A NOTE ON BRIDGELAND’S HALL ALGEBRAS HAICHENG ZHANG 7 Abstract. In this note, let be a finitary hereditary abelian category with enough A 1 projectives. By using the associativity formula of Hall algebras, we give a new and 0 2 simpleproofofthemaintheoremin[17],whichstatesthattheBridgeland’sHallalgebra n of2-cycliccomplexesofprojectiveobjectsin isisomorphictotheDrinfelddoubleHall A a algebra of . In a similar way, we give a simplification of the key step in the proof of J A Theorem 4.11 in [5]. 5 ] T R 1. Introduction . h t Ringel [9] introduced the Hall algebra of a finite dimensional algebra over a finite field. a m By the works of Ringel [9, 10, 11] and Green [3], the twisted Hall algebra, called the [ Ringel–Hall algebra, of a finite dimensional hereditary algebra provides a realization of 1 the positive (negative) part of the corresponding quantum group. In order to obtain a v 2 Hall algebra description of the entire quantum group, one considers the Hall algebras of 0 2 triangulated categories (for example, [4], [14], [16]). In [15], Xiao gave a realization of the 1 whole quantum group by constructing the Drinfeld double of the extended Ringel–Hall 0 . algebra of any hereditary algebra. 1 0 In 2013, Bridgeland [1] considered the Hall algebra of 2-cyclic complexes of projective 7 1 modules over a finite dimensional hereditary algebra A, and achieved an algebra, called : v the (reduced) Bridgeland’s Hall algebra of A, by taking some localization and reduction. i X He proved that there is an algebra embedding from the Ringel–Hall algebra of A to r a its Bridgeland’s Hall algebra. Moreover, the quantum group associated with A can be embedded into the reduced Bridgeland’s Hall algebra of A. This provides a realization of the full quantum group by Hall algebras. In [1], Bridgeland stated without proof that the Bridgeland’s Hall algebra of each finite dimensional hereditary algebra is isomorphic to the Drinfeld double of its extended Ringel–Hall algebra. Later on, Yanagida proved this statement in [17]. With the purpose of generalizing Bridgeland’s construction to a bigger class of exact categories, Gorsky [2] defined the so-called semi-derived Hall algebra of the category of bounded complexes of for each exact category satisfying certain E E finiteness conditions. In particular, if every object in has finite projective resolution, he E gave a similar construction to the category of 2-cyclic complexes of . Recently, inspired E 2010 Mathematics Subject Classification. 16G20, 17B20, 17B37. Key words and phrases. Bridgeland’s Hall algebras; Drinfeld double Hall algebras; Modified Ringel– Hall algebras. 