A new perspective on the Frenkel-Zhu fusion rule theorem
aa r X i v : . [ m a t h . R T ] O c t A New Perspective on the Frenkel-Zhu Fusion Rule Theorem
Alex J. Feingold and Stefan Fredenhagen
Abstract.
In this paper we prove a formula for fusion coefficients of affineKac-Moody algebras first conjectured by Walton [
Wal2 ], and rediscovered in[ Fe ]. It is a reformulation of the Frenkel-Zhu affine fusion rule theorem [ FZ ],written so that it can be seen as a beautiful generalization of the classicalParasarathy-Ranga Rao-Varadarajan tensor product theorem [ PRV ]. Contents
1. Introduction 12. Definition of Fusion Algebra 23. Background and Notation for Finite Dimensional Lie Algebras 34. Notation for Affine Lie Algebras 65. Tensor Product Decompositions 86. The Frenkel-Zhu Theorem and its reformulation 107. Review of the proof of the PRV theorem 138. Conclusion of the proof 16References 18
1. Introduction
Fusion rules play a very important role in conformal field theory [ Fu ], in therepresentation theory of vertex operator algebras [ FLM, FHL, FZ ], and in quite afew other areas. For example, fusion rules were used in [ FS ] to obtain informationon D-brane charge groups in string theory which on the other hand correspond tocertain twisted K-groups. This line of research found a mathematical culminationin the theorem by Freed, Hopkins and Teleman [ FHT ], showing that twisted equi-variant K-theory can be identified with a fusion ring. In [
AFW ] a connection wasfound between the fusion rules for the Virasoro minimal models and elementary
Mathematics Subject Classification.
Primary 17B67, 17B65, 81T40;Secondary 81R10,05E10.
Key words and phrases.
Fusion Rules, Affine Kac-Moody Algebras.AJF wishes to thank the Albert Einstein Institute for its wonderful hospitality and supportwhich helped complete this work. c (cid:13) abelian 2-groups. Further work in [ FW ] extended this idea to find a connectionbetween the fusion rules for type A and A on all levels, and elementary abelian2-groups and 3-groups. This was extended as far as was possible in [ Sal1, Sal2 ] tothe case of A ℓ for any rank ℓ and any level.In [ Fe ] an introduction was given to the subject with major focus on the al-gorithmic aspects of computing fusion rules for affine Kac-Moody algebras. Inparticular, it was emphasized that the Kac-Walton algorithm [ Kac, Wal ] for fu-sion coefficients is closely related to the Racah-Speiser algorithm for tensor productdecompositions, which was the subject of earlier work [
F1, F2 ]. [ Fe ] included a con-jecture on fusion coefficients which restates the Frenkel-Zhu theorem [ FZ ] in a formwhich shows it to be a beautful generalization of the classical Parasarathy-RangaRao-Varadarajan tensor product theorem [ PRV ]. That conjecture had alreadybeen made by Walton [
Wal2 ] in 1994, but we believe that it has not been provenup until now.An outline of the organization of the paper is as follows. We give the definitionof a fusion algebra in section two, then we give notation and background aboutfinite dimensional simple Lie algebras in section three. This includes facts aboutirreducible representations, contravariant Hermitian forms on them, special resultsfor sl and its representations, and projection operators. In section four we brieflygive notations about affine algebras leading to the level k fusion algebra associatedwith simple Lie algebra g . In section five we discuss tensor products of irreduciblefinite dimensional modules for g and the PRV theorem. In section six we statethe Frenkel-Zhu fusion rule theorem, the Walton conjecture, what it says in thespecial case when g = sl , and a corollary relating fusion coefficients to tensorproduct multiplicities. We begin the proof of the Walton conjecture by rewritingthe Frenkel-Zhu theorem in several ways. In section seven we review the proof ofthe PRV theorem and refine it to help find a relationship between the spaces whichoccur in the Frenkel-Zhu theorem and the Walton conjecture. In section eight weput all these pieces together to finish the proof of the Walton conjecture.
2. Definition of Fusion Algebra
Let us begin with the definition of fusion algebra given by J. Fuchs [ Fu ]. Afusion algebra F is a finite dimensional commutative associative algebra over Q with some basis B = { x a | a ∈ A } so that the structure constants N ca,b defined by x a · x b = X c ∈ A N ca,b x c are non-negative integers. There must be a distinguished index Ω ∈ A with thefollowing properties. It is required that the matrix C = [ C a,b ] = [ N Ω a,b ]satisfies C = I . Because 0 ≤ N ca,b ∈ Z , either C = I or C must be an order 2permutation matrix, that is, there is a permutation σ : A → A with σ = 1 and C a,b = δ a,σ ( b ) . NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 3
Write σ ( a ) = a ∗ and call x a ∗ the conjugate of x a . Use it to define the non-negativeintegers N a,b,c = N c ∗ a,b which, by commutativity and associativity of the algebra product, are completelysymmetric in a , b and c . Using this we also find that x Ω is a multiplicative identityelement in F and Ω ∗ = Ω.In this paper we are interested in the structure constants of fusion algebrasthat are associated to affine Lie algebras.
3. Background and Notation for Finite Dimensional Lie Algebras
Now we will introduce notations and review some basic results needed later.Let g be a finite dimensional simple Lie algebra of rank ℓ with Cartan matrix A = [ a ij ] and Cartan subalgebra H . The simple roots and the fundamental weightsof g are linear functionals α , · · · , α ℓ and λ , · · · , λ ℓ , respectively, in the dual space H ∗ . Let the integral weight lattice P be the Z -spanof the fundamental weights, and let P + = { n λ + · · · + n ℓ λ ℓ | ≤ n , · · · , n ℓ ∈ Z } be the set of dominant integral weights of g , and let θ = ℓ X i =1 θ i α i be the highest root of g . The symmetric bilinear form ( · , · ) on H ∗ is determined by a ij = h α i , α j i = 2( α i , α j )( α j , α j ) , ≤ i, j ≤ ℓ and the normalization ( θ, θ ) = 2. The fundamental weights are determined by theconditions h λ i , α j i = δ ij for 1 ≤ i, j ≤ ℓ , and the special “Weyl vector” ρ = ℓ X i =1 λ i will play an important role in several formulas. It is useful to defineˇ λ = 2 λ ( λ, λ ) for any 0 = λ ∈ H ∗ , so we can write ( λ i , ˇ α j ) = δ ij and a ij = ( α i , ˇ α j ). We may also express θ = ℓ X i =1 ˇ θ i ˇ α i so ˇ θ i = θ i ( α i , α i )2 . The Weyl group W of g is defined to be the group of endomorphisms of H ∗ gener-ated by the simple reflections corresponding to the simple roots, r i ( λ ) = λ − ( λ, ˇ α i ) α i , ≤ i ≤ ℓ. ALEX J. FEINGOLD AND STEFAN FREDENHAGEN
