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ON THE QUANTUM LORENTZ GROUP M. Lagraa 9 9 9 1 Laboratoire de Physique Théorique n Université d’Oran Es-Sénia, 31100, Algérie a J and 6 Laboratoire de Physique Théorique et Hautes énergies1, Batiment 211, Université Paris XI, 94405 ORSAY,France ] A Q . h t a m [ 1 v 0 2 0 Abstract : The quantum analogues of Pauli matrices are introduced and investigated. From 1 thesematricesandanappropriatetraceoverspinorialindicesweconstructaquantumMinkowsky 0 9 metric. In this framwork we show explicitly the correspondence between the SL(2,C) and 9 Lorentz quantum groups. The R matrices of the quantum Lorentz group are constructed in / h terms of the R matrices of SL(2,C) group. These R matrices satisfy adequate properties as t a Yang-Baxter equations, Hecke relations and quantum symmetrization of the metric. It is also m shown that the Minkowsky metric leads to an invariant and central norm. : v i X r a 1Laboratoire associé au Centre National de la Recherche Scientifique - URA D0063 1 1 Introduction The Lorentz group plays a fundamental role in physics. First, it constitutes the homogeneous part of Poincaré group which is intrinsically connected to the geometry of the space-time and leaves invariant all physical systems discribed by the special theory of relativity. Second, the different representations of Lorentz group are field discribing particles which constitute the physical systems. For these reasons, it is especially interesting to study the noncommutative version of the Lorentz group. The other reason which makes the study of the quantum Lorentz group interesting is due to the fact that in quantum field theory based on a classical space-time and a classical lorentz group there exist difficulties tied to small space-time distances. One may hope to solve these difficulties by new tools provided by the noncommutative geometry [1-4]. The construction of quantum Lorentz group has been considered by many authors, either di- rectly [5-6] or in the context of the quantum poincaré group [7-8]. Despite intensive efforts, these studies do not completly describe the quantum Lorentz group. In this paper one constructs quantum Lorentz group out of the quantum SL(2,C) group by showing how all properties of the former can be deduced from the later which are well known. The paper is organized in the following way. In Sect. 2 we recall the well known results provided by the bicovariant calculus over SL(2,C) and SU(2) quantum groups. We shall assume that the undotted (conjugate) and dotted generators of quantum SL(2,C) group satisfy the commutation rules of the quantum SU(2) group. In Sect. 3 the construction of quantum Lorentz group is carried out of the quantum SL(2,C) groupfollowingtheanalogueofthehomomorphismfortheclassicalgroupSO(1,3) ∼ SL(2,C)\Z . 