Vertex operator representations of quantum tori at roots of unity ^{1}

Y. Billig & K. Zhao Department of Mathematics and Statistics Carleton University Ottawa, Ontario, K1S 5B6 Canada Email: billig@math.carleton.ca and Institute of Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080, P. R. China Email: kzhao@mail2.math.ac.cn

ABSTRACT. As Lie algebra, we add the center c_{1} (and the outer derivation d_{1}) to the quantum torus C_{q} to give the extended torus Lie algebra Ĉ_{q} (and C̃_{q} respectively). Before the present paper, only some level 1 vertex operator representations for some Ĉ_{q} (and C̃_{q}) were constructed. In this paper, we first give vertex operator representations for gl_{I×∞}(C) where I is an arbitrary index set. By embedding some Ĉ_{q} into gl_{I×∞}(C), we obtain a series of higher level vertex operator representations for Ĉ_{q} and C̃_{q}. Most of these vertex operator representations yield irreducible highest weight modules over these C̃_{q}. Also their character formulas follow directly.

§1. Introduction

Let q = (q_{i,j})_{i,j=0}^{n} be an (n+1)×(n+1) matrix over the complex number field C satisfying

q_{i,i} = 1, q_{i,j} = q_{j,i}^{−1},  (1.1)

where n is a positive integer. The q-quantum torus C_{q} which was studied in [MP] is the unital associative algebra over C generated by t_{0}^{±1}, ..., t_{n}^{±1} and subject to the defining relations

t_{i}t_{j} = q_{i,j}t_{j}t_{i}.  (1.2)

1AMS Subject Classification: 17B20, 17B65, 17B67, 17B68, 16A40, 16A68.

This work is supported by NSERC, Hundred Talents Program of Chinese Academy of Sciences and by NSF of China.

For any a ∈ ZZ^{n+1}, we always write a = (a_{0}, ..., a_{n}), and denote t^{a} = t_{0}^{a_{0}}...t_{n}^{a_{n}}. For any a, b ∈ ZZ^{n+1}, we define the functions σ(a, b) and f(a, b) by

t^{a}t^{b} = σ(a, b)t^{a+b}, t^{a}t^{b} = f(a, b)t^{b}t^{a}.  (1.3)

We also define |α| = α_{0} + α_{1} + ... + α_{n}. Then

σ(a, b) = ∏_{0≤i<j≤n} q_{j,i}^{a_{j}b_{i}}, f(a, b) = ∏_{i,j=0}^{n} q_{j,i}^{a_{j}b_{i}},  (1.4)

and f(a, b) = σ(a, b)σ(b, a)^{−1}. We define radf = {a ∈ ZZ^{n}|f(a, ZZ^{n}) = 1} and the Kronecker delta

δ_{α,radf} = \{ \begin{matrix}1, if α ∈ radf \\ 0, otherwise. \end{matrix}

For properties of C_{q}, f, σ, please refer to [BGK] or [Z].

In most part of this paper we assume that

q_{i,j} = 1, ∀ 1 ≤ i, j ≤ n.  (1.5)

Note that (1.5) implies σ(α, ZZ^{n+1}) = 1 for all α ∈ radf. Under this assumption we have the 1-dimensional central extension Ĉ_{q} = C_{q} ⊕ Cc of C_{q} with

[t^{α}, t^{β}] = t^{α}t^{β} − t^{β}t^{α} + δ_{α_{0},−β_{0}}δ_{α+β,radf}σ(α, β)α_{0}c, ∀ α, β ∈ ZZ^{n+1}.  (1.6)

We extend Ĉ_{q} by a derivation d_{0} with

[d_{0}, t^{α}] = α_{0}t^{α}, ∀ α ∈ ZZ^{n+1}

to give the Lie algebra C̃_{q} = Ĉ_{q} ⊕ Cd_{0}. For any nonnegative integer m, and l ∈ {1, 2, ..., n}, we modify (1.6) into

[t^{α}, t^{β}] = t^{α}t^{β} − t^{β}t^{α} + δ_{α_{0},−β_{0}}δ_{α+β,radf}δ_{−β̄_{l},ᾱ_{l}}σ(α, β)α_{0}c, ∀ α, β ∈ ZZ^{n+1},  (1.6′)

where β̄_{l}, ᾱ_{l} ∈ ZZ_{m}, to get Lie algebras Ĉ_{q}^{(l)}(m) and C̃_{q}^{(l)}(m) correspondingly. In some cases, C̃_{q}^{(l)}(m) ≃ C̃_{q}^{(l)}(m′) for some different m and m′. For example, if n = 1 and q_{1,0}^{k} ≠ 1 for all k ∈ IN, then C̃_{q}^{(1)}(m) ≃ C̃_{q} for all m ∈ ZZ_{+}. We know that Ĉ_{q}, C̃_{q}, C̃_{q}^{(l)}(m) have a ZZ-gradation with respect to d_{0}:

C̃_{q} = C̃_{q}^{(l)}(m) = ⊕_{k∈ZZ}L_{k},  (1.7)

where L_{k} = ⊕_{α∈ZZ^{n}}Ct_{0}^{k}t_{1}^{α_{1}}...t_{n}^{α_{n}} ⊕ δ_{k,0}(⊕Cc ⊕ Cd_{0}). For a ZZ-graded module V = ⊕_{i∈ZZ}V_{k}, its character is defined as

chV = ∑_{k∈ZZ}(dim V_{k})x^{−k}.  (1.8)

In [BGT, BS, G1, G2, GL], level 1 vertex operator representations of the Lie algebras C̃_{q} were constructed, where they assumed q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}} ≠ 1 if α ≠ 0. In this paper, we give not only higher level vertex operator representations of these Lie algebras C̃_{q} (Theorems 2.2, 3.4), but also construct higher level vertex operator representations of the Lie algebras C̃_{q} with q_{1,2} being root of unity and all other q_{i,j} being 1 for i, j ∈ 1, 2, ..., n (Theorems 4.1 and 4.5). Most of these vertex operator representations constructed are irreducible (Theorems 2.2, 3.6, 4.4, 4.6). The character formulas of these representations follow easily. All irreducible vertex operator representations constructed in this paper are highest weight modules (Theorem 5.1), thus the isomorphisms between these modules are clear.

§2. Principal vertex operator representations for C̃_{q}

In this section we shall embed some Ĉ_{q} in gl_{∞}(C) to give principal vertex operator representations for Ĉ_{q} and C̃_{q}. Let us first recall a vertex operator representation theorem from [DJKM]. Let

Ā_{∞} = {(a_{i,j})_{i,j∈ZZ} | a_{i,j} ∈ C, and a_{i,j} = 0 if |i − j| ≫ 0}

be the infinite matrix Lie algebra. We use E_{i,j} to denote the matrix unit, i.e., the matrix with 1 in (i,j)-entry, and 0 elsewhere. Let A_{∞} = Ā_{∞} + Cc be the 1-dimensional central extension with

[X, Y] = XY − Y X + ϕ(X, Y)c, ∀ X, Y ∈ Ā_{∞}

where the 2-cocycle ϕ is given by

ϕ(E_{i,j}, E_{k,l}) = \{ \begin{matrix}δ_{i,l}δ_{j,k}, if i ≤ 0, j ≥ 1 \\ −δ_{i,l}δ_{j,k}, if j ≤ 0, i ≥ 1, \\ 0, otherwise. \end{matrix}

Theorem 2.1 [DJMK] For any m ∈ ZZ, the Lie algebra A_{∞} has an irreducible vertex operator representation R_{m} on the Fock space B = C[x_{1}, x_{2}, x_{3}, ...] so that

R_{m}( ∑_{i,j∈ZZ} u^{i}v^{−j}E_{i,j})

= \frac{1}{1 − v/u} [(u/v)^{m} exp ( ∑_{j∈IN}(u^{j} − v^{j})x_{j}) exp ( − ∑_{j∈IN} \frac{(u^{−j} − v^{−j})}{j} \frac{∂}{∂x_{j}} ) − 1] ,  (2.1)

R_{m}(c) = 1, R_{m}( ∑_{j∈ZZ} E_{j,i+j}) = \{ \begin{matrix}\frac{∂}{∂x_{i}} , if i > 0, \\ −ix_{−i}, if i < 0, \\ m, if i = 0, \end{matrix}

where IN is the set of natural numbers.

Part of the following theorem (q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}} ≠ 1 if α ≠ 0) was proved in [G1] and [GL]. The proof here is slightly different from that in [GL].