This is a finite group of isometries which preserve P . There is a partial order definedon H ∗ defined by λ ≤ µ if and only if µ − λ = ℓ X i =1 k i α i for some 0 ≤ k i ∈ Z . For λ ∈ P + let V λ denote the finite dimensional irreducible g -module withhighest weight λ . It has the weight space decomposition V λ = L β ∈ H ∗ V λβ , where V λβ = { v ∈ V λ | h · v = β ( h ) v, ∀ h ∈ H } is the β weight space of V λ . Of course, there are only finitely many β ∈ H ∗ such that V λβ is nonzero, and we denote by Π λ that finite set of such β . Since dim( V λλ ) = 1,a nonzero highest weight vector v λλ ∈ V λλ is determined up to a scalar. The dualspace ( V λ ) ∗ = Hom ( V λ , C ) is also an irreducible highest weight g -module, calledthe contragredient module of V λ . The action of g on ( V λ ) ∗ is given by( x · f )( v ) = − f ( x · v ) for x ∈ g , f ∈ ( V λ ) ∗ , v ∈ V λ . The highest weight of ( V λ ) ∗ is denoted by λ ∗ , and equals the negative of the lowestweight of V λ . For example, in the case when g is of type A ℓ , if λ = P ℓi =1 n i λ i then λ ∗ = P ℓi =1 n ℓ +1 − i λ i .On V λ with a chosen highest weight vector, v λλ ∈ V λλ , we have a positivedefinite contravariant Hermitian form [ Kac ] ( · , · ) : V λ × V λ → C determined bythe following conditions: (1) ( v λλ , v λλ ) = 1, (2) For any v, v ′ ∈ V λ , and any x ∈ g ,we have ( x · v, v ′ ) = − ( v, x † · v ′ ), where the map x → x † is the Chevalley involutiveautomorphism of g determined by its action on the generators e † i = − f i , f † i = − e i , h † i = − h i , ≤ i ≤ ℓ. Note that for any v ∈ V λβ , v ′ ∈ V λβ ′ , we have β ( h i )( v, v ′ ) = ( h i · v, v ′ ) = − ( v, − h i · v ′ ) = β ′ ( h i )( v, v ′ )so 0 = ( β − β ′ )( h i )( v, v ′ ) for any Cartan generator h i . This means that if β = β ′ then ( v, v ′ ) = 0 so different weight spaces are orthogonal. Let P roj λβ : V λ → V λβ denote the orthogonal projection operator.If V λ and V µ are two irreducible highest weight modules with chosen highestweight vectors and positive definite contravariant Hermitian forms as above, thenwe have a positive definite contravariant Hermitian form on the tensor product V λ ⊗ V µ given by ( v λ ⊗ v µ , v λ ⊗ v µ ) = ( v λ , v λ )( v µ , v µ ) . If V ν is an irreducible submodule of V λ ⊗ V µ then its orthogonal complement( V ν ) ⊥ = { v ∈ V λ ⊗ V µ | ( v, V ν ) = 0 } is clearly a g -submodule since( x · v, V ν ) = − ( v, x † · V ν ) = 0 , for all x ∈ g , v ∈ ( V ν ) ⊥ . This shows that when the tensor product V λ ⊗ V µ is decomposed into a direct sumof irreducible g -modules, the distinct modules obtained are mutually orthogonalwith respect to the contravariant Hermitian form. Let P roj λ,µV ν : V λ ⊗ V µ → V ν denote the orthogonal projection operator from the tensor product to a particularirreducible submodule V ν .We will use certain facts about the representation theory of the simple Liealgebra g = sl of type A whose standard basis { e, f, h } has the brackets [ h, e ] = 2 e , NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 5 [ h, f ] = − f and [ e, f ] = h . An irreducible finite dimensional sl -module V λ isdetermined by its highest weight, the non-negative integer λ ( h ) = m , so we write V λ = V ( m ). If v is a highest weight vector then a basis of V ( m ) can be writtenas { v i | ≤ i ≤ m } where v i = i ! f i v and the action of g is given by the formulas: h · v i = ( m − i ) v i , f · v i = ( i + 1) v i +1 , e · v i = ( m − i + 1) v i − for 0 ≤ i ≤ m with the understanding that v j = 0 for j outside that range. For anyinteger p ≥
0, we understand the operators e p and f p on V ( m ) to mean p repetitionsof the operators e and f , respectively. It is easy to see that the contravariant formhas values ( v i , v j ) = δ i,j (cid:0) mi (cid:1) , for 0 ≤ i = j ≤ m , so the form is positive definite. Lemma 3.1.
Let g = sl and V ( m ) be the irreducible finite dimensional sl -modulewith highest integral weight m ≥ . Then for any integer p ≥ , with respect tothe positive definite contravariant Hermitian form on V ( m ) , we have an orthogonaldirect sum decomposition V ( m ) = ker ( f p ) ⊕ Im ( e p ) . Proof.
From the explicit formulas for the action, it is clear that ker ( f p )is the subspace of the p lowest weight spaces with basis { v m − p +1 , · · · , v m } andthat Im ( e p ) = ( ker ( f p )) ⊥ is the subspace of all other weight spaces with basis { v , · · · , v m − p } . (cid:3) We now go back to the general case of any finite dimensional simple g . Let V λ be an irreducible g -module, α any root of g , and let g α be the correspondingsubalgebra of g isomorphic to sl with standard basis { e α , f α , h α } . The Chevalleyinvolution acts on g α by e † α = − f α , f † α = − e α and h † α = − h α . The completereducibility of finite dimensional sl -modules gives a direct sum decomposition V λ = M i V λγ i ( m i )into irreducible g α -modules, where V λγ i ( m i ) has g -highest weight γ i , and g α -highestweight γ i ( h α ) = m i . If V λγ ( m ) is one of these, then its orthogonal complement isclearly a g α -submodule by the same argument as given above for the decompositionof a tensor product. It means that this decomposition is an orthogonal direct sumdecomposition with respect to the contravariant Hermitian form on V λ . Lemma 3.2.
Let V λ be an irreducible g -module, α any root of g , and g α be thecorresponding subalgebra of g isomorphic to sl . Let β ∈ Π λ be any weight of V λ .Then, for any integer p ≥ such that p + h β, α i ≥ , we have { v ∈ V λβ | e pα ( v ) = 0 } = { v ∈ V λβ | f p + h β,α i α ( v ) = 0 } . Proof.
The Weyl group reflection r α acts on the weights Π λ and r α ( β ) = β − h β, α i α . It is also well-known that the operator R α = ( exp ( f α ))( exp (( − e α ))( exp ( f α )) ∈ GL ( V λ )satisfies R α ( V λβ ) = V λr α ( β ) for any weight β ∈ Π λ . It is clear from the definition of R α that it acts on each of the g α submodules in the decomposition of V λ given inthe paragraph above the lemma. For any 0 = v ∈ V λβ we have 0 = R α ( v ) ∈ V λr α ( β ) .We can write v = P i v i where v i ∈ V λγ i ( m i ), and e pα ( v ) = 0 iff e pα ( v i ) = 0 for each i , ALEX J. FEINGOLD AND STEFAN FREDENHAGEN so we may assume v is in one such irreducible g α -module. The condition e pα ( v ) = 0means v is in one of the top p weight spaces of its irreducible g α -module. This isequivalent to saying that R α ( v ) is in one of the bottom p weight spaces, that is, f pα ( R α ( v )) = 0.If h β, α i ≥ R α ( v ) = cf h β,α i α ( v ) for some nonzero scalar c , which means0 = f pα ( cf h β,α i α ( v )) = cf p + h β,α i α ( v ).If h β, α i < p + h β, α i ≥ R α ( v ) = ce −h β,α i α ( v ) for some nonzeroscalar c which means 0 = f pα ( ce −h β,α i α ( v )) = df p + h β,α i α ( v ) for a nonzero scalar d . (cid:3) If V is any finite dimensional vector space with a positive definite Hermitianform and W is any subspace of V then W has an orthogonal complement W ⊥ = { v ∈ V | ( v, w ) = 0 , ∀ w ∈ W } such that V = W ⊕ W ⊥ . Let P W : V → W be theorthogonal projection of V onto W defined by P W ( v ) = w where v = w + w ′ isthe unique expression for v ∈ V with w ∈ W and w ′ ∈ W ⊥ . If L : V → V is anylinear transformation, there is a unique adjoint linear transformation L † : V → V determined by the conditions( L ( v ) , v ′ ) = ( v, L † ( v ′ )) , for all v, v ′ ∈ V. We call L self-adjoint when L = L † . Note that any orthogonal projection mapis self-adjoint because if v = w + w ′ and v = w + w ′ for w , w ∈ W and w ′ , w ′ ∈ W ⊥ , then( P W ( v ) , v ) = ( w , w + w ′ ) = ( w , w ) = ( w + w ′ , w ) = ( v , P W ( v ))so P † W = P W . Also, it is clear that P W = P W .Finally, later we will need the following lemma. Lemma 3.3.
