An Algebraic Construction Leading to Quantum Invariants of 3-manifolds
AAn Algebraic Construction Leading to QuantumInvariants of 3-manifolds
Mucyo Karemera
Abstract
The notion of ˆΨ-system in linear monoidal categories was introduced byGeer, Kashaev and Turaev. They showed that, under additional assumptions,a ˆΨ-system gives rise to invariants of 3-manifolds. They conjectured that allquantum groups at odd roots of unity give rise to a ˆΨ-system and verified thisconjecture in the case of the Borel subalgebra of U q ( sl ). In this paper weconstruct a ˆΨ-system in the category of modules of a quantum group relatedto U q ( sl ) leading to a family of 3-manifolds invariants. These invariants areconstructed using the quantum dilogarithm defined by Faddeev and Kashaevand allow an interpretation in terms of shapes variables of ideal hyperbolictetrahedra. Turaev and Viro first observed in [18] that the category of representations of thequantum group U q ( sl ) gives rise to topological invariants of 3-manifolds. Theinvariants are obtained as state sums on triangulated 3-manifolds. The key ingredi-ents of the state sums are the 6j-symbols. The 6 j -symbols are naturally associatedwith combinatorial tetrahedra. In the case of finite dimensional representationtheory of the angular momentum, as well as of its q -deformation, the numericalvalues of 6j-symbols are specified by six irreducible representations associated withthe edges of the tetrahedron. In particular, they realize explicitly the tetrahedralsymmetries. In the Turaev-Viro theory, a specification of the deformation parameterto roots of unity allows to separate a sector in the representation category with afinite set of irreducible representations, and the 3-manifold invariant is obtained bysumming over all labelings of the edges of a triangulation by representations fromthis finite set.A related construction to the Turaev-Viro invariants was done by Kashaevin [10]. His invariants were defined as state sums on triangulations of the pair1 a r X i v : . [ m a t h . QA ] O c t M, L ), where M is a 3-manifold and L ⊂ M is a link, using charged versionsof 6 j -symbols associated to certain modules of the Borel subalgebra of U q ( sl ).These invariants led him to the famous volume conjecture [11]. The particularityof his 6 j -symbols, constructed using the quantum dilogarithm function defined byFaddeev and Kashaev in [5], is that they depend on a continuous variable and thetetrahedral symmetries are realized implicitly through non-trivial transformationmatrices.These invariants have been generalized in two ways. A topological generalizationhas been done by Baseilhac and Benedetti in [2] where they define quantumhyperbolic invariants of 3-manifolds using the same charged 6 j -symbols. Theyinterpreted the complexified continuous parameters, entering the 6j-symbols, asshape variables of ideal tetrahedra in Thurston’s gluing equations. This enabledthem to construct quantum invariants which are functions on deformation varietiesof cusped 3-manifolds.This topological framework has been used, to some extend, in the other general-ization done by Geer, Kashaev and Turaev in [7]. This construction is a categoricalgeneralization of the Kashaev-Baseilhac-Benedetti construction. It introduces thenotion of a ˆΨ-system in a monoidal abelian category which provides a generalframework for charged 6 j -symbols. However, in this general context, the depen-dence of 6j-symbols on continuous variables does not necessarily reduce only toone variable, and thus the interpretation in terms of the shapes of hyperbolic idealtetrahedra is not evident.The main result of this paper is the construction of a ˆΨ-system in the categoryof modules of a quantum group related to U q ( sl ) leading to a family of 3-manifoldsinvariants. Although the 6 j -symbols involved in this construction come fromanother quantum group than U q ( sl ), they are similar to the one used in theKashaev-Baseilhac-Benedetti theory in the sense that they are also constructedusing the quantum dilogarithm. They also only depend on one parameter allowinginterpretation in terms of shapes variables of ideal hyperbolic tetrahedra.The paper is organized as follows. In Section 2, our main result (Theorem 2.4)is stated and the family of 3-manifolds invariants is defined. In Section 3, usingthe quantum dilogarithm function, we construct a Ψ-system. Finally, in Section 4,we extend it to a ˆΨ-system and prove Theorem 2.4. H -triangulation of ( M, L ) Let M be a closed connected oriented 3-manifold. A quasi-regular triangulation T of M is a decomposition of M as a union of embedded tetrahedra (3-simplices)2uch that the intersection of any two tetrahedra is a union (possibly, empty) ofseveral of their vertices (0-simplices), edges (1-simplices) and faces (2-simplices).Quasi-regular triangulations differ from the usual triangulations in that they mayhave tetrahedra meeting along several vertices, edges, and faces. Note that eachedge of a quasi-regular triangulation has two distinct endpoints. In the sequel, wedenote Λ i ( T ) the set of i -simplices of T for i ∈ { , , , } .Consider a non-empty link L ⊂ M . An H -triangulation of ( M, L ) is a pair( T , L ) where T is a quasi-regular triangulation of M and L ⊂ Λ ( T ) is such thateach vertex of T belongs to exactly two edges of L and L is the union of theelements of L . Proposition 2.1 ([2], Proposition 4.20) . For any non-empty link L in M , the pair ( M, L ) admits an H -triangulation. H -triangulations of ( M, L ) can be related by elementary moves of two types,the H -bubble moves and the H -Pachner moves . The positive H -bubble move onan H -triangulation ( T , L ) starts with a choice of a face F = v v v ∈ Λ ( T ), where v , v , v ∈ Λ ( T ), such that at least one of its edges, say v v , is in L . Considertwo tetrahedra of T meeting along F . We unglue these tetrahedra along F andinsert a 3-ball between the resulting two copies of F . We triangulate this 3-ballby adding a vertex v at its center and three edges connecting v to v , v , and v .The edge v v is removed from L and replaced by the edges v v and v v . Thismove can be visualized as the transformation v v v v v v v where the blue edges belong to L . The inverse move is the negative H -bubble move .The positive H-Pachner move can be visualized as the transformation3 v v v v v v v v v This transformation preserve the set L . The inverse move is the negative H -Pachnermove ; it is allowed only when the edge common to the three tetrahedra on theright is not in L . Proposition 2.2 ([2], Proposition 4.23) . Let L be a non-empty link in M . Any two H -triangulations of ( M, L ) can be related by a finite sequence of H -bubble movesand H -Pachner moves in the class of H -triangulations of (M,L). ( T , L ) and R -coloring of T A charge on T ∈ Λ ( T ) is a map c : Λ ( T ) → Z such that1. c ( e ) = c ( e ) if e, e are opposite edges,2. c ( e ) + c ( e ) + c ( e ) = if e , e , e are edges of a face of T .We denote Λ ( T ) = { ( T, e ) | T ∈ Λ ( T ) , e ∈ Λ ( T ) } and consider the obvious pro-jection (cid:15) T : Λ ( T ) → Λ ( T ). For any edge e of T , the set (cid:15) − T ( e ) has n elements,where n is the number of tetrahedron adjacent to e .A charge on ( T , L ) is a map c : Λ ( T ) → Z such that1. the restriction of c to any tetrahedron T of T is a charge on T ,2. for each edge e of T not belonging to L we have P e ∈ (cid:15) − T ( e ) c ( e ) = 1,3. for each edge e of T belonging to L we have P e ∈ (cid:15) − T ( e ) c ( e ) = 0.Each charge c on ( T , L ) determines a cohomology class [ c ] ∈ H ( M ; Z / Z ). Proposition 2.3 ([2], Lemma 4.10) . Let ( T , L ) and ( T , L ) be H -triangulationsof ( M, L ) such that ( T , L ) is obtained from ( T , L ) by an H -Pachner move or an H -bubble move. Let c be a charge on ( T , L ) . Then there exists a charge c on T , L ) such that c equals c on all pairs ( T, e ) ∈ Λ ( T ) such that T is not involvedin the move and for any common edge e of T and T , X a ∈ (cid:15) − T ( e ) c ( a ) = X a ∈ (cid:15) − T 0 ( e ) c ( a ) . Moreover, [ c ] = [ c ] . A R -coloring of T is a map Φ from the oriented edges of T to R ∗ such that1. Φ( − e ) = − Φ( e ) for any oriented edge e of T , where − e is e with the oppositeorientation,2. Φ( e )+Φ( e )+Φ( e ) = 0 if e , e , e are edges of a face of T with the followingorientations: e e e A R -coloring Φ represents a cohomology class [Φ] ∈ H ( M, R ). The main result of this paper is the construction of a ˆΨ -system in a particularmonoidal abelian category. This allows us to construct a state sum invariant ofany tuple (
M, L, [Φ] , [ c ]). The key ingredients in this state sum are the charged6j-symbols S ( z | a, c ) ± ∈ A ⊗ , where A = End( V ) and V = C N ⊗ C N , and moreprecisley their matrix elements. We give explicit expression of the matrix elementsusing a suitable basis { u α } α ∈ Z N of V , where Z N = Z /N Z , and its dual basis { ¯ u α } α ∈ Z N (defined in Lemma 3.18).Fix an integer t , a natural number N / ∈ N which divides t + t + 1 and aprimitive N -th root of unity ω . Remark that N is odd. Therefore, for any a ∈ Z ,we will write ω a instead of ω a ( N +1) . For any z ∈ R =0 , = R \ { , } , any a, c ∈ Z and any α, β, µ, ν ∈ Z N we have (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z | a, c ) (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z ) (cid:12)(cid:12)(cid:12) u α ⊗ u ( β − ta,β ) (cid:69) × ( z a (1 − z ) c ) − N ) N ω a (2 c − tβ − β )+2( t +1) { a ( a + µ )+ c ( β − ν ) } +(2 t +1) { a ( µ +1)+ c ( β − ν ) } (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z | a, c ) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = (cid:68) ¯ u µ ⊗ ¯ u ( ν +2 ta,ν ) (cid:12)(cid:12)(cid:12) S ( z ) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) × ( z a (1 − z ) c ) − N ) N ω a (2 c − tν − ν )+2( t +1) { a ( a − α )+ c ( β − ν ) } +(2 t +1) { a ( α +1)+ c ( ν − β ) } where h ¯ u µ ⊗ ¯ u ν | S ( z ) | u α ⊗ u β i and h ¯ u µ ⊗ ¯ u ν | S ( z ) − | u α ⊗ u β i are given in Proposition3.20 and where the Nth root is chosen to be the unique real root.Let ( T , L ) be an H -triangulation of ( M, L ). Fix a total order on Λ ( T ) andconsider a R -coloring Φ of T , a charge c on ( T , L ) and a map α : Λ ( T ) → Z N .From this data, we define the state sum as follows. Let T be a tetrahedron of T with order vertices v , v , v , v . We say that T is right oriented if the vertices v , v and v go round in the counter-clockwise direction when we look at them from v in the increasing order. Otherwise, T is left oriented . Set p = Φ( −−→ v v ) , q = Φ( −−→ v v ) , r = Φ( −−→ v v )where −−→ v i v j is the oriented edge of T going from v i to v j . Then set z = pr ( p + q )( q + r )and denote c ij = c ( v i v j ) and α i = α ( v j v k v l ) for { i, j, k, l } = { , , , } . Weassociate the following matrix element to the tetrahedron TT (Φ , c, α ) = (cid:40) (cid:68) ¯ u α ⊗ ¯ u α (cid:12)(cid:12)(cid:12) S ( z | c , c ) (cid:12)(cid:12)(cid:12) u α ⊗ u α (cid:69) if T is right oriented , (cid:68) ¯ u α ⊗ ¯ u α (cid:12)(cid:12)(cid:12) S ( z | c , c ) − (cid:12)(cid:12)(cid:12) u α ⊗ u α (cid:69) if T is left oriented . (2.1)and we define the state sum as follows K N ( T , L , Φ , c ) = N − | Λ ( T ) | X α Y T ∈T T (Φ , c, α ) ∈ C . (2.2) Theorem 2.4.