Theorem 2.2([GL]) Suppose q = (q_{i,j})_{i,j=0}^{n} satisfies (1.1) and (1.5). Then Ĉ_{q} has an irreducible vertex operator representation R_{m} for any m ∈ ZZ on the Fock space B = C[x_{1}, x_{2}, x_{3}, ...] so that for any α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}, we have

R_{m}( ∑_{j∈ZZ} t_{0}^{j}t^{α}z^{−j}) = \frac{q^{−mα}}{1 − q^{α}} exp ( ∑_{j∈IN}(1−q^{jα})x_{j}z^{j}) exp (− ∑_{j∈IN} \frac{1 − q^{−jα}}{j} \frac{∂}{∂x_{j}} z^{−j}), if q^{α} ≠ 1,

(2.2)

R_{m}(c) = 1, R_{m}(t_{0}^{i}t^{α}) = \{ \begin{matrix}\frac{∂}{∂x_{i}} , if i > 0, q^{α} = 1, \\ −ix_{−i}, if i < 0, q^{α} = 1, \\ m, if i = 0, q^{α} = 1, \end{matrix}

where t^{α} = t_{1}^{α_{1}}...t_{n}^{α_{n}}, q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}. In (2.2) q^{−mα} can be replaced by any multiplicative function γ(α) ∈ C^{∗}. If we define

R_{m}(d_{0})(x_{1}^{k_{1}}...x_{l}^{k_{l}}) = −(k_{1} + 2k_{2} + ... + lk_{l})(x_{1}^{k_{1}}...x_{l}^{k_{l}}),

then B becomes a C̃_{q}-module with character formula

ch(B) = \frac{1}{φ(x^{−1})} = \frac{1}{∏_{i∈IN} (1 − x^{−i})}.  (2.3)

Proof. It is straightforward to verify that the following linear map is a Lie algebra homomorphism:

η : Ĉ_{q} → A_{∞},
t_{0}^{i}t^{α} → ∑_{j∈ZZ} q^{−jα}E_{j−i,j} + δ_{i,0}(1 − δ_{q^{α},1})\frac{c}{1 − q^{α}} , ∀ (i, α) ∈ ZZ^{n+1},  (2.4)
c → c.

Since R_{m} is a product of R_{0} and a Lie automorphism, it suffices to show the theorem for only R_{0}. Letting u = z, v = q^{α}z in (2.1) we deduce that (for q^{α} ≠ 0 and R_{0})

\frac{1}{1 − q^{α}} exp ( ∑_{j∈IN}(1 − q^{jα})x_{j}z^{j}) exp ( − ∑_{j∈IN} \frac{(1 − q^{−jα})}{j} \frac{∂}{∂x_{j}} z^{−j})

= R_{0}( ∑_{i,j∈ZZ} q^{−jα}z^{i−j}E_{i,j}) + \frac{1}{1 − q^{α}} = R_{0}( ∑_{l,j∈ZZ} q^{−jα}z^{−l}E_{j−l,j}) + \frac{1}{1 − q^{α}}

= R_{0}( ∑_{l∈ZZ}( ∑_{j∈ZZ} q^{−jα}E_{j−l,j} + δ_{l,0} \frac{c}{1 − q^{α}})z^{−l})

= R_{0}( ∑_{l∈ZZ} η(t_{0}^{l}t^{α})z^{−l}).

The other parts of the theorem are quite clear.

Remark 2.3. The embedding (2.4) without center can be regarded as the following string representation of C_{q} on C^{∞} = ⊕_{i∈ZZ}Cv_{i} given by

(t_{0}^{k}t^{α})v_{i} = q^{−iα}v_{i−k}, ∀ i, k ∈ ZZ, α ∈ ZZ^{n}.

When G_{q} = ZZ^{n} ⊕ ZZ_{m}, Theorem 2.2 is one of the main theorems in [G1], although they are slightly different in appearance.

Remark 2.4. If n=1 and q_{1,0} is a root of unity of order r, then representation R_{0} is the level 1 representation of the affine Lie algebra ĝl_{r}.

§3. Homogeneous vertex operator representations for Ĉ_{q} and C̃_{q}

In this section we shall first construct a vertex operator representation for gl_{I×∞}(C) where I is an index set, then embed Ĉ_{q} in gl_{I×∞}(C) to give a series of higher level vertex operator representations for Ĉ_{q} and C̃_{q} satisfying (1.5).

Let

Ā_{I×∞} = {(a_{i,j}^{k,l}) | i, j ∈ I, k, l ∈ ZZ, a_{i,j}^{k,l} ∈ C, and a_{i,j}^{k,l} = 0

if |k − l| ≫ 0, and for fixed (k, l) a_{i,j}^{k,l} ≠ 0 only for finitely many (i, j)}

be the infinite matrix Lie algebra. Note that we allow I to be an infinite set. We use E_{i,j}^{k,l} to denote the matrix unit, i.e., the matrix with 1 in (k,l)-(i,j)-entry, and 0 elsewhere. The products are given by

E_{i,j}^{k,l}E_{i′,j′}^{k′,l′} = δ_{j,i′}δ_{l,k′}E_{i,j′}^{k,l′},

[E_{i,j}^{k,l}, E_{i′,j′}^{k′,l′}] = δ_{j,i′}δ_{l,k′}E_{i,j′}^{k,l′} − δ_{j′,i}δ_{l′,k}E_{i′,j}^{k′,l}.

Let A_{I×∞} = Ā_{I×∞} + Cc be the 1-dimensional central extension with

[X, Y] = XY − Y X + ϕ(X, Y)c,  (3.1)

where the 2-cocycle ϕ is given by

ϕ(E_{i,j}^{k,l}, E_{i′,j′}^{k′,l′}) = \{ \begin{matrix}δ_{i,j′}δ_{l′,k}δ_{i′,j}δ_{l,k′}, if k ≤ 0, l ≥ 1 \\ −δ_{i,j′}δ_{l′,k}δ_{i′,j}δ_{l,k′}, if l ≤ 0, k ≥ 1, \\ 0, otherwise. \end{matrix}  (3.2)

Let P_{I} = ⊕_{i∈I}ZZϵ_{i} be a free abelian additive group with a symmetric ZZ-bilinear form (·|·) defined by (ϵ_{i}|ϵ_{j}) = δ_{i,j} for all i, j ∈ I. Let

Q_{I} = ⊕_{i,j∈I}ZZ(ϵ_{i} − ϵ_{j})  (3.3)

be a subgroup of P_{I}. We see that

(α|β) ∈ 2ZZ, ∀ α, β ∈ Q_{I}.  (3.4)

Let ε : Q_{I} × Q_{I} → {±1} be a bimultiplicative function in the sense that

ε(α + β, γ) = ε(α, γ)ε(β, γ), ε(α, β + γ) = ε(α, β)ε(α, γ)  (3.5)

for all α, β, γ ∈ Q_{I}, and with the property

ε(α, α) = (−1)^{(α|α)/2}, ∀ α ∈ Q_{I}.  (3.6)

It follows from (3.6) that

ε(α, β)ε(β, α) = (−1)^{(α|β)}, ∀ α, β ∈ Q_{I}.  (3.7)

Set H_{I} = P_{I} ⊗ZZ C, and extend the ZZ-bilinear form (·|·) to get a symmetric C-bilinear form (·|·) on H_{I}. Let Ĥ_{I} = H_{I} ⊗ C[t, t^{−1}] ⊕ Cc ⊕ Cd_{0} be the Heisenberg Lie algebra with

[α(m), β(n)] = δ_{m,−n}m(α, β)c, ∀ α, β ∈ H_{I}, m, n ∈ ZZ,  (3.8)
[d_{0}, c] = 0, [d_{0}, α(m)] = mα(m), ∀ α ∈ H_{I}, m ∈ ZZ,

where α(n) = α ⊗ t^{n}. Let Ĥ_{I}^{+} = ∑_{i>0} H_{I} ⊗ t^{i}, Ĥ_{I}^{−} = ∑_{i<0} H_{I} ⊗ t^{i}, then we have a subalgebra

H_{I} = Ĥ_{I}^{+} ⊕ Ĥ_{I}^{−} ⊕ Cc ⊕ Cd_{0}.  (3.9)

Let

S(Ĥ_{I}^{−}) = C[ϵ_{i}(−n)|i ∈ I, n ∈ IN].  (3.10)

Define the degree deg(ϵ_{i_{1}}(−n_{1})ϵ_{i_{2}}(−n_{2})...ϵ_{i_{r}}(−n_{r})) to be

deg(ϵ_{i_{1}}(−n_{1})ϵ_{i_{2}}(−n_{2})...ϵ_{i_{r}}(−n_{r})) = −(n_{1} + n_{2} + ... + n_{r}).

Then H_{I} has the natural representation on S(Ĥ_{I}^{−}) in the sense that ϵ_{i}(n) acts as multiplication operator for n < 0, ϵ_{i}(n) acts as differential operator \frac{n∂}{∂ϵ_{i}(−n)} for n > 0, while c acts as the identity, and d_{0} acts as the degree operator.

Let

C[Q_{I}] = ⊕_{α∈Q_{I}}Ce^{α}  (3.11)

be the group algebra of Q_{I}. For any β ∈ Q_{I}, define e_{β} ∈ EndC[Q_{I}] by

e_{β}e^{α} = ε(β, α)e^{α+β}, ∀ α ∈ Q_{I}.  (3.12)

We see that

e_{α}e_{β} = ε(α, β)e_{α+β}, ∀ α, β ∈ Q_{I}.  (3.13)

For any β ∈ H, define β(0) ∈ EndC[Q_{I}] by

β(0)e^{α} = (β|α)e^{α}, ∀ α ∈ Q_{I}.  (3.14)

It is clear that

[β(0), e_{α}] = (β|α)e_{α}, ∀ α ∈ Q_{I}.  (3.15)

We define

d_{0}e^{α} = −\frac{(α|α)}{2}e^{α}, ∀ α ∈ Q_{I}.  (3.16)

Then

[d_{0}, e_{α}] = e_{α}(−α(0) − \frac{(α|α)}{2}), ∀ α ∈ Q_{I}.  (3.17)

Set

V_{Q_{I}} = S(H_{I}^{−}) ⊗C C[Q_{I}],  (3.18)

be the Fock space. We embed EndS(H_{I}^{−}) and EndC[Q_{I}] (respectively (EndS(H_{I}^{−}))[[z]] and (EndC[Q_{I}])[[z]]) canonically into EndV_{Q_{I}} (respectively (EndV_{Q_{I}})[[z]]), for instance,

β(l) = \{ \begin{matrix}β(l) ⊗ 1, if l ≠ 0, \\ 1 ⊗ β(l), if l = 0 \end{matrix} , ∀ β ∈ H_{I}.

