ON THE CLASSIFICATION OF G-GRADED TWISTED ALGEBRAS 3 1 JUAN D.VE´LEZ, LUIS A.WILLS,NATALIAAGUDELO 0 2 n a J Abstract. LetGdenoteagroupandletW beanalgebraoveracom- mutativeringR. WewillsaythatW isaG-gradedtwistedalgebra(not 3 2 necessarilycommutative,neitherassociative)ifthereexistsaG-grading W =⊕g∈GWg where each summand Wg is a free rank one R -module, ] andW hasnomonomialzerodivisors(foreachpairofnonzeroelements A wa,wb en Wa and Wb their product is not zero, wawb 6= 0). It is also R assumed that W has an identity element. . In this article, methods of group cohomology are used to study the h general problem of classification under graded isomorphisms. We give t a a full description of these algebras in the associative cases, for complex m and real algebras. In the nonassociative case, an analogous result is obtained under a symmetry condition of the corresponding associative [ function of the algebra, and when the group providing the grading is 1 finite cyclic. v 4 5 1. Introduction 6 5 G-graded twisted algebras were introduced in [5], and independently in . 1 [14], as distinguished mathematical structures which arise naturally in theo- 0 retical physics [15], [16], [17], [18], [19], and [6]. By one of these algebras we 3 1 meanthefollowing: LetGdenoteagroup. AnR-algebraW (notnecessarily : commutative, neither associative) will be called a G-graded twisted algebra v Xi if there exists a G-grading, i.e., W = ⊕g∈GWg, with WaWb ⊂ Wab, in which each summand W is an R -module of free rank one. We assume that W g r a has an identity element 1 ∈ We, where We denotes the graded component corresponding to the identity element e of G. We also require that W has no monomial zero divisors, i.e., for each pair of nonzero elements w ∈ W , a a andw ∈ W ,their productmustbenon zero, w w 6= 0 (for ageneral study b b a b of nonassociative algebras, the reader may consult [9].) Besides its interest for physicists, these algebras are natural objects of studyfor mathematicians, since they are related to generalizations of Lie al- gebrasthatincludetransformationparameterswithnoncommutativeand/or nonassociativeproperties,asdefinedin[4],wheregeneralizations ofsomere- sults of Scheunert [10] on epsilon or color (super)Lie algebras are discussed. One important case, that of G-graded twisted division algebras (in which every nonzero element has a left and a right inverse, but where the left and right inverses do not necessarily coincide) already includes the classical 1 2 JUAND.VE´LEZ,LUISA.WILLS,NATALIAAGUDELO quaternion and octonion division algebras. They are a new remarkable class of (noncommutative and nonassociative) division algebras over the reals. Their classification is addressed in [11], [12]. For the general case of G-graded twisted algebras over the complex num- bers, first attempts towards their classification appear in [3], where rudi- mentary techniques of group cohomology were introduced. Those methods are exploited in this article to obtain necessary and sufficient conditions for two algebras to be isomorphic under graded isomorphisms (see Theorem 4 and its corollaries). On the other hand, classical techniques of group rep- resentation theory readily give a complete