Up to multiplication by integer powers of ω , the state sum K N ( T , L , Φ , c ) depends only on the isotopy class of L in M and the cohomology classes [Φ] ∈ H ( M, R ) and [ c ] ∈ H ( M, Z / Z ) (and does not depend on the choice of Φ and c in their cohomology classes, the H -triangulation ( T , L ) of ( M, L ) , and the orderingof the vertices of T ). The statement of our Theorem is a direct adaptation of Theorem 29 of [7]. Thereason why K N ( T , L , Φ , c ) is invariant only up to multiplication by integer powersof ω is explained in Subsection 4.3. 6 Construction of a Ψ -system A ω,t and its reduced cyclic modules The two-paramerter quantum groups U r,s ( sl n ) have been introduced by Takeuchiin [17]. In [4], r and s are non zero elements in a filed K such that r = s and U r,s ( sl n ) is defined as the unital associative algebra over K generated by elements k ± i , ( k i ) ± , e i , f i (1 ≤ i < n ) which satisfy the following relations(R1) The k ± i , ( k j ) ± all commutes with one another and k i k − i = k j ( k j ) − = 1,(R2) k i e j = r δ i,j − δ i,j +1 s δ i +1 ,j − δ i,j e j k i and k i f j = r δ i,j +1 − δ i,j s δ i,j − δ i +1 ,j f j k i ,(R3) k i e j = r δ i +1 ,j − δ i,j s δ i,j − δ i,j +1 e j k i and k i f j = r δ i,j − δ i +1 ,j s δ i,j +1 − δ i,j f j k i ,(R4) [ e i , f j ] = δ i,j r − s ( k i − k i ),(R5) [ e i , e j ] = [ f i , f j ] = 0 if | i − j | > e i e i +1 = ( r + s ) e i e i +1 e i + rse i +1 e i , e i e i +1 = ( r + s ) e i +1 e i e i +1 + rse i +1 e i ,(R7) f i f i +1 = ( r − + s − ) f i f i +1 f i + ( rs ) − f i +1 f i , f i f i +1 = ( r − + s − ) f i +1 f i f i +1 + ( rs ) − f i +1 f i .As a Hopf algebra, the coproduct ∆ : U r,s ( sl n ) → U r,s ( sl n ) ⊗ U r,s ( sl n ) is such that k ± i , ( k i ) ± are group-like elements, and the remaining coproducts are determinedby ∆( e i ) = e i ⊗ k i ⊗ e i , ∆( f i ) = 1 ⊗ f i + f i ⊗ k i . Therefore, the counit ε : U r,s ( sl n ) → K and the antipode γ : U r,s ( sl n ) → U r,s ( sl n )are given by ε ( k i ) = ε ( k i ) = 1 , γ ( k i ) = k − i , γ ( k i ) = ( k i ) − ,ε ( e i ) = ε ( f i ) = 0 , γ ( e i ) = − k − i e i , γ ( f i ) = − f i ( k i ) − . When r = q and s = q − , U r,s ( sl n ) modulo the (Hopf) ideal generated by theelements k i − k − i , ≤ i < n , is U q ( sl n ). 7he algebra A ω,t that we are about to introduce, is obtained as a certainquotient of the Borel subalgebra BU r,s ( sl ) of U r,s ( sl ) generated by k ± i , e i , for acertain choice of r, s ∈ C . A ω,t We fix an integer t , a natural number N / ∈ N which divides t + t + 1 and aprimitive N -th root of unity ω . Remark that N is odd and N and t are relativelyprime.The Hopf algebra A ω,t is defined as the quotient of BU ω t ,ω t +1 ( sl ) by the Hopfideal generated by k N − k − k t . Therefore, the generators k , e and e satisfy the following relations k N = 1 , k e = ω − e k , k e = ω t +1 e k ,e e = ( ω t + ω t +1 ) e e e + ω t +1 e e ,e e = ( ω t + ω t +1 ) e e e + ω t +1 e e . (3.1)The coproduct ∆ : A ω,t → A ω,t ⊗ A ω,t is given by∆( k ) = k ⊗ k , ∆( e ) = e ⊗ k ⊗ e , ∆( e ) = e ⊗ k t ⊗ e , (3.2)the counit ε : A ω,t → C by ε ( k ) = 1 , ε ( e ) = 0 , ε ( e ) = 0 , (3.3)and the antipode γ : A ω,t → A ω,t by γ ( k ) = k − , γ ( e ) = − k − e , γ ( e ) = − k − t e . (3.4) A ω,t -modules In what follows, all the vector spaces are finite dimensional C -vector spaces.Let { v α } α ∈ Z N be the canonical basis of C N , where the index α taking its valuesin the set { , , . . . , N − } is interpreted as an element of Z N . Let X , Y ∈ End( C N )be the invertible operators defined by X v α = ω − α v α , Y v α = v α +1 . (3.5) Lemma 3.1.
The algebra
End( C N ) is generated by { X , Y } . In particular, A isgenerated by X , X , Y , Y where X = X ⊗ Id C N , Y = Y ⊗ Id C N , X = Id C N ⊗ X , Y = Id C N ⊗ Y . roof. We consider the canonical basis { v α } α ∈ Z N of C N and its dual basis { ¯ v α } α ∈ Z N .The set { E i,j } i,j ∈ Z N ⊂ End( C N ) defined by h ¯ v β | E i,j | v α i = δ α,j δ β,i , ∀ i, j, α, β ∈ Z N , where δ is Kronecker’s delta, is a linear basis of End( C N ). It is easy to check thatwe have the following relation E i,j E k,l = δ j,k E i,l , ∀ i, j, k, l ∈ Z N . (3.6)First, we compute, for i, α, β ∈ Z N ,1 N X j ∈ Z N (cid:10) ¯ v β | ω ij X j | v α (cid:11) = 1 N X j ∈ Z N ω ij ω − αj δ α,β = 1 N X j ∈ Z N ω j ( i − α ) δ α,β = 1 N N δ i,α δ α,β = h ¯ v β | E i,i | v α i . Hence, for all i ∈ Z N we have E i,i = N P j ∈ Z N ω ij X j .Now, using relation (3.6) and since Y = P j ∈ Z N E j +1 ,j , we can compute, for all j, k ∈ Z N E k,k Y E i,i = E k,k X j ∈ Z N E j +1 ,j E i,i = E k,k X i ∈ Z N δ i,j E j +1 ,i = E k,k E i +1 ,i = δ k,i +1 E i +1 ,i . Hence, for all i ∈ Z N , E i +1 ,i = E i +1 ,i +1 Y E i,i .Finally, for any i, j ∈ Z N we have E i,i − E i − ,i − · · · E i − ( i − j )+1 ,i − ( i − j ) = E i,i − E i − ,i − · · · E j +1 ,j = E i,j . Therefore, the operators X and Y generate End( C N ).We now consider the following operators in A X = X , Y = Y , U = Y − t X t X − t , V = Y − t X t +11 Y . (3.7)We can easily see that we have X N = Y N = U N = V N = Id V . Moreover, byLemma 3.1, A is generated by { X, Y, U, V } since X = X, Y = Y, X = ( X − t Y t U ) t +1 , Y = X − t − Y t V. Proposition-Definition 3.2.
For any p ∈ R =0 , let V p be the space V providedwith a A ω,t -module structure π p : A ω,t → End( V p ) , defined by π p ( k ) = X, π p ( e ) = p N Y,π p ( e ) = (cid:0) p (cid:1) N ( U + V ) Y − , (3.8) where the N th root is chosen to be the unique real root. Furthermore, V p is cyclic,i.e. the operators π p ( k ) , π p ( e ) and π p ( e ) are invertible. We call V p a reducedcyclic A ω,t -module of parameter p . emark 3.3. The reason why we have chosen to use the word “reduced” to namethe cyclic A ω,t -modules V p is because the set { V p } p ∈ R =0 is included in a much largerset of cyclic A ω,t -modules. Indeed, for any p = ( p , p , p ) ∈ C such that p ∈ C ∗ and p / ∈ {− ω i p | i ∈ Z N } , the space V can be provided with a cyclic A ω,t -modulestructure π p : A ω,t → End( V ) , defined by π p ( k ) = X, π p ( e ) = p Yπ p ( e ) = p − t − ( p U + p V ) Y − . The set of reduced cyclic A ω,t -module is determined by the triples of the form p = (cid:16) p N , (cid:0) p (cid:1) N , (cid:0) p (cid:1) N (cid:17) ∈ R where p ∈ R =0 .Proof of Proposition-Definition 1.2. Using equalities (3.5), one can check that XY = ω − YX . Then, using equalities (3.7), one easily sees that π p is an algebra morphism.The operators π p ( k ) and π p ( e ) are clearly invertible since p ∈ R =0 and X , Y ∈ End( C N ) are invertible operators. For the invertibility of π p ( e ) we compute π q ( e ) N . Since U Y − · V Y − = ω − V Y − · U Y − we can compute π q ( e ) N using the q -binomial formula. Indeed, we have π p ( e ) N = p (cid:2) U Y − + V Y − (cid:3) N = p î (cid:0) U Y − (cid:1) N + (cid:0) V Y − (cid:1) N ó = p [Id V + Id V ] = p Id V . (3.9)We conclude that the operator π p ( e ) is invertible since p ∈ R =0 .In order to study the reduced cyclic A ω,t -modules, we need to introduce partic-ular elements of A ω,t . First, we define a , a ∈ A ω,t by ω t (1 − ω ) a = e e − ω t +1 e e , (1 − ω ) a = e e − ω − t e e . (3.10)Using (3.1) and (3.10) one can check that k a = ω t a k , k a = ω t a k ,e a = ω t a e , e a = ω t +1 a e ,e a = ω − t a e , e a = ω − ( t +1) a e ,a a = a a , e e = a + a . (3.11)10nd with (3.2) we have∆( a ) = a ⊗ k t +11 ⊗ a + k t e ⊗ e , ∆( a ) = a ⊗ k t +11 ⊗ a + e k ⊗ e . (3.12)Next, we consider elements c , c ∈ A ω,t defined by c = a k − t e t , c = a k − ( t +1)1 e t . (3.13)By using (3.1) and (3.11), we deduce the following relations k c = c k , k c = c k ,e c = c e , e c = c e , c c = ω t c c (3.14)Finally, for a reduced cyclic A ω,t -module V p , using (3.8), (3.10) and (3.13), wehave π p ( a ) = (cid:0) p (cid:1) N U, π p ( a ) = (cid:0) p (cid:1) N V,π p ( c ) = (cid:0) p t +2 (cid:1) N X − t , π p ( c ) = (cid:0) p t +2 (cid:1) N Y . (3.15)We note that { v α } α ∈ Z N is a basis of common eigenvectors of π p ( k ) and π p ( c ). Lemma 3.4.
A reduced cyclic A ω,t -module V p is simple and V q is equivalent to V p only if p = q .Proof. The simplicity is clear. Indeed, since π p ( k ) = X , π p ( c t +11 ) = (cid:0) p t +2 (cid:1) t +1 N X ,π p ( e ) = p N Y , π p ( c ) = (cid:0) p t +2 (cid:1) N Y , by Lemma 3.1, the only invariant subspaces of V p are { } and V p .If V p ∼ = V q then, by definition, there is an isomorphism S : V p → V q such thatfor all a ∈ A ω,t , Sπ p ( a ) = π q ( a ) S . In particular, we have S ( p Id V ) = Sπ p ( e N ) = π q ( e N ) S = ( q Id V ) S ⇒ p = q. Let Hom A ω,t ( V p , V q ) be the set of morphisms of A ω,t -modules between V p and V q i.e., the set of linear maps f : V p → V q such that for all a ∈ A ω,t , we have π q ( a ) f = f π p ( a ) . Schur’s Lemma implies that 11. if p = q then Hom A ω,t ( V p , V q ) = 0,2. if p = q then Hom A ω,t ( V p , V p ) = End A ω,t ( V p ) = C Id V .Hence, for any f ∈ End A ω,t ( V p ), there is a unique c ∈ C such that f = c Id V . Notation 3.5.
This c is denoted h f i . A ω,t -modules The tensor product V p ⊗ V q is provided with a A ω,t -module structure through( π p ⊗ π q )∆ : A ω,t → End( V p ⊗ V q ) . Definition 3.6.
An admissible pair ( V p , V q ) is a pair of reduced cyclic A ω,t -modulessuch that p = − q . In such case, we will also say that the pair ( p, q ) is admissible. The reason we are interested in admissible pairs is that their tensor productdecomposes as a direct sum of reduced cyclic A ω,t -modules. Lemma 3.7.
We have ( π p ⊗ π q )∆( e ) N = ( p + q ) Id V ⊗ Id V , ( π p ⊗ π q )∆( c ) N = ( π p ⊗ π q )∆( c ) N = ( p + q ) t +2 Id V ⊗ Id V . Proof.