Let V = U ⊕ U be an orthogonal direct sum decomposition of afinite dimensional vector space V with a positive definite Hermitian form, and let W be any subspace of V . Then we have the orthogonal direct sum decomposition of W : W = P W ( U ) ⊕ ( W ∩ U ) . Proof.
Let v ∈ W be in the orthogonal complement of P W ( U ). This meansthat for any u ∈ U , we have0 = ( P W ( u ) , v ) = ( u , P † W ( v )) = ( u , P W ( v )) = ( u , v )which means v ∈ W ∩ U ⊥ = W ∩ U . (cid:3)
4. Notation for Affine Lie Algebras
Let ˆg = g ⊗ C [ t, t − ] ⊕ C c ⊕ C d be the affine algebra constructed from g with derivation d = − t ddt adjoined as usual,and with Cartan subalgebra H = H ⊕ C c ⊕ C d. The simple roots and the fundamental weights of ˆg are linear functionals α , α , · · · , α ℓ and Λ , Λ , · · · , Λ ℓ , respectively, in the dual space H ∗ . The simple roots of g form a basis of H ∗ (asdo the fundamental weights), and we identify them with linear functionals in H ∗ NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 7 having the same values on H ⊆ H and being zero on c and d . Let c ∗ and d ∗ in H ∗ be the functionals which are zero on H and which satisfy c ∗ ( c ) = 1 , c ∗ ( d ) = 0 , d ∗ ( c ) = 0 , d ∗ ( d ) = 1 . Extend the bilinear form ( · , · ) to H ∗ by letting( c ∗ , H ∗ ) = 0 = ( d ∗ , H ∗ ) , ( c ∗ , c ∗ ) = 0 = ( d ∗ , d ∗ ) , and ( c ∗ , d ∗ ) = 1 . Then α = d ∗ − θ andΛ = c ∗ , Λ i = θ i ( α i , α i )2 c ∗ + λ i = ˇ θ i c ∗ + λ i , ≤ i ≤ ℓ, are determined by the conditions h Λ i , α j i = δ ij for 0 ≤ i, j ≤ ℓ . Let the integralweight lattice ˆ P be the Z -span of the fundamental weights, and letˆ P + = { ℓ X i =0 n i Λ i | ≤ n i ∈ Z } be the set of dominant integral weights of ˆg .The affine Weyl group c W of ˆg is the group of endomorphisms of H ∗ generatedby the simple reflections corresponding to the simple roots, r i (Λ) = Λ − (Λ , ˇ α i ) α i , ≤ i ≤ ℓ. This is an infinite group of isometries which preserve ˆ P . The canonical centralelement, c ∈ ˆg acts on an irreducible ˆg -module as a scalar k , called the level of themodule. We will only discuss modules with highest weight Λ ∈ ˆ P + , which are the“nicest” in that they have affine Weyl group symmetry and satisfy the Weyl-Kaccharacter formula. An irreducible highest weight ˆg -module is uniquely determinedby its highest weight Λ = ℓ X i =0 n i Λ i ∈ ˆ P + and, if we define θ = 1 = ˇ θ , then k = Λ( c ) = ℓ X i =0 n i Λ i ( c ) = ℓ X i =0 n i θ i ( α i , α i )2 = ℓ X i =0 n i ˇ θ i . For fixed k there are only finitely many Λ ∈ ˆ P + with Λ( c ) = k , and we denotethat finite set by ˆ P + k . It is easy to see that c W preserves the level k weights { Λ ∈ ˆ P | Λ( c ) = k } . The affine hyperplane determined by the condition Λ( c ) = k canbe projected onto H ∗ and the corresponding action of c W is such that the simplereflections r i for 1 ≤ i ≤ ℓ act as they were defined originally on H ∗ , as isometriesgenerating the finite Weyl group W of g . But the new affine reflection r acts as r ( λ ) = λ − ( λ, θ ) θ + kθ = r θ ( λ ) + kθ , the composition of reflection r θ and thetranslation by kθ , which is not an isometry on H ∗ .Irreducible ˆg -modules ˆ V Λ of level k ≥ P + k , but we can alsoindex them by certain weights of g as follows. From the formulas above we canwrite Λ = ℓ X i =0 n i Λ i = kc ∗ + ℓ X i =1 n i λ i . ALEX J. FEINGOLD AND STEFAN FREDENHAGEN
So there is a bijection between ˆ P + k and the set of weights λ = P ℓi =1 n i λ i such that k = n + ℓ X i =1 n i θ i ( α i , α i )2 = n + ℓ X i =1 n i ˇ θ i = n + h λ, θ i . Since n ≥
0, this is equivalent to the “level k condition” h λ, θ i = ℓ X i =1 n i ˇ θ i ≤ k. Define the set P + k = { λ = ℓ X i =1 n i λ i ∈ P + | h λ, θ i ≤ k } and let the index set A (as in the fusion algebra definition) be P + k . Then we seethat irreducible modules on level k correspond to λ ∈ P + k . Fix level k ≥ λ ] · [ µ ] = X ν ∈ P + k N ( k ) νλ,µ [ ν ] . The distinguished identity element, [0], corresponds to Λ = kc ∗ , and for each [ λ ]there is a distinguished conjugate [ λ ∗ ] such that N ( k )0 λ,µ = δ µ,λ ∗ . Knowing N ( k ) νλ,µ isequivalent to knowing the completely symmetric coefficients N ( k ) λ,µ,ν = N ( k ) ν ∗ λ,µ . Let F ( g , k ) denote this fusion algebra.