1 2 HAICHENGZHANG by the works of Bridgeland and Gorsky, Lu and Peng [5] have generalized Bridgeland’s constructiontoanyhereditaryabeliancategory whichmaynothaveenoughprojectives, A and defined an algebra for the category of 2-cyclic complexes of , called the modified A Ringel–Hall algebra of . They also proved that the resulting algebra is isomorphic to A the Drinfeld double Hall algebra of . A The key step in the proof that the Bridgeland’s Hall algebra or modified Ringel–Hall algebraofahereditaryabeliancategory isisomorphictoitsDrinfelddoubleHallalgebra A is to check the (Drinfeld) commutator relations. The method used in [17] is to make the summations on the left-hand and right-hand sides of the commutator relations symmetric by some analysis of the structure of the category of 2-cyclic complexes of projectives in , as well as some complicated calculations. It seems that the process of the proof is A not too intuitive. While Lu and Peng gave a characterization of some coefficients in the commutator relations by introducing two sets, and obtained the proof by means of the coincidence of the cardinalities of these two sets. Nevertheless, their characterization is a bit complicated. In this note we use the associativity formula of Hall algebras to give a more intuitive and simpler proof that the Bridgeland’s Hall algebra of is isomorphic A to its Drinfeld double Hall algebra. Similarly, based on [5], we give a simplification of the key step in the proof of Theorem 4.11 in [5]. Explicitly, we prove the commutator relations therein by using the associativity formula of Hall algebras rather than Lemma 4.10 in [5]. Let us fix some notations used throughout the paper. k is always a finite field with q elements and set v = √q. is always a finitary hereditary abelian k-category with A enough projectives unless otherwise stated, we also assume that the image Mˆ of M in the Grothendieck group K ( ) is nonzero for any nonzero object M in (cf. [1, 17]). 0 A A We denote by Iso( ) the set of isoclasses (isomorphism classes) of objects in . The A A subcategory of consisting of projective objects is denoted by P. For a complex M = • A ( M di+1 M ) in , its homology is denoted by H (M ). For a finite set i+1 i ∗ • ··· → −−→ → ··· A S, we denote by S its cardinality. For an object M , we denote by Aut (M) the A | | ∈ A automorphism group of M, and set a = Aut (M) . M A | | 2. Preliminaries Inthissection, we collect somenecessary definitions andproperties. All ofthematerials can be found in [1], [12] and [17]. 2.1. 