Let ( V p , V q ) be a pair of reduced cyclic A ω,t -modules. For ( π p ⊗ π q )∆( e )we compute( π p ⊗ π q )∆( e ) N = ( π p ⊗ π q ) (cid:0) ∆( e ) N (cid:1) = ( π p ⊗ π q ) (cid:0) ( e ⊗ k ⊗ e ) N (cid:1) = ( π p ⊗ π q ) (cid:0) e N ⊗ k N ⊗ e N (cid:1) = pY N ⊗ Id V + qX N ⊗ Y N = ( p + q ) Id V ⊗ Id V where we used the q -binomial formula for the third equality.For ( π p ⊗ π q )∆( c ) we have( π p ⊗ π q )∆( c ) N = ( π p ⊗ π q )∆ (cid:0) ( a k − t e t ) N (cid:1) = ( π p ⊗ π q )∆ (cid:0) a N k − Nt e Nt (cid:1) = ( π p ⊗ π q )∆( a N )( π p ⊗ π q )∆( k N ) − t ( π p ⊗ π q )∆( e N ) t = ( π p ⊗ π q )∆( a ) N (Id V ⊗ Id V ) − t ( p + q ) t Id V ⊗ Id V = ( π p ⊗ π q )∆( a ) N ( p + q ) t Id V ⊗ Id V where we used (3.1) and (3.11) for the second equality. Using (3.11) again, onecan use the q -binomial formula to compute ( π p ⊗ π q )∆( a ) N since t and t + 1 areinvertible in Z N . Using (3.12), we have 12 π p ⊗ π q )∆( a ) N = ( π p ⊗ π q ) (cid:0) a ⊗ k t +11 ⊗ a + k t e ⊗ e (cid:1) N = ( π p ⊗ π q ) Ä a N ⊗ k N ( t +1)1 ⊗ a N + k Nt e N ⊗ e N ä = p U N ⊗ Id V + q Id V ⊗ U N + pq Id V ⊗ Id V = ( p + q ) Id V ⊗ Id V where we used (3.1), (3.11) and the q -binomial formula for the second equality,(3.9) and (3.15) for the third one and (3.1) and (3.8) for the last one. Hence wehave ( π p ⊗ π q )∆( c ) N = ( p + q ) t +2 Id V ⊗ Id V . Finally the computation of ( π p ⊗ π q )∆( c ) N is similar to the previous one. Wehave( π p ⊗ π q )∆( c ) N = ( π p ⊗ π q )∆ Ä ( a k − ( t +1)1 e t ) N ä = ( π p ⊗ π q )∆ Ä a N k − N ( t +1)1 e Nt ä = ( π p ⊗ π q )∆( a ) N ( p + q ) t Id V ⊗ Id V = ( π p ⊗ π q )( a ⊗ k t +11 ⊗ a + e k ⊗ e ) N ( p + q ) t Id V ⊗ Id V = ( π p ⊗ π q )( a N ⊗ k N ( t +1)1 ⊗ a N + e N k N ⊗ e N )( p + q ) t Id V ⊗ Id V = ( p + q ) ( t +2) Id V ⊗ Id V . Proposition 3.8.
Let ( V p , V q ) be an admissible pair. Then the A ω,t -module V p ⊗ V q is equivalent to the direct sum of N copies of the reduced cyclic A ω,t -module V p + q .Proof. In order to find the submodules of V p ⊗ V q , we only consider the action of k , c , e and c on V p ⊗ V q .First, ( π p ⊗ π q )∆( k ) = X ⊗ X is clearly diagonalizable and its spectrum is theset of all N -th roots of unity { ω α | α ∈ Z N } . We write V p ⊗ V q = M α ∈ Z N W α where W α = Ker (( π p ⊗ π q )∆( k ) − ω α Id V ⊗ Id V ). We have dim W α = N for all α ∈ Z N .Now we show that for each α ∈ Z N , we can decompose W α in the following way W α = M β ∈ Z N W ( α,β ) W ( α,β ) = Ker (cid:0) ( π p ⊗ π q )∆( c ) | Wα − sω tβ Id W α (cid:1) and s = (cid:0) ( p + q ) t +2 (cid:1) N ∈ R ∗ . By Lemma 3.7, we have(( π p ⊗ π q )∆( c )) N = ( π p ⊗ π q )∆( c ) N = s N Id V ⊗ Id V . This means that the minimal polynomial of ( π p ⊗ π q )∆( c ) divides x N − s N = Y β ∈ Z N ( x − sω β )This implies that the minimal polynomial of ( π p ⊗ π q )∆( c ) has only simple zeros,which means that ( π p ⊗ π q )∆( c ) is diagonalizable. Moreover, it also implies that thespectrum of ( π p ⊗ π q )∆( c ) is contained in (cid:8) sω β | β ∈ Z N (cid:9) . Actually, its spectrumis exactly (cid:8) sω β | β ∈ Z N (cid:9) , since ( π p ⊗ π q )∆( c ) is invertible, c c = ω t c c and t isinvertible in Z N . Futhermore, since k c = c k , k c = c k and that ( π p ⊗ π q )∆( c )and ( π p ⊗ π q )∆( c ) are invertible, we have( π p ⊗ π q )∆( k )( W α ) = ( π p ⊗ π q )∆( c )( W α ) = W α , ∀ α ∈ Z N , which implies that the spectrum of ( π p ⊗ π q )∆( c ) | Wα is (cid:8) sω β | β ∈ Z N (cid:9) . This allowsus to define the eigenspaces W ( α,β ) for all α, β ∈ Z N as announced.Now, one can see that for all α, β ∈ Z N , we have dim W ( α,β ) = N . Indeed, thiscomes from the fact that dim W α = N , ( π p ⊗ π q )∆( c ) | Wα is invertible, c c = ω t c c and t is invertible in Z N .Let { u i } i ∈ Z N be a basis of W , and consider for all α, β ∈ Z N and all i ∈ Z N ξ ( α,β ) ,i = 1 r α s β ( π p ⊗ π q )∆( e α c β ) u i ∈ W ( α,β ) , where r = ( p + q ) N ∈ R =0 . By construction, we have for all α, β ∈ Z N and all i ∈ Z N ( π p ⊗ π q )∆( k ) ξ ( α,β ) ,i = ω − α ξ ( α,β ) ,i , ( π p ⊗ π q )∆( c ) ξ ( α,β ) ,i = sω tβ ξ ( α,β ) ,i , ( π p ⊗ π q )∆( e ) ξ ( α,β ) ,i = rξ ( α +1 ,β ) ,i , ( π p ⊗ π q )∆( c ) ξ ( α,β ) ,i = sξ ( α,β +1) ,i . This clearly shows that for each i ∈ Z N , the subspace Ξ i ⊂ V p ⊗ V q generated by (cid:8) ξ ( α,β ) ,i (cid:9) α,β ∈ Z N is an irreducible submodule. Furthermore, Ξ i is a reduced cyclic A ω,t -module of parameter p + q . By Lemma 3.4, the reduced cyclic A ω,t -modulesΞ i are all equivalent. Remark 3.9.
The dual space V ∗ p of a reduced cyclic A ω,t -module V p can be providedwith a A ω,t -module structure π ∗ p : A ω,t → End( V ∗ p ) defined, for all a ∈ A ω,t and all α, β ∈ Z N , by (cid:0) π ∗ p ( a ) v ∗ α (cid:1) v β = v ∗ α ( π p ( γ ( a )) v β ) . This A ω,t -module is actually equivalent to the reduced cyclic A ω,t -module V − p . .2 A Ψ -system in the category of A ω,t -modules Since A ω,t is a Hopf algebra over C , the category of A ω,t -modules, as a subcategoryof the category of C -vector spaces, is a monoidal abelian category. This is theframework required to construct a Ψ-system in a category.In order to make this construction, we need to introduce certain sets of mor-phisms called the multiplicity spaces. We have previously have shown that1. V p ∼ = V q if and only if p = q ,2. if p = q then End A ω,t ( V p ) = C Id V ,3. if p = q then Hom A ω,t ( V p , V q ) = 0,4. If ( V p , V q ) is an admissible pair then V p ⊗ V q ∼ = ( V p + q ) ⊕ N . Definition 3.10.
Let ( V p , V q ) be an admissible pair, the following vector spaces oflinear maps are called multiplicity spaces H p,q = { f : V p + q → V p ⊗ V q | f π p + q ( a ) = ( π p ⊗ π q )∆( a ) f, ∀ a ∈ A ω,t } ¯ H p,q = { f : V p ⊗ V q → V p + q | π p + q ( a ) f = f ( π p ⊗ π q )∆( a ) , ∀ a ∈ A ω,t } . In other words, the elements of the multiplicity spaces are morphisms in thecategory of A ω,t -modules, H p,q = Hom A ω,t ( V p + q , V p ⊗ V q ) and ¯ H p,q = Hom A ω,t ( V p ⊗ V q , V p + q ) . By Proposition 3.8 and Schur’s lemma, we clearly have dim H p,q = dim ¯ H p,q = N . In order to construct bases for the multiplicity spaces, we start by introducing twooperator valued functions.Let U ∈ A ⊗ be an operator such that U N = − Id V ⊗ and x ∈ R =0 , . The firstfunction is the quantum dilogarithm Ψ x ( U ) introduced by Faddeev and Kashaev [5].It is defined as a solution of the functional equationΨ x ( ω − U )Ψ x ( U ) = (1 − x ) N − x N U . (3.16)15ollowing [7], we can write Ψ x ( U ) = X α ∈ Z N ψ x,α ( − U ) α where ψ x,α ∈ C . For all α ∈ Z N , (3.16) implies that ψ x,α = x N ω − α − (1 − x ) N ψ x,α − . Then, for all α ∈ Z N , we write ψ x,α = ψ x, α Y j =1 x N ω − j − (1 − x ) N , (3.17)where ψ x, ∈ C ∗ is chosen so that det (Ψ x ( U )) = 1. Using the notation of [13], wehave ψ x,α = ψ x, ω α ( α +1) x αN w Ä (1 − x ) N (cid:12)(cid:12) α ä (3.18)where w ( x | α ) = α Y j =1 − xω j , is defined for all x ∈ C such that x N = 1 and all α ∈ { , · · · , N − } ⊂ Z .The computation of the determinant of Ψ x ( U ) N is made possible due to thefollowing formula shown in [9].Ψ x ( U ) N = ψ Nx, (1 − x ) − N D (1) D Ä (1 − x ) − N ä D Å U ω Ä x − x ä N ã (3.19)where D ( x ) = N − Y j =1 (1 − xω j ) j . (3.20) Lemma 3.11.