5. Tensor Product Decompositions
There is a close relationship between the product in fusion algebras associatedwith an affine Kac-Moody algebra ˆg and tensor product decompositions of irre-ducible g -modules. Let V λ be the irreducible finite dimensional g -submodule ofˆ V Λ generated by a highest weight vector. In the special case when Λ = k Λ = kc ∗ ,that finite dimensional g -module is V , the one dimensional trivial g -module. Since g is semisimple, any finite dimensional g -module is completely reducible. Therefore,we can write the tensor product of irreducible g -modules V λ ⊗ V µ = X ν ∈ P + M ult νλ,µ V ν as the direct sum of irreducible g -modules, including multiplicities. This decompo-sition is independent of the level k and is part of the basic representation theoryof g . The fusion products [ λ ] · [ µ ] are obtained by a subtle truncation of the abovesummation.The Racah-Speiser algorithm gives the formula M ult νλ,µ = X w ∈ W ǫ ( w ) M ult λ ( w ( ν + ρ ) − µ − ρ )where W is the Weyl group of g , ǫ ( w ) = ( − length ( w ) is the sign of w , the Weylvector ρ = P λ i is the sum of the fundamental weights of g , and M ult λ ( β ) = dim ( V λβ ) is the inner multiplicity of the weight β in V λ . Recall that Π λ = { β ∈ NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 9 H ∗ | dim ( V λβ ) > } denotes the set of all weights of V λ . In fact, the only weights ν for which M ult νλ,µ may be nonzero are those of the form ν = β + µ where β ∈ Π λ .This algorithm assumes that you can already produce the weight diagram ofany irreducible module, V λ , so we should have discussed that first, but in factthe special case of the Racah-Speiser algorithm when µ = 0 gives a recursion forthe inner multiplicities of V λ . Since V is the trivial one-dimensional module, V λ ⊗ V = V λ , so M ult νλ, = δ λ,ν and therefore0 = X w ∈ W ǫ ( w ) M ult λ ( w ( ν + ρ ) − ρ )for ν = λ . One knows that M ult λ ( wλ ) = 1 and M ult λ ( wν ) = M ult λ ( ν ) for all w ∈ W , so the above formula implies that M ult λ ( ν ) = − X = w ∈ W ǫ ( w ) M ult λ ( ν + ρ − wρ )for ν = λ . Since ρ > wρ in the partial ordering on weights, this gives an effectiverecursion for M ult λ ( ν ).In [ F1, F2 ] Feingold studied certain patterns which occur in the tensor productdecomposition of a fixed irreducible g -module, V λ , with all other modules V µ . Forfixed λ , as µ varies there are only a finite number of different patterns of outermultiplicities which can occur, and there are sets of values for µ for which thepattern is constant, called zones of stability for tensor product decompositions. Wehave the following precise result from [ F2 ] about when a particular weight β of V λ ,reaches the zone of stability. Theorem 5.1.
Let λ, µ ∈ P + and β ∈ Π λ be such that β + µ ∈ P + . Let β − r β,j α j , · · · , β, · · · β + q β,j α j be the α j weight string through β . If h µ, α j i ≥ q β,j then M ult β + µλ,µ = M ult β + µ + λ j λ,µ + λ j . Since h µ + λ j , α j i = h µ, α j i + 1, it is clear that h µ, α j i ≥ q β,j implies M ult β + µλ,µ = M ult β + µ + mλ j λ,µ + mλ j for all m ≥ . This result shows that for fixed λ ∈ P + and fixed β ∈ Π λ , the tensor productmultiplicities M ult β + µλ,µ have zones of stability as µ varies, and it is sufficient tostudy the finite number of µ such that h µ, α j i ≤ q β,j for 1 ≤ j ≤ ℓ .There is another important result about tensor product coefficients which playeda role in [ F1, F2 ]. In 1977 Prof. Bertram Kostant drew the attention of Feingoldto the following beautiful result of Parthasarathy, Ranga Rao and Varadarajan[
PRV ], which is here rewritten slightly.
Theorem 5.2. [ PRV ] Let λ, µ ∈ P + and β ∈ Π λ be such that β + µ ∈ P + . Let ℓ = rank ( g ) and let = e j ∈ g α j be a root vector corresponding to the simple root α j for ≤ j ≤ ℓ . Then M ult β + µλ,µ = dim { v ∈ V λβ | e h µ,α j i +1 j v = 0 , ≤ j ≤ ℓ } .
6. The Frenkel-Zhu Theorem and its reformulation
Now let us turn to the Frenkel-Zhu fusion rule theorem for affine Kac-Moodyalgebras. (Note that this is closely related to results of Gepner-Witten [ GW ], whichappeared much earlier in the physics literature. Also, see Haisheng Li [ Li ].) Theorem 6.1. [ FZ ] Let λ, µ, ν ∈ P + k , and let = e θ ∈ g θ be a root vector of g inthe θ root space of g . Let v νν ∈ V ν be a highest weight vector and write H ′ = Hom g ( V λ ⊗ V µ ⊗ V ν , C ) . Then the level k fusion coefficient N ( k ) λ,µ,ν , which is completely symmetric in λ , µ and ν , equals the dimension of the vector space F Z k ( λ, µ, ν ) = { f ∈ H ′ | f ( e k −h ν,θ i +1 θ V λ ⊗ V µ ⊗ v νν ) = 0 } . We now state the main result of this paper, the theorem, conjectured by Walton,which is a blending of the PRV and FZ theorems, showing that the FZ theorem isactually a beautiful generalization of the PRV theorem.
Theorem 6.2.
For λ, µ ∈ P + k , β ∈ Π λ such that β + µ ∈ P + k , we have N ( k ) ( β + µ ) λ,µ equals the dimension of the space W + k ( λ, β, µ ) = { v ∈ V λβ | e h µ,α j i +1 j v = 0 , ≤ j ≤ ℓ, and e k −h β + µ,θ i +1 θ v = 0 } . In [
Wal2 ] the statement of the conjecture is slightly different from above, withthe condition e k −h β + µ,θ i +1 θ v = 0 replaced by the condition f k −h µ,θ i +1 θ v = 0. Theequivalence of these two conditions is precisely the content of Lemma 3.2.Theorem 6.2 implies the following result, which tells the level k at which thefusion coefficient associated with a single weight β ∈ Π λ equals the tensor productmultiplicity associated with that weight. Corollary 6.3.
Suppose λ, µ ∈ P + k , and β ∈ Π λ is such that β + µ ∈ P + k . Let the θ weight string through β in Π λ be β − rθ, · · · , β, · · · , β + qθ . Then k ≥ h µ, θ i + r implies N ( k ) ( β + µ ) λ,µ = M ult β + µλ,µ . Before starting the proof of the theorem, we will show how it reproduces thewell known fusion coefficients in the special case when g = sl , where ℓ = 1, θ = α ,and P + k = { n λ | n ∈ Z , ≤ n ≤ k } . In this case we use the notation [ n ] insteadof n λ , so V [ n ] = V ( n ) is the irreducible g -module with highest weight [ n ]. Theweights of V [ n ] are { β = [ n − i ] | ≤ i ≤ n } and each weight space V [ n ][ n − i ] isone-dimensional. For 0 ≤ n ≤ n ∈ Z , the tensor product decomposition V [ n ] ⊗ V [ n ] = n M i =0 V [ n + n − i ] is well-known. If [ n ] , [ n ] ∈ P + k then the fusion product corresponds to a truncationof this tensor product, so that only terms [ n + n − i ] ∈ P + k could appear,with coefficients no larger than 1. Note that the following Corollary 6.4 says thetruncation is somewhat stronger than that, requiring n + n − i ≤ k − i . Sincethere is a symmetry between n and n , it is not surprising to also find the condition i ≤ n symmetric to the assumption i ≤ n . NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 11
Corollary 6.4.