We define the action of d_{0} on V_{Q_{I}} by

d_{0} = d_{0} ⊗ 1 + 1 ⊗ d_{0}.  (3.19)

Recall that z^{α} ∈ EndC[Q_{I}])[[z]] is given by

z^{α}e^{β} = z^{(α|β)}e^{β}, ∀ α, β ∈ Q_{I}.  (3.20)

Thus we have

[α(0), z^{β}] = 0, z^{α}e_{β} = e_{β}z^{α+(α|β)}, ∀ α, β ∈ Q_{I}.  (3.21)

Similarly, for any nonzero complex number a, we have the evaluation a^{α} of z^{α} so that a^{α} ∈ End(C[Q_{I}]).

Now we modify A_{I×∞} to give the Lie algebra A_{I×∞}(ε) as follows. As vector spaces, A_{I×∞} = A_{I×∞}(ε). But the Lie bracket is given by

[Ė_{i,j}^{k,l}, Ė_{i′,j′}^{k′,l′}] = ε(ϵ_{i} − ϵ_{j}, ϵ_{i′} − ϵ_{j′})δ_{j,i′}δ_{l,k′} Ė_{i,j′}^{k,l′} − ε(ϵ_{i′} − ϵ_{j′}, ϵ_{i} − ϵ_{j})δ_{j′,i}δ_{l′,k}Ė_{i′,j}^{k′,l}
+ε(ϵ_{i} − ϵ_{j}, ϵ_{i′} − ϵ_{j′})ϕ(E_{i,j}^{k,l}, E_{i′,j′}^{k′,l′})c, ∀ , i, j, i′, j′ ∈ I, k, l, k′, l′ ∈ ZZ.  (3.22)

Here we used Ė_{i,j}^{k,l} (which is viewed as matrix unit in A_{I×∞}(ε)) to distinguish the matrix unit with the ordinary one E_{i,j}^{k,l} which is viewed as matrix unit in A_{I×∞}). We know that the Lie algebras A_{I×∞} and A_{I×∞}(ε) are isomorphic.

The following theorem is inspired by Theorem 2.1, [Theorem 1.2, TV], and the results in [F].

Theorem 3.1 The Lie algebra A_{I×∞}(ε) has an irreducible vertex operator representation R_{m} for any m ∈ ZZ on the Fock space V_{Q_{I}} given by

R_{m}( ∑_{k,l∈ZZ} Ė_{i,j}^{k,l}z_{1}^{k}z_{2}^{−l}) = −\frac{δ_{i,j}}{(1 − z_{2}/z_{1})} + e_{ϵ_{i}−ϵ_{j}}z_{1}^{ϵ_{i}+1+m−δ_{i,j}}z_{2}^{−ϵ_{j}−m}(1 − z_{2}/z_{1})^{−δ_{i,j}}·

· exp ( − ∑_{p∈−IN} \frac{z_{1}^{−p}ϵ_{i}(p) − z_{2}^{−p}ϵ_{j}(p)}{p}) exp ( − ∑_{p∈IN} \frac{z_{1}^{−p}ϵ_{i}(p) − z_{2}^{−p}ϵ_{j}(p)}{p}), ∀ i, j ∈ I, (3.23)

R_{m}(c) = 1, R_{m}( ∑_{k∈ZZ} Ė_{i,i}^{k,k+l}) = ϵ_{i}(l), ∀ l ∈ ZZ.

Proof. Since the proof of this theorem is quite standard (although onerous), instead of giving the detailed proof we shall only sketch the proof.

We can verify (via a lot of computations) that

[R_{m}( ∑_{k,l∈ZZ} Ė_{i,j}^{k,l}z_{1}^{k}z_{2}^{−l}), R_{m}( ∑_{k,l∈ZZ} Ė_{j,j′}^{k,l}z_{3}^{k}z_{4}^{−l})] = R_{m}([ ∑_{k,l∈ZZ} Ė_{i,j}^{k,l}z_{1}^{k}z_{2}^{−l}, ∑_{k,l∈ZZ} Ė_{j,j′}^{k,l}z_{3}^{k}z_{4}^{−l}]),

for all cases: (1) i ≠ j ≠ j′ ≠ i, (2) i ≠ j = j′, (3) j′ = i ≠ j, (4) i = j = j′ separately by using formula (3.37). So the first and the second formulas in (3.23) follow. The third formulas in (3.23) with l ≠ 0 are clear from Theorem 2.1. The third formulas in (3.23) with l = 0 follow from the facts that [R_{m}(∑_{k∈ZZ} Ė_{i,i}^{k,k}), R_{m}(∑_{k∈ZZ} Ė_{i,i}^{k,k+l})] = 0 for all l ∈ ZZ, and

[R_{m}( ∑_{k∈ZZ} Ė_{i,i}^{k,k}), R_{m}( ∑_{k,l∈ZZ} Ė_{i′,j′}^{k,l}z_{1}^{k}z_{2}^{−l})] = (δ_{i,i′} − δ_{i,j′})R_{m}( ∑_{k,l∈ZZ} Ė_{i′,j′}^{k,l}z_{1}^{k}z_{2}^{−l})].

The irreducibility follows from the last equation of (3.23) and the action of e_{ϵ_{i}−ϵ_{j}}.

Theorem 3.2 Suppose q satisfies

q_{i,0} = q_{0,i}^{−1}, q_{0,0} = q_{i,j} = q_{j,i} = 1, ∀ i, j ∈ {1, 2, ..., n}.  (3.24)

Let ĝl_{m}(C_{q}, ε) = gl_{m}(C_{q}) ⊕ Cc be the 1-dimensional central extension with

[Ė_{i,j}(t^{α}), Ė_{i′,j′}(t^{β})] = δ_{j,i′}ε(ϵ_{i} − ϵ_{j}, ϵ_{i′} − ϵ_{j′})Ė_{i,j′}(t^{α}t^{β}) − δ_{j′,i}ε(ϵ_{i′} − ϵ_{j′}, ϵ_{i} − ϵ_{j})Ė_{i′,j}(t^{β}t^{α})
−ε(ϵ_{i} − ϵ_{j}, ϵ_{i′} − ϵ_{j′})δ_{j,i′}δ_{j′,i}δ_{α_{0}+β_{0},0}δ_{q^{−β},q^{α}}σ(α, β)α_{0}c, ∀ α, β ∈ Z^{n+1}, i, j, i′, j′ ∈ I.

Then ĝl_{m}(C_{q}, ε) has an irreducible vertex operator representation R on the Fock space V_{Q_{I}} with I = {1, 2, ..., m} so that for any α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}, any i ≠ j ∈ I, we have

R( ∑_{p∈ZZ} Ė_{i,j}(t_{0}^{p}t^{α})z^{−p}) = e_{ϵ_{i}−ϵ_{j}}z^{ϵ_{i}−ϵ_{j}+1−δ_{i,j}}(q^{α})^{−ϵ_{j}}(1 − q^{α})^{−δ_{i,j}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{i}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{i}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}), if q^{α} ≠ 1,

(3.25)
R(c) = 1, R(Ė_{i,i}(t_{0}^{l}t^{α})) = ϵ_{i}(l), if q^{α} = 1.

If we extend ĝl_{m}(C_{q}, ε) by d_{0} to get the Lie algebra g̃l_{m}(C_{q}, ε) with

[d_{0}, Ė_{i,j}(t^{α})] = α_{0}Ė_{i,j}(t^{α}), ∀ α ∈ ZZ^{n+1},

then V_{Q_{I}} is a module over g̃l(C_{q}) via (3.19), and

chV_{Q_{I}} = ( ∑_{α∈Q_{I}} x^{−(α|α)/2})ϕ(x^{−1})^{−m}.  (3.26)

Proof. It is straightforward to verify that the following linear map is a Lie algebra homomorphism:

τ : ĝl_{m}(C_{q}, ε) → A_{I×∞}(ε),
Ė_{i,j}(t_{0}^{k}t^{α}) → ∑_{p∈ZZ} q^{−pα}Ė_{i,j}^{p−k,p} + \frac{δ_{k,0}δ_{i,j}(1 − δ_{1,q^{α}})c}{1 − q^{α}}, ∀ (k, α) ∈ ZZ^{n+1}, i, j ∈ I  (3.27)
c → c.

Letting z_{1} = z, z_{2} = q^{α}z in (2.23) we deduce that (for q^{α} ≠ 1 and R_{0})

e_{ϵ_{i}−ϵ_{j}}z^{ϵ_{i}−ϵ_{j}+1−δ_{i,j}}(q^{α})^{−ϵ_{j}}(1 − q^{α})^{−δ_{i,j}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{i}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{i}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p})

= R_{0}( ∑_{k,l∈ZZ} q^{−lα}z^{k−l}Ė_{i,j}^{k,l}) + \frac{δ_{i,j}}{1 − q^{α}}

= R_{0}( ∑_{l,k∈ZZ} q^{−lα}z^{−k}Ė_{i,j}^{l−k,l}) + \frac{δ_{i,j}}{1 − q^{α}}

= R_{0}(( ∑_{k∈ZZ}( ∑_{l∈ZZ} q^{−lα}Ė_{i,j}^{l−k,l}) + δ_{k,0} \frac{δ_{i,j}c}{1 − q^{α}})z^{−k})

= R_{0}( ∑_{k∈ZZ} τ(Ė_{i,j}(t_{0}^{k}t^{α})z^{−k})).

