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REAL FORMS OF COMPLEX QUANTUM

arXiv:hep-th/9108018v1 23 Aug 1991REAL FORMS OF COMPLEX QUANTUMANTI-DE-SITTER ALGEBRA Uq(Sp(4; C))AND THEIR CONTRACTION SCHEMESJerzy Lukierski∗†, Anatol Nowicki∗‡and Henri Ruegg∗D´epartement de Physique Th´eorique,Universit´e de Gen`eve, 24 quai Ernest-Ansermet,1211 Gen`eve 4, SwitzerlandAbstractWe describe four types of inner involutions of the Cartan-Weyl basis pro-viding (for |q| = 1 and q real) three types of real quantum Lie algebras:Uq(O(3, 2)) (quantum D = 4 anti-de-Sitter), Uq(O(4, 1)) (quantum D = 4de-Sitter) and Uq(O(5)). We give also two types of inner involutions of theCartan-Chevalley basis of Uq(Sp(4; C)) which can not be extended to innerinvolutions of the Cartan-Weyl basis.

We outline twelve contraction schemesfor quantum D = 4 anti-de-Sitter algebra.All these contractions providefour commuting translation generators, but only two (one for |q| = 1, secondfor q real) lead to the quantum Poincar´e algebra with an undeformed spacerotations O(3) subalgebra.∗Partially supported by the Swiss National Science Foundation†On leave of absence from the Institute for Theoretical Physics, University of Wroc law, ul.Cybulskiego 36, 50205 Wroc law, Poland‡On leave of absence from the Institute of Physics, Pedagogical University, Plac S lowia´nski 6,65020 Zielona G´ora, Poland1

1IntroductionRecently the formalism of quantum groups and quantum Lie algebras1) (see e.g. [1-6]) has been applied to physical space-time symmetries.Several authors havelooked for q-deformations of D = 4 Lorentz group and D = 4 Lorentz algebra [7-11].

In other papers the contraction schemes have been used to obtain the quantumdeformation of semisimple Lie algebras describing Minkowski or Euclidean group ofmotions. In particular there were obtained:a) quantum deformations of D = 2 and D = 3 Euclidean and Minkowski geometries,described as quantum Lie algebra or as quantum group [12,13]b) quantum deformation of D = 4 Poincar´e algebra [14]Quite recently also several deformations of D = 2 supersymmetry algebra in itsEuclidean [15-17] as well as Minkowski [15,17] version were obtained.In our recent paper [14] we obtained the quantum deformation of D = 4 Poincar´ealgebra by contracting the Cartan-Weyl basis of a particular real form Uq(SO(3, 2))(|q| = 1) of the second rank quantum Lie algebra Uq(Sp(4; C)).

Performing thesecalculations we realized that the chosen scheme is not unique. We found that thereare at least two questions which should be answered in detail for the physical appli-cations of any quantum Lie algebra Uq(ˆg) of rank ≥2:a) The classification of involutions of the quantum Lie algebra (Uq(ˆg), ∆, S, E), where∆denotes comultiplication (∆: Uq(ˆg) →Uq(ˆg)⊗Uq(ˆg)), S - the antipode (”quantuminverse” ) (S : Uq(ˆg) →Uq(ˆg)) and ǫ is the counit (ǫ : Uq(ˆg) →C)2).

In principleone can introduce four types of involutions (see Sect. 2).b) The extension of inner involutions of Cartan-Chevalley basis to the full Lie al-gebra basis, described by the Cartan-Weyl generators.

It appears that for physicalapplications one should ”solve” the Serre relations, and study the reality conditionfor the deformed Lie algebra generators.In this paper we address these questions for the simple case of rank 2 quantum Liealgebra Uq(Sp(4)), and provide the answers. It appears that different real structuresimply different signatures as well as different restrictions on q (|q| = 1 or q real).

