russia

Received July 27, 1999