The other formulas of (3.25) are quite clear. To see that V_{Q_{I}} is a module over g̃l(C_{q}), we need to show that

[R(d_{0}), R( ∑_{p∈ZZ} Ė_{i,j}(t_{0}^{p}t^{α})z^{−p})] = R( ∑_{p∈ZZ} Ė_{i,j}(t_{0}^{p}t^{α})pz^{−p}) = −\frac{z∂}{∂z} R( ∑_{p∈ZZ} Ė_{i,j}(t_{0}^{p}t^{α})z^{−p}),

which can be verified by using (3.17) and (the action on C[Q_{I}])

z \frac{∂}{∂z} z^{α} = α(0)z^{α}, ∀ α ∈ Q_{I}.

Other parts of the theorem follows easily. Thus we complete the proof of this theorem.

Remark 3.3. The embedding (3.27) without center is inspired by the following string representation of gl_{m}(C_{q}) on C^{m×∞} = ⊕_{i∈ZZ^{m}, j∈ZZ}Cv_{i,j} given by

(E_{k,l}(t_{0}^{p}t^{α}))v_{i,j} = δ_{i,l}q^{−jα}v_{k,j−p}.

When G_{q} = ZZ^{2} or ZZ^{3}, I is finite, and q_{1,0}^{i}q_{2,0}^{j} ≠ 1 for (i, j) ≠ (0, 0), Theorem 3.2 is one of the main theorems in [G2], although they are slightly different in appearance.

Theorem 3.4 Let q satisfy (3.24). For any positive integer m, let ω_{m} be the primitive root of unity of order m, I = ZZ/mZZ = {1, 2, ..., m}. Then Ĉ_{q}^{(1)}(m) has a vertex operator representation R_{m} on the Fock space S(Ĥ_{I}^{−}) so that for any α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}, we have

R_{m}( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}) = ∑_{j=1}^{m} \frac{ω_{m}^{−jα_{1}}}{(1 − q^{α})} · exp ( − ∑_{p∈−IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p})·

· exp ( − ∑_{p∈IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p}), if q^{α} ≠ 1,  (3.28)

R_{m}(c) = m, R_{m}(t_{0}^{l}t^{α}) = ∑_{j=1}^{m} ω_{m}^{−jα_{1}}ϵ_{j}(l), if q^{α} = 1.

Proof. It is straightforward to verify that the following linear map is a Lie algebra homomorphism:

τ : Ĉ_{q}^{(1)}(m) → A_{I×∞}(ε),
t_{0}^{i}t^{α} → ∑_{j=1}^{m} ω_{m}^{−jα_{1}} ∑_{k∈ZZ} q^{−kα}Ė_{j,j}^{k−i,k} + \frac{δ_{i,0}δ_{\overline{α_{1}},0}(1 − δ_{q^{α},1})mc}{1 − q^{α}},  (3.29)
c → mc,

where \overline{α_{1}} ∈ ZZ_{m}. If q^{α} ≠ 1, using R_{0} in Theorem 3.1 we deduce that

R_{m}( ∑_{k∈ZZ}(t_{0}^{k}t^{α})z^{−k}) := R_{0}( ∑_{k∈ZZ} τ(t_{0}^{k}t^{α})z^{−k})

= ∑_{k∈ZZ} z^{−k}R_{0} (∑_{j=1}^{m} ω_{m}^{−jα_{1}} ∑_{p∈ZZ} q^{−pα}Ė_{j,j}^{p−k,p} + \frac{δ_{k,0}δ_{\overline{α_{1}},0}mc}{1 − q^{α}} )

= ∑_{j=1}^{m} ω_{m}^{−jα_{1}} ∑_{k∈ZZ} z^{−k} (R_{0}( ∑_{p∈ZZ} q^{−pα}Ė_{j,j}^{p−k,p}) + \frac{δ_{k,0}}{1 − q^{α}} )

(letting z_{1} = z, z_{2} = q^{α}z in (3.23) with i = j, acting on S(Ĥ_{I}^{−}) ⊗ 1)

= ∑_{j=1}^{m} \frac{ω_{m}^{−jα_{1}}}{(1 − q^{α})} · exp ( − ∑_{p∈−IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p}).

The other parts of the theorem are quite clear.

The following Lemma is very useful to further study on the structure of the module in Theorem 3.4.

Lemma 3.5. Let ω_{m} be given as in Theorem 3.4, q_{1,0}, ..., q_{n,0} be nonzero numbers, W ⊂ V be vector spaces. Suppose the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α ∈ ZZ^{n}} is infinite. Let f_{j}(q^{α}) = ∑_{i} q^{iα}v_{i}^{(j)} ∈ V for all j = 1, 2..., m, α ∈ ZZ^{n}. If ∑_{j=1}^{m} ω_{m}^{jα_{1}}f_{j}(q^{α}) ∈ W for all α ∈ ZZ^{n}, then f_{j}(q^{α}) ∈ W for all j and all α ∈ ZZ^{n}.

Proof. We know that

∑_{i} q^{iα} ∑_{j=1}^{m} ω_{m}^{jα_{1}}v_{i}^{(j)} = ∑_{j=1}^{m} ω_{m}^{jα_{1}}f_{j}(q^{α}) ∈ W, ∀ α ∈ ZZ^{n}.  (3.30)

By letting α_{1} = km + i_{0} where i_{0} = 0, 1, ..., m − 1 and k ∈ ZZ, and changing other α_{i} ∈ ZZ in (3.30), we obtain that

∑_{i} q^{iα} ∑_{j=1}^{m} ω_{m}^{ji_{0}}v_{i}^{(j)} ∈ W, ∀ α ∈ ZZ^{n} with α_{1} = km + i_{0}.

From the hypothesis we know that the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α ∈ ZZ^{n} with a_{1} = i_{0} mod m} is infinite. We deduce that

∑_{j=1}^{m} ω_{m}^{ji_{0}}v_{i}^{(j)} ∈ W, ∀ i_{0} ∈ {0, 1, ..., m − 1}, i ∈ ZZ.

It follows that v_{i}^{(j)} ∈ W, ∀ i, j ∈ ZZ. The lemma follows.

Theorem 3.6 S(Ĥ_{I}^{−}) in Theorem 3.4 is a C̃_{q}^{(1)}(m)-module. If, in addition to the assumptions in Theorem 3.4, we assume that the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α ∈ ZZ^{n}} is infinite, or there exists α ∈ ZZ^{n} such that (α_{1}, m) = 1 and q^{α} = 1, then C̃_{q}^{(1)}(m)-module S(Ĥ_{I}^{−}) is irreducible, and

ch(S(Ĥ_{I}^{−})) = φ(x^{−1})^{−m}.  (3.31)

Proof. It is clear that S(Ĥ_{I}^{−}) is a C̃_{q}^{(1)}(m)-module. If there exists α ∈ ZZ^{n} such that (α_{1}, m) = 1 and q^{α} = 1, from the third equation of (3.28) we see that

ϵ_{j}(l) ∈ R_{m}(C̃_{q}^{(1)}(m)), ∀ j ∈ I, l ∈ ZZ.  (3.32)

Thus C̃_{q}^{(1)}(m)-module S(Ĥ_{I}^{−}) in Theorem 3.4 is irreducible. Next we suppose {q^{α}|α ∈ ZZ^{n}} is infinite. For q^{α} ≠ 1, let

f_{j}(q^{α}, z) = \frac{1}{(1 − q^{α})} exp ( − ∑_{p∈−IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(1 − q^{−pα})ϵ_{j}(p)}{p}z^{−p}).

For any v ∈ S(Ĥ_{I}^{−}), from (3.28) we know that

∑_{j=1}^{m} ω_{m}^{−jα_{1}}f_{j}(q^{α}, z)v ∈ (R_{m}(C̃_{q}^{(1)}(m))v)[z, z^{−1}], ∀ α ∈ ZZ^{n} with q^{α} ≠ 1.  (3.33)

Using Lemma 3.5, we see that, for any j,

f_{j}(q^{α}, z)v ∈ (R_{m}(C̃_{q}^{(1)}(m))v)[z, z^{−1}], ∀ v ∈ S(Ĥ_{I}^{−}), α ∈ ZZ^{n} with q^{α} ≠ 1.  (3.34)

It is clear that, for any j,

[f_{j}(q^{α}, z_{1}), f_{j}(q^{−α}, z_{2})]v ∈ (R_{m}(U(C̃_{q}^{(1)}(m)))v)[z_{1}^{±1}, z_{2}^{±1}],
∀ v ∈ S(Ĥ_{I}^{−}), α ∈ ZZ^{n} with q^{α} ≠ 1.  (3.35)

By using the following well known formulas

f(z_{1}, z_{2})Dδ(\frac{z_{2}}{q^{α}z_{1}} ) = f(z_{1}, q^{α}z_{1})Dδ(\frac{z_{2}}{q^{α}z_{1}} ) − q^{α}z_{1}(\frac{∂}{∂z_{2}} f(z_{1}, z_{2}))δ(\frac{z_{2}}{q^{α}z_{1}} ),  (3.36)
f(z_{1}, z_{2})δ(\frac{z_{2}}{q^{α}z_{1}} ) = f(z_{1}, q^{α}z_{1})δ(\frac{z_{2}}{q^{α}z_{1}} ),  (3.37)

where

δ(z) = ∑_{p∈ZZ} z^{p}, (Dδ)(z) = ∑_{p∈ZZ} pz^{p},  (3.38)

it is standard to show that, if q^{α} ≠ 1, then

[f_{j}(q^{α}, z_{1}), f_{j}(q^{−α}, z_{2})] = ∑_{p,p′∈ZZ} q^{−pα}(q^{(p+p′)α} − 1)ϵ_{j}(p + p′)z_{1}^{−p}z_{2}^{−p′} + (Dδ)(\frac{z_{2}}{q^{α}z_{1}} ).