classification in the associative case (Corollaries 1 and 3.) This last result is achieved via Theorem 1, which allows to represent these algebras as quotients of certain group rings. A similar description is possible in the general, not associative case, where in- stead ofgroupringsthecorrespondingstructuresarequotients ofloop rings, objects, notwithstanding, poorly studied in the literature. As for general nonassociative algebras, we deal in this article only with the simplest cases. We study algebras over the complex and real numbers which are graded by finite abelian groups, and whose associativity function –see (2.1) below– satisfies certain symmetry condition (Theorems 7 and 8.) 2. Definitions and basic concepts Definition 1. Let G denote a group and let W be an algebra over a commutative ring R. We will say that W is a G-graded twisted algebra (not necessarily commutative neither associative) if thereexists a G-grading W = ⊕g∈GWg, with WaWb ⊂ Wab, where each summand Wg is a free rank one R -module. It is required that W has no monomial zero divisors. This condition means that for each pair of nonzero elements w ∈ W and a a w ∈ W , w w 6= 0. It is also assumed that W has an identity element b b a b 1= w ∈ W , where e denotes the identity element of G. e e Since each graded component W is a free rank one R -module, we may g choose w ∈ W such that B = {w : g ∈ G} is a basis for W as an g g g R-module. For any such fixed basis B we may associate a function (the ∗ structure constant with respect to B) C : G×G → R , which takes values B ∗ in R = R−{0}, and such that for any pair of elements w ∈ W , w ∈ W , a a b b w w = C (a,b)w . We notice that w w = C (a,e)w , implies C (a,e) = a b B ab a e B a B 1, for all a in G. The fact that W has no monomial zero divisors implies that C must take values in a a subdomain A of the multiplicative group B ∗ R . If we are interested in stressing this fact, we will write C : G×G → A. B On the other hand, if B is understood, we will simply write C :G×G → A, omitting the subscript. If R is a field, and A ⊂ R, a subfield, the commutative and associative properties of W can be understoodby means of the following two functions, ON THE CLASSIFICATION OF G-GRADED TWISTED ALGEBRAS 3 q :G×G → A, and r :G×G×G→ A, defined as: q(a,b) = C(a,b)C(b,a)−1 (2.1) r(a,b,c) = C(b,c)C(ab,c)−1C(a,bc)C(a,b)−1 IfGisabelian,itholdsthatw (w w ) = r(a,b,c)(w w )w ;andthatw w = a b c a b c a b q(a,b)w w . b a Definition 2. By amorphism between two G-graded twisted algebras W = ⊕g∈GWg and V = ⊕g∈GVg we mean an unitarian homomorphism of R- algebras φ : W → V. If it also preserves the grading, i.e., φ(W ) ⊂ V , we g g say the morphism is graded. Remark 1. Let us fix B = {w : g ∈ G} and B = {v : g ∈ G} bases for 1 g 2 g W and V, respectively. Defining a graded morphism φ :W → V amounts to giving a function ϕ : G → R such that φ(w ) = ϕ(g)v . Clearly, φ will be g g an isomorphism if and only if for all g, ϕ(g) is a unit in R. The classification problem of G-graded twisted algebras admits at least two versions. (1) W and V can be isomorphic as graded algebras, that is, there are graded isomorphisms φ :W → V and ψ : V → W such that φψ and ψφ is the identity. (2) W and V can be isomorphic as R-algebras, without taking into con- sideration the grading. In this article we will restrict to the case where R is a field, that we will denote by k. It will also be assumed that G is a finite group. 