Inparticular in the case of Uq(O(3, 2)) the Cartan subalgebra of O(3, 2) can be chosenmaximally compact (O(2) ⊕O(2)) or maximally noncompact (O(1, 1) ⊕O(1, 1)),and two triples (ei, e−i, hi) (i = 1, 2) describing Cartan-Chevalley basis of Sp(4) fordifferent real structures describe different real forms of SL(2, C) (O(3) or O(2, 1)).If we follow the method ref. [14] and perform the contraction procedure for differentreal structures we obtain different deformations of the four-dimensional inhomoge-neous space-time algebras: the quantum Poincar´e algebras with Minkowski signature(3,1) or the ones describing four-dimensional quantum geometry with the signature(2,2).

Further, the ”physical” quantum Poincar´e algebra (with signature (3,1)) canbe deformed in two different ways:a) with undeformed O(3) subalgebra of D = 3 space rotations.b) with undeformed O(2, 1) subalgebra.The first case was obtained in our recent paper [14], with |q| = 1. If we consider2

all possible real structures, both cases a) and b) can be accompanied with the realityconditions |q| = 1 as well as q = ¯q (q real). As a consequence we conclude that inthe deformation formulae given in [14] for the quantum Poincar´e algebra also thepurely imaginary values of deformation parameter κ may appear.

We obtain thatmodulo some numerical coefficients, depending on the choice of the involution, thesinP0κ and cos P0κ terms can be replaced by sinh P0κ and cosh P0κ terms, providingagreement with earlier results for three - dimensional case [12,13].The plan of our paper is the following. In Sect.

2 we shall rewrite the Hopfbialgebra Uq(Sp(4; C)) in Cartan-Weyl basis, with comultiplication table, antipodes,counits and Serre relations replaced by bilinear algebraic relations. In Sect.

3 weshall consider two types of the Hopf algebra involutions, satisfying respectively theconditionsi)S((S(a∗))∗) = a ⇔S ◦∗= ∗◦S−1(1a)ii)S−1((S(a⊗))⊗) = a ⇔S ◦⊗= ⊗◦S(1b)We show that there are four different ways of introducing in total 16 real structures ofCartan-Weyl basis of Uq(Sp(4)). (eight of type 1a) and eight of type 1b)).

Expressingthe Cartan-Weyl generators of Uq(Sp(4)) as quantum deformation of O(3, 2) Liealgebra basis, one can relate the choice of real structures with the choice of signatureof real D = 5 orthogonal algebras O(5−k, k) (k = 0, 1, 2). The results are presentedin Table 1.For completeness we shall present also the involutions ∗s = S ◦∗which due to the relation (1a) satisfy ∗2s = 1.

They provide the examples of innerinvolutions of Cartan-Chevalley basis which however cannot be extended to Cartan-Weyl basis. In Sect.

4 we discuss the contraction schemes of different real formsof Uq(Sp(4)), which are obtained for 16 choices of real structures (12 for quantumanti-de-Sitter, 2 for quantum de-Sitter, and 2 for Uq(O(5)), which can be treated asquantum D = 4 Euclidean de-Sitter algebra). Finally in Sect.

5 we present generalremarks and comments.2Uq(Sp(4)) in Cartan-Weyl basisLet us recall that the Lie algebra C2 ≡Sp(4) can be described by the followingstandard choice of simple roots (see e.g. [19])α1 = 1√2, −1√2!α2 = (0,√2)(2.1)which leads to the following symmetrized Cartan matrix αij =< αi, αj > (i = 1, 2):α = 1−1−12(2.2)The Drinfeld-Jimbo q-deformation of Uq(Sp(4)) is described by the following defor-mation of Cartan-Chevalley basis (ei, e−i, hi), corresponding to simple roots (2.1) (3

[x] ≡(q −q−1)−1 · (qx −q−x))[ei, e−j] = δij[hi]q[hi, e±j] = ±αijej[hi, hj] = 0restricted also by the q-Serre relationse±α1he±α1 [e±α1, e±α2]q±1iq±1q±1 = 0he±α2 [e±α2, e±α1]q∓1iq∓1 = 0(2.4)where [eα, eβ]q = eαeβ −q−<α,β>eβeα The coproduct and antipodes are given by theformulae∆(hi) = hi ⊗1 + 1 ⊗hi∆(e±i) = e±i ⊗qhi2 + q−hi2 ⊗e±i(2.5)andS(hi) = −hiS(ei±) = −q± 12diei±(2.6)where di =< αi, αi >= (1, 2).In order to write the Cartan-Weyl basis one has to introduce the generators corre-sponding to nonsimple roots. (α1 + α2, 2α1 + α2).