Since {q^{α}|α ∈ ZZ^{n}} is infinite, we see that for any i ≠ 0, there exists α ∈ ZZ^{n} such that q^{iα} ≠ 1. Thus

ϵ_{j}(p) ∈ R_{m}(U(C̃_{q}^{(1)}(m))), ∀ p ∈ ZZ \ {0}, j ∈ I.

The irreducibility of the module S(Ĥ_{I}^{−}) follows. The character formula (3.31) is clear.

Remark 3.7. R_{1} in Theorem 3.4 is R_{0} in Theorem 2.2. If q_{1,0}^{k} ≠ 1 for all k ∈ IN, then the conditions in Theorem 3.6 are satisfied. If the conditions in Theorem 3.6 are not satisfied, we do not know when the C̃_{q}^{(1)}(m)-module S(Ĥ_{I}^{−}) is reducible except for some special cases. For example, if n = 1, and q_{1,0} is a primitive root of unity of order m ≥ 2, we know that S(Ĥ_{I}^{−}) is a highest weight module of level m over the affine algebra ĝl_{m} with character formula (3.31) which is the m-power of the character of level one highest weight module over ĝl_{m}. Clearly S(Ĥ_{I}^{−}) is a reducible module over ĝl_{m} (also over C̃_{q}^{(1)}(m) = C̃_{q}).

§4. Vertex operator representations of quantum tori at root of unity The quantum tori C_{q} studied in Theorems 2.2, 3.2, 3.4, 3.5 (also in [BGT, BS, G1, G2, GL])

have only t_{0} which does not commute with the remaining commuting variables. Next we shall present vertex operator representations for quantum tori which have more non-commuting variables.

From now on in this section, we fix I = ZZ/mZZ = {1, 2, ..., m} where m ∈ IN, and ε in (3.5) so that

ε(ϵ_{i} − ϵ_{i+1}, ϵ_{j} − ϵ_{j+1}) = \{ \begin{matrix}−1, if j = i or i − 1, \\ 1 otherwise, \end{matrix}  (4.1)

where i = 1, 2, ..., m − 1. Let

η(ϵ_{i} − ϵ_{j}) = \{ \begin{matrix}−1, if j < i, \\ 1 otherwise, \end{matrix}  (4.2)

where i, j ∈ {1, 2, ..., m}. It is easy to see that

η(ϵ_{i} − ϵ_{j})Ė_{i,j}^{k,l}η(ϵ_{j} − ϵ_{j′})Ė_{j,j′}^{l,l′} = η(ϵ_{i} − ϵ_{j′})Ė_{i,j′}^{k,l′},  (4.3)
η(ϵ_{i} − ϵ_{j})e_{ϵ_{i}−ϵ_{j}}η(ϵ_{j} − ϵ_{j′})e_{ϵ_{j}−ϵ_{j′}} = η(ϵ_{i} − ϵ_{j′})e_{ϵ_{i}−ϵ_{j′}}.

Also from now on we fix q = (q_{i,j})_{i,j=0}^{n} satisfying n > 1,

\{ \begin{matrix}q_{i,j} = q_{j,i}^{−1}, q_{i,i} = 1, ∀ i, j ∈ \{0, 1, ..., n\}, \\ q_{i,j} = 1, ∀ i, j ∈ \{2, ..., n\}, \\ q_{1,i} = 1, ∀ i ∈ \{3, ..., n\}, \\ and q_{1,2}^{−1} = q_{2,1} = ω_{m} is a primitive root of unity of order m. \end{matrix}  (4.4)

For this q, generally, (1.6) and (1.6’) do not define Lie algebras. We have to modify them to give Lie algebras. For this q and any nonnegative integer r we define Ĉ_{q}^{(l)}(r) and C̃_{q}^{(l)}(r) by modifying (1.6’) into

[t^{α}, t^{β}] = t^{α}t^{β} − t^{β}t^{α}

+δ_{α_{0},−β_{0}}δ_{ᾱ_{1},−β̄_{1}}δ_{ᾱ_{2},−β̄_{2}}δ_{α+β,radf}δ_{−β̄_{l},ᾱ_{l}}σ(α, β)α_{0}c, ∀ α, β ∈ ZZ^{n+1},  (4.5)

where β̄_{l}, ᾱ_{l} ∈ ZZ_{r}, β̄_{1}, ᾱ_{1}, β̄_{2}, ᾱ_{2} ∈ ZZ_{m}.

Theorem 4.1 Suppose q = (q_{i,j})_{i,j=0}^{n} satisfies (4.4). Set I = ZZ/mZZ = {1, 2, ..., m}. Then Ĉ_{q} = Ĉ_{q}^{(1)}(0) has a vertex operator representation R on the Fock space V_{Q_{I}} so that for any α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}, we have

R( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}) = ∑_{j=1}^{m} η(ϵ_{j−α_{1}} − ϵ_{j})ω_{m}^{−jα_{2}}[e_{ϵ_{j−α_{1}}−ϵ_{j}}z^{ϵ_{j−α_{1}}−ϵ_{j}+1−δ_{ᾱ_{1},0}}(q^{α})^{−ϵ_{j}}(1 − q^{α})^{−δ_{ᾱ_{1},0}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p})],

if q^{α} ≠ 1 or ᾱ_{1} ≠ 0,  (4.6)

R(c) = m, R(t_{0}^{l}t^{α}) = ∑_{j=1}^{m} ω_{m}^{−jα_{2}}ϵ_{j}(l), if q^{α} = 1 and ᾱ_{1} = 0,

where ᾱ_{1} ∈ ZZ_{m}. Via (3.19), V_{Q_{I}} is a weight module over C̃_{q} and

chV_{Q_{I}} = ( ∑_{α∈Q_{I}} x^{−(α|α)/2})ϕ(x^{−1})^{−m}.  (4.7)

Proof. It is straightforward to verify that the following linear map is a Lie algebra homomorphism:

τ : Ĉ_{q} → A_{I×∞}(ε),
t_{0}^{i}t^{α} → ∑_{j=1}^{m} ω_{m}^{−jα_{2}}η(ϵ_{j−α_{1}} − ϵ_{j}) ∑_{k∈ZZ} q^{−kα}Ė_{j−α_{1},j}^{k−i,k} + \frac{δ_{i,0}δ_{\overline{α_{1}},0}δ_{\overline{α_{2}},0}(1 − δ_{q^{α},1})mc}{1 − q^{α}},  (4.8)
c → mc.

If q^{α} ≠ 1, using R_{0} in Theorem 3.1 we deduce that

R( ∑_{k∈ZZ}(t_{0}^{k}t^{α})z^{−k}) := R_{0}( ∑_{k∈ZZ} τ(t_{0}^{k}t^{α})z^{−k})

= ∑_{k∈ZZ} z^{−k}R_{0} (∑_{j=1}^{m} ω_{m}^{−jα_{2}}η(ϵ_{j−α_{1}} − ϵ_{j}) ∑_{p∈ZZ} q^{−pα}Ė_{j−α_{1},j}^{p−k,p} + \frac{δ_{k,0}δ_{\overline{α_{1}},0}δ_{\overline{α_{2}},0}mc}{1 − q^{α}} )

= ∑_{k∈ZZ} z^{−k} ∑_{j=1}^{m} ω_{m}^{−jα_{2}}η(ϵ_{j−α_{1}} − ϵ_{j}) (R_{0}( ∑_{p∈ZZ}(q^{−pα}Ė_{j−α_{1},j}^{p−k,p}) + \frac{δ_{k,0}δ_{\overline{α_{1}},0}c}{1 − q^{α}}))

(letting z_{1} = z, z_{2} = q^{α}z in (3.23) for R_{0})

= ∑_{j=1}^{m} ω_{m}^{−jα_{2}}η(ϵ_{j−α_{1}} − ϵ_{j})[e_{ϵ_{j−α_{1}}−ϵ_{j}}z^{ϵ_{j−α_{1}}−ϵ_{j}+1−δ_{ᾱ_{1},0}}(q^{α})^{−ϵ_{j}}(1 − q^{α})^{−δ_{ᾱ_{1},0}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p})].

The other formulas of (4.6) are quite clear. To see that V_{Q_{I}} is a weight module over C̃_{q}, we need to show that

[R(d_{0}), R( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p})] = R( ∑_{p∈ZZ}(t_{0}^{p}t^{α})pz^{−p}) = −\frac{z∂}{∂z} R( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}),

similar to Theorem 3.4, which can be verified by using (3.17) and

z \frac{∂}{∂z} z^{α} = α(0)z^{α}, ∀ α ∈ Q_{I}.

Other parts of the theorem follows easily. Thus we complete the proof of this theorem.

Remark 4.2. The embedding (4.8) without center is inspired by the following string representation of C_{q} on C^{m×∞} = ⊕_{i∈ZZ_{m}, j∈ZZ}Cv_{i,j} given by

(t_{0}^{p}t^{α})v_{i,j} = ω_{m}^{−iα_{2}}q^{−jα}v_{i−α_{1},j−p}.  (4.9)

Remark 4.3. The Lie algebras considered in Theorems 3.2 and 4.1 are not isomorphic. Indeed, it follows by examining the maximal abelian subalgebras M satisfying c ∉ [Ĉ_{q}, M]. But some of them have isomorphic quotients.

