MINIMAL MODEL FUSION RULES FROM 2-GROUPS

Füsun Akman Alex J. Feingold^{1} Michael D. Weiner

Department of Mathematics Cornell University Ithaca, New York 14853-7901

Department of Mathematical Sciences The State University of New York Binghamton, New York 13902-6000

Department of Mathematics Colgate University Hamilton, NY 13346

Abstract

The fusion rules for the (p, q)-minimal model representations of the Virasoro algebra are shown to come from the group G = Z_{2}^{p+q−5} in the following manner. There is a partition G = P_{1} ∪ ... ∪ P_{N} into disjoint subsets and a bijection between {P_{1}, ..., P_{N}} and the sectors {S_{1}, ..., S_{N}} of the (p, q)-minimal model such that the fusion rules S_{i} ∗ S_{j} = ∑_{k} D(S_{i}, S_{j}, S_{k})S_{k} correspond to P_{i} ∗ P_{j} = ∑_{k∈T (i,j)} P_{k} where T (i, j) = {k|∃a ∈ P_{i}, ∃b ∈ P_{j}, a + b ∈ P_{k}}.

The Virasoro algebra Vir is the infinite dimensional Lie algebra with basis {c, L_{m} | m ∈ Z} and brackets

[L_{m}, L_{n}] = (m − n)L_{m+n} + \frac{m^{3} − m}{12}δ_{m,−n}c

where c is central. A highest weight representation [KR] of Vir is a module M containing a vector v_{0} such that L_{m}v_{0} = 0 for all m ≥ 1, L_{0}v_{0} = hv_{0} for some h ∈ C, and M is generated by v_{0}. It means that M is spanned by vectors of the form

L_{−n_{1}}L_{−n_{2}} … L_{−n_{r}}v_{0}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 17B68, 81R10; Secondary 81T40.

1Partially supported by the National Security Agency under Grant number MDA904-94-H-2019. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon.

This paper is in final form and no version of it will be submitted for publication elsewhere.

Revised: Jan. 3, 1996

for n_{1} ≥ n_{2} ≥ … ≥ n_{r} ≥ 1 with r ≥ 0. Since c is central, its value on M is a scalar, which we continue to denote by c. The Verma module M(c, h) is the module, determined up to isomorphism by the values of c and h, such that the above vectors form a basis. For certain values of these parameters the Verma module is not irreducible, but contains nonzero null vectors, that is, vectors which generate proper highest weight submodules. Taking the quotient of M(c, h) by its maximal proper submodule, consisting of the sum of all submodules generated by null vectors, yields the irreducible Vir-module denoted by V (c, h). Note that in order to determine if a vector v_{0} such that L_{0}v_{0} = hv_{0} is null, it suffices to check whether L_{1} and L_{2} kill v_{0}, since the bracket formula shows that brackets of these generate all L_{m} for m ≥ 1.

In [FZ] it is shown that for any c ∈ C, V (c, 0) has the structure of a Vertex Operator Algebra (VOA) [FLM] and each V (c, h) is a module for that VOA. Various special classes of Vir-modules have been investigated and classified. A highest weight module with c and h real admits a Hermitian form (·, ·) determined by the conditions

(v_{0}, v_{0}) = 1  and  (L_{m}v, w) = (v, L_{−m}w)

for any v, w in the module and any integer m. An important class of highest weight modules consists of the unitary modules for which the Hermitian form is positive definite. If c ≥ 1 and h ≥ 0 are real then V (c, h) is unitary, but there is also a discrete series of unitary modules with c < 1 and only a finite number of corresponding special h values. The c values are given by the formula

c = 1 − \frac{6}{p(p + 1)} for p ≥ 2

and the h values are given by

h_{m,n} = \frac{(n(p + 1) − mp)^{2} − 1}{4p(p + 1)} for 0 < m < p + 1, 0 < n < p.

A larger discrete series of modules was investigated in [BPZ], and called the minimal models. For integers p, q ≥ 2 with gcd(p, q) = 1, the (p, q)-minimal model consists of the Virasoro modules V (c, h_{m,n}) whose values of c and h_{m,n} are given by

c = 1 − \frac{6(p − q)^{2}}{pq}

and

h_{m,n} = \frac{(np − mq)^{2} − (p − q)^{2}}{4pq} for 0 < m < p, 0 < n < q.