Introducing the normal order ofroots (α1, α4 = 2α1 +α2, α3 = α1 +α2, α2) and using the following defining relationsfor nonsimple generators staying in normal order (α, α + β, β) [20,21]eα+β = eαeβ −q−<α,β>eβeαe−α−β = e−βe−α −q<α,β>e−αe−β(2.7)one obtains ([A, B]q ≡AB −qBA)e3 = [e1, e2]qe−3 = [e−2, e−1]q−1e4 = [e1, e3]e−4 = [e−3, e−1](2.8)The complete set of commutation relations for Uq(C2) in q-deformed Cartan-Weylbasis was given in [14] . We recall that[ei, e−j] = δij[hi]q[hi, hj] = 0[hi, e±j] = ±αije±j[e3, e−3] = [h3]qh3 = h1 + h2[e4, e−4] = [h4]qh4 = h1 + h3(2.9a))4

The set of bilinear relations equivalent to q-Serre relations takes the form:[e1, e4]q−1 = 0,[e−2, e−3]q = 0[e4, e3]q−1 = 0,[e−3, e−4]q = 0[e3, e2]q−1 = 0,[e−4, e−1]q = 0(2.9b)The formulae for coproduct and antipode for the generators (2.8) look as follows:∆(e3) = e3 ⊗q12h3 + q−12 h3 ⊗e3 + (q−1 −q) q−12 h2e1 ⊗e2q12 h1∆(e−3) = e−3 ⊗q12 h3 + q−12h3 ⊗e−3 + (q −q−1) q−12 h1e−2 ⊗e−1q12h1(2.10a)∆(e4) = e4 ⊗q12h4 + q−12 h4 ⊗e4 + (q −q−1) {(1 −q−1)·q−12h2e21 ⊗e2qh1 −q−12 h3e1 ⊗e3q12 h1o∆(e−4) = e−4 ⊗q12h4 + q−12 h4 ⊗e4 + (q −q−1) {(q −1)·q−h1e−2 ⊗e2−1q12h2 + q−12h1e−3 ⊗e−1q12h3o(2.10b)andS(e3) = −q12e3 + q12(1 −q2)e1e2S(e−3) = −q−12e−3 + q−12 (1 −q−2) e−2e−1S(e4) = −qe4 + (1 −q2) {(q −1)e21e2 + e1e3}S(e−4) = −q−1e−4 + (1 −q−2)n(q−1 −1) e−2e2−1 + e−3e−1o(2.11)The description of quantum algebra in Cartan-Weyl basis is an alternative descrip-tion to the standard one, given by (2.3 - 2.6). There are two advantages of thisapproach:a) The nonlinear Serre constraints are replaces by bilinear relations (see (2.9b))b) In the limit q →1 one obtains the ordinary Lie algebra relations.In comparison with [14] we shall propose here more general relations between the”root” generators ea, e−a, hi, (a = 1·; i = 1, 2) and the ”physical” rotation generatorsMAB = −MBA (A, B = 0, 1, 2, 3, 4, ), allowing for the q-dependent rescaling of the”root” generatorse±a →E±a = C±a(q)e±a(2.12)where C±(1) = 1.In fact a nontrivial choice of the numerical coefficients C±ais necessary if we wish to obtain the reality conditions for MAB from the innerinvolution formulae for the Cartan-Chevalley basis, satisfying the condition (1.1a).5

3Real forms of Uq(Sp(4))Let us consider the Hopf algebra A over C with comultiplication ∆and antipodeS. We shall distinguish the following four types of involutive homomorphisms in A:i) The + - involution, which is an anti-automorphism in the algebra sector, andautomorphism in the coalgebra sector, i.e.