For any x ∈ R =0 , and any operator U ∈ A ⊗ such that there existsan invertible opertator M ∈ A ⊗ such that U = − M ( X ⊗ Id V ) M − , we have ψ Nx, = (1 − x ) N − D Ä (1 − x ) − N ä D (1) − ⇒ det (cid:0) Ψ x ( U ) N (cid:1) = 1 . (3.21)16 roof. First we compute det Å D Å X ω Ä xx − ä N ãã using the matrix form of X in thecanonical basis { v α } α ∈ Z N of C N . D Å X ω Ä xx − ä N ã = N − Y j =1 Å Id C N − Ä xx − ä N ω j +1 X ã j = N − Y j =1 − Ä xx − ä N ω j +1 − Ä xx − ä N ω j +2 . . . 1 − Ä xx − ä N ω j + N j Hence we havedet Å D Å X ω Ä xx − ä N ãã = N − Y j =1 N Y i =1 Å − Ä xx − ä N ω j + i ã j = N − Y j =1 Ç − ÅÄ xx − ä N ω j ã N å j = N − Y j =1 Ä − Ä xx − ää j = N − Y j =1 Ä − x ä j = Ä − x ä N ( N − Since U = − M ( X ⊗ Id V ) M − = − M ( X ⊗ Id C N ⊗ Id V ) M − , we havedet Å D Å U ω Ä x − x ä N ãã = det Å D Å X ω Ä xx − ä N ãã N = (1 − x ) − N N − . Therefore, we havedet (cid:0) Ψ x ( U ) N (cid:1) = det Ç ψ Nx, (1 − x ) − N D (1) D Ä (1 − x ) − N ä − D Å U ω Ä x − x ä N ã − å = (cid:16) ψ Nx, (1 − x ) − N D (1) D Ä (1 − x ) − N ä − (cid:17) N det Å D Å U ω Ä x − x ä N ãã − = (cid:16) ψ Nx, (1 − x ) − N D (1) D Ä (1 − x ) − N ä − (1 − x ) N − (cid:17) N = (cid:16) ψ Nx, (1 − x ) − N D (1) D Ä (1 − x ) − N ä − (cid:17) N . So if ψ Nx, = (1 − x ) N − D Ä (1 − x ) − N ä D (1) − then det (cid:0) Ψ x ( U ) N (cid:1) = 1.From now on, for all x ∈ R =0 , , we assume that ψ x, satisfies (3.21) and is suchthat det (Ψ x ( U )) = 1. 17he inverse of Ψ x ( U ) that we will denote ¯Ψ x ( U ) satisfies¯Ψ x ( U )¯Ψ x ( ω − U ) = (1 − x ) N − x N U and Ψ x ( U ) ¯Ψ x ( U ) = 1 . Setting ¯Ψ x ( U ) = X α ∈ Z N ¯ ψ x,α ( − U ) α we find, in a similar way, that for all α ∈ Z N ¯ ψ x,α = ¯ ψ x, Ä xx − ä αN ω α w Ä (1 − x ) − N (cid:12)(cid:12) α ä (3.22)where ¯ ψ x, ∈ C ∗ is chosen according to ψ x, ∈ C ∗ . In order to make this choice, weuse a formula similar to the formula (3.19). This formula and the following Lemmawere shown in [9].¯Ψ x ( U ) N = ¯ ψ Nx, N − N ( − x ) − N − D (1) D Ä (1 − x ) N ä D (cid:16) U − (cid:0) − xx (cid:1) N (cid:17) (3.23) Lemma 3.12.
There exists α ∈ Z N such that for any x ∈ R =0 , and any operator U ∈ A ⊗ such that U N = − Id V ⊗ we have ¯ ψ x, = ω α % Ä (1 − x ) N ä ψ − x, ⇒ Ψ x ( U ) ¯Ψ x ( U ) = 1 (3.24) where % ( x ) = N − − x N − x . (3.25)From now on, for all x ∈ R =0 , , we assume that ¯ ψ x, satisfies (3.24). Remark 3.13.
The fact that our invariant is defined up to multiplication by aninteger power of ω is independent from the fact that we have determined ¯ ψ x, upto multiplication by an integer power of ω . In other words, even if we had exactlydetermined ¯ ψ x, , our invariant would still be defined up to multiplication by aninteger power of ω . This will be explained in Subsection 4.3. The second operator valued function, is defined by L ( U , V ) = 1 N X α,β ∈ Z N ω − αβ U α ⊗ V β . where U , V ∈ A satisfy U N = V N = Id V .18 emma 3.14. Let α, β ∈ Z N , we have L ( ω α U , ω β V ) = L ( U , V ) ω αβ U β ⊗ V α . Proof. L ( ω α U , ω β V ) = 1 N X γ,δ ∈ Z N ω − γδ + αγ + βδ U γ ⊗ V δ = 1 N X γ,δ ∈ Z N ω − γ ( δ − α )+ βδ U γ ⊗ V δ − α + α = 1 N X γ,δ ∈ Z N ω − γδ + β ( δ + α ) U γ ⊗ V δ V α = 1 N X γ,δ ∈ Z N ω − ( γ − β ) δ + αβ U γ − β + β ⊗ V δ V α = 1 N X γ,δ ∈ Z N ω − γδ U γ ⊗ V δ ω αβ U β ⊗ V α = L ( U , V ) ω αβ U β ⊗ V α . Lemma 3.15. If U ∈ A is an operator such that U N = Id V , then det ( L ( U , X )) = 1 . (3.26) Proof.
We consider the operators of A in their matrix form in the basis { v α } α ∈ Z N of V . For any β ∈ Z N , we compute X β = X β ⊗ Id C N = à Id C N ω β Id C N . . . ω β ( N − Id C N í . Thereby, for any α ∈ Z N , we have X β ∈ Z N ω − αβ X β = X β ∈ Z N â ω − αβ Id C N ω β (1 − α ) Id C N . . . ω β ( N − − α ) Id C N ì = N à δ α, Id C N δ α, Id C N . . . δ α,N − Id C N í . M ∈ A ⊗ be the permutation matrix such that for all V , W ∈ A we have W ⊗ V = MV ⊗ WM − . Then we compute M L ( U , X ) M − = 1 N X α,β ∈ Z N ω − αβ X β ⊗ U α = X α ∈ Z N Ç N X β ∈ Z N ω − αβ X β ⊗ U α å = X α ∈ Z N à δ α, Id C N ⊗ U α δ α, Id C N ⊗ U α . . . δ α,N − Id C N ⊗ U α í = à Id C N ⊗ Id C N Id C N ⊗ U . . . Id C N ⊗ U N − í . Finally we havedet ( L ( U , X )) = det (cid:0) M L ( U , X ) M − (cid:1) = det (cid:16) U N ( N − (cid:17) = 1 . S ( x )The operator S ( x ), where x ∈ R =0 , , that we are now going to define is central forour theory. Indeed, the bases for the multiplicity spaces as well as the 6 j -symbols,which are key ingredients to construct our invariant, will be expressed through thisoperator. Notation 3.16.
Let U and V be two operators such that UV = ω α VU , where α ∈ Z N . We will write this relation in the following way , , U V U V U V if α = 0 if α = 1 if α = − U V α if α ∈ Z N \ { , ± } Note that in the case of operators
X, Y, U and V , we have the following relations20 VYU t tt + 1 t We define, for any x ∈ R =0 , , the following invertible operator valued function S ( x ) = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X )where E = − Y − X Y , F = U − X t +11 V E − , G = U V − F, H = U V − E. (3.27)and the subscripts show how the operators are embedded in A ⊗ . These operatorshave the following commutation relations H EGF
Note that
E, F, G and H satisfy the condition of Lemma 3.12. Thereby, usingLemma 3.15, we see that the operator S ( x ) is invertible since det (cid:0) S ( x ) N (cid:1) = 1.Moreover, since L ( U t V, X ) − = L ( U − t V − , X ) , we have S ( x ) − = L ( U − t V − , X ) ¯Ψ x ( H ) ¯Ψ x ( G ) ¯Ψ x ( F ) ¯Ψ x ( E ) . Proposition 3.17.
For any admissible pair ( p, q ) ∈ ( R =0 ) , the following equationis satisfied ( π p ⊗ π q )∆( a ) = S Ä qp + q ä ( π p + q ( a ) ⊗ Id A ) S Ä qp + q ä − , ∀ a ∈ A ω,t (3.28)21 roof. Since e e = a + a , it is enough to check equation (3.28) for a ∈{ k , e , a , a } . In the following computaions, we set x = qp + q ∈ R =0 , and wewill thus write q − xqx instead of p and qx instead of p + q .For a = k : Since XU t V = ω − U t V X , we have, by Lemma 3.14, L ( U t V, X ) X = 1 N X α,β Z N ω − αβ X ( ωU t V ) α X β = X L ( ωU t V, X )= X X L ( U t V, X ) . Hence, since X X commutes with E, F, G and H , we have, using (3.8) S ( x )( π qx ( k ) ⊗ Id A ) = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X ) X = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) X X L ( U t V, X )= X X Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X )= ( π q − xqx ⊗ π q )∆( k ) S ( x ) . For a = e : Since Y commutes with F, G, H and U t V and that Y E = ωEY we have, using (3.8) and equation (3.16) S ( x )( π qx ( e ) ⊗ Id A ) = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X ) (cid:0) qx (cid:1) N Y = (cid:0) qx (cid:1) N Y Ψ x ( ω − E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X )= (cid:0) qx (cid:1) N Y Ä (1 − x ) N − x N E ä S ( x ) = (cid:16)(cid:0) q − xqx (cid:1) N Y + q N X Y (cid:17) S ( x )= ( π q − xqx ⊗ π q )∆( e ) S ( x ) . For a = a : Since U commutes with U t V and has the following commutationrelations H EGF U we have, S ( x )( π qx ( a ) ⊗ Id A ) = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X ) Ä q x ä N U = Ä q x ä N U Ψ x ( E )Ψ x ( ω − F )Ψ x ( ω − G )Ψ x ( H ) L ( U t V, X ) . x ( E )Ψ x ( ω − F )Ψ x ( ω − G ) = Ψ x ( E )Ψ x ( ω − F ) Ä (1 − x ) N − x N G ä Ψ x ( G )= Ψ x ( E ) Ä (1 − x ) N Ψ x ( ω − F ) − x N G Ψ x ( F ) ä Ψ x ( G )= Ä (1 − x ) N Ä (1 − x ) N − x N F ä Ψ x ( E ) − x N G Ψ x ( ω − E ) ä Ψ x ( F )Ψ x ( G )= Ä (1 − x ) N − ( x − x ) N F − x N G Ä (1 + x ) N − x N E ää Ψ x ( E )Ψ x ( F )Ψ x ( G )= Ä (1 − x ) N + x N GE − ( x − x ) N ( F + G ) ä Ψ x ( E )Ψ x ( F )Ψ x ( G ) . Therefore, we finally have, using (3.27) and (3.15) S ( x )( π qx ( a ) ⊗ Id V )= Ä q x ä N U Ψ x ( E )Ψ x ( ω − F )Ψ x ( ω − G )Ψ x ( H ) L ( U t V, X )= Ä q x ä N U Ä (1 − x ) N + x N GE − ( x − x ) N ( F + G ) ä S ( x )= Å (cid:0) ( q − xqx ) (cid:1) N U + (cid:0) q (cid:1) N U GE − Ä q − xq x ä N U ( F + G ) ã S ( x )= Å (cid:0) ( q − xqx ) (cid:1) N U + (cid:0) q (cid:1) N X t +11 U − Ä q − xq x ä N X t Y ( U + V ) Y − ã S ( x )= ( π q − xqx ⊗ π q ) (cid:0) a ⊗ k t +11 ⊗ a + k t e ⊗ e (cid:1) S ( x ) = ( π q − xqx ⊗ π q )∆( a ) S ( x ) . For a = a : Since V commutes with U t V and has the following commutationrelations H EGF V we have, S ( x )( π qx ( a ) ⊗ Id V ) = Ψ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H ) L ( U t V, X ) Ä q x ä N V = Ä q x ä N V Ψ x ( ω − E )Ψ x ( F )Ψ x ( G )Ψ x ( ω − H ) L ( U t V, X )= Ä q x ä N V Ψ x ( ω − E )Ψ x ( F )Ψ x ( ω − H )Ψ x ( G ) L ( U t V, X ) . x ( F )Ψ x ( ω − H ) = Ψ x ( F ) Ä (1 − x ) N − x N H ä Ψ x ( H )= Ä (1 − x ) N Ψ x ( F ) − x N H Ψ x ( ω − F ) ä Ψ x ( H )= Ä (1 − x ) N Ψ x ( F ) − x N H Ä (1 − x ) N − x N F ä Ψ x ( F ) ä Ψ x ( H )= Ä (1 − x ) N + x N HF − ( x − x ) N H ä Ψ x ( F )Ψ x ( H ) . Hence, using equality (3.16) again, we haveΨ x ( ω − E )Ψ x ( F )Ψ x ( ω − H )= Ä (1 − x ) N Ä (1 − x ) N − x Y N E ä + x N HF − ( x − x ) N H ä Ψ x ( E )Ψ x ( F )Ψ x ( H )= Ä (1 − x ) N + x N HF − ( x − x ) N ( E + H ) ä Ψ x ( E )Ψ x ( F )Ψ x ( H ) . Therefore, using (3.27), (3.15) and the fact that Ψ x ( H )Ψ x ( G ) = Ψ x ( G )Ψ x ( H ), wefinally have S ( x )( π qx ( a ) ⊗ Id V )= Ä q x ä N V Ψ x ( ω − E )Ψ x ( F )Ψ x ( ω − H )Ψ x ( G ) L ( U t V, X )= Ä q x ä N V Ä (1 − x ) N + x N HF − ( x − x ) N ( E + H ) ä S ( x )= Å (cid:0) ( q − xqx ) (cid:1) N V + (cid:0) q (cid:1) N V HF − Ä q − xq x ä N V ( E + H ) ã S ( x )= Å (cid:0) ( q − xqx ) (cid:1) N V + (cid:0) q (cid:1) N X t +11 V − Ä q − xq x ä N ( U + V ) Y − X Y ã S ( x )= ( π q − xqx ⊗ π q ) (cid:0) a ⊗ k t +11 ⊗ a + e k ⊗ e (cid:1) S ( x ) = ( π q − xqx ⊗ π q )∆( a ) S ( x ) . From now on, for technical reasons, we will use the following particular basis of V instead of the canonical basis { v α } α ∈ Z N . Lemma 3.18.