2 We shall start by investigating the quantum analogues of the Pauli matrices from which we construct an adequate Minkowsky metric. An example of a two parameters deformation is given and the completeness relations are established. From the properties of the quantum Pauli matrices and the generators of SL(2,C), we construct the generators of the quantum Lorentz group. We show that they satisfy the axiomatic structure of the Hopf algebras and the orthogonality relations. We also construct the R matrices of the Lorentz group out of those of SL(2,C) group. These R matrices satisfy the Yang-Baxter equations, the Hecke relations and exhibit the quantum symetrization properties of the Minkowsky metric. In Sect. 4 we investigate the properties of the Minkowsky space. In particular, we show that the Minkowsky metric induces an invariant norm which commutes with the Hopf algebra A generated by the quantum SL(2,C) group generators, the undotted and dotted basis (spinors) of the bicovariant bimodule over A and the quantum coordinates of the Minkowsky space. 2 Bicovariant Calculus On SL(2,C) and SU(2) Quantum Groups Before we start to construct the quantum Lorentz group out of the quantum SL(2,C) group, let us recall some results provided by the bicovariant calculus over the SL(2,C) and SU(2) 2 quantum groups. Let an unital ⋆-algebra A generated by M β(α,β = 1,2) which preserves a α nondegenerate bilinear form ε ε M αM β = ε I , εγδM αM β = εαβI , εαγε = δα = ε εγα αβ γ δ γδ A γ δ A γβ β βγ which are the unimodularity conditions. The nondegenerate bilinear form ε is considered as a quantum spinor metric, I being the unity of A. To preserve these conditions under A the antimultiplicative involution ⋆ : A → A, the spinor metric must satisfy the condition (ε )⋆ = λε with λλ⋆ = 1 leading to: αβ β˙α˙ ε M α˙M β˙ = ε I , εγ˙δ˙M α˙M β˙ = εα˙β˙I , εα˙γ˙ε = δα˙ = ε εγ˙α˙ α˙β˙ γ˙ δ˙ γ˙δ˙ A γ˙ δ˙ A γ˙β˙ β˙ β˙γ˙ where M β˙ = (M β)⋆. For convenience, we take λ = 1. A carries a structure of a ⋆-Hopf α˙ α algebra with a coaction ∆ : A → A ⊗ A, a counit ε : A → C and an antipode S : A → A defined on the generators by ∆(M β) = M γ ⊗M β, ε(M β) = δβ and S(M β) = ε M γεδβ. α α γ α α α αγ δ On the dotted copy, we have (∆(M β))⋆ = ∆((M β)⋆) = ∆(M β˙) = M γ˙ ⊗M β˙, ε(M β˙) = δβ˙ α α α˙ α˙ γ˙ α˙ α˙ and S(M β˙) = ε M γ˙εδ˙β˙. the involution ⋆ acts on the antipode as (S(M β))⋆ = εβ˙δ˙M γ˙ε = α˙ α˙γ˙ δ˙ α δ˙ γ˙α˙ S−1(M β˙). α˙ It is known [9] that the generators of a such system satify the noncommutativity relations R±αβM σM ρ = M αM βR±σρ where the forms of the R matrices are given by R±αβ = σρ γ δ σ ρ γδ γδ δαδβ + a±1εαβε satifying R±αβR∓σρ = δαδβ with a + a−1 + εαβε = 