(ai ∈A):(a1 · a2)+ = a+2 a+1(∆(a))+ = ∆(a+)(3.1)ii) The ∗-involution, which is an automorphism in the algebra sector, and anantiautomorphism in the coalgebra sector, i.e. (a1 · a2)∗= a∗1a∗2(∆(a))∗= ∆′(a∗)(3.2)where ∆′ = τ∆= R∆R−1 (τ-flip automorphism in A ⊗A).We assume further that the bialgebra involutions i), ii) satisfy the relation (1a).iii) The ⊕- involution, which is an antiautomorphism in both algebra and coal-gebra sectors, i.e.

(a1 · a2)⊕= (a2)⊕(a1)⊕(∆(a))⊕= ∆′ a⊕(3.3)iv) The O* - involution, which is an automorphism in both algebra and coalgebrasectors, i.e. (a1 · a2)O* = (a1)O* · (a2)O*(∆(a))O* = ∆aO*(3.4)We assume for the bialgebra involutions iii), iv) the relation (1b).

We would liketo recall that the standard real structure of the Hopf algebra A is given by theinvolution i) (see e.g. [2]).Now we shall present 16 involutions of Uq(Sp(4)) -four in every category i) - iv),distinguished by two parameters λ and ǫ (we choose further ǫ = ±1, λ = ±1) - whichare inner on Cartan - Weyl basis(k = q12h):i) + - involutions (|q| = 1)k+i = ki⇔h+i = −hie+±1 = λe±1e+±2 = ǫe±2e+±3 = −λǫq∓1e±3e+±4 = ǫq∓1e±4(3.5)We define the rescaled generators E±3, E±4 by introducing in the formula (2.12)C±3(q) = C±4(q) = q∓12(3.6)satisfying the relationsE+±3 = −λǫE±3E+±4 = ǫE±46

We define the rotation generators as follows:M12 = h1M34 =1√2(E−3 −E3)M23 =1√2 (e1 + e−1)M24 = −12 (e2 + e−2 + E4 + E−4)M31 =1√2 (e1 −e−1)M14 = −12 (e2 −e−2 −E4 + E−4)M04 = h3M03 =1√2 (E3 + E−3)M02 = 12 (e2 −e−2 + E4 −E−4)M01 = 12 (e2 + e−2 −E4 −E−4)(3.8)giving for ǫ = λ = −1 the conditionMAB = −M+AB(3.9)withA, B = 0, 1, 2, 3, 4and for q = 1[MAB, MCD] = gBCMAD + gADMBC −gACMBD −gBDMAC(3.10)whereǫ = λ = −1 :gAB = diag(−+ −+ +)(3.11a)By proper choice of ”i” factors in the formulae (3.8) one can achieve the condition(3.9) also for other values of ǫ and λ. One gets then the relations (3.10) with thefollowing set of metrics:λ = 1, ǫ = −1 :gAB = diag(−+ −−+)(3.11b)λ = −1, ǫ = 1 :gAB = diag(+ + −+ −)(3.11c)λ = 1, ǫ = 1 :gAB = diag (+ + −−−)(3.11d)The involution (3.5) with ǫ = λ = 1 corresponds in dual picture to the one pro-posed in [1] for the real form Spq(2n; R) of the complex quantum group Spq(2n; C).ii) ∗- involution (q real)k∗i = k−1i↔h∗i = −hie∗±1 = λe∓1e∗±2 = ǫe∓2e∗±3 = −λǫq±1e∓3e∗±4 = ǫq±1e∓4(3.12)7