3. G-graded twisted algebras and Loop rings ∗ Let W be a G-graded twisted algebra, and let C : G×G → A ⊂ k be the structure constant with respect to a fixed basis B. The function C gives rise to an extension E of A by G, that will be denoted by A× G, defined C as follows: E = {(α,g) : α ∈ A,g ∈ G} ⊂ A×G. We endow this set with W the following operation: (α,a).(β,b) = (αβC(a,b),ab). It is easy to see that whenever W is associative, E is a group, actually W an extension of A by G. In general, if W is not associative, then E turns W out to be a loop, a structure which is almost a group, except that its binary operation is not necessarily associative, but where there is still an identity element, and where each element has a right and a left inverse, not neces- sarily equal. As in the case of an ordinary group, the loop ring R = k[E ] W may be defined analogously. The structure of any associative G-graded twisted algebra can be under- stood using the following fundamental result. 4 JUAND.VE´LEZ,LUISA.WILLS,NATALIAAGUDELO Theorem 1. Let W = ⊕g∈GWg, and let C : G×G → A be the structure constant related to a fixed choice of basis for W. Let A × G denote the C extension of A by G defined above, and let R = k[A× G] be the Loop ring C over A × G. Let us define the vector subspace I of R generated by all C elements of the form I = h(α,g)−α(1,g) : α∈ A,g ∈ Gi. Then I ⊂ R is a bilateral ideal and R/I is a k-algebra isomorphic to W. Proof. It is an elementary fact that I is a bilateral ideal of R. ∗ Now, for (α,a) ∈ A× G let us define ϕ : A× G → W as the func- C C ∗ tion which sends (α,a) to αv ,where W denotes the invertible elements of a W. Clearly, ϕ((α,a)(β,b)) = ϕ(α,a)ϕ(β,b), and consequently ϕ can be ex- tendedtoak-linearmap(thatwewilldenotebythesameletter)ϕ : R → W. An easy computation shows that I ⊂ Ker(ϕ), and consequently ϕ de- scends to the quotient ϕ : R/I → W, sending the class of λ (α,a) (α,a) into λ αv . An inverse map can be defined explicitly by extending (α,a) a P linearly the map φ : W → R/I that sends each element v to the class of a P (1,a). φ is indeed a homomorphism of k algebras: (”−” denotes the class of an element) φ( λ v λ v ) = φ( λ λ C(a,b)v ) a a b b a b ab a∈G b∈G a,b∈G P P P = φ( ( C(a,b)λ λ )v ) a b ab g∈G a+b=g P P − (3.1) = C(a,b)λ λ (1,g) a b g∈G a+b=g ! P P A similar computation shows that (3.1) is equal to φ( λ v )φ( λ v ). a a b b a∈G b∈G (cid:3) P P 4. Associative G-graded algebras over C Standard techniques of group representation theory allows us to com- pletely classify all G-graded twisted associative C-algebras. For this, let us fix a basis B and let C : G × G → A ⊂ C∗ be the structure constant of W with respect to B. As observed above, whenever W is associative E = A× G is not only a loop, but a group. Let R = C[A× G] be its W C C associated group ring. It is a standard result that R is the regular repre- sentation of E , and that if R decomposes into irreducible representations W R = Va1 ⊕···⊕Var, with V 