For ≤ n , n ≤ k , ≤ i ≤ n with ≤ n − i + n ≤ k , the sl fusion coefficient N ( k ) [ n + n − i ][ n ] , [ n ] equals if i ≤ n and n + n − i ≤ k − i , zerootherwise. Proof.
For 1 ≤ i ≤ n , the raising operator e = e θ sends V [ n ][ n − i ] isomorphi-cally onto V [ n ][ n − i +2] , and kills the highest weight space V [ n ][ n ] . This means that for v ∈ V [ n ][ n − i ] and p ≥ e p +11 v = 0 iff n < n − i + 2( p + 1) iff i ≤ p. The conditions on v in the Walton space W + k ([ n ] , [ n − i ] , [ n ]) = { v ∈ V [ n ][ n − i ] | e n +11 v = 0 and e k − ( n + n − i )+11 v = 0 } are then i ≤ n and n + n − i ≤ k − i . When these are satisfied, we have W + k ([ n ] , [ n − i ] , [ n ]) = V [ n ][ n − i ] so the sl fusion coefficient N ( k ) [ n + n − i ][ n ] , [ n ] = 1, and otherwise, it is zero. (cid:3) In order to prove Theorem 6.2 we must understand the connection between thePRV theorem, the statement of the theorem and the FZ theorem. We begin byrewriting the FZ theorem in a slightly different form. We can define a g -modulemap Φ : Hom ( V λ ⊗ V µ , V ν ∗ ) → Hom ( V λ ⊗ V µ ⊗ V ν , C )by (Φ f )( v λ ⊗ v µ ⊗ v ν ) = ( f ( v λ ⊗ v µ ))( v ν ) . It is easy to check that this is a g -module map and an isomorphism. In general,for V and W any two g -modules, Hom ( V, W ) is a g -module under the action,( x · L )( v ) = x · ( L ( v )) − L ( x · v ) for any v ∈ V and any L ∈ Hom ( V, W ). Itmay be helpful to use the notations π V : g → End ( V ), π W : g → End ( W ), and π : g → End ( Hom ( V, W )) to distinguish the representations of g on these threespaces. Then the above equation is saying that π ( x )( L ) = π W ( x ) ◦ L − L ◦ π V ( x ).We also have the definition of the space of g -module maps from V to W , Hom g ( V, W ) = { L ∈ Hom ( V, W ) | π ( x )( L ) = 0 , ∀ x ∈ g } = { L ∈ Hom ( V, W ) | π W ( x ) ◦ L = L ◦ π V ( x ) , ∀ x ∈ g } . If v ∈ V β is a weight vector of weight β , that is, for any h ∈ H , π V ( h ) v = β ( h ) v ,and L is any g -module map, then π W ( h ) L ( v ) = L ( π V ( h ) v ) = L ( β ( h ) v ) = β ( h ) L ( v )shows that L ( V β ) ⊆ W β . If P roj Vβ : V → V β and P roj Wβ : W → W β are the orthog-onal projection operators, then it is easy to see that L ( P roj Vβ ( v )) = P roj Wβ ( L ( v ))for any v ∈ V .Since Φ is a g -module isomorphism, it is clear that it restricts to an isomorphismΦ : Hom g ( V λ ⊗ V µ , V ν ∗ ) → Hom g ( V λ ⊗ V µ ⊗ V ν , C ) . We wish to describe the preimage of the space
F Z k ( λ, µ, ν ) under Φ. Since Φis an isomorphism, f ∈ F Z k ( λ, µ, ν ) is of the form Φ g for a unique element g ∈ Hom g ( V λ ⊗ V µ , V ν ∗ ). The conditions on f mean that( g ( e k −h ν,θ i +1 θ V λ ⊗ V µ ))( v νν ) = 0 . This allows us to rewrite the FZ theorem as follows.
Theorem 6.5. [ FZ ] Let λ, µ, ν ∈ P + k , and let = e θ ∈ g θ be a root vector of g inthe θ root space of g . Let v νν ∈ V ν be a highest weight vector and write H = Hom g ( V λ ⊗ V µ , V ν ∗ ) . Then the level k fusion coefficient N ( k ) λ,µ,ν equals the dimension of the space (6.1) F Z ′ k ( λ, µ, ν ) = { g ∈ H | g ( e k −h ν,θ i +1 θ V λ ⊗ V µ )( v νν ) = 0 } . There is a natural isomorphism of g -modules(6.2) Ψ : Hom ( V ∗ , W ) → W ⊗ V which is defined as follows. For any L ∈ Hom ( V ∗ , W ),Ψ( L ) = d X j =1 L ( v ∗ j ) ⊗ v j where d = dim ( V ) = dim ( V ∗ ), { v , · · · , v d } is any basis of V and { v ∗ , · · · , v ∗ d } isthe dual basis of V ∗ , that is, the basis such that v ∗ i ( v j ) = δ ij . The inverse mapsends a basic tensor w ⊗ v ∈ W ⊗ V to the element in Hom ( V ∗ , W ) which sendsany f ∈ V ∗ to f ( v ) w ∈ W . We will always choose the basis of V to consist ofweight vectors, and if v j has weight µ j , so that for any h ∈ H , π V ( h ) v j = µ j ( h ) v j ,then it is easy to see that the weight of the dual vector v ∗ j is − µ j . Namely, by thedefinition of the representation of g on the dual space V ∗ , for 1 ≤ i ≤ d we have( π V ∗ ( h ) v ∗ j )( v i ) = − v ∗ j ( π V ( h ) v i ) = − v ∗ j ( µ i ( h ) v i ) = − µ i ( h ) v ∗ j ( v i )= − µ i ( h ) δ ij = − µ j ( h ) δ ij = − µ j ( h ) v ∗ j ( v i )which says that π V ∗ ( h ) v ∗ j = − µ j ( h ) v ∗ j . So Π λ ∗ = − Π λ . This means that a highestweight vector v νν ∈ V νν has a dual lowest weight vector v ν ∗ − ν ∈ V ν ∗ − ν , and all otherweight vectors of V ν ∗ with weights above − ν are zero on v νν . In other words, withrespect to the positive definite Hermitian form on the irreducible module V ν ∗ , theorthogonal complement of the lowest weight space V ν ∗ − ν is the subspace of linearfunctionals in V ν ∗ that send v νν to 0. We now see that(6.3) F Z ′ k ( λ, µ, ν ) = { g ∈ H | g ( e k −h ν,θ i +1 θ V λ ⊗ V µ ) ∈ ( V ν ∗ − ν ) ⊥ } . For any g ∈ H we know that Im ( g ) is a submodule of V ν ∗ , so if g = 0 then g issurjective. Also, g sends weight vectors to weight vectors of the same weight, and g sends highest (resp., lowest) weight vectors to highest (resp., lowest) weight vectors. V ν ∗ has a one dimensional highest weight space in which we have chosen a basisvector v ν ∗ ν ∗ ∈ V ν ∗ ν ∗ . V ν ∗ also has a one dimensional lowest weight space in which wehave chosen a basis vector v ν ∗ − ν ∈ V ν ∗ − ν . The tensor product V λ ⊗ V µ decomposesinto the direct sum of irreducible modules, but g must send any highest (resp.,lowest) weight vector whose weight is not ν ∗ (resp., not − ν ) to zero, so it sendsall irreducible