From the above formula, which we use to define h_{m,n} for the rest of the paper, we see that h_{m,n} = h_{p−m,q−n}, and the number of distinct such h_{m,n} is N = \frac{1}{2}(p − 1)(q − 1). These modules are of special interest from the viewpoint of VOAs because V (c, 0) has the structure of a rational VOA whose only irreducible VOA-modules are the modules V (c, h_{m,n}). In that case, one may hope to completely analyze the spaces of intertwining operators between modules (see [FHL]) and understand the fusion rules which give the dimensions of those spaces. In [W] the fusion rules for all minimal models are proved. The result is as follows.

Definition. For integer p ≥ 2, we say that the triple of integers (m, m′, m′′) is padmissible when 0 < m, m′, m′′ < p, the sum m + m′ + m′′ < 2p is odd, and the “triangle” inequalities m < m′ + m′′, m′ < m + m′′, and m′′ < m + m′ are satisfied. For p, q ≥ 2 with gcd(p, q) = 1, we say that ((m, n), (m′, n′), (m′′, n′′)) is (p,q)-admissible when (m, m′, m′′) is p-admissible and (n, n′, n′′) is q-admissible.

Lemma 1. For 0 < m′ ≤ m < p, the set of all m′′ such that (m, m′, m′′) is p-admissible equals {m − m′ + 1 + 2i | 0 ≤ i ≤ min(m′, p − m) − 1}.

Proof. If 0 < m′ ≤ m < p then m′′ = m−m′+1+2i means m+m′+m′′ = 2m+1+2i is odd. Two of the triangle inequalities are satisfied: m′ + m′′ ≥ m′ + (m − m′ + 1) = m + 1 > m, and m + m′′ ≥ m + (m − m′ + 1) ≥ m′ + 1 > m′. We consider the two cases where (1) m′ ≤ p − m, and (2) m′ > p − m.

Case 1: Suppose that 1 ≤ m − m′ + 1 ≤ m′′ ≤ m + m′ − 1 and m + m′ + m′′ is odd. Then m + m′ > m + m′ − 1 ≥ m′′. We have m′′ < p because m′′ ≤ m + m′ − 1 ≤ m + (p − m) − 1 = p − 1 < p, so that m + m′ + m′′ ≤ p + m′′ < 2p.

Case 2: Let 1 ≤ m − m′ + 1 ≤ m′′ ≤ 2p − (m + m′) − 1 with m + m′ + m′′ odd. We have m′′ ≤ 2p − (m + m′) − 1 < 2p − p − 1 < p so m + m′ > p > m′′. We also have m + m′ + m′′ ≤ m + m′ + 2p − (m + m′) − 1 < 2p.

We now have to show that (m, m′, m′′) is not p-admissible if m′′ < m − m′ + 1 or if m′′ > m − m′ − 1 + 2 min(m′, p − m). If m′′ < m − m′ + 1, then m′ + m′′ < m + 1, so m′ + m′′ ≤ m, violating a triangle inequality. For large values of m′′ we again consider the two cases above. In case (1) the condition m′′ > m + m′ − 1 implies m′′ ≥ m + m′. In case (2) if m′′ > 2p − (m + m′) − 1 then m + m′ + m′′ > 2p − 1, so m + m′ + m′′ ≥ 2p. ■

Definition. Let N_{(m′,n′),(m′′,n′′)}^{(m,n)} = 1 if ((m, n), (m′, n′), (m′′, n′′)) is (p, q)-admissible, 0 otherwise.

Theorem. [Wang] The fusion rules for the minimal models are

V (c, h_{m′,n′}) × V (c, h_{m′′,n′′}) = ∑_{(m,n)} N_{(m′,n′),(m′′,n′′)}^{(m,n)}V (c, h_{m,n}).

Note that the definition of p-admissible is totally symmetric in the three integers, so the definition of (p, q)-admissible has an S_{3} × S_{3} symmetry. While h_{m,n} = h_{p−m,q−n}, the definition of admissibility does not allow the arbitrary replacement of a pair (m, n) by (p − m, q − n). In fact, if the triple ((m, n), (m′, n′), (m′′, n′′)) is (p, q)-admissible, then ((m, n), (m′, n′), (p − m′′, q − n′′)) is not, as m + m′ + m′′ and n + n′ + n′′ are odd, while m + m′ + p − m′′ and n + n′ + q − n′′ cannot both be odd (p, q cannot have a common factor of 2). Therefore the sum in the Theorem is meant to be over all pairs (m, n), but only one of V (c, h_{m,n}) and V (c, h_{p−m,q−n}) may have a nonzero coefficient.