Theorem 4.4 In addition to the assumptions in Theorem 4.1, if one of the following holds then C̃_{q}-module V_{Q_{I}} in Theorem 4.1 is irreducible:

(1) {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α = (α_{1}, α_{2}, ..., α_{n}) ∈ ZZ^{n}} is infinite;

(2) there exists α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} such that m|α_{1}, (α_{2}, m) = 1 and q^{α} = 1.

Proof. Claim 1. ϵ_{j}(l) ∈ U(R_{m}(C̃_{q})), the universal enveloping algebra of R_{m}(C̃_{q}), ∀ j ∈ I, l ∈ ZZ \ {0}.

If Condition (2) holds, Claim 1 follows from the third equation of (4.6), furthermore, ϵ_{j}(0) ∈ U(R_{m}(C̃_{q})), ∀ j ∈ I. Next we suppose Condition (1) holds. Fix l ∈ ZZ. Then the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with m|α_{1} − l} is infinite. For each α = (α_{1}, ...α_{n}) ∈ ZZ^{n} with m|α_{1} − l, let

f_{j}(q^{α}, l, z) = η(ϵ_{j−α_{1}} − ϵ_{j})e_{ϵ_{j−α_{1}}−ϵ_{j}}z^{ϵ_{j−α_{1}}−ϵ_{j}+1−δ_{ᾱ_{1},0}}(q^{α})^{−ϵ_{j}}(1 − q^{α})^{−δ_{ᾱ_{1},0}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−pα}ϵ_{j}(p))}{p}z^{−p}),

where q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}} and we require q^{α} ≠ 1 if ᾱ_{1} = 0. For any v ∈ V_{Q_{I}}, from (4.6) we know that

∑_{j=1}^{m} ω_{m}^{−jα_{2}}f_{j}(q^{α}, l, z)v ∈ (R_{m}(C̃_{q})v)[z, z^{−1}], ∀ l ∈ ZZ, α ∈ ZZ^{n}.

Note that f_{j}(q^{α}, l, z)v ∈ V_{Q_{I}}[z, z^{−1}] is indeed a vector valued function of q^{α}. Thus using Lemma 3.5, we see that, for any j,

f_{j}(q^{α}, l, z)v ∈ (R_{m}(C̃_{q})v)[z, z^{−1}], ∀ l ∈ ZZ, α ∈ ZZ^{n}, v ∈ V_{Q_{I}}.  (4.10)

It is clear that, for any j,

[f_{j}(q^{α}, 0, z_{1}), f_{j}(q^{−α}, 0, z_{2})]v ∈ (R_{m}(U(C̃_{q}))v)[z_{1}^{±1}, z_{2}^{±1}],
∀ α ∈ ZZ^{n} with m|α_{1}, q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}} ≠ 1, ∀ v ∈ V_{Q_{I}}.  (4.11)

As in the proof of Theorem 3.6, we have

[f_{j}(q^{α}, 0, z_{1}), f_{j}(q^{−α}, 0, z_{2})] = ∑_{p,p′∈ZZ} q^{−pα}(q^{(p+p′)α} − 1)ϵ_{j}(p + p′)z_{1}^{−p}z_{2}^{−p′} + (Dδ)(\frac{z_{2}}{q^{α}z_{1}} ).

Since Condition (1) holds, then

ϵ_{j}(p) ∈ R_{m}(U(L)), ∀ p ∈ ZZ \ {0}, j ∈ I.  (4.12)

Claim 1 follows as well.

Claim 2. e_{ϵ_{i}−ϵ_{j}}z^{ϵ_{i}−ϵ_{j}}q_{1,0}^{(i−j)ϵ_{j}} ∈ U(R_{m}(C̃_{q}))[z, z^{−1}], ∀ i, j ∈ I.

Equation (4.10) with ᾱ_{1} ≠ 0 holds even when Condition (2) is true. Indeed, it follows from the fact

ad(ϵ_{j}(0))^{k}R( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}) ∈ (U(R_{m}(C̃_{q}))v)[z, z^{−1}], ∀ k ∈ IN, α ∈ ZZ^{n} with ᾱ_{1} ≠ 0.

From (4.10) with ᾱ_{1} ≠ 0 and Claim 1, we know that, if ᾱ_{1} ≠ 0, for any v ∈ V_{Q_{I}},

e_{ϵ_{j−α_{1}}−ϵ_{j}}z^{ϵ_{j−α_{1}}−ϵ_{j}}q_{1,0}^{−α_{1}ϵ_{j}}v = η(ϵ_{j−α_{1}} − ϵ_{j})z^{−1} exp ( ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−p(α_{1},0...0)}ϵ_{j}(p))}{p}z^{−p})·

·f_{j}(q^{(α_{1},0...0)}, α_{1}, z) exp ( ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}(p) − q^{−p(α_{1},0...0)}ϵ_{j}(p))}{p}z^{−p})v ∈ (U(R_{m}(C̃_{q}))v)[z, z^{−1}].

Thus

e_{ϵ_{j−α_{1}}−ϵ_{j}}z^{ϵ_{j−α_{1}}−ϵ_{j}}q_{1,0}^{−α_{1}ϵ_{j}} ∈ (U(R_{m}(C̃_{q})))[z, z^{−1}].

Claim 2 follows.

Now we are ready to show the irreducibility of C̃_{q}-module V_{Q_{I}}. Suppose W is a nonzero C̃_{q}-submodule of V_{Q_{I}}. Then d_{0} acts diagonally on W. From Claim 1 we can choose a nonzero weight element in W of the form

w = ∑_{i=1}^{r} u_{i} ⊗ e^{γ_{i}} ∈ W,  (4.13)

where u_{i} ∈ S(Ĥ_{I}^{−}), γ_{i} ∈ Q_{I}, such that r is minimal.

Suppose r ≥ 2. We see that any two of γ_{i}’s are distinct and

deg(u_{1}) − \frac{(γ_{1}|γ_{1})}{2} = deg(u_{i}) − \frac{(γ_{i}|γ_{i})}{2}, ∀ i = 1, ..., r.

Since γ_{1} ≠ γ_{2}, there exist k, l ∈ I such that

(ϵ_{k} − ϵ_{l}|γ_{1}) ≠ (ϵ_{k} − ϵ_{l}|γ_{2}).

From Claim 2, we deduce that

e_{ϵ_{k}−ϵ_{l}}z^{ϵ_{k}−ϵ_{l}}q_{1,0}^{(l−k)ϵ_{j}}w = ∑_{i=1}^{r} ε(ϵ_{k} − ϵ_{l}, γ_{i})z^{(ϵ_{k}−ϵ_{l}|γ_{i})}q_{1,0}^{(l−k)(ϵ_{j}|γ_{i})}u_{i} ⊗ e^{ϵ_{k}−ϵ_{l}+γ_{i}} ∈ W [z, z^{−1}]. (4.14)

Since the powers of z in the right hand side of (4.13) are not the same, we get a nonzero element in W which has expression like in (4.13) with smaller r. This contradicts the minimality of r. Consequently r = 1. From Claims 1 and 2, we know that any nonzero element u ⊗ e^{γ} can generate the whole C̃_{q}-module V_{Q_{I}}, where u ∈ S(Ĥ_{I}^{−}), γ ∈ Q, thus W = V_{Q_{I}}. This completes the proof of this theorem.

Next we shall construct level mr vertex operator representations for the algebra Ĉ_{q}^{(l)}(r) for any positive integer r defined in (4.5). For any k ∈ {1, 2, ..., r}, let I = I_{k} = {1, 2, ..., m}. For each I_{k}, along from (3.3) through (3.21), similarly we define P_{I_{k}} = ⊕_{j∈I}ZZϵ_{j}^{(k)} with the standard bilinear form, Q_{I^{(k)}} = ⊕_{i,j∈I_{i}}ZZ(ϵ_{i}^{(k)} − ϵ_{j}^{(k)}), H_{I^{(k)}} and Ĥ_{I^{(k)}} (with the same c, d_{0}), the operators e_{β} ∈ EndC[Q_{I^{(k)}}] for any β ∈ Q_{I^{(k)}}, the operators β(0) ∈ EndC[Q_{I^{(k)}}] for any β ∈ H_{I^{(k)}}, the operators z^{β} ∈ EndC[Q_{I^{(k)}}][[z]] for any β ∈ Q_{I^{(k)}} (operators for different k are commutative), and the vector space

V_{Q_{I^{(k)}}} = C[Q_{I^{(k)}}] ⊗ S(H_{I^{(k)}}^{−}).

Then we have the tensor product

(V_{Q_{I}})^{r} = ⊗_{i=1}^{r}V_{Q_{I^{(k)}}}.  (4.15)

Thus we have the representation R^{(k)} of Ĉ_{q}^{(l)}(0) on each V_{Q_{I^{(k)}}} in the sense of Theorem 4.1.