The set { u α } α ∈ Z N ⊂ V where u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) v ( α ,β ) , for all α ∈ Z N , is a basis of V . Its dual { ¯ u α } α ∈ Z N ⊂ V ∗ is given, for all α ∈ Z N ,by ¯ u α = 1 N X β ∈ Z N ω tβ ( β +1) − β ( α − α + ) ¯ v ( α ,β ) , roof. For any α ∈ Z N we have v α = 1 N ω tα ( α +1)+ α ( α − ) X β ∈ Z N ω − α β u ( α ,β ) and ¯ v α = ω − tα ( α +1) − α ( α − ) X β ∈ Z N ω α β ¯ u ( α ,β ) . This shows that { u α } α ∈ Z N , and { ¯ u α } α ∈ Z N , are generating sets of V and V ∗ re-spectively. Hence these sets are bases since their cardinality is the same as thedimension of V and V ∗ .A straightforward computation shows that for all α, β ∈ Z N we have¯ u β u α = δ α,β which implies that { u α } α ∈ Z N , and { ¯ u α } α ∈ Z N are dual bases.The basis { u α } α ∈ Z N satisfies the following relations. Lemma 3.19.
For all α ∈ Z N and all m ∈ Z N X m u α = ω − mα u α , Y m u α = u ( α + m,α + m ) ,U m u α = ω − m ( m − t +1) − mtα u ( α − mt,α ) ,V m u α = ω − m ( m − t +1) − m ( tα + α + ) u ( α − mt,α ) . In particular, we have for all α ∈ Z N and all m ∈ Z N ( U V − ) m u α = ω m ( α + ) u α , ( U t V ) m u α = ω m ( m − m ( α − α + t ) u ( α + m,α ) . Proof.
For any α ∈ Z N we have, by the equalities (3.7), Xv α = ω − α v α , Y v α = v ( α +1 ,α ) ,U v α = ω − t ( α − α ) v ( α − t,α ) ,V v α = ω − ( t +1) α v ( α − t,α +1) . Then we can easily derive the following equalities
U V − v α = ω α + tα v ( α ,α − ,U t V v α = ω − ( t +1)( α + ) v ( α +1 ,α +1) . u α for any α ∈ Z N .For X , we have Xu α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) Xv ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) ω − α v ( α ,β ) = ω − α u α . For Y , we have Y u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) Y v ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) v ( α +1 ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( ( α +1) − ( α +1)+ ) v ( α +1 ,β ) = u ( α +1 ,α +1) . For U , we have U u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) U v ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) ω − t ( α − β ) v ( α − t,β ) = ω − tα X β ∈ Z N ω − tβ ( β +1)+ β ( α − ( α − t )+ ) v ( α − t,β ) = ω − tα u ( α − t,α ) . For V , we have V u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) V v ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) ω − ( t +1) α v ( α − t,β +1) = ω − ( tα + α + ) X β ∈ Z N ω − t ( β +1)( β +2)+( β +1) ( α − ( α − t )+ ) v ( α − t,β +1) = ω − ( tα + α + ) u ( α − t,α ) . For
U V − , we have U V − u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) U V − v ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) ω α + tβ v ( α ,β − = ω α + X β ∈ Z N ω − t ( β − β +( β − ( α − α + ) v ( α ,β − = ω α + u α . U t V , we have U t V u α = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) U t V v ( α ,β ) = X β ∈ Z N ω − tβ ( β +1)+ β ( α − α + ) ω − ( t +1) ( β + ) v ( α +1 ,β +1) = ω α − α + t X β ∈ Z N ω − t ( β +1)( β +2)+( β +1) ( α − ( α +1)+ ) v ( α +1 ,β +1) = ω α − α + t u ( α +1 ,α ) . The Lemma follows easily from the previous equalities.Now we can give an explicit expression of matrix elements of S ( x ) and itsinverse using two functions defined in [13]. Let us recall what are these functions.The first one is defined by f ( x, y | z ) = N − X m =0 w ( x | m ) w ( y | m ) z m , (3.29)where x, y, z ∈ C are such that (cid:8) x N , y N (cid:9) ⊂ C \ { } and z N = 1 − x N − y N . (3.30)The last condition provides periodicity on (3.29) on the variable m of period N .The second function is a generalisation of the previous one, namely F Å x uy v (cid:12)(cid:12)(cid:12)(cid:12) z ã = N − X m =0 w ( x | m ) w ( u | m ) w ( y | m ) w ( v | m ) z m , (3.31)where x, y, u, v, z ∈ C are such that (cid:8) x N , y N , u N , v N (cid:9) ⊂ C \ { } and z N = (1 − x N )(1 − u N )(1 − y N )(1 − v N ) . (3.32)The last condition provides periodicity on (3.31) on the variable m of period N ,just as for the function f ( x, y | z ).The explicit matrix elements of S ( x ) and S ( x ) − with respect to { u α } α ∈ Z N and { ¯ u α } α ∈ Z N were computed in [9] and are given in the following proposition. Proposition 3.20 ([9], Proposition 2.11) . For all x ∈ R =0 , and all α, β, µ, ν ∈ Z N ,set r = ( t + 1)( ν − β + µ − α ) and s = ( t + 1)( ν − β ) + t ( µ − α ) . hen we have (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( x ) (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = δ µ + ν ,α δ µ + ν ,α + β × ω − ( t +1)( α − µ )( α − µ − t − t +1) α − β + ν +1)) × ω − ( t +1) { β ( β − tα + α − ν ))+ ν (( t +1) ν +2( µ + ν +1)) } × ψ x, ψ x,r ψ x,s f Ä (1 − x ) N , (1 − x ) − N ω − r − (cid:12)(cid:12) ( x − N ω β + ä × F Ç (1 − x ) N ( x − − N ω β − r − (1 − x ) − N ω − s − ( x − N ω β − (cid:12)(cid:12)(cid:12)(cid:12) − ω α + å and (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( x ) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = δ µ ,α + β δ µ + ν ,α + β × ω − ( t +1)( α − µ )( α − µ − t − α + tµ + ν +1)) × ω ( t +1) { β (( t − β +2( tµ + ν +1))+ ν ( ν − − t +1) µ − tµ − ν − t )) } × ¯ ψ x, ¯ ψ x,r ¯ ψ x,s f Ä (1 − x ) − N , (1 − x ) N ω − r − (cid:12)(cid:12) ( x − − N ω ν + r + ä × F Ç (1 − x ) − N ( x − N ω ν − r − (1 − x ) N ω − s − ( x − − N ω − ν − (cid:12)(cid:12)(cid:12)(cid:12) − ω α + å . For all x ∈ R =0 , , we define e α ( x ) = S ( x )(Id V ⊗ u α ) ¯ e α ( x ) = (Id V ⊗ ¯ u α ) S ( x ) − . (3.33)If ( p, q ) ∈ ( R =0 ) is an admissible pair, then by equation (3.28), e α Ä qp + q ä ∈ H p,q and ¯ e α Ä qp + q ä ∈ ¯ H p,q , ∀ α ∈ Z N . Moreover, ¶ e α Ä qp + q ä© α ∈ Z N and ¶ ¯ e α Ä qp + q ä© α ∈ Z N form dual basis of H p,q and ¯ H p,q respectively, where the duality is reflected by the relations¯ e β Ä qp + q ä e α Ä qp + q ä = δ α,β Id V , (3.34)and X α ∈ Z N e α Ä qp + q ä ¯ e α Ä qp + q ä = Id V ⊗ Id V (3.35)28 emark 3.21. As we have just shown, the non-trivial elements of the multiplicityspaces H p,q and ¯ H p,q depend only on qp + q ∈ R =0 , . We will call this property thescaling invariance property of the multiplicity spaces. In order to keep track ofthe difference between H p,q and H λp,λq and the difference between ¯ H p,q and ¯ H λp,λq respectively, where λ ∈ R =0 , we define, for all admissible pair ( p, q ) ∈ ( R =0 ) , thefollowing isomorphism h p,q : V → H p,q and ¯ h p,q : V ∗ → ¯ H p,q by h p,q ( u α ) = e α Ä qp + q ä , ¯ h p,q (¯ u α ) = ¯ e α Ä qp + q ä , ∀ α ∈ Z N Ψ -systemDefinition 3.22. A Ψ -system in the category of A ω,t -modules consists of1. a distinguished set of simple objects { V i } i ∈ I such that Hom( V i , V j ) = 0 for all i = j ,2. an involution, I → I , i i ∗
3. two families of morphisms { b i : C → V i ⊗ V i ∗ } i ∈ I and { d i : V i ⊗ V i ∗ → C } i ∈ I such that for all i ∈ I (Id V i ⊗ d i ∗ )( b i ⊗ Id V i ) = Id V i = ( d i ⊗ Id V i )(Id V i ⊗ b i ∗ ) (3.36)
4. Let H i,jk and H ki,j be Hom A ω,t ( V k , V i ⊗ V j ) and Hom A ω,t ( V i ⊗ V j , V k ) respectively.For any i, j ∈ I such that H i,jk = 0 for some k ∈ I , the identity morphism Id V i ⊗ V j is in the image of the linear map M k ∈ I H i,jk ⊗ H ki,j −→ End A ω,t ( V i ⊗ V j ) x ⊗ y x ◦ y Note that C is provided with an A ω,t -module structure through the counit ε : A ω,t → End( C ) ∼ = C . If the A ω,t -module structure of V i and V i ∗ is provided by π i : A ω,t → End( V i ) and π i ∗ : A ω,t → End( V i ∗ ) respectively, then the morphisms b i and d i satisfy b p ( k ) ε ( a ) = ( π p ⊗ π p ∗ )∆( a ) b p ( k ) ε ( a ) d p ( u ⊗ v ) = d p ( π p ⊗ π p ∗ )∆( a )( u ⊗ v )for all a ∈ A ω,t , all k ∈ C and all u ⊗ v ∈ V i ⊗ V i ∗ .In our case, the set I is R =0 and the involution is given by p ∗ = − p .29 emma 3.23. For any p ∈ R =0 , the morphism d p is a solution of the followingsystem of homogeneous linear equations d p = d p ( X ⊗ X ) d p = d p (cid:0) XY − ⊗ Y (cid:1) d p = d p Ä V − ⊗ X − ( t +1) U ä d p = ω − d p (cid:0) U V − ⊗ U V − (cid:1) . Proof.