For a fixed (p, q)-minimal model, we will use the notation [h] = V (c, h) to write the fusion rules more briefly. We may give these rules in a table, which may be interpreted as the multiplication table for the associated Verlinde algebra. The fusion rules involving [0] are just [0] × [h] = [h], but we include these in our tables anyway.

Definition. The Verlinde algebra V associated with a fixed (p, q)-minimal model is the associative algebra with basis indexed by the distinct modules [h_{m,n}] and with product given by the fusion rules.

In the case of p = 3, q = 4, one gets c = \frac{1}{2}. The values of h_{m,n} = h_{3−m,4−n} for 0 < m < 3 and 0 < n < 4 are given in the following table.

h_{m,n} n = 1 n = 2 n = 3
m = 1  0  ^{\frac{1}{16}}  ^{\frac{1}{2}}
m = 2  ^{\frac{1}{2}}  ^{\frac{1}{16}}  0

The c = \frac{1}{2} fusion rule table is as follows:

[h] × [h′] [0]  [\frac{1}{16}]  [\frac{1}{2}]
[0]  [0]  [\frac{1}{16}]  [\frac{1}{2}]
[\frac{1}{16}]  [\frac{1}{16}] [0] + [\frac{1}{2}] [\frac{1}{16}]
[\frac{1}{2}]  [\frac{1}{2}]  [\frac{1}{16}]  [0]

This well-known case, called the Ising model, was studied in great detail in [FRW] using the spinor construction [FFR] from one fermion. The objective was to unify the superVOA V (\frac{1}{2}, 0) ⊕ V (\frac{1}{2}, \frac{1}{2}) (the Neveu-Schwarz sector) with its module V (\frac{1}{2}, \frac{1}{16}) ⊕ V (\frac{1}{2}, \frac{1}{16}) (the Ramond sector), and to find the generalization of the Jacobi-Cauchy identity obeyed by the intertwining operators from the Ramond sector. The spinor construction naturally provided two copies of the h = \frac{1}{16} module, and the above Ising fusion rules were replaced by fusion rules given by the group Z_{4}. It meant that each vector in the Ramond sector corresponded to a unique intertwining operator. The question remained whether that was a special situation, or if other fusion rules could be rewritten in terms of a group by taking multiple copies of certain sectors. While the Ising fusion rules could also have come from the group Z_{2} × Z_{2}, it was not checked if that choice would be consistent with the VOAmodule structure and the generalization of the Jacobi-Cauchy identity for intertwiners.

Let us explain further what we mean by the replacement of these Ising fusion rules by fusion rules given by the group Z_{4} = {0, 1, 2, 3} under addition modulo 4. We associate these four group elements with the four irreducible Vir-modules from the fermionic construction as follows:

0 ↔ [0], 1 ↔ [\frac{1}{16}]_{1}, 2 ↔ [\frac{1}{2}], 3 ↔ [\frac{1}{16}]_{2}.

Then the addition in Z_{4} would give the Ising fusion rules if we could not distinguish between the two copies of the [\frac{1}{16}]-module. Below we will give a precise definition of when a finite abelian group covers the fusion rules of a (p, q)-minimal model.

The next example to examine is p = 4, q = 5, so c = \frac{7}{10}, h ∈ \{0, \frac{3}{5}, \frac{7}{16}, \frac{3}{80}, \frac{1}{10}, \frac{3}{2}\}. The values of h_{m,n} = h_{4−m,5−n} for 0 < m < 4 and 0 < n < 5 are given in the following table.

h_{m,n} n = 1 n = 2 n = 3 n = 4
m = 1  0  ^{\frac{1}{10}}  ^{\frac{3}{5}}  ^{\frac{3}{2}}
m = 2  ^{\frac{7}{16}}  ^{\frac{3}{80}}  ^{\frac{3}{80}}  ^{\frac{7}{16}}
m = 3  ^{\frac{3}{2}}  ^{\frac{3}{5}}  ^{\frac{1}{10}}  0

The c = \frac{7}{10} fusion rule table is as follows:

[h] × [h′] [0]  [\frac{3}{5}]  [\frac{7}{16}]  [\frac{3}{80}]  [\frac{1}{10}]  [\frac{3}{2}]
[0]  [0]  [\frac{3}{5}]  [\frac{7}{16}]  [\frac{3}{80}]  [\frac{1}{10}]  [\frac{3}{2}]
[\frac{3}{5}]  [\frac{3}{5}] [0] + [\frac{3}{5}]  [\frac{3}{80}]  [\frac{3}{80}] + [\frac{7}{16}]  [\frac{1}{10}] + [\frac{3}{2}] [\frac{1}{10}]
[\frac{7}{16}]  [\frac{7}{16}]  [\frac{3}{80}]  [0] + [\frac{3}{2}]  [\frac{1}{10}] + [\frac{3}{5}]  [\frac{3}{80}]  [\frac{7}{16}]
[\frac{3}{80}]  [\frac{3}{80}] [\frac{3}{80}] + [\frac{7}{16}] [\frac{1}{10}] + [\frac{3}{5}] [0] + [\frac{3}{5}] + [\frac{3}{2}] + [\frac{1}{10}] [\frac{7}{16}] + [\frac{3}{80}] [\frac{3}{80}]
[\frac{1}{10}]  [\frac{1}{10}] [\frac{1}{10}] + [\frac{3}{2}]  [\frac{3}{80}]  [\frac{7}{16}] + [\frac{3}{80}]  [0] + [\frac{3}{5}] [\frac{3}{5}]
[\frac{3}{2}]  [\frac{3}{2}]  [\frac{1}{10}]  [\frac{7}{16}]  [\frac{3}{80}]  [\frac{3}{5}]  [0]

We would like to point out an interesting feature of the above fusion rule table. If you examine the number of times that an entry appears on each row, you find that [0] and [\frac{3}{2}] each appear at most once, [\frac{3}{5}], [\frac{7}{16}] and [\frac{1}{10}] each appear at most twice, and [\frac{3}{80}] appears at most four times. In the cyclic group Z_{12} we find a matching list of repetitions of elements of orders dividing 12. Only 0 has order 1 and only 6 has order 2, but 4 and 8 have order 3, 3 and 9 have order 4, 2 and 10 have order 6, and 1, 5, 7 and 11 each have order 12. If we take multiple copies of the modules, the multiplicity given by the number of repetitions, we can get a perfect correspondence with the group Z_{12} so that the fusion rules are replaced by the group law of addition modulo 12. The correspondence is

\begin{matrix}0 ↔ [0], 1 ↔ [\frac{3}{80}]_{1}, 2 ↔ [\frac{1}{10}]_{1}, 3 ↔ [\frac{7}{16}]_{1}, 4 ↔ [\frac{3}{5}]_{1}, 5 ↔ [\frac{3}{80}]_{2}, \\ 6 ↔ [\frac{3}{2}], 7 ↔ [\frac{3}{80}]_{3}, 8 ↔ [\frac{3}{5}]_{2}, 9 ↔ [\frac{7}{16}]_{2}, 10 ↔ [\frac{1}{10}]_{2}, 11 ↔ [\frac{3}{80}]_{4}. \end{matrix}

We found that while some other minimal model fusion rules could be similarly covered by finite cyclic groups, in order to cover all of them we had to use certain 2-groups. We suspect that this may be explained by the Goddard-Kent-Olive coset construction [GO]. This investigation is a first step toward a uniform method of constructing all the minimal models, including the intertwining operators, in such a way that each intertwining operator corresponds to a unique vector in some module. The fusion rules will be replaced by a group law, so the Verlinde algebra will be replaced by a group algebra. (This does not mean there is an algebra map from the new group algebra to the old Verlinde algebra.) We believe this will make possible a construction of an algebraic system containing the direct sum of V (c, 0) and copies of each of its modules V (c, h), so that the intertwiners are in one-to-one correspondence with the states in the modules, and they obey a matrix generalization of the Jacobi-Cauchy identity.

Fix p, q ≥ 2 with gcd(p, q) = 1. Let

S(p, q) = {(m, n) | 0 < m < p, 0 < n < q}

and let the set of distinct modules in the (p, q)-minimal model be denoted by

V = V (p, q) = {[h_{m,n}] | (m, n) ∈ S(p, q)}.

Definition. Let G be a finite abelian group. We say a map Φ : G → V (p, q) covers the fusion rules of the (p, q)-minimal model if the following conditions are satisfied:

(1) If g_{i} ∈ G and Φ(g_{i}) = [h_{m_{i},n_{i}}] for 1 ≤ i ≤ 2, then Φ(g_{1} + g_{2}) = [h_{m_{3},n_{3}}] for some (m_{3}, n_{3}) ∈ S(p, q) with ((m_{1}, n_{1}), (m_{2}, n_{2}), (m_{3}, n_{3})) (p, q)-admissible.