2-cyclic complexes. Let ( ) be the abelian category of 2-cyclic complexes over 2 C A . The objects of this category consist of diagrams A dM 1 // M = M oo M • 1 0 dM 0 A NOTE ON BRIDGELAND’S HALL ALGEBRAS 3 in such that dM dM = 0, i Z . A morphism s : M N consists of a diagram A i+1 ◦ i ∈ 2 • • → • dM 1 // M oo M 1 0 dM 0 s1 s0 (cid:15)(cid:15) dN (cid:15)(cid:15) 1 // N oo N 1 0 dN 0 withs dM = dN s , i Z . Two morphisms s ,t : M N aresaidtobehomotopic i+1◦ i i ◦ i ∈ 2 • • • → • if there are morphisms h : M N , i Z , such that t s = dN h +h dM, i i i → i+1 ∈ 2 i− i i+1◦ i i+1◦ i ∈ Z . For an object M ( ), we define its class in the Grothendieck group K ( ) to be 2 • 2 0 ∈ C A A Mˆ := Mˆ Mˆ K ( ). • 0 1 0 − ∈ A Denote by ( ) the homotopy category obtained from ( ) by identifying homotopic 2 2 K A C A morphisms. Denote by (P) ( ) the full subcategory whose objects are complexes 2 2 C ⊂ C A of projectives in , and by (P) its homotopy category. The shift functor of complexes 2 A K induces an involution of ( ). This involution shifts the grading and changes the signs 2 C A of differentials as follows dM −dM M = M oo 1 // M ∗ M∗ = M oo 0 // M . • 1 0 ←→ • 0 1 dM −dM 0 1 Let b( ) be the bounded derived category of , with the suspension functor [1]. D A A Let ( ) = b( )/[2] be the orbit category, also known as the root category of . 2 R A D A A The category b( ) is equivalent to the bounded homotopy category Kb(P), since is D A A hereditary. In this case, we can equally well define ( ) as the orbit category of Kb(P). 2 R A Lemma 2.1. ([7], [1, Lemma 3.1]) There is an equivalence D : ( ) (P) sending 2 2 R A → K a bounded complex of projectives (Pi)i∈Z to the 2-cyclic complex // P oo P . Li∈Z 2i+1 Li∈Z 2i Lemma 2.2. ([1, Lemma 3.3]) If M ,N (P), then there exists an isomorphism of • • 2 ∈ C vector spaces Ext1 (N ,M ) = Hom (N ,M∗). C2(A) • • ∼ K2(A) • • A complex M ( ) is called acyclic if H (M ) = 0. Each object P P determines • 2 ∗ • ∈ C A ∈ acyclic complexes 1 // 0 // K = ( P oo P ), K∗ = ( P oo P ). P P 0 1 Lemma 2.3. ([1, Lemma 3.2]) For each acyclic complex M (P), there are objects • 2 ∈ C P,Q P, unique up to isomorphism, such that M = K K∗. ∈ • ∼ P L Q 4 HAICHENGZHANG 2.2. Hall algebras. Given objects L,M,N , let Ext1 (M,N) Ext1 (M,N) be ∈ A A L ⊂ A the subset consisting of those equivalence classes of short exact sequences with middle term L. Definition 2.4. TheHall algebra ( )of isthevectorspaceoverCwithbasiselements H A A [M] Iso( ), and with multiplication defined by ∈ A Ext1(M,N) [M] [N] = X | A L|[L]. ⋄ Hom (M,N) A [L]∈Iso(A) | | By [9], the above operation defines on ( ) the structure of a unital associative ⋄ H A algebra over C, and the class [0] of the zero object is the unit. Remark 2.5. Given objects L,M,N , set ∈ A gL = N′ L N′ = N,L/N′ = M . MN |{ ⊂ | ∼ ∼ }| By Riedtmann-Peng formula [8, 6], Ext1 (M,N) a gL = | A L| L . MN Hom (M,N) · a a A M N | | Thus in terms of alternative generators [[M]] = [M], the product takes the form aM [[M]] [[N]] = X gL [[L]], ⋄ MN [L]∈Iso(A) which is the definition used, for example, in [9, 12]. The associativity of Hall algebras amounts to the following formula X gM gL = X gL gN (=: gL ), (2.1) XY MZ XN YZ XYZ [M]∈Iso(A) [N]∈Iso(A) for any objects L,X,Y,Z . ∈ A For objects M,N , let ∈ A M,N := dim Hom (M,N) dim Ext1 (M,N), h i k A − k A and it descends to give a bilinear form , : K ( ) K ( ) Z, 0 0 h· ·i A × A −→ known as the Euler form. We also consider the symmetric Euler form ( , ) : K ( ) K ( ) Z, 0 0 · · A × A −→ defined by (α,β) = α,β + β,α for all α,β K ( ). The Ringel–Hall algebra ( ) 0 tw h i h i ∈ A H A of is the same vector space as ( ), but with multiplication defined by A H A [M] [N] = vhMˆ,Nˆi [M] [N]. ∗ · ⋄ A NOTE ON BRIDGELAND’S HALL ALGEBRAS 5 The extended Ringel–Hall algebra ˜( ) of is defined as an extension of ( ) by tw H A A H A adjoining symbols K for α K( ), and imposing relations α ∈ A K K = K , K [M] = vhα,Mˆi[M]K , α β α+β α α for α,β K ( ) and [M] Iso( ). 0 ∈ A ∈ A ˜ By Green [3] and Xiao [15], the extended Ringel–Hall algebra ( ) is a topological H A bialgebra (see [12]) with comultiplication ∆ and counit ǫ defined by ∆([L]Kα) = X vhMˆ,NˆigMLN[M]Kα+Nˆ ⊗[N]Kα and ǫ([L]Kα) = δL,0. [M],[N]∈Iso(A) It is well known that there exists a nondegenerate symmetric bilinear ϕ( , ) : ˜( ) ˜( ) C, − − H A ×H A −→ defined by ϕ([M]K ,[N]K ) = δ a v(α,β). α β [M],[N] M This is a Hopf pairing (see for example [3, 12, 15]). Then the Drinfeld double Hall algebra D( ) of is by definition the free product ˜( ) ˜( ) divided out by the commutator A A H A ∗H A relations (with a,b ˜( )) ∈ H A Xϕ(a(2),b(1)) a(1) b(2) = Xϕ(a(1),b(2))(1 b(1)) (a(2) 1). (2.2) · ⊗ ⊗ ◦ ⊗ Here we use Sweedler’s notation: ∆(a) = a a . P (1) (2) ⊗ 2.3. Bridgeland’s Hall algebras. Let ( ( )) be the Hall algebra of the abelian 2 H C A category ( ) defined in Definition 2.4 and ( (P)) ( ( )) be the subspace 2 2 2 C A H C ⊂ H C A spanned by the isoclasses of complexes of projective objects. Define ( (P)) to be tw 2 H C the same vector space as ( (P)) with “twisted” multiplication defined by 2 H C [M ] [N ] := vhMˆ0,Nˆ0i+hMˆ1,Nˆ1i [M ] [N ]. • • • • ∗ · ⋄ Then ( (P)) is an associative algebra (see [1]). tw 2 H C We have the following simple relations for the acyclic complexes K and K∗. P P Lemma 2.6. ([1, Lemma 3.5]) For any object P P and any complex M (P), we • 2 ∈ ∈ C have the following relations in ( (P)) tw 