components whose highest weight is not ν ∗ to zero. The dimensionof the space of highest (resp., lowest) weight vectors in V λ ⊗ V µ of weight ν ∗ (resp., − ν ) is the tensor product multiplicity M = M ult ν ∗ λ,µ , so we may choose abasis { u , · · · , u M } of that HWV space U + (resp., LWV space U − ) and determine g i ∈ H uniquely by the conditions g i ( u j ) = δ i,j v ν ∗ ν ∗ (resp., g i ( u j ) = δ i,j v ν ∗ − ν ) for1 ≤ i, j ≤ M . Then { g , · · · , g M } is a basis of H . Let us denote by U ( g ) theuniversal enveloping algebra of g . It is clear that g i takes the submodule U ( g ) u i NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 13 isomorphically to V ν ∗ and sends all other irreducible submodules U ( g ) u j , j = i ,of the tensor product to zero, so it is essentially an orthogonal projection from thetensor product to one of its components followed by an isomorphism. Let P roj λ,µU + be the orthogonal projection from V λ ⊗ V µ to the subspace of highest weight vectorsof weight ν ∗ , and let P roj λ,µU − be the orthogonal projection from V λ ⊗ V µ to thesubspace of lowest weight vectors of weight − ν . Then for any v ∈ V λ ⊗ V µ , write v = u + v ′ + v ′′ where u = P roj λ,µU − ( v ) ∈ U − , v ′ is of weight − ν but is orthogonal to U − so is not a lowest weight vector and must be a sum of vectors from irreduciblecomponents whose highest weights are not ν ∗ , and v ′′ is a sum of vectors of weightsnot − ν . Then g ( v ) = g ( u ) + g ( v ′ ) + g ( v ′′ ) with g ( u ) ∈ V ν ∗ − ν , and g ( v ′ ) = 0 and g ( v ′′ )is a sum of vectors of weights not − ν , so P roj ν ∗ − ν ( g ( v )) = g ( u ) = g ( P roj λ,µU − ( v )). Asimilar argument applies to U + , so we have shown that for any g ∈ H we have g ◦ P roj λ,µU + = P roj ν ∗ ν ∗ ◦ g , (6.4) g ◦ P roj λ,µU − = P roj ν ∗ − ν ◦ g. (6.5)But this means that we can rewrite the Frenkel-Zhu space in (6.3) as F Z ′ k ( λ, µ, ν ) = { g ∈ H | P roj ν ∗ − ν g ( e k −h ν,θ i +1 θ V λ ⊗ V µ ) = 0 } (6.6) = { g ∈ H | g ( P roj λ,µU − ( e k −h ν,θ i +1 θ V λ ⊗ V µ )) = 0 } . (6.7)
7. Review of the proof of the PRV theorem
Now we will review the proof of the PRV theorem and see if it allows us to findan isomorphism between the Frenkel-Zhu space
F Z ′ k ( λ, µ, ν ) and the Walton space W + k ( λ, β, µ ) when ν ∗ = β + µ .In the proof of the PRV theorem one looks at the g -module V = Hom ( V µ ∗ , V λ ),where π : g → End ( V ) denotes the representation. As noted above (see eq. (6.2)), V ∼ = V λ ⊗ V µ , and this isomorphism is given by the map Ψ which sends irreduciblecomponents in V to isomorphic irreducible components in V λ ⊗ V µ . The proofbegins by considering the subspace of all lowest weight vectors (LWVs) in V , U = { L ∈ V | π ( f i ) L = 0 , ≤ i ≤ ℓ } where ℓ = rank ( g ) and e i , f i , h i are the generators of g with the usual Serrerelations. Then L ∈ U iff π λ ( f i ) ◦ L = L ◦ π µ ∗ ( f i ) , for 1 ≤ i ≤ ℓ. It is clear that U is invariant under the operators π ( h j ), so it has a weight spacedecomposition U = r M m =1 U m where U m = { L ∈ U | π ( h ) L = − ν m ( h ) L, ∀ h ∈ H } is the − ν m -weight space, − ν , · · · , − ν r are the distinct lowest weights of irreducible components in V whosecorresponding highest weights are ν ∗ , · · · , ν ∗ r . Furthermore, dim ( U m ) = M ult ν ∗ m λ,µ is the multiplicity of V ν ∗ m in the tensor product V λ ⊗ V µ because the independentvectors in U m each generate a distinct irreducible component in V . Let v ∗ = v µ ∗ µ ∗ be a highest weight vector (HWV) in V µ ∗ of weight µ ∗ dual to v = v µ − µ ∗ a LWV in V µ of weight − µ ∗ . The key step in the proof of the PRV theorem is the followinglemma. Lemma 7.1.
Define the linear map ξ : U → V λ by ξ ( L ) = L ( v ∗ ) , ∀ L ∈ U. Then ξ is injective and the range of ξ equals V ′ = { v ∈ V λ | π λ ( f i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ } . Proof.
Because the highest weight vector v ∗ ∈ V µ ∗ satisfies π µ ∗ ( f i ) h µ ∗ ,α i i +1 v ∗ = 0for 1 ≤ i ≤ ℓ , we have π λ ( f i ) h µ ∗ ,α i i +1 L ( v ∗ ) = L ( π µ ∗ ( f i ) h µ ∗ ,α i i +1 v ∗ ) = 0so ξ ( U ) ⊆ V ′ . Let g = g − ⊕ H ⊕ g + be the triangular decomposition of g , where g − is the Lie subalgebra of g generated by the negative root vectors, that is, thespan of f , · · · , f ℓ and all their multibrackets, and similarly g + is generated bythe positive root vectors. Let U ( g ) be the universal enveloping algebra of g andextend the meaning of any representation of g to include the representation ofthe associative algebra U ( g ). We may also have use for the universal envelopingalgebras U ( g − ) and U ( g + ). It is well known that U ( g − ) is spanned by all productsof the form y = f i · · · f i s for any s ≥ ≤ i j ≤ ℓ for 1 ≤ j ≤ s , and that V µ ∗ = U ( g − ) v ∗ is spanned by all vectors of the form π µ ∗ ( y ) v ∗ = π µ ∗ ( f i ) · · · π µ ∗ ( f i s ) v ∗ for y as above. If L ( v ∗ ) = 0 for some L ∈ U then we get0 = π λ ( y ) L ( v ∗ ) = L ( π µ ∗ ( y ) v ∗ )showing that L = 0 and therefore ξ is injective. Let v ∈ V ′ be arbitrary and try todefine L ∈ V by L ( π µ ∗ ( y ) v ∗ ) = π λ ( y ) v for any y ∈ U ( g − ). If π µ ∗ ( y ) v ∗ = 0 then it is known that y can be written y = ℓ X i =1 y i f h µ ∗ ,α i i +1 i for some y i ∈ U ( g − ), so π λ ( y ) v = 0. This means that L is well-defined on π µ ∗ ( U ( g − )) v ∗ = V µ ∗ . By the definition of the linear map L we have( π λ ( f i ) ◦ L )( π µ ∗ ( y ) v ∗ ) = π λ ( f i y ) v = L ( π µ ∗ ( f i y ) v ∗ ) = ( L ◦ π µ ∗ ( f i ))( π µ ∗ ( y ) v ∗ )which shows that π λ ( f i ) ◦ L = L ◦ π µ ∗ ( f i ) so L ∈ U . This completes the argumentthat ξ is an isomorphism from U to V ′ . (cid:3) Now suppose that L ∈ U m for some 1 ≤ m ≤ r , so π ( h ) L = − ν m ( h ) L for any h ∈ H . But π ( h ) L = π λ ( h ) ◦ L − L ◦ π µ ∗ ( h ) so ξ ( L ) ∈ V λµ ∗ − ν m has weight µ ∗ − ν m because π λ ( h )( Lv ∗ ) = L ( π µ ∗ ( h ) v ∗ ) − ν m ( h ) Lv ∗ = L ( µ ∗ ( h ) v ∗ ) − ν m ( h ) Lv ∗ = ( µ ∗ − ν m )( h ) Lv ∗ . NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 15