6= V , then the exponents a can be computed 1 r i j i in terms of the characters χ and χ as R Vi 1 a = χ ,χ = χ (g)χ (g) = dimV . i R Vi |A× G| R Vi i C g∈G (cid:10) (cid:11) P (See[7],[8]). Hence,thehomomorphismϕ : C[A×C G] → ⊕ri=1HomC(Vi,Vi) that send each element s ∈ C[A× G] into (ψ ) , where ψ : V → V C i i=1,....r i i i ON THE CLASSIFICATION OF G-GRADED TWISTED ALGEBRAS 5 denotes multiplication by s, is in fact an isomorphism of C-algebras (no grading involved) [8]. With notation as in Theorem 1: Theorem2. Thereisanisomorphism (notnecessarilygraded)ofC-algebras ϕ: C[A×C G]/I → ⊕ri=1HomC(Vi,Vi)/Ji [s] 7→ (ψ ) i i=1,....r where J = ϕ(I) is isomorphic to a product J ×···×J , of bilateral ideals 1 n Ji ⊂ HomC(Vi,Vi). Proof. The ideal J decomposes as a product of bilateral ideals J = J × 1 ···×Jr, with Ji ⊂ HomC(Vi,Vi). Hence, W ∼= R/I = ⊕ri=1HomC(Vi,Vi)/Ji. Buteach oneofthealgebras HomC(Vi,Vi)issimpleandconsequentlyJi = 0 or Ji = HomC(Vi,Vi). Thus, W ∼= ⊕iHomC(Vi,Vi) where the sum occurs only for those i such that J = (0). (cid:3) i Corollary1. LetW beaG-graded twisted associative C-algebra. ThenW is isomorphic asaC-algebratoafiniteproduct ofmatrixalgebras Matni×ni(C), where n = dimV .The algebra W is commutative if and only if n = i i i dimV = 1, and therefore, if and only if W ≃ C×···×C. i 4.1. Associative G-graded algebras over R. For the real case we pro- ceed in a similar manner as in the last section. For any group G, the group ring R[G] is isomorphic to a finite direct sum of R-algebras of the form Hom (V ,V ) where D is a division algebra over the reals. Moreover, Di i i i Di = HomR(Vi,Vi)G (see [8]). This immediately yields the following: Theorem 3. Let W = ⊕g∈GWg be a G-graded, twisted, associative R- algebra. Then W is isomorphic to a direct sum ⊕r Hom (V ,V ), where i=1 Di i i D denotes one of the division rings R,C, or H, the quaternions. i Proof. We already know that W ≃ R[A× G]/I, for some bilateral ideal I. C From the remark above (4.1) it follows that R[A× G]/I ≃ ⊕n Hom (V ,V )/J , C i=1 Di i i i for bilateral ideals Ji of HomDi(Vi,Vi), where Di = HomR(Vi,Vi)A×CG is a division (associative) algebra over the reals. But it is a well known fact that D mustequalto oneof thealgebras R,Cor H. Thus, theresultfollows. (cid:3) i 5. Graded Morphisms and group Cohomology In this section we study the problem of determining when two G-graded twisted k-algebras are isomorphic under a graded isomorphism. The follow- ing theorem provides necessary and sufficient conditions for two algebras to be graded-isomorphic in terms of the second group cohomology H2(G,k∗). For the basic notions about group cohomology, the reader may consult [1], [2]. 6 JUAND.VE´LEZ,LUISA.WILLS,NATALIAAGUDELO Theorem 4. Let V = ⊕g∈GVg and W = ⊕g∈GWg two G-graded k-algebras. Let us fix bases B and B for V and W, respectively, and let C and C be 1 2 1 2 the associated structure constants. Then V is isomorphic to W if and only if the function C C−1 is in the kernel of d2 : C2(G,k∗) → C3(G,k∗) and 1 2 the class [C C−1] is trivial in H2(G,k∗). 