2 H C [K ] [M ] = vhPˆ,Mˆ•i[K M ], [M ] [K ] = v−hMˆ•,Pˆi[K M ]; (2.3) P • P • • P P • ∗ ⊕ ∗ ⊕ [K∗] [M ] = v−hPˆ,Mˆ•i[K∗ M ], [M ] [K∗] = vhMˆ•,Pˆi[K∗ M ]; (2.4) P ∗ • P ⊕ • • ∗ P P ⊕ • [K ] [M ] = v(Pˆ,Mˆ•)[M ] [K ], [K∗] [M ] = v−(Pˆ,Mˆ•)[M ] [K∗]. (2.5) P ∗ • • ∗ P P ∗ • • ∗ P In particular, for P,Q P, we have ∈ [K ] [K ] = [K K ], [K ] [K∗] = [K K∗]; (2.6) P ∗ Q P ⊕ Q P ∗ Q P ⊕ Q [[K ],[K ]] = [[K ],[K∗]] = [[K∗],[K∗]] = 0. (2.7) P Q P Q P Q 6 HAICHENGZHANG By Lemmas 2.3 and 2.6, the acyclic elements of ( (P)) satisfy the Ore conditions tw 2 H C and thus we have the following definition from [1]. Definition 2.7. The Bridgeland’s Hall algebra of , denoted by ( ), is the localiza- A DH A tion of ( (P)) with respect to the elements [M ] corresponding to acyclic complexes tw 2 • H C M . In symbols, • ( ) := ( (P))[ [M ]−1 H (M ) = 0 ]. tw 2 • ∗ • DH A H C | As explained in [1], this is the same as localizing by the elements [K ] and [K∗] for all P P objects P P. Writing α K ( ) in the form α = Pˆ Qˆ for some objects P,Q P, 0 ∈ ∈ A − ∈ one defines K = [K ] [K ]−1,K∗ = [K∗] [K∗]−1. Note that the equalities in (2.5) α P ∗ Q α P ∗ Q continue to hold with the elements [K ] and [K∗] replaced by K and K∗, respectively, P P α α for any α K ( ). 0 ∈ A For each object M , by [1, Lemma 4.1], we fix a minimal projective resolution1 of ∈ A the form 0 Ω δM P M 0. (2.8) M M −→ −→ −→ −→ Set δM // C := Ω oo P . M M M 0 Since the minimal projective resolution of M is unique up to isomorphism, the complex C is well-defined up to isomorphism. M Lemma 2.8. ([1, Lemma 4.2]) Each object L (P) has a direct sum decomposition • 2 ∈ C L = C C∗ K K∗. • M ⊕ N ⊕ P ⊕ Q Moreover, the objects M,N and P,Q P are uniquely determined up to isomor- ∈ A ∈ phism. As in [1], we have an element E in ( ) defined by M DH A E := vhΩˆM,Mˆi K [C ] ( ). M · −ΩˆM ∗ M ∈ DH A It is easy to see that the shift functor defines an algebra involution on ( ). Set ∗ DH A F = E∗ for any object M . M M ∈ A Theorem 2.9. ([1, Lemmas 4.6,4.7]) The maps Ie : ˜( ) ֒ ( ), [M] E , K K ; + H A → DH A 7→ M α 7→ α Ie : ˜( ) ֒ ( ), [M] F , K K∗ − H A → DH A 7→ M α 7→ α are both embeddingsof algebras. Moreover, the multiplication map m : a b Ie(a) Ie(b) ⊗ 7→ + ∗ − defines an isomorphism of vector spaces m : ˜( ) ˜( ) ≃ // ( ). H A ⊗H A DH A 1The notations PM and ΩM will be used throughout the paper. A NOTE ON BRIDGELAND’S HALL ALGEBRAS 7 3. Main Theorem In this section, we first present the main theorem which was stated by Bridgeland in [1], proved by Yanagida in [17], and generalized by Lu and Peng in [5]. Then we provide a new and succinct proof by using the associativity