If we rescale the generators e±3, e±4 using the scaling factor (3.6) and introduce insuitable way the rotation generators (3.8) with proper ”i-factors” , in order to satisfythe conditionMAB = M∗AB(3.13)one can show that the choices of λ and ǫ in (3.12) implies the following D = 5orthogonal metrics3)λ = −1, ǫ = −1 :gAB = diag(+ + + + +)(3.14a)λ = 1, ǫ = −1 :gAB = diag(+ + + −+)(3.14b)λ = −1, ǫ = 1 :gAB = diag(−+ + + −)(3.14c)λ = 1, ǫ = 1 :gAB = diag(−+ + −−)(3.14d)We see therefore that we obtained for six involutions (3.11a-d) and (3.14c-d) theO(3, 2) metric, for the choice (3.14b) - the O(4, 1) metric, and for (3.14a) - the O(5)metric. All eight involutions (3.5) and (3.12) satisfy the condition (1.1a).iii) ⊕- involution (|q| = 1)k⊕i = k−1i↔h⊕i = hie⊕±1 = λe∓1e⊕±2 = ǫe∓2e⊕±3 = λǫe∓3e⊕±4 = ǫe∓4(3.15)In the case of the involutions (3.15) the rescaling of ”root” generators is notneeded.

If one choose ”i - factors” in the formula (3.8) properly, the relations (3.15)will imply the reality conditionsM⊕AB = −MAB(3.16)and the following choices of the metric in the formula ( 3.10):λ = 1, ǫ = 1 :gAB = diag(+, +, +, +, +)(3.17a)λ = 1, ǫ = −1 :gAB = diag(−, +, +, +, −)(3.17b)λ = −1, ǫ = 1 :gAB = diag(−, −, −, +, −)(3.17c)λ = −1, ǫ = −1 :gAB = diag(+, −, −, +, +)(3.17d)We would like to mention here that in our previous paper [14] we considered the con-traction scheme for the involution (3.17b); the involution (3.17a) was also presentedin [14].8

iv) O*- involution (qreal)kO*l = ki ⇔hO*i = hieO*±1 = λe±1eO*±2 = ǫe±2eO*±3 = λǫe±3eO*±4 = ǫe±4(3.18)The choice of the involutions (3.18) implies for real generators.MO*AB = MAB(3.19)the following choices of metric:λ = 1, ǫ = 1 :gAB = diag(−+ −+ +)λ = 1, ǫ = −1 :gAB = diag(+ + −+ −)λ = −1, ǫ = 1 :gAB = diag(−+ −−+)λ = −1, ǫ = −1 :gAB = diag(+ + −−−)(3.20)By considering the involutions (3.15) and (3.18) we see again that we obtained sixinvolutions providing O(3, 2) metric, one - the O(4, 1) metric, and one giving O(5)metric. All eight involutions (3.15) and (3.18) satisfy the condition (1.1b).In order to complete the discussion of involutions for Uq(Sp(4)) we should con-sider also two types of involutions, obtained by the composition of antipode mapand the involutions i), ii).

Using the formulae for the antipode one gets the followingtwo new involutions:v) ∗s = S · ∗Because the antipode S is an antiautomorphism of algebra as well as coalgebra,the involution ∗s is a standard real structure of Hopf bialgebra behaving as the + -involution (see i)), and satisfies the relations [1]:(a1a2)∗s = (a2)∗s · (a1)∗s(∆(a))∗s = ∆(a∗s)(3.21)On Uq(Sp(4)) the relations (3.21) imply that q should be real. Besides the involution∗s takes the following explicite form:k∗si = ki⇔h∗si = hie∗s±1 = −λq∓12e∓1e∗s±2 = −ǫq±1e∓2e∗s±3 = −λǫq∓12 ˜e∓3e∗s±4 = ǫq∓1˜e∓4(3.22)9