This shows that ξ provides an isomorphism between each subspace U m and V ′ µ ∗ − ν m = { v ∈ V λµ ∗ − ν m | π λ ( f i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ } . The PRV notation for this subspace is V − ( λ ; µ ∗ − ν m , µ ∗ ) and their result is theformula for the tensor product multiplicity M ult ν ∗ m λ,µ = dim ( V − ( λ ; µ ∗ − ν m , µ ∗ )) . Replacing f i by e i in the definition of the space V − ( λ ; γ, µ ∗ ) one gets another space, V + ( λ ; γ, µ ∗ ) = { v ∈ V λγ | π λ ( e i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ } . In the proof of the PRV theorem it is shown that dim ( V − ( λ ; γ, µ ∗ )) = dim ( V + ( λ ; − γ ∗ , µ ))by using an automorphism coming from the longest element of the Weyl group, W .Then the final result of the PRV theorem is that M ult ν ∗ m λ,µ = dim ( V + ( λ ; ν ∗ m − µ, µ )) . To understand this we must discuss the longest element and a little bit ofthe theory of Lie groups. First it is necessary to know that the elements of theWeyl group are in one-to-one correspondence with the Weyl chambers in H ∗ . Thedominant chamber, P + , corresponding to the identity element in W , is also asso-ciated with a choice of simple roots, ∆ = { α , · · · , α ℓ } , or with a choice of positiveroots, R + , by the condition λ ∈ P + iff h λ, α i i ≥
0, for 1 ≤ i ≤ ℓ . The oppositechamber − P + defined by the conditions h λ, α i i ≤ P + by a uniqueelement w ∈ W such that w ( P + ) = − P + , which means w (∆) = − ∆, and w ( R + ) = R − . This is the longest element whose length is the number of positiveroots and whose order is 2. For example, in type A , w = r r r = r θ , but intype B , w = r r r r = r θ . Since w (∆) = − ∆, there is an order 2 permutation σ ∈ S ℓ such that w ( α i ) = − α σ ( i ) for 1 ≤ i ≤ ℓ . If ν ∈ P + then w ( ν ) = − ν ∗ isthe lowest weight in Π ν , so we have h ν, α i i = h w ( ν ) , w ( α i ) i = h− ν ∗ , − α σ ( i ) i = h ν ∗ , α σ ( i ) i . We use ν ∗ = − w ( ν ) to extend the definition of dual weight to any ν ∈ H ∗ . Notethat θ is the highest weight of the adjoint representation and − θ = w ( θ ) = − θ ∗ isthe lowest weight, so θ ∗ = θ . Therefore, for any ν ∈ H ∗ we have h ν, θ i = h w ( ν ) , w ( θ ) i = h− ν ∗ , − θ i = h ν ∗ , θ i . We say π V : g → End ( V ) is an integrable representation when π V ( H ) actsdiagonalizably on V and all π V ( e i ) and π V ( f i ) are locally nilpotent on V . This iscertainly true for V any finite dimensional g -module, including the adjoint repre-sentation, g itself, so that exp ( π V ( x )) ∈ GL ( V ) and exp ( ad ( x )) ∈ Aut ( g ) for all x = e i , x = f i and x = h ∈ H . It is not hard to check that( exp ( π V ( x ))) π V ( y ) ( exp ( π V ( x ))) − = π V ( exp ( ad ( x )) y )for all y ∈ g . Of particular interest are the elements r π V i = ( exp ( π V ( f i )))( exp ( π V ( − e i )))( exp ( π V ( f i ))) ∈ GL ( V ) for 1 ≤ i ≤ ℓ . It is known [ Kac ] that r π V i ( V µ ) = V r i ( µ ) for any weight µ of V , and r adi ( g α ) = g r i ( α ) for any root α of g . If the longest element is written as a productof simple reflections, w = r i · · · r i s , then we have corresponding elements w π V = r π V i · · · r π V i s ∈ GL ( V ) and w ad = r adi · · · r adi s ∈ Aut ( g )such that w π V ◦ π V ( y ) ◦ ( w π V ) − = π V ( w ad ( y ))so using y = h ∈ H we can get w π V ( V µ ) = V w ( µ ) and w ad ( g α ) = g w ( α ) . In particular, this means that for 1 ≤ i ≤ ℓ , we have w ad ( e i ) ∈ g w ( α i ) = g − α σ ( i ) so w ad ( e i ) = c i f σ ( i ) for some 0 = c i ∈ C and w ad ( f i ) = c − i e σ ( i ) . Then we have w π V ◦ π V ( f i ) = π V ( w ad ( f i )) ◦ w π V = c − i π V ( e σ ( i ) ) ◦ w π V and for any power, p i , w π V ◦ π V ( f i ) p i = c − p i i π V ( e σ ( i ) ) p i ◦ w π V . Using p i = h µ ∗ , α i i + 1 and V = V λ , we see that w π V provides an isomorphismbetween V − ( λ ; γ, µ ∗ ) = { v ∈ V λγ | π λ ( f i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ } and V + ( λ ; − γ ∗ , µ ) = { v ∈ V λ − γ ∗ | π λ ( e i ) h µ,α i i +1 v = 0 , ≤ i ≤ ℓ } . Since w ad ( g θ ) = g − θ we also have w ad ( e θ ) = c f θ for some 0 = c ∈ C and for anypower, p , w π V ◦ π V ( f θ ) p = c − p π V ( e θ ) p ◦ w π V . Applying w π λ to the space W + k ( λ, β, µ ) in Theorem 6.2 gives the isomorphic space W − k ( λ, − β ∗ , µ ∗ ) = { v ∈ V λ − β ∗ | π λ ( f j ) h µ ∗ ,α j i +1 v = 0 , ≤ j ≤ ℓ, and π λ ( f θ ) k −h β + µ,θ i +1 v = 0 } . (7.1)It is clear that W − k ( λ, − β ∗ , µ ∗ ) is a subspace of V − ( λ ; − β ∗ , µ ∗ ), W − k ( λ, − β ∗ , µ ∗ ) = { v ∈ V − ( λ ; − β ∗ , µ ∗ ) | π λ ( f θ ) k −h β + µ,θ i +1 v = 0 } which corresponds by ξ to a subpace of U . Our next step is to find the conditionon L ∈ U which corresponds to this subspace.