1 2 Proof. Let us suppose that V and W are isomorphic as k-algebras under a grading preserving isomorphism φ : V → W. This implies that there ∗ exists ϕ : G → k which sends each vector v into ϕ(g)w . But φ a homo- g g morphism implies that φ(C (a,b)v ) = ϕ(a)w ϕ(b)w , and consequently, 1 a+b a b C (a,b)ϕ(ab)w = ϕ(a)ϕ(b)C (a,b)w . From this we obtain 1 a+b 2 a+b (5.1) C (a,b)C−1(a,b) = ϕ(a)ϕ(ab)−1ϕ(b). 1 2 Notice that d1ϕ(a,b) = ϕ(b)ϕ−1(ab)ϕ(a), and therefore C C−1 belongs 1 2 to the image of d1 : C1(G,k∗) → C2(G,k∗). Thus, d2(C C−1) = 1, and 1 2 [C C−1] = 1 in H2(G,k∗). 1 2 Reciprocally, if d2(C C−1) = 1,and[C C−1]= 1 inH2(G,k∗), thenthere 1 2 1 2 exists ϕ : G → k∗ such that d1ϕ = C C−1 and consequently equation (5.1) 1 2 holds. It then follows that the function φ : V → W defined on the basis B as φ(v ) = ϕ(g)w is a homomorphism of k-algebras, which is injective, 1 g g since so it is φ. But V are W are k-vector spaces of the same dimension (equal to |G|). Hence, φ is an isomorphism. (cid:3) Remark 2. If V and W are associative, then d2C = d2C = 1, since both 1 2 these terms are equal to the associativity function in (2.1). In this case, the class of each C is an element of H2(G,k∗) and the condition [C C−1] = 1 i 1 2 is equivalent to [C ]= [C ] in H2(G,k∗). 1 2 Corollary 2. Let W = ⊕g∈GWg be a G-graded k-algebra, and let be B and ′ ′ B be bases for W, with associated constant structures C and C . Let r and ′ ′ r be the corresponding functions as defined above. Then r = r . In other words, the associativity function of W does not depend on any chosen basis. Proof. The identity isomorphism I : W → W is trivially graded. By the previous theorem, [C′C−1] = 1 and consequently C′C−1 ∈ im(d1). Hence, d2(C′C−1) = 1, and therefore r′ =d2(C′) = d2(C)= r. (cid:3) ∗ Let us notice now that if C and C take values in a subdomain A ⊂ k , 1 2 then C C−1 ∈ C2(G,A). Hence, if d2(C C−1) = 1, it makes sense to talk 1 2 1 2 abouttheclass [C C−1] ∈ H2(G,A). Thefollowing theoremgives acriterion 1 2 intermsofH2(G,A) todeterminewhenV andW areisomorphic. Thismay ∗ be useful in many cases where A is a finite subgroup of k . Theorem5. φ :V → W isa(graded)isomorphism ifandonlyifd2(C C−1) = 1 2 1, and [C C−1] ∈ ker(i ), where i : H2(G,A) → H2(G,k∗) denotes the ho- 1 2 2 2 ∗ momorphism in cohomology induced by the inclusion i: A→ k . ON THE CLASSIFICATION OF G-GRADED TWISTED ALGEBRAS 7 Proof. The short exact sequence of groups i π ∗ 1 → A→ k → k /A → 1. induces an exact sequence of complexes ((*)) 1 → C•(G,A) →i• C•(G,k∗)→π• C•(G,k∗/A) → 1, • ∗ • • ∗ where we may identify the quotient C (G,k )/C (G,A) with C (G,k /A) ∗ via the isomorphism that sends the class of h : G → k into π ◦ h. By Theorem 4, V and W are graded isomorphic, if and only if d2(C C−1) = 1, 1 2 and [C C−1] = 1 in H2(G,k∗). But looking at the long exact sequence for 1 2 cohomology ··· → H1(G,k∗) →π1 H1(G,k∗/A) →δ H2(G,A) →i2 H2(G,k∗)→ ··· we see that this occurs precisely when i ([C C−1])= 1. (cid:3) 2 1 2 Example 1. IfGdenotesacyclic groupofordern,thenitis wellknowthat H2(G,C∗) = C∗/(C∗)n = {1}. Hence, if V and W are G-graded associative ∗ algebras with structure constants given by C ,C : G×G → A ⊂ C , then 1 2 [C ][C ]−1 = 1and consequently they are isomorphic. Itreadily follows that 1 2 C[t]/(tn−1) = ⊕n−1Ctr is arepresentative of theuniqueisomorphismclass. r=0 Remark 3. If k = R, then H2(G,R∗) = {1}, if n is odd, and it is equal to {1,−1}, if n is even. In the first case, there exists a unique real associative algebra R[t]/(tn − 1). In the second case, there are exactly two algebras, given by R[t]/(tn − 1), and R[t]/(tn + 1). On the other hand, if V and W have structure constants C ,C : G × G → A, where |A| and |G| are 1 2 relatively primeintegers, then H2(G,A) = {1}. From theprevious theorem, if d2(C C−1) = 1 then V and W are isomorphic as graded algebras. In 1 2 the general case d2(C C−1) = d2(C )d2(C )−1 = r r−1, where r is the 1 2 1 2 1 2 i associativity function of C . Thus, r = r if and only if V and W are i 1 2 isomorphic. This proves the following. Theorem 6. Let W1 = ⊕g∈GWg1, and W2 = ⊕g∈GWg2, be G-graded k- algebras over a finite group G. Let r r : G3 → A be the corresponding 1, 2 associativity functions. If |A| and |G| are relatively prime integers, then W1 and W2 are isomorphic as graded algebras if and only if r = r . In 1 2 particular, if W1 and W2 are associative, and if |A| and |G| are relatively prime integers, then W1 and W2 are isomorphic as graded algebras. 6. Some classification results in the nonassociative case In this section we shall give a complete classification under graded iso- morphisms of all G-graded twisted algebras, when G is a finite cyclic group, andundertheconditionthattheirassociative functionsatisfiesthefollowing symmetric condition: r(a,b,c) = r(b,a,c), for all a,b,c ∈ G . 8 JUAND.VE´LEZ,LUISA.WILLS,NATALIAAGUDELO We start with a general discussion. Let G be any abelian group G, and let W = ⊕g∈GWg be a G-graded twisted k-algebra. Let us fix a basis B = {v : g ∈ G}, and denote by T : W → W the k-linear map defined by g g multiplying (on the left) by v . If a and b are arbitrary elements of G, then g for any v ∈ B g T (T (v )) = T (v v ) a b g a b g = v (v v ) a b g = r(a,b,g)(v v )v a b g = r(a,b,g)C(a,b)v v ab g = r(a,b,g)C(a,b)T (v ), ab g and consequently, (6.1) T (v ) =r(a,b,g)−1C(a,b)−1T (T (v )). ab g a b g Similarly, (6.2) T (v ) =r(b,a,g)−1C(b,a)−1T (T (v )). ba g b a g Since G is abelian, T = T . Therefore, ab ba T (T (v )) = r(a,b,g)C(a,b)r(b,a,g)−1C(b,a)−1T (T (v )). a b g b a g Using the symmetry condition (6) r(a,b,g) = r(b,a,g), it follows that T (T (v )) = q(a,b)T (T (v )). a b g b a g Hence, for any x ∈ W (not necessarily homogeneous) (6.3) T (T (x)) = q(a,b)T (T (x)) a b b a In a similar way, (6.4) T (T (x)) = q(b,a)T (T (x)). b a a b Let G be a cyclic group of order n. Fix any generator a of G. Let w be a any non zero element in W . Define inductively w = w · w , where a ak a ak−1 w = α, for some arbitrary fixed α 6= 0 in C. Define v = βkw , where β a0 ak ak denotes any primitive n-th root of unity of α−1, (βn = 1/α). If k = n, we see that v = βnw = βnα = 1, and a0 a0 v v = βkw w = βkw = v . a ak−1 a ak−1 ak ak The basis B = {1,va,...,van−1} will be called the standard basis for W. In whatfollows, thestructureconstantC (ar,ak) willbedenotedby C(ar,ak). B Notice that C(a,ar) = 1, for all r = 0,...n−1. SinceT sendsv intov ,thelinear mapT mustpermutethebasisB a ak ak+1 a cyclically, andconsequentlytheminimalpolynomialforT mustbeYn−1 = a 0. Hence, the eigenvalues of T are precisely the n-th complex roots of unity a {ω ,...,ω }. For each 1 ≤ j ≤ n, let z = n−1ωkv . 