Theorem 4.12 Suppose ω_{m}, q = (q_{i,j})_{i,j=0}^{n} are given as in Theorem 3.8, l ∈ {1, 2, ..., n}. For any positive integer r, let I_{i} = ZZ/mZZ = {1, 2, ..., m} for i ∈ {1, 2, ..., r}, let ω_{r} be the primitive root of unity of order r. Then Ĉ_{q}^{(l)}(r) has a vertex operator representation R_{r}^{(l)} on the Fock space (4.15) so that for any α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}, we have

R_{r}^{(l)}( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}) = ∑_{k=1}^{r} ω_{r}^{−kα_{l}} ∑_{j=1}^{m} η(ϵ_{j−α_{1}}^{(k)} − ϵ_{j}^{(k)})ω_{m}^{−jα_{2}}·

·[e_{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}+1−δ_{ᾱ_{1},0}}(q^{α})^{−ϵ_{j}^{(k)}}(1 − q^{α})^{−δ_{ᾱ_{1},0}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p})],

if q^{α} ≠ 1 or ᾱ_{1} ≠ 0,  (4.16)

R_{r}^{(l)}(c) = mr, R_{r}^{(l)}(t_{0}^{i}t^{α}) = ∑_{k=1}^{r} ω_{r}^{−kα_{l}} ∑_{j=1}^{m} ω_{m}^{−jα_{2}}ϵ_{j}^{(k)}(i), if q^{α} = 1 and ᾱ_{1} = 0,

where ᾱ_{1} ∈ ZZ_{m}. Via (3.19), (V_{Q_{I}})^{r} is a weight module over C̃_{q}^{(l)}(r) and

ch(V_{Q_{I}})^{r} = ( ∑_{α∈Q_{I}} x^{−(α|α)/2})^{r}ϕ(x^{−1})^{−mr}.  (4.17)

Proof. We denote by R^{(k)} the vertex operator representation of Ĉ_{q} = Ĉ_{q}^{(1)}(0) on V_{Q_{I^{(k)}}} in the sense of Theorem 4.1. We still use R^{(k)} to denote the extended representation of Ĉ_{q} on ⊗_{i=1}^{r}V_{Q_{I^{(k)}}}, for example,

R^{(1)}(t^{α}) = R^{(1)}(t^{α}) ⊗ 1 ⊗ ... ⊗ 1,
R^{(2)}(t^{α}) = 1 ⊗ R^{(1)}(t^{α}) ⊗ 1 ⊗ ... ⊗ 1.

Thus we have

[R^{(k)}(t^{α}), R^{(j)}(t^{α′})] = δ_{k,j}[(σ(α, α′) − σ(α′, α))R^{(k)}(t^{α+a′})

+δ_{α+α′,radf}δ_{α_{0},−α′_{0}}δ_{ᾱ_{1},−ᾱ′_{1}}δ_{ᾱ_{2},−ᾱ′_{2}}σ(α, α′)mα_{0}], ∀ α, α′ ∈ ZZ^{n+1}.

From (4.6) we know that

R_{r}(t^{α}) = ∑_{k=1}^{r} ω_{r}^{−kα_{l}}R^{(k)}(t^{α}) ∀ α ∈ ZZ^{n+1}.

Then for any α, α′ ∈ ZZ^{n+1} we deduce that

[R_{r}(t^{α}), R_{r}(t^{α′})] = [∑_{k=1}^{r} ω_{r}^{−kα_{l}}R^{(k)}(t^{α}), ∑_{k=1}^{r} ω_{r}^{−kα′_{l}}R^{(k)}(t^{α′})]

= ∑_{k=1}^{r} ω_{r}^{−k(α_{l}+α′_{l})}[R^{(k)}(t^{α}), R^{(k)}(t^{α′})]

= ∑_{k=1}^{r} ω_{r}^{−k(α_{l}+α′_{l})}[(σ(α, α′) − σ(α′, α))R^{(k)}(t^{α+a′}) + δ_{α+α′,radf}δ_{α_{0},−α′_{0}}δ_{ᾱ_{1},−ᾱ′_{1}}δ_{ᾱ_{2},−ᾱ′_{2}}σ(α, α′)mα_{0}]

= (σ(α, α′) − σ(α′, α))R_{r}^{(l)}(t^{α+a′}) + δ_{α+α′,radf}δ_{ᾱ_{l},−ᾱ′_{l}}δ_{α_{0},−α′_{0}}δ_{ᾱ_{1},−ᾱ′_{1}}δ_{ᾱ_{2},−ᾱ′_{2}}σ(α, α′)mrα_{0}
= R_{r}^{(l)}((σ(α, α′) − σ(α′, α))t^{α+a′} + δ_{α+α′,radf}δ_{ᾱ_{l},−ᾱ′_{l}}δ_{α_{0},−α′_{0}}δ_{ᾱ_{1},−ᾱ′_{1}}δ_{ᾱ_{2},−ᾱ′_{2}}σ(α, α′)α_{0}c)
= R_{r}^{(l)}([t^{α}, t^{α′}]),

where ᾱ_{l}, −ᾱ′_{l} ∈ ZZ_{r} ᾱ_{1}, ᾱ′_{1}, ᾱ_{2}, −ᾱ′_{2} ∈ ZZ_{m} and the last bracket is in C̃_{q}^{(l)}(r). So R_{r}^{(l)} is a representation of C̃_{q}^{(l)}(r). The character formula (4.17) is clear. Thus we proved the theorem.

Like in Remark 3.7 for R_{m} in Theorem 3.4, some representations R_{r}^{(l)} of C̃_{q}^{(l)}(r) are irreducible. Now we give some sufficient conditions for R_{r}^{(l)} to be irreducible.

Theorem 4.13 The vertex operator representation R_{r}^{(l)} of C̃_{q}^{(l)}(r) in Theorem 4.12 is irreducible if one of the following holds:

(1) (m, r) = 1 and there exists α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} such that m|α_{1}, (α_{l}, r) = (α_{2}, m) = 1, and q^{α} = 1;

(2) l > 2 and the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}} is infinite;

(3) l ∈ {1, 2}, (m, r) = 1 and the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α = (α_{1}, ..., α_{n}) ∈ ZZ^{n}} is infinite. Proof. The proof of this theorem is very similar to that of Theorem 3.6. So here we only

outline the proof for Cases (1) and (2).

(1) From the hypothesis and the third equation of (4.16), we deduce that ϵ_{j}^{(k)}(i) ∈ U(R_{r}^{(l)}(C̃_{q}^{(l)}(r))), the universal enveloping algebra of R_{r}^{(l)}(C̃_{q}^{(l)}(r)), ∀ j ∈ I, i ∈ ZZ, k ∈ {1, 2, ..., r}. From

ad(ϵ_{j}^{(k)}(0))^{i}R_{r}^{(l)}( ∑_{p∈ZZ}(t_{0}^{p}t^{α})z^{−p}) ∈ U(R_{r}^{(l)}(C̃_{q}^{(l)}(r)))[z, z^{−1}], ∀ i ∈ IN, α ∈ ZZ^{n} with α_{1} ≠ 0,

we deduce that, if ᾱ_{1} ≠ 0, then

f_{j}^{(k)}(α, z) = e_{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}+1}(q^{α})^{−ϵ_{j}^{(k)}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p})

∈ U(R_{r}^{(l)}(C̃_{q}^{(l)}(r)))[z, z^{−1}];

e_{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}}q_{1,0}^{−α_{1}ϵ_{j}^{(k)}} = z^{−1} exp ( ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q_{1,0}^{−pα_{1}}ϵ_{j}^{(k)}(p))}{p}z^{−p})·

·f_{j}^{(k)}((α_{1}, 0, ..., 0), z) exp ( ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q_{1,0}^{−pα_{1}}ϵ_{j}^{(k)}(p))}{p}z^{−p}) ∈ U(R_{m}(C̃_{q}))[z, z^{−1}].

Thus

e_{ϵ_{i}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{i}^{(k)}−ϵ_{j}^{(k)}}q_{1,0}^{(i−j)ϵ_{j}^{(k)}} ∈ U(R_{r}^{(l)}(C̃_{q}^{(l)}(r)))[z, z^{−1}], ∀ ī ≠ j̄ ∈ ZZ_{m}.

Suppose W is a nonzero C̃_{q}^{(l)}(r)-submodule of (V_{Q_{I}})^{r}. Then d_{0} acts diagonally on W. We can choose a nonzero weight element in W of the form

w = ∑_{i=1}^{r} u_{i} ⊗ e^{γ_{i}} ∈ W,  (4.18)

where u_{i} ∈ S(⊗_{j=1}^{r}Ĥ_{I_{j}}^{−}), γ_{i} ∈ ⊗_{j=1}^{r}Q_{I^{(j)}}, such that r is minimal.

Repeat the last paragraph of the proof of Theorem 4.12 we see that W = (V_{Q_{I}})^{r}.

(2) From the hypothesis we know that for any fixed s_{1}, s_{2}, s_{l}, the set {q^{α} = q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}}|α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with α_{1} = s_{1}(mod m), α_{2} = s_{2}(mod m), α_{l} = s_{l}(mod r)} is infinite. Using Lemma 3.5 we deduce that

f_{j}^{(k)}(α, z) = e_{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{j−α_{1}}^{(k)}−ϵ_{j}^{(k)}+1−δ_{ᾱ_{1},0}}(q^{α})^{−ϵ_{j}^{(k)}}(1 − q^{α})^{−δ_{ᾱ_{1},0}}·

· exp ( − ∑_{p∈−IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p}) exp ( − ∑_{p∈IN} \frac{(ϵ_{j−α_{1}}^{(k)}(p) − q^{−pα}ϵ_{j}^{(k)}(p))}{p}z^{−p})

∈ U(R_{r}^{(l)}(C̃_{q}))[z, z^{−1}], ∀ α ∈ ZZ^{n} with q^{α} ≠ 1 or ᾱ_{1} ≠ 0.