0, and a 6= 0. These γ δ γδ σρ γδ γ δ αβ R matrices satisfy the Yang-Baxter equation, the Hecke equations (R± + a±2)(R± − 1) and ε R±αλR±βρ = a∓1ε δρ. αβ σγ λδ γδ σ Now, we consider a right-invariant basis θ of the bicovariant bimodule Γ over A on which the α right coaction acts as ∆ (θ ) = θ ⊗I, ∆ (θ ) = θ ⊗I and the left coaction acts as R α α R α˙ α˙ ∆ (θ ) = M β ⊗θ , ∆ (θα) = S(M α)⊗θβ L α α β L β ∆ (θ ) = M β˙ ⊗θ , ∆ (θα˙) = S−1(M α˙)⊗θβ˙ (1) L α˙ α˙ β˙ L β˙ where (θ )⋆ = θ and the spinorial indices are lowered and raised as θ = θβε ,θα = θ εβα, α α˙ α βα β θα˙ = εα˙β˙θ and θ = ε θβ˙. From the bicovariance properties of the bimodule A− Γ [4] we β˙ α˙ α˙β˙ can show the existence of functionals f : A → C satisfying the following properties θ a = (a⋆f β)θ , θαa = (a⋆f˜α)θβ (2) α α β β θ a = (a⋆f β˙)θ , θα˙a = (a⋆f˜α˙)θβ˙ (3) α˙ α˙ β˙ β˙ aθ = θ (a⋆f β ◦S) , aθα = θβ(a⋆f˜α ◦S) (4) α β α β aθ = θ (a⋆f β˙ ◦S) , aθα˙ = θβ˙(a⋆f˜α˙ ◦S) (5) α˙ β˙ α˙ β˙ f β(ab) = f γ(a)f β(b) , f˜β(ab) = f˜β(a)f˜γ(b), (6) α α γ α γ α 3 f β˙(ab) = f γ˙(a)f β˙(b) , f˜β˙(ab) = f˜β˙(a)f˜γ˙(b) (7) α˙ α˙ γ˙ α˙ γ˙ α˙ f β(I) = δβ = f˜β(I) , f β˙(I) = δβ˙ = f˜β˙(I) (8) α α α α˙ α˙ α˙ M γ(f β ⋆a) = (a⋆f γ)M β , S(M α)(f˜γ ⋆a) = (a⋆f˜α)S(M γ), (9) α γ α γ γ β γ β M γ˙(f β˙ ⋆a) = (a⋆f γ˙)M β˙ , S−1(M α˙)(f˜γ˙ ⋆a) = (a⋆f˜α˙)S−1(M γ˙) (10) α˙ γ˙ α˙ γ˙ γ˙ β˙ γ˙ β˙ where the convolution product is defined by a ⋆ f = (f ⊗ I)∆(a) for any a ∈ A. Setting a = S(a) (S−1(a)) into the right undotted (dotted) relation of (9)((10)), then applying S−1(S) and comparing with the corresponding left undotted (dotted) relation of (9) ((10)), we get f β = f˜β ◦S ,f β˙ = f˜β˙ ◦S−1. (11) α α α˙ α˙ For the generators of A, the left relation (9) gives M γM δf β(M σ) = f γ(M δ)M σM β α ρ γ δ α ρ δ γ which shows that there exist two functionals f α(M ρ) proportional to the R±ρα matri- ±γ δ γδ ces. Applying these functionals on both sides of the unimodularity condition, we obtain: f α(M ρ) = a∓1R±ρα and f α(S(M ρ)) = a±1R∓αγ [10]. The same procedure can be used ±γ δ 2 γδ ±γ δ 2 δγ for the dotted copy of A generators. Then there exist two basis θ corresponding to the ±α functionals f β. ±α Applying the ⋆ involution on both sides of (2) and (3), we get respectively (θ a)⋆ = (θ )⋆(f β(a ))⋆a⋆ = a⋆(θ )⋆, (θαa)⋆ = (θβ)⋆(f˜ α(a ))⋆a⋆ = a⋆(θα)⋆, ±α ±β ±α (1) (2) ±α ± ± ±β (1) (2) ± (θ a)⋆ = (θ )⋆(f β˙(a ))⋆a⋆ = a⋆(θ )⋆ and (θα˙a)⋆ = (θβ˙)⋆(f˜ α˙(a ))⋆a⋆ = a⋆(θα˙)⋆(12) ±α˙ ±β˙ ±α˙ (1) (2) ±α˙ ± ± ±β˙ (1) (2) ± where a and a denote elements of ∆(a) = a ⊗a . In the other hand, for a = a⋆, (4) (1) (2) (1) (2) and (5) give respectively a⋆θ = θ f β(S(a⋆ ))a⋆ , a⋆θα = θβf˜ α(S(a⋆ ))a⋆ ±α ±β ±α (1) (2) ± ± ±β (1) (2) a⋆θ = θ f β˙(S(a⋆ ))a⋆ , a⋆θα˙ = θβ˙f˜ α˙(S(a⋆ ))a⋆ . (13) ±α˙ ±β˙ ±α˙ (1) (2) ± ± ±β˙ (1) (2) But for a = M ρ, we have (f β(M δ))⋆ = (a∓1R±δβ)⋆ = a∓1R±β˙δ˙ = f β˙(S(M δ˙)) for a real. σ ±α σ 2 ασ 2 σ˙α˙ ∓α˙ σ˙ Therefore (12) and (13) are consistent if (θ )⋆ = θ and (θα)⋆ = θα˙ yielding ±α ∓α˙ ± ∓ (f β(a))⋆ = f β˙(S(a⋆)), (f˜ β(a))⋆ = f˜ β˙(S(a⋆)), ±α ∓α˙ ±α ∓α˙ (f β˙(a))⋆ = f β(S(a⋆)) and (f˜ β˙(a))⋆ = f˜ β(S(a⋆)). (14) ±α˙ ∓α ±α˙ ∓α Finally, by using the spinor metric to raise the indices of the right invariant basis into (2) and (3), we may also show that f β = εβδf˜ γε and εβ˙δ˙f γ˙ε = f˜ β˙. (15) ±α ±δ γα ±δ˙ γ˙α˙ ±α˙ Inthisstage,wehavenoindicationontheexplicit formsoff α(M ρ˙)orf α˙(M ρ)tocontrolthe ±γ δ˙ ±γ˙ δ noncommutativity between undotted and dotted generators of the quantum SL(2,C) group. 4 To carry this point we assume either the generators M β commute with M β˙ or are controled α α˙ by the R matrices satisfying the properties of the quantum SU(2) group. In the following we assume the later possibility. To reflect the specific properties of the quantum SU(2) group, we have to add the unitarity conditiononthegeneratorsasM β˙ = S(M α)andM β = S−1(M α˙). Applying S onbothsides α˙ β α β˙ of the unimodularity condition, we may show that the unitarity condition yields ε = λεβ˙α˙ αβ with λλ⋆ = 1. In the following we take λ = −1. Then to be consistent with the quantum SU(2) group, the spinor metric must satisfy (ε )⋆ = ε = −εαβ and (εαβ)⋆ = εβ˙α˙ = −ε . (16) αβ β˙α˙ αβ It is easy to see, by using the unitarity condition of the generators into (1), that θ = θα and α˙ θ = θα˙ implying conditions on the functionals α f β˙ = f˜ α , f˜ β˙ = f α (17) ±α˙ ±β ±α˙ ±β As stated above, we assume that the functionals of the quantum SL(2,C) group satisfy the same properties as the SU(2) ones. Therefore, if for example we set a = M ρ˙ into the left σ˙ relation of (2), we get M γM δ˙f β(M ρ˙) = f γ(M δ˙)M ρ˙M β or M γM δ˙R±ρ˙β = R±δ˙γ M ρ˙M β (18) α σ˙ ±γ δ˙ ±α σ˙ δ˙ γ α σ˙ γδ˙ ασ˙ δ˙ γ where f γ(M δ˙) = R±δ˙γ = f γ(S(M σ)) = a±1R∓γσ. we have also objects of the form ±α σ˙ ασ˙ ±α δ 2 δα f γ(S(M δ˙)) = R∓γδ˙ = f γ(S(S(M σ))) = ε f γ(S(M λ))ενσ = ±α σ˙ σ˙α ±α δ δλ ±α ν a±12εδλR∓νγαλενσ = εσ˙ρ˙f±αγ(Mγ˙ρ˙)εγ˙δ˙ = εσ˙ρ˙R±ρα˙γλ˙ελ˙δ˙. From (6) and (7), we get respectively f γ(M δ˙)f β(S(M ρ˙)) = δβδρ˙ = R±δ˙γ R∓βρ˙ (19) ±α σ˙ ±γ δ˙ α σ˙ ασ˙ δ˙γ f γ(S(M δ˙))f β(M ρ˙) = δβδρ˙ = R∓γδ˙ R±ρ˙β (20) ±α σ˙ ±γ δ˙ α σ˙ σ˙α γδ˙ 3 The quantum Lorentz group To have a correspondence between SL(2,C) and Lorentz quantum groups, we must construct the quantum analogues of the Pauli matrices. Let us consider an element X as a tensor αβ˙ product of an undotted and dotted elements of right invariant basis of the bimodule A-Γ (bispinor). X can always be expanded on a system of four independent 2 × 2 matrices αβ˙ σI (I = 0,..,3) as X σI . X tranforms as αβ˙ I αβ˙ αβ˙ ∆ (X ) = M σM ρ˙ ⊗X , ∆ (X ) = X ⊗I. (21) L αβ˙ α β˙ σρ˙ R αβ˙ αβ˙ 5 Then, we have Proposition (3,1): a) There exist four 2 ×2 matrices σIα˙β given by ± σIα˙β = εα˙λ˙R∓σρ˙ενβσI (22) ± λ˙ν σρ˙ such that X σIα˙β = X α˙β transforms under the left coaction as I ± ± ∆ (X α˙β) = S−1(M α˙)S(M β)⊗X σ˙ρ. L ± σ˙ ρ ± b) σIα˙β are hermitean iff σI are. In this case X are real. ± αβ˙ I Proof.: a) Setting into (21) M σM ρ˙ = R±γ˙δM µ˙M νR∓σρ˙, obtained by multiplying from α β˙ αβ˙ γ˙ δ µ˙ν the right both sides of (18) by R∓ξτ˙ and by using (19), we obtain ρ˙β ∆ (X ) = R±γ˙δM µ˙M νR∓σρ˙ ⊗X . L αβ˙ αβ˙ γ˙ δ µ˙ν σρ˙ Multiplying from the left both sides by R∓αβ˙ and using (20), we deduce λ˙ν ∆ (X R∓αβ˙σI ) = M µ˙M τ ⊗X R∓σρ˙σI = L I λ˙ν αβ˙ λ˙ ν I µ˙τ σρ˙ ε S−1(M γ˙)εδ˙µ˙ετξS(M κ)ε ⊗X R∓σρ˙σI λ˙γ˙ δ˙ ξ κν I µ˙τ σρ˙ yielding ∆ (X εα˙λ˙R∓σρ˙ενβσI ) = S−1(M α˙)S(M β)⊗X εδ˙λ˙R∓σρ˙ενγσI , L I λ˙ν σρ˙ δ˙ γ I λ˙ν σρ˙ which can be written under the form ∆ (X α˙β) = ∆ (X σIα˙β) = S−1(M α˙)S(M β) ⊗ X δ˙γ L ± L I ± δ˙ γ ± with σIα˙β = εα˙λ˙R∓σρ˙ενβσI ± λ˙ν σρ˙ from which we obtain ε σIα˙βε = R∓σρ˙σI . Multiplying from the right both sides by R±λ˙ν λ˙α˙ ± βν λ˙ν σρ˙ γτ˙ and using (19), we get σI = ε R±λ˙νε σIγ˙µ. (23) αβ˙ λ˙γ˙ αβ˙ µν ± b) Under theconditions (16)onthespinor metric we have (R±αρ)⋆ = R±σν, forareal, implying σν αρ (R∓αλ˙βν˙)⋆ = (a±21εβρR∓ασρνεσλ)⋆ = a±12εβρR∓σανρεσλ. 6 In the other hand, by using (11) and (15), we obtain a±21εβρR∓σανρεσλ = f∓αν(εβρMρσεσλ) = f∓αν(S−1(Mλβ)) = f˜∓αν(Mλβ) = εασf∓δσ(Mλβ)εδν = a±21εασR∓βδλσεδν = R∓βν˙λα˙ yielding (σIα˙β)⋆ = εβ˙ν˙R∓ρσ˙ ελα(σI )⋆. Therefore, if (σI )⋆ = σI then ± ν˙λ σρ˙ σρ˙ ρσ˙ (σIα˙β)⋆ = εβ˙ν˙R∓ρσ˙ ελασI = σIβ˙α. ± ν˙λ ρσ˙ ± The same procedure can be applyied to (23) to show the converse. It is now easy to see by applying the ⋆ involution on both sides of (21) that (X )⋆ = X implying X σI = (X σI )⋆ = αβ˙ βα˙ I βα˙ I αβ˙ (X )⋆σI which shows that X are real. Q.E.D. I βα˙ I To define a quantum metric of the space M spanned by X , we have to define an ade- I quate trace [10-11] over the spinorial indices which makes invariant this metric under quantum SL(2,C) group. Proposition (3,2): M is endowed with a metric GIJ given by 1 1 1 1 G IJ = Tr(σIσJ) = εανσI σJβ˙γε = Tr(σI σJ) = ε σIγ˙ασJ εν˙β˙ (24) ± Q ± Q αβ˙ ± γν Q ± Q ν˙γ˙ ± αβ˙ such that a) G IJX X are invariant under quantum SL(2,C) group. ± I J b) G IJ are hermitean if the matrices σI are. ± αβ˙ Proof: a) To show the invariance of G under the quantum SL(2,C) group, we consider the norm of X as G IJX X = 1εανX X β˙γε which transforms under SL(2,C) as ± I J Q αβ˙ ± γν 1 ∆ (G IJX X ) = εανM σM ρ˙S−1(M β˙)S(M γ)ε ⊗X X δ˙λ = L ± I J Q α β˙ δ˙ λ γν σρ˙ ± 1 1 εανM σS(M γ)ε ⊗X X δ˙λ = εανM