1 n j k=0 j ak If t ≡ s mod n, then ωtjvat = ωsjvas, sincePt = s+nq implies ωtjvat = ωnjqωsjvas = 1·ωsjvas. ON THE CLASSIFICATION OF G-GRADED TWISTED ALGEBRAS 9 Consequently, the sum n−1ωkv can be also be written as ωkv . k=0 j ak k∈Zn j ak Now, ω −1 is an eigenvalue associated to z because j j P P T (z ) = ωkv v = ω −1 ω k+1v a j j a ak j j ak+1 kX∈Zn k+X1∈Zn n = ωj−1 ωsjvas = ωj−1zj. sX∈Zn On the other hand, by (6.4) T (T (z )) = q(b,a)T (T (z )) b a j a b j T (ω −1z ) = q(b,a)T (T (z )) b j j a b j ω −1T (z ) = q(b,a)T (T (z )) j b j a b j q(a,b)ω −1T (z ) = T (T (z )). j b j a b j Hence, T (z ) is a eigenvector of the eigenvalue q(a,b)ω −1. Since T has n b j j a differenteigenvalues, theremustexistω andz suchthatω −1 = q(a,b)ω −1 i i i j and T (z ) = η z , for some η 6= 0 en C. b j b,j i b,j If we take b = ar we obtain (6.5) Tar(zj) = Tar( ωkjvak)= ωkjC(ar,ak)var+k k∈Zn k∈Zn Pn−1 P (6.6) = ω−r ωk+rC(ar,ak)v j j ar+k k+r∈Zn (6.7) = ω−j r PωsjC(ar,as−r)vas, s∈Zn P (s = k+r). On the other hand, (6.8) Tar(zj) = ηar,jzi =ηar,j ωsivas sX∈Zn (6.9) = ηar,jq(ar,a)sωsjvas. sX∈Zn Comparing coefficients in (6.7) and (6.8) we see that ωs−rC(ar,as−r) = η q(ar,a)sωs j ar,j j and consequently (6.10) C(ar,as−r) = η q(ar,a)sω r. ar,j j In particular, if s = r, we see that 1 = C(ar,a0) = η q(ar,a)rω r. ar,j j Thus, η = q(a,ar)rω −r. Substituting in (6.10) we get C(ar,as−r) = ar,j j q(a,ar)rω−rq(ar,a)sωr, and consequently C(ar,as−r) = q(ar,a)s−r. j j If we let k = s−r we see that C(ar,ak)= q(ar,a)k = C(ar,a)kC(a,ar)−k = C(ar,a)k, 10 JUAND.VE´LEZ,LUISA.WILLS,NATALIAAGUDELO (using that C(a,ar) = 1). Thus, (6.11) r(ar,as,at) = C(as,a)tC(ar+s,a)−tC(ar,a)s+tC(ar,a)−s (6.12) = (C(as,a)C(ar,a)C(ar+s,a)−1)t Let G be a cyclic group of order n and let W = ⊕g∈GWg, a G-graded twisted over C. Let us assume that r is symmetric in the first two entries. If B denotes the standard basis for W, then W is completely determined by the function f : G → A, given by f(ar) = C(ar,a), where a is any fixed generator of G. Hence, there are precisely |A|n−2 non isomorphic G-graded twisted algebras. Example 2. Let Z4 = a,a2,a3,a4 = 1 and W = ⊕3r=0War, a C-algebra. Let A = {1,−1}. The minimal polynomial for T is Y4 − 1 = 0, with a (cid:8) (cid:9) eigenvalues ω = −i,ω = −1,ω = i,ω = 1. It is easy to see that the 1 2 3 4 corresponding eigenvectors are z = 1+iv−v −iv3 1 2 z = 1−v+v2−v3 2 z = 1−iv−v2+iv3 3 z = 1+v+v2+v3. 4 Hence, T (z ) = −iz a 1 1 T (z ) = −z a 2 2 T (z ) = iz a 3 3 T (z ) = z a 4 4 Case 1 : Suppose C(a2,a) = −1 and C(a3,a) = 1. Therefore T (T (z )) = −T (T (z )) a a2 1 a2 a 1 T (T (z )) = iT (z ) a a2 1 a2 1 Then, T (z ) is an eigenvector associated to the eigenvalue i; that is, a2 1 T (z ) = β(a2,1)z . Hence, a2 1 3 β(a2,1) = q(a,a2)2ω−2 = (−1)2(i)−2 = −1 1 and the structure constants are given by C(a2,a2) = q(a2,a)2 = 1 C(a2,a3) = q(a2,a)3 = −1 C(a2,a) = q(a2,a) = −1 Similarly,T (z )isaneigenvectoroftheeigenvalue1,i.e.,T (z )= β(a2,2)z , a2 2 a2 2 4 with β(a2,2) = 1. And T (z ) is an eigenvector of the eigenvalue −i, i.e., a2 3 T (z ) = β(a2,3)z ,withβ(a2,3) = −1. Finally, T (z )is aneigenvector of a2 3 1 a2 4