(2) If ((m_{1}, n_{1}), (m_{2}, n_{2}), (m_{3}, n_{3})) is (p, q)-admissible then for 1 ≤ i ≤ 3 there exist elements g_{i} ∈ G such that Φ(g_{i}) = [h_{m_{i},n_{i}}] and g_{1} + g_{2} = g_{3}.

This definition can also be described in the following alternative way. Let N be the number of distinct irreducible Virasoro modules in the (p, q)-minimal model. Let S_{1}, ..., S_{N} be labels for these N sectors, with S_{1} corresponding to h_{1,1} = 0, and let the fusion rules be denoted by D(S_{i}, S_{j}, S_{k}). Then the Verlinde algebra V is an N-dimensional vector space over Q with basis {S_{1}, ..., S_{N}} and with product S_{i}∗S_{j} = ∑_{k} D(S_{i}, S_{j}, S_{k})S_{k}. Let (G, +, 0) be a finite abelian group and let G = P_{1} ∪ ... ∪ P_{N} be a partition into N disjoint subsets with P_{1} = {0}. Let W be an N-dimensional vector space over Q with basis {P_{1}, ..., P_{N}} and give W an algebra structure by defining the product P_{i} ∗ P_{j} = ∑_{k∈T (i,j)} P_{k} where T (i, j) = {k|∃a ∈ P_{i}, ∃b ∈ P_{j}, a + b ∈ P_{k}}. We say that this partition of G gives the fusion rules of the (p, q)-minimal model if the bijection S_{i} ↔ P_{i} determines an algebra isomorphism between V and W.

Let r = p + q − 4, and define the abelian group H to be Z_{2}^{r}. For x = (x_{1}, ⋯, x_{r}) ∈ H define supp(x), the support of x, to be the set of coordinates of x which are 1, and define wt(x) = |supp(x)|, the weight of x, to be the number of coordinates of x which are 1. Of course, H also has a Boolean ring structure with the product given by coordinate-wise multiplication.

Define the subgroups A = {x ∈ H | supp(x) ⊆ {1, …, p − 2}} and B = {x ∈ H | supp(x) ⊆ {p − 1, …, p + q − 4}} in H. Note that in H, x + y = z is equivalent to x + y + z = 0, which has an obvious S_{3} symmetry. Also, if a_{i} ∈ A, b_{i} ∈ B, 1 ≤ i ≤ 3, then (a_{1} +b_{1})+(a_{2} +b_{2}) = a_{3} +b_{3} iff (a_{1} +b_{1})+(a_{2} +b_{2})+(a_{3} +b_{3}) = 0 iff a_{1} +a_{2} +a_{3} = 0 and b_{1} + b_{2} + b_{3} = 0, which has an S_{3} × S_{3} symmetry. The symmetric group S_{p−2} acts as a group of automorphisms of A by permuting the coordinates, as S_{q−2} acts on B. For 0 < m < p and 0 < n < q, let

A_{m} = {x ∈ A | wt(x) = m − 1} and B_{n} = {x ∈ B | wt(x) = n − 1}. It is clear that these subsets are the orbits into which A and B are partitioned under the actions of S_{p−2} and S_{q−2}, respectively.

Lemma 2. For any x, y ∈ H we have

supp(x + y) = supp(x) ∪ supp(y) − supp(x) ∩ supp(y)

is the symmetric difference of the supports of x and y. Furthermore, we have supp(x) ∩ supp(y) = supp(x · y), so that

wt(x + y) = wt(x) + wt(y) − 2 wt(x · y).

The following Lemma has an analog for 0 < n_{2} ≤ n_{1} < q.

Lemma 3. For 0 < m_{2} ≤ m_{1} < p, we have

A_{m_{1}} + A_{m_{2}} = ⋃_{m_{3}∈M} A_{m_{3}}

where M = {m_{3} | (m_{1}, m_{2}, m_{3}) is p-admissible}.

Proof. To prove that A_{m_{1}} + A_{m_{2}} is contained in the given union, we will prove that

{wt(a_{1} + a_{2}) | a_{1} ∈ A_{m_{1}}, a_{2} ∈ A_{m_{2}}} = {m_{1} − m_{2} + 2i | 0 ≤ i ≤ min(m_{2}, p − m_{1}) − 1}.