where ˜e±a = τe±a (a = 3, 4; τ −flip automorphism), i.e.˜e3 = [e2, e1]q˜e4 = [˜e3, e1]˜e−3 = [e−1, e−2]q−1˜e−4 = [e−1, ˜e−3](3.23)We easily see from the formula (3.22) that the involution ∗s does not have an ex-tension to the Cartan-Chevalley basis.vi) +s = S · +In similar way one can introduce for |q| = 1 behaving as the ∗-involution (seeii)). It is given by the following outer involution of the Cartan-Chevalley basis:k+si= k−1l⇔hi = h+sie+s±1 = −λq± 12e±1e+s±2 = −ǫq±1e±2e+s±3 = −λǫq± 12 ˜e±3e+s±4 = ǫq±1˜e±4(3.24)4Different contraction schemesIn our previous paper [14] we have chosen the ⊕-involution (3.15) with the choiceof the parameters λ = 1, ǫ = −1 (see 3.17b), providing the real quantum Lie alge-bra Uq(O(3, 2)).

In this paper we have obtained twelve real quantum Lie algebrasUq(O(3, 2)) (see table 1)In particular Uq(Sp(4)) as a second rank quantum algebra contains two quantumUq(Sl(i)(2, C)) subalgebras (i = 1, 2), with generators (ei, e−i, hi). Because we havechosen the assignment provided by the formulae (3.8) we obtain thatSL(1)(2; C) = (M12, M23, M31)(4.1a)SL(2)(2; C) = M04, 1√2 (M02 + M14) , 1√2 (M01 −M24)!

(4.1b)Our contraction is defined by rescaling the generators (µ, ν = 1, 2, 3, 4) [14]Pµ = 1RMµ0Mµν unchanged(4.2)the rescaling of the deformation parameter [12-14]lnq =1κRfor q realiκRfor |q| = 1(4.3)and performing the limit R →∞. Following [14] we obtain in this limit that10

12345678Root ma-TypealgebraCo-MetricsCartanNondeformedD = 4pings;ofalgebra(g00, g11, g22, g33, g44)subalgebrasubalgebrasubalgebravalues of qinvolutionD = 3(q = 1)∆± →∆±+anti-aut. (+ + −−−)NC ⊕NCO(2, 1)O(3, 1)aut.

(−+ −−−)NC ⊕NCO(2, 1)O(3, 1)|q| = 1(+ + −+ −)NC ⊕NCO(2, 1)O(2, 2)(−+ −+ +)NC ⊕NCO(2, 1)O(2, 2)∆± →∆∓anti-(+ + + + +)C ⊕CO(3)O(4)q real*aut.aut. (+ + + −+)C ⊕CO(2, 1)O(3, 1)(−+ + + −)C ⊕CO(3)O(3, 1)(−+ + −−)C ⊕CO(2, 1)O(2, 2)∆± →∆∓anti-anti-(+ + + + +)C ⊕CO(3)O(4)|q| = 1⊕aut.aut.

(+ + + −+)C ⊕CO(2, 1)O(3, 1)(+ −−−+)C ⊕CO(3)O(3, 1)(+ −−+ +)C ⊕CO(2, 1)O(2, 2)∆± →∆±(−+ −+ +)NC ⊕NCO(2, 1)O(3, 1)|q| realO*aut.aut. (+ + −+ −)NC ⊕NCO(2, 1)O(2, 2)(−+ −−+)NC ⊕NCO(2, 1)O(2, 2)(+ + −−−)NC ⊕NCO(2, 1)O(3, 1)Table 1: DESCRIPTION OF REAL STRUCTURES AND CONTRACTIONS OFUq(Sp(4; C)) (Aut.

≡automorphism, antiaut. ≡antiautomorphism, C - compactabelian (O(2)), NC - noncompact abelian (O(1, 1)); ∆+(∆−) - set of positive (neg-ative) roots)i) The quantum subalgebra Uq(SL(1)(2, C)) becomes a classical one, i.e.Uq(SL(1)(2; C))−→R→∞U(SL1(2; C)(4.4)ii) The generators Pµ = limR→∞Pµ commute[Pµ, Pν] = 0(4.5)however their primitive coproduct is modifiediii) If we put all Pµ = 0, we obtain the quantum deformation of six generators,describing the quantum counterpart of Lorentz algebra.