For any α ∈ ZZ^{n} with α_{1} = 0(mod m) and q^{α} ≠ 1, similar to (4.12) by computing

[f_{j}^{(k)}(α, z_{1}), f_{j}^{(k)}(−α, z_{2})] ∈ U(R_{r}^{(l)}(C̃_{q}))[z, z^{−1}],

we deduce that

ϵ_{j}^{(k)}(p) ∈ R_{m}(U(L)), ∀ p ∈ ZZ \ {0}, j, k.  (4.19)

Similar to Claim 2 in the proof of Theorem 4.11, we can also show that

e_{ϵ_{i}^{(k)}−ϵ_{j}^{(k)}}z^{ϵ_{i}^{(k)}−ϵ_{j}^{(k)}}q_{1,0}^{(i−j)ϵ_{j}^{(k)}} ∈ U(R_{r}^{(l)}(C̃_{q}))[z, z^{−1}], ∀ i, j ∈ I, ∀ k.

Then we repeat the remaining discussion for Condition (1) to see the irreducibility of the representation.

Remark 4.14 If we assume that q_{1,0}^{α_{1}}...q_{n,0}^{α_{n}} = 1 implies α = (α_{1}, α_{2}, ..., α_{n}) = 0, then

C̃_{q}^{(l)}(r) ≃ C̃_{q}^{(l)}(r) ≃ C̃_{q} and Ĉ_{q}^{(l)}(r) ≃ Ĉ_{q}^{(l)}(r) ≃ Ĉ_{q} for all the algebras discussed in this section, and all the modules constructed in this paper for these algebras are irreducible.

§5. Highest weight modules

For brevity, we denote L = Ĉ_{q} , C̃_{q}, Ĉ_{q}^{(l)}(r) , C̃_{q}^{(l)}(r), defined by (1.6) and (1.6’) under the condition (1.5), or defined by (4.5) under the condition (4.4). With respect to d_{0}, L has the ZZ-gradation (1.7). Let L_{+} = ⊕_{i∈ZZ_{+}}L_{i}, L_{−} = ⊕_{i<0}L_{i}.

Now we define highest weight modules over L. Suppose ċ, ḋ ∈ C, λ = (λ_{α}) where λ_{α} ∈ C, α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} (λ_{α} ∈ C can be arbitrary if q^{α} = 1). We can define the 1-dimensional L_{+}-module Cv_{0} via

L_{i}v_{0} = 0, if i > 0; cv_{0} = ċv_{0}, dv = ḋv_{0}, t^{α}v_{0} = λ_{α}v_{0}, ∀ α ∈ ZZ^{n}.  (5.1)

This implies that [L_{0}, L_{0}]v_{0} = 0. Then we have the induced L-module

V̄ (λ, ċ, ḋ) = Ind_{L}^{L_{+}}Cv_{0} = U(L) ⊗_{U(L_{+}+L_{0})} Cv_{0},

where U(L) is the universal enveloping algebra of the Lie algebra L. It is clear that, as vector spaces, V̄ (λ, ċ, ḋ) ≃ U(L_{−}). The module V̄ (λ, ċ, ḋ) has a unique maximal proper submodule J. Then we obtain the irreducible module

V (λ, ċ, ḋ) = \frac{V̄ (λ, ċ, ḋ)}{J}.  (5.2)

It is clear that V (λ, ċ, ḋ) is uniquely determined by the parameters λ, ċ, ḋ. Since the structure of V (λ, ċ, ḋ) is independent of ḋ, so we shall always assume that ḋ = 0, and simply denote V (λ, ċ, ḋ) by V (λ, ċ). Generally, not all weight spaces of V (λ, ċ, ḋ) are finite-dimensional. Recently, the necessary and sufficient conditions for V (λ, ċ) to have finite-dimensional weight spaces were determined for n = 1 in [RZ]. All the modules constructed in Sections 2 and 3 have finite-dimensional weight spaces. The following result is quite clear.

Theorem 5.1 (a) In Theorem 2.2, B ≃ V (λ, 1, 0), where λ_{α} = \frac{q^{−mα}}{1−q^{α}} for all α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with q^{α} ≠ 1, and λ_{α} = m otherwise. The highest weight vector in B is 1.

(b) In Theorem 3.6 (or 3.4), S(Ĥ_{I}^{−}) ≃ V (λ, m, 0), where

λ_{α} = ∑_{j=1}^{m} \frac{ω_{m}^{−jα_{1}}}{(1 − q^{α})} = δ_{ᾱ_{1},0} \frac{m}{(1 − q^{α})}, ∀ α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with q^{α} ≠ 1,

and λ_{α} = 0 otherwise. The highest weight vector in S(Ĥ_{I}^{−}) is 1.

(c) In Theorem 4.4 (or 4.1), V_{Q_{I}} ≃ V (λ, m, 0), where

λ_{α} = δ_{ᾱ_{1},0} ∑_{j=1}^{m} \frac{ω_{m}^{−jα_{2}}}{(1 − q^{α})} = δ_{ᾱ_{1},0}δ_{ᾱ_{2},0} \frac{m}{(1 − q^{α})}, ∀ α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with q^{α} ≠ 1,

and λ_{α} = 0 otherwise. The highest weight vector in V_{Q_{I}} is 1 ⊗ 1.

(d) In Theorem 4.6 (or 4.5), (V_{Q_{I}})^{r} ≃ V (λ, mr, 0), where

λ_{α} = δ_{ᾱ_{1},0} ∑_{i=1}^{r} ω_{r}^{−iα_{l}} ∑_{j=1}^{m} \frac{ω_{m}^{−jα_{2}}}{(1 − q^{α})}

= δ_{ᾱ_{1},0}δ_{ᾱ_{2},0}δ_{ᾱ_{l},0} \frac{mr}{(1 − q^{α})}, ∀ α = (α_{1}, ..., α_{n}) ∈ ZZ^{n} with q^{α} ≠ 1,

and λ_{α} = 0 otherwise. The highest weight vector in (V_{Q_{I}})^{r} is 1 = (1 ⊗ 1)^{r}.

We cannot ignore the fact that the character formulas in Theorems 3.2 and 4.1 are the same. This coincidence suggests that some isomorphism may exist. This is indeed the case. In Theorem 3.2 we know that V_{Q_{I}} is an irreducible highest weight module with highest weight vector 1 ⊗ 1 by the following sense

\{ \begin{matrix}(E_{i,j}(t_{0}^{k}t^{a}))(1 ⊗ 1) = 0, if k ∈ IN, \\ (E_{i,j}(t_{0}^{0}t^{a}))(1 ⊗ 1) = 0, if i ≠ j, \\ c(1 ⊗ 1) = 1, \\ (E_{i,i}(t_{0}^{0}t^{a}))1 ⊗ 1 = \{ \begin{matrix}\frac{1⊗1}{(1−q^{α})}, if q^{α} ≠ 1, \\ 0, if q^{α} = 1, \end{matrix} \end{matrix}  (5.3)

for all α = (α_{1}, α_{2}, ..., α_{n}) ∈ ZZ. Now we construct a new quantum torus C_{q′} from C_{q} in Theorem 3.2 as follows: C_{q′} is generated by generators t_{0}^{±1}, E^{±1}, F^{±1}, t_{1}^{±1}, ...t_{n}^{±1}, where t_{0}, t_{1}, ...t_{n} has the same relations as in C_{q}, E and F commute with t_{0}, t_{1}, ...t_{n}, but EF = ω_{n}F E. Homogeneous elements in C_{q′} are written as t_{0}^{α_{0}}E^{α_{1}}F^{α_{2}}t_{1}^{α_{3}}...t_{n}^{α_{n+2}} for α = (α_{0}, α_{1}, α_{2}, α_{3}, ...α_{n+2}) ∈ ZZ^{n+3}. Then we have the irreducible highest weight C̃_{q′}-module with highest weight vector 1 ⊗ 1 in Theorem 4.1 such that

\{ \begin{matrix}c(1 ⊗ 1) = m, (t_{0}^{α_{0}}E^{α_{1}}F^{α_{2}}t_{1}^{α_{3}}...t_{n}^{α_{n+2}})(1 ⊗ 1) = 0, if α_{0} ∈ IN, \\ (t_{0}^{0}E^{α_{1}}F^{α_{2}}t_{1}^{α_{3}}...t_{n}^{α_{n+2}})1 ⊗ 1 = \{ \begin{matrix}δ_{ᾱ_{1},0}δ_{ᾱ_{2},0} \frac{m}{(1−q^{α})}, if q^{α} ≠ 1, \\ 0, if q^{α} = 1, \end{matrix} \end{matrix}  (5.4)

where q^{α} = q_{1,0}^{α_{3}}...q_{n,0}^{α_{n+3}}. Via the Lie homomorphism

\{ \begin{matrix}g̃l_{m}(C_{q}) → C̃_{q′}, \\ c → \frac{c}{m}, d_{0} → d_{0}, \\ E_{i,j}(t_{0}^{k}t^{a}) → \frac{1}{m}t_{0}^{k}E^{j−i}(ω_{m}^{−j}F + (ω_{m}^{−j}F)^{2} + ... + (ω_{m}^{−j}F)^{m})t_{1}^{α_{1}}...t_{n}^{α_{n}}, \end{matrix}  (5.5)

one can verify that the modules defined in (5.3) and (5.4) are the same. Thus the module in Theorem 3.2 is a module in Theorem 4.1 with q_{1,0} = q_{2,0} = 1. Thus we have proved

Theorem 5.2 The module in Theorem 3.2 is a module in Theorem 4.1 with q_{1,0} = q_{2,0} = 1.

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