8. Conclusion of the proof
The root vector f θ ∈ g − θ can be expressed as some multibracket of the simpleroot vectors f , · · · , f ℓ , so L ∈ U implies that π ( f θ ) L = 0 so π λ ( f θ ) ◦ L = L ◦ π µ ∗ ( f θ ).Furthermore, since − θ is the lowest root of g , [ f θ , f i ] = 0 for 1 ≤ i ≤ ℓ , so in anyrepresentation of g , the representatives of these root vectors commute. For any p ≥ V ′ V ′ ( p ) = { v ∈ V λ | π λ ( f i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ, π λ ( f θ ) p v = 0 } . NEW PERSPECTIVE ON THE FRENKEL-ZHU FUSION RULE THEOREM 17
Then for any L ∈ U , ξ ( L ) ∈ V ′ ( p ) iff π λ ( f θ ) p L ( v µ ∗ µ ∗ ) = 0 iff π λ ( y ) π λ ( f θ ) p L ( v µ ∗ µ ∗ ) = 0for all y ∈ U ( g − ). But since π λ ( y ) commutes with π λ ( f θ ), and since π λ ( y ) L ( v µ ∗ µ ∗ ) = L ( π µ ∗ ( y ) v µ ∗ µ ∗ ) and U ( g − ) v µ ∗ µ ∗ = V µ ∗ , so ξ ( L ) ∈ V ′ ( p ) iff π λ ( f θ ) p L ( V µ ∗ ) = 0 iff L ( V µ ∗ ) ⊆ Ker ( π λ ( f θ ) p ) . Then ξ provides an isomorphism from the subspace U ( p ) = { L ∈ U | π λ ( f θ ) p L ( V µ ∗ ) = 0 } = { L ∈ U | L ( V µ ∗ ) ⊆ Ker ( π λ ( f θ ) p ) } to V ′ ( p ). Let − ν be one of the weights − ν m which occur in the weight spacedecomposition of U , corresponding to a highest weight module V ν ∗ where ν ∗ = β + µ so h β + µ, θ i = h ν ∗ , θ i = h ν, θ i . We have seen that ξ provides an isomorphismbetween U − ν and V ′ µ ∗ − ν = V − ( λ ; µ ∗ − ν, µ ∗ ) = V − ( λ ; − β ∗ , µ ∗ ), so it also providesan isomorphism between corresponding weight spaces U − ν ( p ) = { L ∈ U − ν | π λ ( f θ ) p L ( V µ ∗ ) = 0 } = { L ∈ U − ν | L ( V µ ∗ ) ⊆ Ker ( π λ ( f θ ) p ) } and V ′− β ∗ ( p ) = { v ∈ V λ − β ∗ | π λ ( f i ) h µ ∗ ,α i i +1 v = 0 , ≤ i ≤ ℓ, π λ ( f θ ) p v = 0 } , which will equal the Walton space W − k ( λ, − β ∗ , µ ∗ ) when p = k − h ν, θ i + 1. Lemma 8.1.
For any integer p ≥ we have Ψ( U − ν ( p )) = ( Ker ( π λ ( f θ ) p ) ⊗ V µ ) ∩ Ψ( U − ν ) and we have the orthogonal direct sum decomposition Ψ( U − ν ) = Ψ( U − ν ( p )) ⊕ P roj λ,µ Ψ( U − ν ) ( Im ( π λ ( e θ ) p ) ⊗ V µ ) . Proof.
Apply the isomorphism Ψ to U − ν ( p ) to get the subspaceΨ( U − ν ( p )) = { Ψ( L ) ∈ V λ ⊗ V µ | L ∈ U − ν ( p ) } of certain lowest weight vectors of weight − ν in V λ ⊗ V µ . Recall the definitionΨ( L ) = d X j =1 L ( v ∗ j ) ⊗ v j where d = dim ( V µ ) = dim ( V µ ∗ ), { v , · · · , v d } is a basis of V µ and { v ∗ , · · · , v ∗ d } isthe dual basis of V µ ∗ . Then we see thatΨ( L ) ∈ Ker ( π λ ( f θ ) p ) ⊗ V µ , for all L ∈ U − ν ( p )since L ( v ∗ j ) ∈ Ker ( π λ ( f θ ) p ) for 1 ≤ j ≤ d . Of course, Ψ( L ) ∈ Ψ( U − ν ), so weget containment in one direction. Now suppose that Ψ( L ) ∈ Ψ( U − ν ) and Ψ( L ) ∈ Ker ( π λ ( f θ ) p ) ⊗ V µ , so for 1 ≤ j ≤ d we have L ( v ∗ j ) ∈ Ker ( π λ ( f θ ) p ), giving L ∈ U − ν ( p ) so Ψ( L ) ∈ Ψ( U − ν ( p )).Let g θ ∼ = sl be the subalgebra with basis e θ , f θ and h θ = [ e θ , f θ ]. As mentionedin Section 3, V λ has a decomposition into the orthogonal direct sum of irreducible g θ -modules, V λ = M i V λγ i ( m i ) where dim ( V λγ i ( m i )) = m i + 1 and the highest weight of V λγ i ( m i ) is γ i ∈ Π λ so m i = γ i ( h θ ). Also recall from Section 3 that from the representation theory of sl ,on each irreducible component we have the orthogonal decomposition V λγ i ( m i ) = Ker ( π λ ( f θ ) p ) ⊕ Im ( π λ ( e θ ) p )into the p lowest h θ weight spaces and the rest. So we also get the orthogonaldecomposition V λ = Ker ( π λ ( f θ ) p ) ⊕ Im ( π λ ( e θ ) p ) . Of course, in the first equation above we mean the kernel and image of thoseoperators restricted to each irreducible component. This gives an orthogonal de-composition V λ ⊗ V µ = Ker ( π λ ( f θ ) p ) ⊗ V µ ⊕ Im ( π λ ( e θ ) p ) ⊗ V µ . Lemma 3.3 applied to this decomposition of the tensor product gives the orthogonaldirect sum decomposition of the subspace Ψ( U − ν ) as stated. (cid:3) Let { Ψ( L ) , · · · , Ψ( L d p ) } be a basis of the first summand Ψ( U − ν ( p )) in theabove decomposition of Ψ( U − ν ), and let { Ψ( L d p +1 ) , · · · , Ψ( L M ) } be a basis of thesecond summand, where M = M ult ν ∗ λ,µ = dim ( U − ν ) = dim (Ψ( U − ν )). Then thereis a basis, { g , · · · , g d p , · · · , g M } of H = Hom g ( V λ ⊗ V µ , V ν ∗ ) determined by theconditions g i (Ψ( L j )) = δ i,j v ν ∗ − ν for v ν ∗ − ν a lowest weight vector in V ν ∗ . The subspace H ( K p ) = { g ∈ H | g (Ψ( U − ν ( p ))) = 0 } of elements of H that vanish on the first summand, has basis { g d p +1 , · · · , g M } andthe subspace H ( I p ) = { g ∈ H | g ( P roj λ,µ Ψ( U − ν ) ( Im ( π λ ( e θ ) p ) ⊗ V µ )) = 0 } of elements of H that vanish on the second summand, has basis { g , · · · , g d p } so d p = dim ( H ( I p )). Remember that the dimension of the Walton space W − k ( λ, − β ∗ , µ ∗ )is d p when p = k − h ν, θ i + 1. But in that case, H ( I p ) equals the Frenkel-Zhu space F Z ′ k ( λ, µ, ν ) = { g ∈ H | g ( P roj λ,µ Ψ( U − ν ) ( e k −h ν,θ i +1 θ V λ ⊗ V µ )) = 0 } so we have completed the proof of Theorem 6.2. References [AFW] F. Akman, A. Feingold, M. Weiner,
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