In table 1 we list thesealgebras in column 8 in the free limit κ →∞.The choice of a real form for Sp(4) leads to the choice of a real form of SL(1)(2, C).These real three - dimensional algebras are listed in table 1 in column 7, and due tothe relation (4.4) they are not deformed.We see from table 1 that out of 12 real quantum algebras Uq(O(3, 2)) only twoseem to be physically interesting - one considered in [14] and second one given by ∗- involution (3.12) with λ = −1 and ǫ = 1. Because the rescaling (3.6) for R →∞disappears, in the contraction limit we obtaina) The algebra considered in [14], with the generators satisfying the reality con-dition corresponding to the notion of Hermitean operators in quantum theoryb) The algebra from [14] with imaginary parameter κ and the reality condition forgenerators corresponding to the notion of real number (or real function) in classicaltheory.

In such a case we obtain the cos and sin deformation terms replaced by coshand sinh terms.11

We would like to recall that the conventional Poincar´e algebra is obtained in thelimit κ →∞.5Final RemarksThis paper has been written in a search for other contractions describing quantumdeformation of Poincar´e algebra. We keep the contraction scheme the same as pro-posed in [14] but we apply it to all possible real forms of quantum de-Sitter algebraUq(Sp(4)).

As a result we obtain two cases which are interesting as candidates forquantum deformation of the Poincar´e algebra: one corresponding to quantum, andsecond to classical realization of quantum algebra.In our paper we restricted ourselves to the involutions satisfying the conditions(1a-b) - describing two types of real structure of Hopf bialgebra over C. If theseconditions are not satisfied, we shall still get real bialgebras. It is possible that otherphysically interesting contractions are of this type.

Also it should be added that wedid not realize the involutions as adjoint operators on the representation spaces ofquantum algebra. In general one can say only that the involutions which are algebraautomorphisms are related with classical realizations, (involution = complex conju-gation), and the involutive antiautomorphisms are good candidates for the adjointoperation defined for quantum realizations (involution = Hermitean conjugation).Finally, we would like to stress that the contraction produces the deformationdescribed by a dimensionfull parameter, in our case mass-like parameter κ.

It wouldbe extremely interesting to find the role in real world for such a mass-like parameter.One of possible applications is the regularization of local fields by the introductionof large but finite value of κ. Unfortunately, in order to make such an idea moreconcrete firstly the κ - deformation of Minkowski geometry should be formulated.AcknowlegmentsOne of us (J.L.) would like to thank Dr. V. Dobrev for discussions.

Two of theauthors (J.L.) and (A.N.) would like to thank the University of Gen`eve for its warmhospitality.FOOTNOTES:1.

Following [1] we distinguish here the quantum group Gq as the q-deformationof the (algebra of functions on the) Lie group, and the quantum Lie algebra Uq(ˆg),described by the q-deformation of the universal enveloping algebra U(ˆg) of the Liealgebra.2. For the discussion of real forms of the complex quantum Lie algebra Uq(SL(2; c))see e.g.

[18]; for discussion of real forms of quantum groups see [1]. In these papersonly the antiautomorphisms of algebra and automorphisms of coalgebra are listed.3.

We recall that ∗-involution is a conjugation of algebra, not changing the order in12

a product of operators, and the condition (3.13) implies the reality of the metric in(3.10).References[1] L. Faddeev, N. Reshetikhin and L. Takhtajan, Algebra i Analiz, 1(1989),178;Engl. translation Leningrad Math.

Journ. 1, 193(1990)[2] L. Woronowicz, Comm.

Math. Phys.

111(1987)613, 122(1989)125[3] W.G. Drinfeld, ”Quantum groups”, Proceedings of International Congress ofMath., Berkeley, USA (1986), p. 703[4] M. Jimbo, Lett.

Math. Phys.

10( 1985), 63; ibid. 11(1986),247[5] Yu.T.

Manin, ”Quantum groups and noncommutative geometry”, Centre deRecherches Math. Univ.

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