σε M ρεµγε ⊗X X δ˙λ = Q α λ γν σδ˙ ± Q α λρ µ γν σδ˙ ± 1 1 I ⊗ εσρX X δ˙λε = I ⊗ εσρσI σJδ˙λε X X = I ⊗G IJX X . Q σδ˙ ± λρ Q σδ˙ ± λρ I J ± I J The same computation may be applyed to show that ε X γ˙αX εν˙β˙ is invariant under quan- ν˙γ˙ ± αβ˙ tum SL(2,C) group. Now, using (22) and the form of the R matrices, we obtain 1 1 G±IJ = QεαξσIαβ˙εβ˙λ˙R∓σλ˙ρν˙ενγσJσρ˙εγξ = Qa±12εαξσIαβ˙εβ˙λ˙ερδR∓σµδνεµλενγσJσρ˙εγξ = 7 1 Qa±21εαξεβ˙λ˙ερδ(δµσδξδ +a∓1εσδεµξ)εµλσIαβ˙σJσρ˙ = 1 −Qa±12εαξεβ˙λ˙ερδ(δµσδξδ +a∓1εσδεµξ)εdotλµ˙σIαβ˙σJσρ˙ = 1 −Qa±21εαξερδ(δµσδξδ +a∓1εσδεµξ)σIαµ˙σJσρ˙ = 1 Q(a±12σIξµ˙σJµξ˙ −a∓21σIξξ˙σJδδ˙) (25) where we have used (16) in the third and fifth line and the σI indices are raised and lowered as for the basis of the bicovariant A−Γ bimodule (σIα = σI ερα, σI β˙ = εβ˙ρ˙σI etc,..). From a β˙ ρβ˙ α αρ˙ similar computation we can show that Tr(σI σJ) gives the same form (25) for G IJ. ± ± b) if σI are hermitean, we have from (16) and the proposition(3,1) αβ˙ 1 1 (G IJ)⋆ = (εανσI σJβ˙γε )⋆ = ε σJγ˙βσI εν˙α˙ = G JI. ± Q αβ˙ ± γν Q ν˙γ˙ ± βα˙ ± which shows that the metric G IJ is hermitean. Q.E.D. ± Note that if σI are hermitean, then as well as X as its norm GIJX X are real. αβ˙ I ± I J Now, we can give an explicit example where we take σI the usual four matrices which are the 2×2 identity matrix σ0 and the three Pauli matrices σi (i = 1,2,3) as: αβ˙ αβ˙ 1 0 0 1 0 −i 1 0 σ0 = , σ1 = , σ2 = , σ3 = . αβ˙ 0 1 αβ˙ 1 0 αβ˙ i 0 αβ˙ 0 −1 ! ! ! ! For the spinorial metric satisfying (16), we may take the most general form as ir −q−1 ir q−1 εαβ = d−12 q21 ir 2 ! , εαβ = d−21 −q12 ir2 ! and −ir q1 −ir −q1 εα˙β˙ = d−21 −q−21 −i2r ! , εα˙β˙ = d−21 q−12 −ir2 ! where q and r 6= ±1 are real, d = −r2 +1 and Q = a+a−1 = d−1(2r +q +q−1) = −ε εαβ. αβ With this choice, the computer MAPLE program gives metrics G IJ of the form ± −A 0 2A rd−1(Q(21)) A d−1Q (1) (1) Q (1) (−)  0 A −iA d−1Q 2iA rd−1(Q(12))  (2) (2) (−) (2) Q  2A rd−1(Q(21)) iA d−1Q A −4A r2d−2((Q(12))2) −2A rd−2Q Q   (1) Q (2) (−) (2) (3) Q (2) (−) (1)   A(1)d−1Q(−) −2iA(2)rd−1(QQ(21)) −2A(2)rd−2Q(−)Q(12) A(2) −A(3)d−2(Q(−))2Q2    8 where A = a∓3, A = a±1, A = a∓1, Q = q1 +q−1 and Q = q−q−1. The inverse (1) 2 (2) 2 (3) 2 (1) 2 2 (−) Q 2 G is given by ±IJ −A Q2 + d−1A (Q )2(Q−2)Q 0 rd−1A Q Q d−1A Q Q2 (4) 4 4 (2) (1) 2 (3) (1) 4 (3) (−)  0 2 Q2A id−1A Q 2Q2 −ird−1A Q Q  4 (3) 4 (3) (−) 2 (3) (1)  rd−1A Q Q −id−1A Q Q2 Q2A 2 0   2 (3) (1) 4 (3) (−) 4 (3)   d−1A Q 2Q2 ird−1A Q Q 0 Q2A   4 (3) (−) 2 (3) (1) 4 (3)   2    where A = a±3. In the classical limit r = 0 and q = 1, these metrics reduce to the (4) 2 classical Minkowsky metric with signature (−,+,+,+). From the computer MAPLE program, we obtain the completeness relations as σI σ ρ˙σ = Qδσε ερ˙δ˙ , σ σIρ˙σ = Qδρ˙ε