formula of Hall algebras. Main Theorem ([1],[17],[5]) The Bridgeland’s Hall algebra ( ) is isomorphic to the DH A Drinfeld double Hall algebra D( ). A In what follows, we will give the proof of Main Theorem. By Theorem 2.9, it suffices to prove that the commutator relation Xϕ(a(2),b(1))I+e(a(1))∗I−e(b(2)) = Xϕ(a(1),b(2))I−e(b(1))∗I+e(a(2)) (3.1) holds in ( ) for each a = [A]K and b = [B]K with α,β K ( ) and [A],[B] α β 0 DH A ∈ A ∈ Iso( ). By writing out the comultiplications ∆([A]K ) and ∆([B]K ), and substituting α β A into (3.1), we find that we only need to prove that Relation (3.1) holds for a = [A] and b = [B]. Since ∆([A]) = X vhA1,A2igAA1A2[A1]KAˆ2 ⊗[A2]; [A1],[A2] ∆([B]) = X vhB1,B2igBB1B2[B1]KBˆ2 ⊗[B2], [B1],[B2] the left hand side of (3.1) becomes LHS of (3.1) = X vhA1,A2i+hB1,B2igAA1A2gBB1B2ϕ([A2],[B1]KBˆ2)EA1KAˆ2FB2 [A1],[A2],[B1],[B2] = X vhA1,A2i+hA2,B2igAA1A2gAB2B2aA2EA1KAˆ2FB2 [A1],[A2],[B2] = X vhA1,A2i+hA2,B2igAA1A2gAB2B2aA2vhΩA1,A1iK−ΩˆA1[CA1]KAˆ2vhΩB2,B2iK−∗ΩˆB2[CB∗2] [A1],[A2],[B2] = X vxgAA1A2gAB2B2aA2KAˆ2−ΩˆA1K−∗ΩˆB2[CA1][CB∗2], [A1],[A2],[B2] where x = A ,A + A ,B + Ω ,A + Ω ,B (A ,A ) (Ω ,A ) h 1 2i h 2 2i h A1 1i h B2 2i− 2 1 − B2 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = A +Ω ,B A + Ω ,A A ,Ω . h 2 B2 2 − 1i h A1 1i−h 1 B2i RHS of (3.1) = X vhA1,A2i+hB1,B2igAA1A2gBB1B2ϕ([A1]KAˆ2,[B2])FB1KB∗ˆ2EA2 [A1],[A2],[B1],[B2] = X vhA′1,A′2i+hB1′,A′1igAA′1A′2gBB1′A′1aA′1FB1′KA∗ˆ′1EA′2. [A′],[A′],[B′] 1 2 1 8 HAICHENGZHANG = [A′],X[A′],[B′]vhA′1,A′2i+hB1′,A′1igAA′1A′2gBB1′A′1aA′1vhΩB1′,B1′iK−∗ΩˆB1′[CB∗1′]KA∗ˆ′1vhΩA′2,A′2iK−ΩˆA′2[CA′2] 1 2 1 = [A′],X[A′],[B′]vx′gAA′1A′2gBB1′A′1aA′1KA∗ˆ′1−ΩˆB1′K−ΩˆA′2[CB∗1′][CA′2], 1 2 1 where x′ =hA′1,A′2i+hB1′,A′1i+hΩB1′,B1′i+hΩA′2,A′2i−(A′1,B1′)−(ΩA′2,B1′) = hAˆ′1 +ΩˆA′2,Aˆ′2 −Bˆ1′i+hΩˆB1′,Bˆ1′i−hBˆ1′,ΩˆA′2i. Lemma 3.1. For any objects X,Y and T,W P. In ( ) we have ∈ A ∈ DH A [C C∗ K K∗ ] = vhWˆ−Tˆ,Xˆ−YˆiK K∗ [C C∗]. X ⊕ Y ⊕ T ⊕ W Tˆ Wˆ X ⊕ Y Proof. By the commutative diagram (1 ) 0 // T W oo T W ⊕ ⊕ (0 ) 1 (a b) a′ b′ c d (cid:16)c′ d′(cid:17) (cid:15)(cid:15) (cid:16)δX 0(cid:17)// (cid:15)(cid:15) Ω P oo P Ω , X Y X Y ⊕ ⊕ 0 (cid:16) −δY (cid:17) we easily obtain that Hom (K K∗ ,C C∗) = qhTˆ,ΩˆX+PˆYi+hWˆ,PˆX+ΩˆYi. | C2(A) T ⊕ W X ⊕ Y | Hence, [K K∗ ][C C∗] = vm[C C∗ K K∗ ], T ⊕ W X ⊕ Y X ⊕ Y ⊕ T ⊕ W where m = Tˆ +Wˆ ,Pˆ +Ωˆ +Pˆ +Ωˆ 2 Tˆ,Ωˆ +Pˆ 2 Wˆ ,Pˆ +Ωˆ X X Y Y X Y X Y h i− h i− h i ˆ ˆ ˆ ˆ = T W,X Y . h − − i (cid:3) For any fixed objects A,B,X,Y , we denote by WXY the set ∈ A AB (f,g,h) 0 // X f // A g // B h // Y // 0 is exact in , { | A} and denote by Hom (A ,B ) the set X A 1 2 Y g g 0 // X // A // B // Y // 0 is exact in . { | A} By [13, (8.8)], |WAXBY| = aXaY XaLgLAXgYBL, [L] A NOTE ON BRIDGELAND’S HALL ALGEBRAS 9 and it is easy to see that WXY Hom (A,B) = | AB |. (3.2) X A Y | | a a X Y Lemma 3.2. (1) For any A ,B . In ( ) we have 1 2 ∈ A DH A [CA1][CB∗2] = X vaaLgLAX1 gYBL2KΩˆA1−ΩˆXKP∗ˆB2−PˆY[CX ⊕CY∗], [X],[Y],[L] where a = P ,Ω Ω ,P + Pˆ +Ωˆ Ωˆ Pˆ ,Xˆ Yˆ ; h A1 B2i−h A1 B2i h B2 X − A1 − Y − i (2) For any A ,B . In ( ) we have 2 1 ∈ A DH A [CB∗1][CA2] = X va′aL′gLB′1YgXA2L′KPˆA2−PˆXKΩ∗ˆB1−ΩˆY[CX ⊕CY∗], [X],[Y],[L′] where a′ = P ,Ω Ω ,P + Pˆ +Ωˆ Pˆ Ωˆ ,Xˆ Yˆ . h B1 A2i−h B1 A2i h X B1 − A2 − Y − i Proof. We only prove (1), since (2) is similar. Ext1C2(A)(CA1,CB∗2) ∼= HomK2(P)(CA1,CB2) = Hom (A ,B ) ∼ R2(A) 1 2 (3.3) ∼= i∈ZHomDb(A)(A1,B2[2i]) ⊕ = Hom (A ,B ), since is hereditary. ∼ A 1 2 A Consider an extension of C by C∗ A1 B2 η : 0 C∗ L C 0. −→ B2 −→ • −→ A1 −→ It induces a long exact sequence in homology H (C∗ ) H (L ) H (C ) H (C∗ ) H (L ) H (C ). 0 B2 −→ 0 • −→ 0 A1 −→ 1 B2 −→ 1 • −→ 1 A1 Writing L = C C∗ K K∗ for some objects X,Y and T,W P, we obtain • X ⊕ Y ⊕ T ⊕ W ∈ A ∈ the following exact sequence δ 0 X A B Y 0, 1 2 −→ −→ −→ −→ −→ where δ is determined by the equivalence class of η via the canonical isomorphism in (3.3) Ext1 (C ,C∗ ) = Hom (A ,B ). (3.4) C2(A) A1 B2 ∼ A 1 2 By considering the kernels of differentials in C∗ ,L and C , we get that B2 • A1 P W = P , and then Ω T = Ω . Y ⊕ ∼ B2 X ⊕ ∼ A1 That is, T and W are uniquely determined by X and Y up to isomorphism, respectively. The canonical isomorphism (3.4) induces an isomorphism Ext1C2(A)(CA1,CB∗2)CX⊕CY∗⊕KT⊕KW∗ ∼= XHomA(A1,B2)Y. 10 HAICHENGZHANG Hence, WXY |Ext1C2(A)(CA1,CB∗2)CX⊕CY∗⊕KT⊕KW∗ | = |aAa1B2| = XaLgLAX1 gYBL2. X Y [L] Thus, [CA1][CB∗2] = X vhPA1,ΩB2i−hΩA1,PB2iaLgLAX1 gYBL2[CX ⊕CY∗ ⊕KT ⊕KW∗ ], [X],[Y],[L] here we have used that |HomC2(A)(CA1,CB∗2)| = qhΩA1,PB2i. By Lemma 3.1, [CA1][CB∗2] = X vaaLgLAX1 gYBL2KTKW∗ [CX ⊕CY∗] [X],[Y],[L] = X vaaLgLAX1 gYBL2KΩˆA1−ΩˆXKP∗ˆB2−PˆY[CX ⊕CY∗], [X],[Y],[L] where a = P ,Ω Ω ,P + Wˆ Tˆ,Xˆ Yˆ h A1 B2i−h A1 B2i h − − i = P ,Ω Ω ,P + Pˆ +Ωˆ Ωˆ Pˆ ,Xˆ Yˆ . h A1 B2i−h A1 B2i h B2 X − A1 − Y − i (cid:3) Proof of Main Theorem LHS of (3.1) = X va+xgAA1A2gAB2B2gLAX1 gYBL2aA2aLKAˆ2−ΩˆA1K−∗ΩˆB2KΩˆA1−ΩˆXKP∗ˆB2−PˆY[CX ⊕CY∗] [A1],[A2],[B2], [L],[X],[Y] = X vhBˆ2+ΩˆX−PˆY−Aˆ2,Xˆ−YˆigAA1A2gAB2B2gLAX1 gYBL2aA2aLKAˆ2−ΩˆXKB∗ˆ2−PˆY[CX ⊕CY∗] [A1],[A2],[B2], [L],[X],[Y] = X vhYˆ+Lˆ+ΩˆX−PˆY−Aˆ2,Xˆ−YˆigAA1A2gAB2B2gLAX1 gYBL2aA2aLKAˆ2−ΩˆXKY∗ˆ+Lˆ−PˆY[CX ⊕CY∗] [A1],[A2],[B2], [L],[X],[Y] = X vhLˆ+ΩˆX−ΩˆY−Aˆ2,Xˆ−YˆigLAXA2gAB2YLaA2aLKAˆ2−ΩˆXKL∗ˆ−ΩˆY[CX ⊕CY∗], [A2],[L], [X],[Y] here we get the last equality by using the associativity formula in (2.1).