By definition, for all a ∈ A ω,t we have ε ( a ) d p = d p ( π p ⊗ π − p )∆( a ) . (3.37)If a = k , we have ε ( k ) = 1 and ( π p ⊗ π − p )∆( k ) = X ⊗ X . Hence, equality(3.37) becomes d p = d p ( X ⊗ X ) . (3.38)If a = e , we have ε ( e ) = 0 and( π p ⊗ π − p )∆( e ) = p N Y ⊗ Id A +( − p ) N X ⊗ Y. Hence, equality (3.37) is equivalent to0 = d p ( Y ⊗ Id A − X ⊗ Y ) ⇔ d p ( Y ⊗ Id A ) = dp ( X ⊗ Y )which leads us to d p = d p ( XY − ⊗ Y ) . (3.39)If a = e , we have ε ( e ) = 0 and( π p ⊗ π − p )∆( e ) = (cid:0) p (cid:1) N Z ⊗ Id A + (cid:0) − p (cid:1) N X t ⊗ Z, where Z = ( U + V ) Y − . A similar computation to the previous one leads us to thefollowing equality d p = d p ( X t Z − ⊗ Z ) (3.40)If a = a , we have ε ( a ) = 0 and( π p ⊗ π − p )∆( a ) = (cid:0) p (cid:1) N U ⊗ Id A + (cid:0) p (cid:1) N X t +1 ⊗ U + (cid:0) − p (cid:1) N X t Y ⊗ Z, d p ( U ⊗ Id A + X t +1 ⊗ U − X t Y ⊗ Z ) = d p ( U ⊗ Id A )+ d p ( X t +1 ⊗ U ) − d p ( X t Y ⊗ Z ) (3.40) = d p ( U ⊗ Id A ) + d p ( X t +1 ⊗ U ) − d p ( ZY ⊗ Id A )= d p ( U ⊗ Id A ) + d p ( X t +1 ⊗ U ) − d p ( U ⊗ Id A ) − d p ( V ⊗ Id A )= d p ( X t +1 ⊗ U ) − d p ( V ⊗ Id A ) (3.38) = d p (Id A ⊗ X − ( t +1) U ) − d p ( V ⊗ Id A ) , which leads to d p = d p ( V − ⊗ X − ( t +1) U ) . (3.41)Finally, by reconsidering the last computation, we get0 = d p ( U ⊗ Id A + X t +1 ⊗ U − X t Y ⊗ Z )= d p ( U ⊗ Id A ) + d p ( X t +1 ⊗ U ) − ω − t d p ( Y X t ⊗ Z ) (3.39) = d p ( U ⊗ Id A ) + d p ( X t +1 ⊗ U ) − ω − t d p ( X t +1 ⊗ Y Z )= d p ( U ⊗ Id A ) + d p ( X t +1 ⊗ U ) − d p ( X t +1 ⊗ U ) − ωd p ( X t +1 ⊗ V )= d p ( U ⊗ Id A ) − ωd p ( X t +1 ⊗ V ) (3.38) = d p ( U ⊗ Id A ) − ωd p (Id A ⊗ X − ( t +1) V ) , which leads to d p = d p ( U − ⊗ X − ( t +1) V ) . (3.42)From there, we use equality (3.41) to get d p ( U − ⊗ X − ( t +1) V ) = d p ( V − ⊗ X − ( t +1) U ) , which is equivalent to d p = ω − d p ( U V − ⊗ U V − ) (3.43) Lemma 3.24.
The duality morphisms b p : C → V p ⊗ V − p and d p : V p ⊗ V − p → C are given by b p (1) = X α,β ∈ Z N b p,α,β u α ⊗ u β where b p,α,β = δ α, − β ω ( ( t +1)( α − α )( α − α +2)+ α + α ) , (3.44) and d p ( u α ⊗ u β ) = δ α, − β ω − ( ( t +1)( α − α )( α − α − α − α ) . (3.45)31 roof. First we compute d p using Lemma 3.23 and Lemma 3.19.We start with the first and the last equality of the Lemma 3.23. We have d p ( u α ⊗ u β ) = d p ( X ⊗ X )( u α ⊗ u β ) = ω − α − β d p ( u α ⊗ u β )and d p ( u α ⊗ u β ) = ω − d p ( U V − ⊗ U V − )( u α ⊗ u β ) = ω α + β d p ( u α ⊗ u β ) . Hence we have d p ( u α ⊗ u β ) = δ α, − β d p ( u α ⊗ u β ) . (3.46)We consider the third equality of the Lemma 3.23 d p = d p ( V − ⊗ X − ( t +1) U ) . Since
X, U, V ∈ A are invertible, this equality is equivalent to d p = d p Ä V − ⊗ X − ( t +1) U ä t +1 A straightforward computation shows that Ä V − ⊗ X − ( t +1) U ä t +1 = ω Ä V − ( t +1) ⊗ X − t U t +1 ä , hence, the following equality holds true d p = ω d p Ä V − ( t +1) ⊗ X − t U t +1 ä . Therefore, using Lemma 3.19, we compute d p ( u α ⊗ u β ) = δ α, − β d p ( u α ⊗ u β ) = δ α, − β ω d p Ä V − ( t +1) ⊗ X − t U t +1 ä ( u α ⊗ u β )= δ α, − β ω d p Ä V − ( t +1) ⊗ X − t U t +1 ä ( u α ⊗ u − α )= δ α, − β ω − ( t +2) α +( t +1) α + t +2 d p ( u ( α − ,α ) ⊗ u ( − α +1 , − α ) )= δ α, − β ω − ( t +2) α ( α +1)+ α ( ( t +1) α + t +2 ) d p ( u ( α − α ,α ) ⊗ u ( − α + α , − α ) )= δ α, − β ω − ( t +2) α +( t +1) α ( α +1) d p ( u (0 ,α ) ⊗ u (0 , − α ) ) . (3.47)Now we use the second equality of Lemma 3.23 in the following way d p ( u (0 ,α ) ⊗ u (0 , − α ) ) = d p ( XY − ⊗ Y )( u (0 ,α ) ⊗ u (0 , − α ) )= ωd p ( u ( − ,α − ⊗ u (1 , − α +1) ) (3.47) = ω − ( t +1) α − t d p ( u (0 ,α − ⊗ u (0 , − α +1) )= ω − ( t +1) α ( α +1) − tα d p ( u (0 ,α − α ) ⊗ u (0 , − α + α ) )= ω − ( t +1) α − (2 t +1) α d p ( u (0 , ⊗ u (0 , ) . (3.48)32sing equalities (3.47) and (3.48), we get d p ( u α ⊗ u β ) = δ α, − β ω − ( t +2) α +( t +1) α ( α +1) d p ( u (0 ,α ) ⊗ u (0 , − α ) )= δ α, − β ω − ( t +2) α +( t +1) α ( α +1) ω − ( t +1) α − (2 t +1) α d p ( u (0 , ⊗ u (0 , )= δ α, − β ω − ( ( t +1)( α − α )( α − α − α − α ) d p ( u (0 , ⊗ u (0 , ) . Finally, we set d p ( u (0 , ⊗ u (0 , ) = 1. The formula for b p is easily computedusing formula (3.36). Theorem 3.25.
In the category of A ω,t -modules, the set of objects { V p } p ∈ R =0 withthe involution p ∗ = − p and the duality morphisms defined in the Lemma 3.24 is a Ψ -system.Proof. By definition, we have to check the following three points :1. Hom( V p , V q ) = 0 for all p = q ,2. The morphisms { b p : C → V p ⊗ V − p } p ∈ R =0 and { d p : V p ⊗ V − p → C } p ∈ R =0 sat-isfy (Id V ⊗ d − p )( b p ⊗ Id V ) = Id V = ( d p ⊗ Id V )(Id V ⊗ b − p ) , ∀ p ∈ R =0 (3.49)3. If ( p, q ) is admissible, then Id V p ⊗ V q is in the image of the linear map H p,q ⊗ ¯ H p,q → End( V p ⊗ V q ) , x ⊗ y xy. Point (1) is clear by Schur’s lemma, point (2) is straightforward using Lemma 3.24and point (3) is given by formula (3.35). ˆΨ -system H We consider the vector space H , called the space of multiplicities , defined by H = M ( p,q ) ∈ ( R =0 ) admissible H p,q ⊕ ¯ H p,q . We are going to determine the key operators in End( H ) that will allow us to extendthe Ψ-system defined in Theorem 3.25 into a ˆΨ-system.33 .1.1 The standard operators Let us first recall that the
Mobius group is defined as the group PGL(2 , C ) actingon C ∪ {∞} as follows Å a bc d ã ( x ) = ax + bcx + d if x ∈ C ac if x = ∞ . The elements of the Mobius group are called
Mobius transformation . Definition 4.1.
We say that f ∈ End( H ) is a standard operator if f is invertibleand if for all admissible pair ( p, q ) ∈ ( R =0 ) , there exists an admissible pair ( r, s ) ∈ ( R =0 ) and a Mobius transformation M ∈ PGL(2 , C ) satisfying the followingtwo conditions :1. f ( H p,q ) = H r,s and f ( ¯ H p,q ) = ¯ H r,s , (4.1) or f ( H p,q ) = ¯ H r,s , and f ( ¯ H p,q ) = H r,s , (4.2) M Ä qp + q ä = sr + s (4.3)The scaling invariance property of the multiplicity spaces extends to the standardoperators in the following sense : if f ∈ End( H ) is a standard operator, then forall α, β ∈ Z N , there exists functions f α,β , ¯ f α,β : R =0 , → C such that f h p,q ( u α ) = X β ∈ Z N f α,β Ä qp + q ä h r,s (¯ u β ) ,f ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ f α,β Ä qp + q ä h r,s ( u β ) , (4.4)34f f satisfies (4.1) and f h p,q ( u α ) = X β ∈ Z N f α,β Ä qp + q ä ¯ h r,s (¯ u β ) ,f ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ f α,β Ä qp + q ä h r,s ( u β ) , (4.5)if f satisfies (4.2). Definition 4.2.
An operator f ∈ End( H ) is grading-preserving if for all admissiblepairs ( p, q ) ∈ ( R =0 ) we have f ( H p,q ) ⊂ H p,q and f ( ¯ H p,q ) ⊂ ¯ H p,q . Clearly, the invertible grading-preserving operators are standard.Let π p,q : H → H p,q , ¯ π p,q : H → ¯ H p,q be the obvious projections. We provide H with a symmetric bilinear pairing h , i : H ⊗ H → C by h u, v i = X ( p,q ) ∈ ( R =0 ) admissible ( h ¯ π p,q ( u ) π p,q ( v ) i + h ¯ π p,q ( v ) π p,q ( u ) i )for any u, v ∈ H . Definition 4.3.
A transpose of f ∈ End( H ) is a map f ∗ ∈ End( H ) such that h f u, v i = h u, f ∗ v i for all u, v ∈ H . We say that f ∈ End( H ) is symmetric if f ∗ = f . Since H p,q and ¯ H p,q are dual vector spaces, the transpose f ∗ of f ∈ End( H ), ifit exists, is unique and ( f ∗ ) ∗ = f .If f ∈ End( H ) is standard, the equalities (4.1) and (4.2) ensure that f ∗ exists.Moreover, in that case, f ∗ is also standard. A and B and their transpose We now define the operators
A, B ∈ End( H ) by Au = X ( p,q ) ∈ ( R =0 ) admissible (Id V ⊗ ¯ π p,q ( u ))( b − p ⊗ Id V ) + ( d − p ⊗ Id V )(Id V ⊗ π p,q ( u )) , (4.6)35 u = X ( p,q ) ∈ ( R =0 ) admissible (¯ π p,q ( u ) ⊗ Id V )(Id V ⊗ b q ) + (Id V ⊗ d q )( π p,q ( u ) ⊗ Id V ) . (4.7)For each u ∈ H , there are only finitely many non-zero terms in these sums, since u has only finitely many non-zero components π p,q ( u ) and ¯ π p,q ( u ).Using (3.36), one can easily prove that the operators A and B are involutive(also see [7, Lemma 3]). Hence, from their definition, we clearly have the followingequalities A ( H p,q ) = ¯ H − p,p + q , A ( ¯ H p,q ) = H − p,p + q ,B ( H p,q ) = ¯ H p + q, − q , B ( ¯ H p,q ) = H p + q, − q . (4.8)Moreover, A and B are both standard operators. Indeed, we have Å ã Ä qp + q ä = p + qq and Å − ã Ä qp + q ä = − qp . (4.9)The equalities (4.8) ensure us that we have the following for A ∗ and B ∗ A ∗ ( H p,q ) = ¯ H − p,p + q , A ∗ ( ¯ H p,q ) = H − p,p + q ,B ∗ ( H p,q ) = ¯ H p + q, − q , B ∗ ( ¯ H p,q ) = H p + q, − q . (4.10)The equalities (4.8) and (4.10) ensure us that for all α, β ∈ Z N there exists functions A α,β , ¯ A α,β , A ∗ α,β , ¯ A ∗ α,β : R =0 , → C and B α,β , ¯ B α,β , B ∗ α,β , ¯ B ∗ α,β : R =0 , → C such that for all admissible pairs ( p, q ) ∈ ( R =0 ) we have Ah p,q ( u α ) = X β ∈ Z N A α,β Ä qp + q ä ¯ h − p,p + q (¯ u β ) ,A ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ A α,β Ä qp + q ä h − p,p + q ( u β ) ,A ∗ h p,q ( u α ) = X β ∈ Z N A ∗ α,β Ä qp + q ä ¯ h − p,p + q (¯ u β ) ,A ∗ ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ A ∗ α,β Ä qp + q ä h − p,p + q ( u β ) . Bh p,q ( u α ) = X β ∈ Z N B α,β Ä qp + q ä ¯ h p + q, − q (¯ u β ) ,B ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ B α,β Ä qp + q ä h p + q, − q ( u β ) ,B ∗ h p,q ( u α ) = X β ∈ Z N B ∗ α,β Ä qp + q ä ¯ h p + q, − q (¯ u β ) ,B ∗ ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ B ∗ α,β Ä qp + q ä h p + q, − q ( u β ) . We define (cid:15) N = ß if N = 1 mod 4 i if N = 3 mod 4and we use the following result. Proposition 4.4 ([9], Proposition 3.4) . For all x ∈ R =0 , and all α, β ∈ Z N wehave A α,β ( x ) ≡ (cid:15) N x N − N δ α , − β N ω − ( t ( α − α − β )( α − α − β − − α ( α − − ( t +1) α − β ( β +2 t +1)) , ¯ A α,β ( x ) ≡ (cid:15) − N x N − N δ α , − β ω ( t ( β + α − α )( β + α − α − β ( β − − ( t +1) α + α ( α +2 t +1)) ,A ∗ α,β ( x ) ≡ (cid:15) N x − N − N δ α , − β N ω − ( t ( β + α − α )( β + α − α − β ( β − − ( t +1) α + α ( α +2 t +1)) , ¯ A ∗ α,β ( x ) ≡ (cid:15) − N x − N − N δ α , − β ω ( t ( α − α − β )( α − α − β − − α ( α − − ( t +1) α − β ( β +2 t +1)) , and B α,β ( x ) ≡ (1 − x ) N − N δ α, − β ω − ( ( t +1)( α − α )( α − α − α − α ) , ¯ B α,β ( x ) ≡ (1 − x ) N − N δ α, − β ω ( ( t +1)( α − α )( α − α +2)+ α + α ) ,B ∗ α,β ( x ) ≡ (1 − x ) − N − N δ α, − β ω − ( ( t +1)( α − α )( α − α +2)+ α + α ) , ¯ B ∗ α,β ( x ) ≡ (1 − x ) − N − N δ α, − β ω ( ( t +1)( α − α )( α − α − α − α ) . L, R and C The operators
L, R and C are defined as follows L = A ∗ A, R = B ∗ B, C = ( AB ) ∈ End( H ) . (4.11)37he operators L, R and C are clearly invertible and by the equalities (4.8) and(4.10), we easily see that they are grading-preserving. Hence, these operators arealso standard.Moreover, these operators are symmetric. It is clear for L and R . For C , weuse ([7, Lemma 5]) which states that( ABA ) ∗ = BAB.