εδσ (26) αβ˙ I α β˙δ˙ ±Iαβ˙ ± β˙ δα or σI β˙σ σ = Qδσδβ˙ , σ α σI ρ˙ = Qδαδρ˙ (27) α Iρ˙ α ρ˙ ±I β˙ ±σ σ β˙ where σ = G σJ . Note that from the computer MAPLE program we can check that ±Iαβ˙ ±IJ αβ˙ σ α˙β = G σJα˙β = G σJα˙β = G σJα˙β and σJα˙βG = σJα˙βG . J +IJ + −IJ − ±IJ ± + +JI − −JI Remark(3,1) : – The metric G can be written as G = G G GKL = G G 1Tr(σKσL) = ±IJ ±IJ ±IL ±JK ± ±IL ±JKQ ± 1Tr(σ σ ) = G G 1Tr(σKσL) = 1Tr(σ σ ). With a way quite analogous that was Q ±J I ±IL ±JKQ ± Q J ±I used to derive (25), we may show that 1 G±IJ = Q(a∓21σIβ˙δσJδ˙β −a±21σIδ˙δσJβ˙β) (28) – The completeness relations (27) may be used to convert a vector to a bispinor and vice versa 1 1 X = X σI ⇔ X = εανX σJβ˙δε G or X = ε σ β˙αX εν˙δ˙. αβ˙ I αβ˙ I Q αβ˙ ± δν ±JI I Q ν˙β˙ I αδ˙ We are now ready to show how two copies of undotted and dotted generators of A may be combined to form generators Λ K of an unital algebra L corresponding to the quantum Lorentz L group. Theorem (3,1): The generators Λ K (L,K = 0,1,2,3) of the Quantum Lorentz group are given by L 1 1 Λ K = ε σ δ˙αM σσK M ρ˙εγ˙β˙ = σ αM σσK ρ˙S−1(M γ˙) (29) L Q γ˙δ˙ L α σρ˙ β˙ Q Lγ˙ α σ ρ˙ 9 or 1 Λ K = εαδM σσK M ρ˙σNβ˙γε G . (30) L Q α σρ˙ β˙ ± γδ ±NL They are real and satisfy the axiomatic structure of Hopf algebras with the relations G Λ NΛ M = G I and G LKΛ NΛ M = G NMI (31) ±NM L K ±LK L ± L K ± L where I = I is the unity of L ⊂ A. L A Proof: Multiplying from the left both sides of (21) by σNδ˙α and making the trace over dotted indices, we get ∆ X Tr(σNδ˙ασL ) = ∆ X QGNL = ε σNδ˙αM σσK M ρ˙εγ˙β˙ ⊗X L L ± αβ˙ L L ± γ˙δ˙ ± α σρ˙ β˙ K yielding 1 ∆ X = ε σ δ˙αM σσKM ρ˙εγ˙β˙ ⊗X = Λ K ⊗X . (32) L L Q γ˙δ˙ L α σρ˙ β˙ K L K We can also multiply from the right both sides of (21) by σNβ˙γ and take trace over undotted ± indices to have (30). To show that (29) is equal to (30), we note that 1 εαδσNβ˙γε G = εαδσNβ˙γε ε σ µ˙ρσ εν˙τ˙ = ± γδ ±NL Q ± γδ ν˙µ˙ L ±Nρτ˙ εαδδβ˙ε εσγε ε σ µ˙ρεν˙τ˙ = ε σ µ˙αεν˙β˙ (33) τ˙ σρ γδ ν˙µ˙ L ν˙µ˙ L where we have used G = Tr(σ σ ) and the completeness relation (27). Substituting this ±NL L ±N equality in (30), we retrieve (29). The reality of the generators is obtained by noticing that (σNβ˙γG )⋆ = G σNγ˙β = σ γ˙β ± ±NL ±LN ± L from which, we get 1 1 (Λ K)⋆ = ( εαδM σσK M ρ˙σNβ˙γε G )⋆ = ε σ γ˙βM ρσK M σ˙εδ˙α˙ = Λ K L Q α σρ˙ β˙ ± γδ ±NL Q δ˙γ˙ L β ρσ˙ α˙ L due to the fact that σI are hermitean. The Hopf structure of the algebra generated by Λ K σρ˙ L is given by: a) Acting the coaction on both sides of (29), we obtain 1 ∆(Λ K) = σ αM δS−1(M γ˙)⊗M σσK ρ˙S−1(M µ˙)δλδν˙. L Q Lγ˙ α ν˙ λ σ ρ˙ δ µ˙ From the completeness relation (27), we deduce 1 1 ∆(Λ K) = σ αM δσI ν˙S−1(M γ˙)⊗ σ δM σσK ρ˙S−1(M µ˙) = Λ I ⊗Λ K. (34) L Q Lγ˙ α δ ν˙ Q Iµ˙ λ σ ρ˙ L I 10