From Lemma 2 we see that the smallest value of wt(a_{1} + a_{2}) occurs when supp(a_{2}) ⊆ supp(a_{1}), in which case wt(a_{1} · a_{2}) = wt(a_{2}) is maximal. The largest value of wt(a_{1} + a_{2}), which occurs when wt(a_{1} · a_{2}) is minimal, is determined in two cases as follows.

Case 1: If wt(a_{1})+wt(a_{2}) ≤ p−2, then the largest value of wt(a_{1}+a_{2}) is wt(a_{1})+wt(a_{2}) when wt(a_{1} · a_{2}) = 0.

Case 2: If wt(a_{1}) + wt(a_{2}) > p − 2 then the largest value of wt(a_{1} + a_{2}) is [(p − 2) − wt(a_{1})] + [(p − 2) − wt(a_{2})] = 2p − 4 − (wt(a_{1}) + wt(a_{2})) when wt(a_{1} · a_{2}) = wt(a_{1}) + wt(a_{2}) − (p − 2) is minimal.

It is clear that the indicated intermediate values of wt(a_{1} +a_{2}) occur as wt(a_{1} ·a_{2}) takes on all intermediate values between its minimum and maximum. So the union over any proper subset of M would not contain A_{m_{1}} + A_{m_{2}}.

Now, for any m_{3} ∈ M, and for any a_{3} ∈ A_{m_{3}}, we wish to find elements in A_{m_{1}} and A_{m_{2}} whose sum is a_{3}. From the last paragraph we see that for some a_{1} ∈ A_{m_{1}} and a_{2} ∈ A_{m_{2}}, a_{1} + a_{2} = a′_{3} ∈ A_{m_{3}}. Let σ ∈ S_{p−2} be a permutation such that σ(a′_{3}) = a_{3}. Then we have σ(a_{1}) + σ(a_{2}) = a_{3} with σ(a_{1}) ∈ A_{m_{1}} and σ(a_{2}) ∈ A_{m_{2}}. ■

Let H_{m,n} = A_{m} + B_{n} = {a + b ∈ H | a ∈ A_{m}, b ∈ B_{n}}. It is clear that

H = ⋃_{0<m<p} ⋃_{0<n<q} H_{m,n}

is a disjoint union. Define a surjective map Φ̃ : H → S(p, q) by Φ̃(h) = (m, n) if h ∈ H_{m,n}. Now let I ≈ Z_{2} be the cyclic subgroup of H generated by (1, 1, ⋯, 1), and let G ≈ Z_{2}^{r−1} be the quotient group H/I. Note that H_{m,n} + I = H_{p−m,q−n} + I, so that we have the induced surjective map Φ : G → V (p, q) given by Φ(h + I) = [h_{m,n}] if h ∈ H_{m,n}.

Theorem. With notation as above, Φ : G → V (p, q) covers the fusion rules of the (p, q)- minimal model.

Proof. For any (m_{1}, n_{1}), (m_{2}, n_{2}) ∈ S(p, q) we first show that

H_{m_{1},n_{1}} + H_{m_{2},n_{2}} = ⋃_{(m_{3},n_{3})∈T} H_{m_{3},n_{3}}

where T = {(m_{3}, n_{3}) ∈ S(p, q) | ((m_{1}, n_{1}), (m_{2}, n_{2}), (m_{3}, n_{3})) is (p, q)-admissible}. By the S_{3} × S_{3} symmetry of addition in H = A × B, and of (p, q)-admissibility, we may assume that 0 < m_{2} ≤ m_{1} < p and 0 < n_{2} ≤ n_{1} < q. The equality is equivalent to showing that

A_{m_{1}} + A_{m_{2}} = ⋃_{m_{3}∈M} A_{m_{3}} and B_{n_{1}} + B_{n_{2}} = ⋃_{n_{3}∈N} B_{n_{3}}

where M = {m_{3}|(m_{1}, m_{2}, m_{3}) is p-admissible} and N = {n_{3}|(n_{1}, n_{2}, n_{3}) is q-admissible}. But these follow immediately from Lemma 3 and its analog for q.

From this equality in H, it is now straightforward to check that Φ covers the fusion rules. ■

The main result in this paper was presented by A. J. F. on November 18, 1995, at the Special Session on Quantum Affine Algebras and Related Topics, during the American Mathematical Society meeting in Greensboro, NC.

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