Hence we have C = ( AB ) = ABABAB = ABA ( ABA ) ∗ . Now we can determine these operators using the functions L α,β , ¯ L α,β , R α,β , ¯ R α,β , C α,β , ¯ C α,β : R =0 , → C satisfying Lh p,q ( u α ) = X β ∈ Z N L α,β Ä qp + q ä h p,q ( u β ) ,L ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ L α,β Ä qp + q ä ¯ h p,q (¯ u β ) ,Rh p,q ( u α ) = X β ∈ Z N R α,β Ä qp + q ä h p,q ( u β ) ,R ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ R α,β Ä qp + q ä ¯ h p,q (¯ u β ) ,Ch p,q ( u α ) = X β ∈ Z N C α,β Ä qp + q ä h p,q ( u β ) ,C ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ C α,β Ä qp + q ä ¯ h p,q (¯ u β ) , for all admissible pairs ( p, q ) ∈ ( R =0 ) . Proposition 4.5.
For all x ∈ R =0 , and all α, β ∈ Z N we have L α,β ( x ) ≡ x N − N δ α +2 t,β δ α ,β ω tα + α − t − , ¯ L α,β ( x ) ≡ x N − N δ α − t,β δ α ,β ω tα + α +1 ,R α,β ( x ) ≡ (1 − x ) N − N δ α,β ω t +1)( α − α )+ α , ¯ R α,β ( x ) ≡ (1 − x ) N − N δ α,β ω t +1)( α − α )+ α ,C α,β ( x ) ≡ δ α,β ω − (2 t +1) α , ¯ C α,β ( x ) ≡ δ α,β ω − (2 t +1) α . roof. Using Proposition 4.4, a straightforward computation leads to the results.
Remark 4.6.
Note that the fact that the operator C is non trivial implies that thecategory of A ω,t -modules is non-pivotal (see [7], Remark 41). L , R and C Now we can fix square roots of the operators
L, R and C . These are the keyoperators mentioned in the beginning of this section that will allow us to extendour Ψ-system into a ˆΨ-system. Proposition 4.7.
The square roots of the operators
L, R and C can be chosen tobe, respectively, the grading-preserving operators given, for all x ∈ R =0 , and all α, β ∈ Z N , by L α,β ( x ) ≡ x N − N δ α + t,β δ α ,β ω (2 tα + α − t − , ¯ L α,β ( x ) ≡ x N − N δ α − t,β δ α ,β ω (2 tα + α − t ) ,R α,β ( x ) ≡ (1 − x ) N − N δ α,β ω ( t +1)( α − α )+ α , ¯ R α,β ( x ) ≡ (1 − x ) N − N δ α,β ω ( t +1)( α − α )+ α ,C α,β ( x ) ≡ δ α,β ω − ( t + ) α , ¯ C α,β ( x ) ≡ δ α,β ω − ( t + ) α . Proof.
Let us write for any L = L L as a product of commuting operators L and L such that, for any admissible pair ( p, q ) ∈ ( R =0 ) we have L | H p,q ⊕ ¯ H p,q ≡ Ä qp + q ä N − N and L N | H = Id H . Since N is odd, we set L = L L N +12 . We do similarly for R and C . Remark 4.8.
In [7, equation (35)] the square root of L is defined by L = BAR − AB (4.12) and it is shown that it implies that ( L ) = L . Although L has not been definedthis way in our case, a straightforward computation shows that equality (4.12) holdstrue. q = R BL BL − C − , acts as a scalar onˇ H = M ( p,q ) ∈ ( R =0 ) admissible H p,q and ˆ H = M ( p,q ) ∈ ( R =0 ) admissible ¯ H p,q . This property is central to define our invariant.
Proposition 4.9.
For all x ∈ R =0 , and all α ∈ Z N , the operator q = R BL BL − C − is given by q e α ( x ) = ω e α ( x ) and q ¯ e α ( x ) = ω − ¯ e α ( x ) . Proof.
A straightforward computation leads to the results. ˆΨ -system in the category of A ω,t -modules Now that we have shown that there is a Ψ-system in the category of A ω,t -modules,we can construct its associated 6 j -symbols. As we have already said, the operator S ( x ) will have an essential role in this construction.With the square roots L , R and C chosen in Proposition 4.7, we are goingto show that the Ψ-system defined in Theorem 3.25 extends to a ˆΨ-system. j -symbols For any x ∈ R =0 , , we define the algebra morphism∆ x : A → A ⊗ A by ∆ x ( a ) = S ( x )( a ⊗ Id V ) S ( x ) − , (4.13)We also consider the following function ∗ : ( R =0 , ) −→ R =0 , ( x, y ) y − xy − xy This function is well defined only for pairs ( x, y ) ∈ ( R =0 , ) such that x = y − .40 efinition 4.10. We say that a pair ( x, y ) ∈ ( R =0 , ) is compatible if x = y − . Lemma 4.11.
For all compatible pairs ( x, y ) ∈ ( R =0 , ) , we have (∆ x ∗ y ⊗ Id V ) ∆ xy = (Id V ⊗ ∆ x ) ∆ y (4.14) Proof.
If ( x, y ) ∈ ( R =0 , ) is a compatible pair, then ∆ xy and ∆ x ∗ y are well definedfunctions. For any x ∈ R =0 , , Proposition 3.17 implies the following equalities∆ x ( X ) = X X , ∆ x ( Y ) = (1 − x ) N Y + x N X Y ∆ x ( U ) = (1 − x ) N U + x N X t +11 U + ( x − x ) N X t Y Z , ∆ x ( V ) = (1 − x ) N V + x N X t +11 V + ( x − x ) N Z X Y , where Z = ( U + V ) Y − . Using these equalities, a straightforward computationshows that equation (4.14) holds true for a ∈ { X, Y, U, V } . This ends the proofsince { X, Y, U, V } is a generating set of A .Let ( x, y ) ∈ ( R =0 , ) be compatible. Using (4.13), we have for all a ∈ A (∆ x ∗ y ⊗ Id V ) ∆ xy ( a ) = S ( x ∗ y ) S ( xy )( a ⊗ Id V ⊗ Id V ) ( S ( x ∗ y ) S ( xy )) − and (Id V ⊗ ∆ x ) ∆ y ( a ) = S ( x ) S ( y )( a ⊗ Id V ⊗ Id V ) ( S ( x ) S ( y )) − . Therefore, by Lemma 4.11, the following equality holds in A ⊗ for all a ∈ A ( S ( x ) S ( y )) − S ( x ∗ y ) S ( xy )( a ⊗ Id V ⊗ Id V )= ( a ⊗ Id V ⊗ Id V ) ( S ( x ) S ( y )) − S ( x ∗ y ) S ( xy )Since the center of A is trivial, the former equality implies the existence of anelement T ( x, y ) ∈ A ⊗ such that( S ( x ) S ( y )) − S ( x ∗ y ) S ( xy ) T ( x, y ) = Id V ⊗ Id V ⊗ Id V . Hence we have S ( x ) S ( y ) = S ( x ∗ y ) S ( xy ) T ( x, y ) . (4.15) Definition 4.12.
The operator T ( x, y ) ∈ A ⊗ and its inverse are called j -symbols. emark 4.13. The operators T ( x, y ) and T ( x, y ) − correspond to the j -symbols(positive and negative respectively) defined in [7, p.13] in the following way : for p, q, r ∈ R =0 such that x = rq + r and y = q + rp + q + r , we have (¯ v ⊗ ¯ u ) T ( x, y )( v ⊗ u ) = ß p q p + qr p + q + r q + r ™ (¯ u ⊗ ¯ v ⊗ u ⊗ v ) ∈ C where ¯ u ⊗ ¯ v ⊗ u ⊗ v ∈ ¯ H p + q,r ⊗ ¯ H p,q ⊗ H q,r ⊗ H p,q + r and (¯ u ⊗ ¯ v ) T ( x, y ) − ( u ⊗ v ) = ß p q p + qr p + q + r q + r ™ − (¯ u ⊗ ¯ v ⊗ u ⊗ v ) ∈ C where ¯ u ⊗ ¯ v ⊗ u ⊗ v ∈ ¯ H p,q + r ⊗ ¯ H q,r ⊗ H p,q ⊗ H p + q,r . Thus, T ( x, y ) and T ( x, y ) − are interpreted as elements of End( H ⊗ ) , or moreprecisely, T ( x, y ) : H p,q + r ⊗ H q,r −→ H p,q ⊗ H p + q,r ,T ( x, y ) − : H p,q ⊗ H p + q,r −→ H p,q + r ⊗ H q,r . j -symbols The following Theorem is the key to determine the 6 j -symbols T ( x, y ). For thesake of simplicity, our statement is adapted to our context. It is therefore slightlydifferent from the original one. Theorem 4.14 ([5], p.5) . Let ( x, y ) ∈ ( R =0 , ) be compatible and U , V be twooperators such that UV = ω − VU and U N = V N = − , then Ψ x ( U )Ψ y ( V ) = Ψ x ∗ y ( V )Ψ xy ( − VU )Ψ y ∗ x ( U ) . Lemma 4.15.
Let U , V ∈ A be such that U N = V N = 1 and UV = ω VU .Then L ( U , V ) L ( U , V ) = L ( U , V ) L ( U , V ) L ( U , V ) . roof. For all i ∈ Z N we have X k ∈ Z N Ç N X j ∈ Z N ω − ij V j å Ç N X l ∈ Z N ω − kl V l å = 1 N X l,k ∈ Z N ω − kl Ç X j ∈ Z N ω − ij V j + l å = 1 N X l,k ∈ Z N ω l ( i − k ) Ç X j ∈ Z N ω − ij V j å = Ç X j ∈ Z N ω − ij V j å N X l,k ∈ Z N ω l ( i − k ) = Ç X j ∈ Z N ω − ij V j å N X k ∈ Z N N δ i,k = 1 N X j ∈ Z N ω − ij V j . Using the identity above and Lemma 3.14, we easily make the following computation L ( U , V ) L ( U , V ) = 1 N X i,j ∈ Z N ω − ij U i V j L ( U , V )= 1 N X i,j ∈ Z N ω − ij L ( U , ω i V ) U i V j = L ( U , V ) 1 N X i,j ∈ Z N ω − ij U i U i V j = L ( U , V ) X i ∈ Z N U i U i X k ∈ Z N Ç N X j ∈ Z N ω − ij V j å Ç N X l ∈ Z N ω − kl V l å = L ( U , V ) X i,k ∈ Z N U i U k Ç N X j ∈ Z N ω − ij V j å Ç N X l ∈ Z N ω − kl V l å = L ( U , V ) Ç N X i,j ∈ Z N ω − ij U i V j å Ç N X k,l ∈ Z N ω − kl U k V l å = L ( U , V ) L ( U , V ) L ( U , V ) . Now we show that T ( x, y ) = S ( y ∗ x ) is a solution of the equation (4.15). Remark 4.16.
Setting z = y ∗ x and using Remark 4.13, we have z = pr ( p + q )( q + r ) , (¯ v ⊗ ¯ u ) S ( z )( v ⊗ u ) = ß p q p + qr p + q + r q + r ™ (¯ u ⊗ ¯ v ⊗ u ⊗ v ) ∈ C where ¯ u ⊗ ¯ v ⊗ u ⊗ v ∈ ¯ H p + q,r ⊗ ¯ H p,q ⊗ H q,r ⊗ H p,q + r and (¯ u ⊗ ¯ v ) S ( z ) − ( u ⊗ v ) = ß p q p + qr p + q + r q + r ™ − (¯ u ⊗ ¯ v ⊗ u ⊗ v ) ∈ C here ¯ u ⊗ ¯ v ⊗ u ⊗ v ∈ ¯ H p,q + r ⊗ ¯ H q,r ⊗ H p,q ⊗ H p + q,r . Thus, S ( z ) and S ( z ) − are interpreted as operators with the following source andtarget spaces S ( z ) : H p,q + r ⊗ H q,r −→ H p,q ⊗ H p + q,r , (4.16) S ( z ) − : H p,q ⊗ H p + q,r −→ H p,q + r ⊗ H q,r . (4.17) Theorem 4.17. S ( x ) S ( y ) = S ( x ∗ y ) S ( xy ) S ( y ∗ x ) .Proof. The following commutation relations hold trueUsing Lemma 3.14 and Lemma 4.15 for U = U t V , we see that the propositionis equivalent to the following equalityΨ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( E )Ψ y ( F )Ψ y ( G )Ψ y ( H )= Ψ x ∗ y ( E )Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( X E )Ψ xy ( X t F ) × Ψ xy ( X t G )Ψ xy ( X H )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H ) . (4.18)Since X E = − E E , X t F = − G F ,X t G = − G G , X H = − H E , the former equality is equivalent toΨ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( E )Ψ y ( F )Ψ y ( G )Ψ y ( H )= Ψ x ∗ y ( E )Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( − E E )Ψ xy ( − G F ) × Ψ xy ( − G G )Ψ xy ( − H E )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H ) .
44n the left hand side, we can apply Theorem 4.14 to E and E . Indeed, E commutes with F , G and H and E E = ω − E E . This givesΨ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( E )Ψ y ( F )Ψ y ( G )Ψ y ( H )= Ψ x ∗ y ( E )Ψ xy ( − E E )Ψ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( F ) × Ψ y ( G )Ψ y ( H ) . On the right hand side, since E E commutes with F , G and H , we haveΨ x ∗ y ( E )Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( − E E )Ψ xy ( − G F ) × Ψ xy ( − G G )Ψ xy ( − H E )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H )= Ψ x ∗ y ( E )Ψ xy ( − E E )Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( − G F ) × Ψ xy ( − G G )Ψ xy ( − H E )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H )Hence, (4.18) is equivalent toΨ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( F )Ψ y ( G )Ψ y ( H )= Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( − G F )Ψ xy ( − G G ) × Ψ xy ( − H E )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H ) (4.19)On the left hand side of (4.19), we haveΨ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ( H )Ψ y ( F )Ψ y ( G )Ψ y ( H )=Ψ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ∗ y ( F )Ψ xy ( − F H )Ψ y ∗ x ( H )Ψ y ( G )Ψ y ( H )=Ψ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ x ∗ y ( F )Ψ xy ( − E G )Ψ y ∗ x ( H )Ψ y ( G )Ψ y ( H )=Ψ x ∗ y ( F )Ψ y ∗ x ( E )Ψ x ( F )Ψ x ( G )Ψ xy ( − E G )Ψ y ( G )Ψ y ( H )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ y ∗ x ( E )Ψ x ( F )Ψ xy ( − E G )Ψ x ∗ y ( G )Ψ xy ( − G G ) × Ψ y ∗ x ( G )Ψ y ( H )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ x ( F )Ψ y ( G )Ψ x ( E )Ψ xy ( − G G )Ψ y ∗ x ( G )Ψ y ( H )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ x ( F )Ψ y ( G )Ψ x ( E )Ψ xy ( − G G )Ψ y ( H )Ψ y ∗ x ( G )Ψ y ∗ x ( H )where we successively1. applied theorem 4.14 to H and F ,2. used the equality F H = E G ,3. used the fact that F commutes with E , F and G , and the fact that H commutes with G and H , 45. used the fact that G commutes with E G , and applied Theorem 4.14 to G and G ,5. used the fact that E commutes with F , and applied Theorem 4.14 to E and G ,6. used the fact that G commutes with H .On the right hand side of (4.19), we haveΨ x ∗ y ( F )Ψ x ∗ y ( G )Ψ x ∗ y ( H )Ψ xy ( − G F )Ψ xy ( − G G ) × Ψ xy ( − H E )Ψ y ∗ x ( E )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ x ∗ y ( G )Ψ xy ( − G F )Ψ xy ( − G G ) × Ψ x ( E )Ψ y ( H )Ψ y ∗ x ( F )Ψ y ∗ x ( G )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ x ( F )Ψ y ( G )Ψ xy ( − G G )Ψ x ( E )Ψ y ( H )Ψ y ∗ x ( G )Ψ y ∗ x ( H )=Ψ x ∗ y ( F )Ψ x ( F )Ψ y ( G )Ψ x ( E )Ψ xy ( − G G )Ψ y ( H )Ψ y ∗ x ( G )Ψ y ∗ x ( H ) . where we successively1. used the fact that H commutes with G F and G G , and appliedTheorem 4.14 to H and E ,2. used the fact that F commutes with H E and G G , and appliedTheorem 4.14 to G and F ,3. used the fact that E commutes with G G . ˆΨ -system Let S ( z ) be as described in Remark 4.16. The Ψ-system of Theorem 3.25 extendsto a ˆΨ-system if the following equations holds true C C S ( z ) = S ( z ) C C (4.20) L R S ( z ) = S ( z ) L R (4.21) R R S ( z ) = S ( z ) R C (4.22) L S ( z ) = C S ( z ) L L (4.23)46 heorem 4.18. The Ψ -system of Theorem 3.25 extends to a ˆΨ -system with L , R and C given in Proposition 4.7.Proof. By defintion of a Ψ-system, these equations hold true without the squareroots (see equations (33a), (33b), (33c) and (33d) of [7]). Therefore, it is easy tosee that they also hold true for L , R and C . j -symbols and the proof of Theorem 2.4 j -symbolsDefinition 4.19. For any z ∈ R =0 , and a, c ∈ Z we define the charged 6 j -symbolsby S ( z | a, c ) = q − ac R c R − a S ( z ) L − a R − c (4.24) S ( z | a, c ) − = q ac R − c L − a S ( z ) − R − a R c (4.25)Explicitly we have (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z | a, c ) (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z ) (cid:12)(cid:12)(cid:12) u α ⊗ u ( β − ta,β ) (cid:69) ( z a (1 − z ) c ) − N ) N × ω a (2 c − tβ − β )+2( t +1) { a ( a + µ )+ c ( β − ν ) } +(2 t +1) { a ( µ +1)+ c ( β − ν ) } (4.26) (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z | a, c ) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) = (cid:68) ¯ u µ ⊗ ¯ u ( ν +2 ta,ν ) (cid:12)(cid:12)(cid:12) S ( z ) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) ( z a (1 − z ) c ) − N ) N × ω a (2 c − tν − ν )+2( t +1) { a ( a − α )+ c ( β − ν ) } +(2 t +1) { a ( α +1)+ c ( ν − β ) } (4.27)where (cid:10) ¯ u µ ⊗ ¯ u ν | S ( z ) | u α ⊗ u β (cid:11) and (cid:10) ¯ u µ ⊗ ¯ u ν | S ( z ) − | u α ⊗ u β (cid:11) are given by Proposition3.20. j -symbols The symmetry relations of the charged 6j-symbols are expressed with the followingsymmetric operators A = AL − and B = BR − in End( H ). For any α, β ∈ Z N ,we consider the functions A α,β , ¯ A α,β , B α,β , ¯ B α,β : R =0 , → C A h p,q ( u α ) = X β ∈ Z N A α,β Ä qp + q ä h p,q ( u β ) , A ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ A α,β Ä qp + q ä ¯ h p,q (¯ u β ) , B h p,q ( u α ) = X β ∈ Z N B α,β Ä qp + q ä h p,q ( u β ) , B ¯ h p,q (¯ u α ) = X β ∈ Z N ¯ B α,β Ä qp + q ä ¯ h p,q (¯ u β ) , for all admissible pairs ( p, q ) ∈ ( R =0 ) . A straightforward computation, usingLemmas 4.4 and 4.7, leads us to the followings results. Lemma 4.20.
The operators A , B ∈ End( H ) are symmetric involutions. For all α, β ∈ Z N , the numbers A α,β ≡ (cid:15) N δ α , − β N ω − ( t ( α − α − β ) + α ( α + t − β ( β + t − − ) , (4.28)¯ A α,β ≡ (cid:15) − N δ α , − β ω ( t ( α − α − β ) + α ( α + t − β ( β + t − − ) , (4.29) B α,β ≡ δ α, − β ω − ( ( t +1)( α − α ) + α ) , (4.30)¯ B α,β ≡ δ α, − β ω ( ( t +1)( α − α ) + α ) , (4.31) satisfy the following equalities for all x ∈ R =0 , A α,β = A α,β ( x ) , ¯ A α,β = ¯ A α,β ( x ) , B α,β ( x ) = B α,β ( x ) , ¯ B α,β = ¯ B α,β ( x ) . Using the previous Lemma, Remark 4.16 and Proposition 4.9, the followingProposition is a direct adaptation of the formulas (50), (51) and (52) of [7].
Proposition 4.21 (The symmetry relations) . The charged 6j-symbols verify thefollowing symmetry relations (cid:68) ¯ u α ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S ( z | a, c ) (cid:12)(cid:12)(cid:12) u µ ⊗ u β (cid:69) ≡ ω a X α ,µ ∈ Z N A α,α ¯ A µ,µ (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S Ä zz − | a, b ä − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) (4.32)48 ¯ u α ⊗ ¯ u µ (cid:12)(cid:12)(cid:12) S ( z | a, c ) (cid:12)(cid:12)(cid:12) u β ⊗ u ν (cid:69) ≡ ω − c X α ,ν ∈ Z N ¯ A ν,ν B α,α (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S (cid:0) z − | b, c (cid:1) − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) (4.33) (cid:68) ¯ u µ ⊗ ¯ u β (cid:12)(cid:12)(cid:12) S ( z | a, c ) (cid:12)(cid:12)(cid:12) u α ⊗ u ν (cid:69) ≡ ω a X β ,ν ∈ Z N B β,β ¯ B ν,ν (cid:68) ¯ u µ ⊗ ¯ u ν (cid:12)(cid:12)(cid:12) S Ä zz − | a, b ä − (cid:12)(cid:12)(cid:12) u α ⊗ u β (cid:69) (4.34) Proof of theorem 2.4.
The statement of our Theorem is an adaptation of the The-orem 29 of [7]. In our case, the former can be expressed as follows.
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