Quantizations of \(D=3\) Lorentz symmetry

The European Physical Journal C, Apr 2017

Using the isomorphism \({\mathfrak {o}}(3;{\mathbb {C}})\simeq {\mathfrak {sl}}(2;{\mathbb {C}})\) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms \({\mathfrak {o}}(3)\) and \({\mathfrak {o}}(2,1)\) of the complex Lie algebra \({\mathfrak {o}}(3;{\mathbb {C}})\) in terms of real forms of \({\mathfrak {sl}}(2;{\mathbb {C}})\): \({\mathfrak {su}}(2)\), \({\mathfrak {su}}(1,1)\) and \({\mathfrak {sl}}(2;{\mathbb {R}})\). We prove that the \(D=3\) Lorentz symmetry \({\mathfrak {o}}(2,1)\simeq {\mathfrak {su}}(1,1)\simeq {\mathfrak {sl}}(2;{\mathbb {R}})\) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard \({\mathfrak {su}}(1,1)\) and \({\mathfrak {sl}}(2;{\mathbb {R}})\) q-analogs and by simple Jordanian \({\mathfrak {sl}}(2;{\mathbb {R}})\) twist deformation. These quantizations are presented in terms of the quantum Cartan–Weyl generators for the quantized algebras \({\mathfrak {su}}(1,1)\) and \({\mathfrak {sl}}(2;{\mathbb {R}})\) as well as in terms of quantum Cartesian generators for the quantized algebra \({\mathfrak {o}}(2,1)\). Finally, some applications of the deformed \(D=3\) Lorentz symmetry are mentioned.

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Quantizations of \(D=3\) Lorentz symmetry

Eur. Phys. J. C Quantizations of D = 3 Lorentz symmetry J. Lukierski 1 V. N. Tolstoy 0 1 0 Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics , Moscow 119991, Russian Federation 1 Institute for Theoretical Physics, University of Wroc?aw , pl. Maxa Borna 9, 50-205 Wroc?aw , Poland Using the isomorphism o(3; C) sl(2; C) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r -matrices) for real forms o(3) and o(2, 1) of the complex Lie algebra o(3; C) in terms of real forms of sl(2; C): su(2), su(1, 1) and sl(2; R). We prove that the D = 3 Lorentz symmetry o(2, 1) su(1, 1) sl(2; R) has three different Hopfalgebraic quantum deformations, which are expressed in the simplest way by two standard su(1, 1) and sl(2; R) qanalogs and by simple Jordanian sl(2; R) twist deformation. These quantizations are presented in terms of the quantum Cartan-Weyl generators for the quantized algebras su(1, 1) and sl(2; R) as well as in terms of quantum Cartesian generators for the quantized algebra o(2, 1). Finally, some applications of the deformed D = 3 Lorentz symmetry are mentioned. - The search for quantum gravity is linked with studies of noncommutative space-times and quantum deformations of space-time symmetries. The considerations of simple dynamical models in quantized gravitational background (see e.g. [1?3]) indicate that the presence of quantum gravity effects generates noncommutativity of space-time coordinates, and as well the Lie-algebraic space-time symmetries (e.g. Lorentz, Poincar?) are modified into quantum symmetries, described by noncocommutative Hopf algebras, named by Drinfeld quantum deformations or quantum group [4]. We recall that in relativistic theories the basic role is played by Lorentz symmetries and the Lorentz algebra, i.e. all aspects of their quantum deformations should be studied in a very detailed and careful way. For classifications, constructions and applications of quantum Hopf deformations of an universal enveloping algebra U (g) of a Lie algebra g, Lie bialgebras (g, ?) play an essential role (see e.g. [4?7]). Here the co-bracket ? is a linear skew-symmetric map g ? g ? g with the relations consistent with the Lie bracket in g: ?([x , y]) = [x ? 1 + 1 ? x , ?(y)] ? [y ? 1 + 1 ? y, ?(x )], (? ? id)?(x ) + cycle = 0 (1.1) for any x , y ? g. The first relation in (1.1) is a condition of the 1-cocycle and the second one is the co-Jacobi identity (see [4,7]). The Lie bialgebra (g, ?) is a correct infinitesimalization of the quantum Hopf deformation of U (g) and the operation ? is an infinitesimal part of the difference between a coproduct and an opposite coproduct ? in the Hopf algebra, ?(x ) = h?1( ? ? ) mod h where h is a deformation parameter. Any two Lie bialgebras (g, ?) and (g, ? ) are isomorphic (equivalent) if they are connected by a gautomorphism ? satisfying the condition for any x ? g. Of special interest here are the quasitriangle Lie bialgebras (g, ?(r)) := (g, ?, r ), where the co-bracket ?(r) is given by the classical r -matrix r ? g ? g as follows: It is easy to see from (1.2) and (1.3) that two quasitriangular Lie bialgebras (g, ?(r)) and (g, ?(r )) are isomorphic iff the classical r -matrices r and r are isomorphic, i.e. (? ? ?)r = r . Therefore for a classification of all non-equivalent quasitriangular Lie bialgebras (g, ?(r)) of the given Lie algebra g we need find all non-equivalent (nonisomorphic) classical r -matrices. Because non-equivalent quasitriangular Lie bialgebras uniquely determine non-equivalent quasitriangular quantum deformations (Hopf algebras) of U (g) (see [4,5]) therefore the classification of all non-equivalent quasitriangular Hopf algebras is reduced to the classification of the non-equivalent classical r -matrices. In this paper we investigate the quantum deformations of D = 3 Lorentz symmetry. Firstly, following [8], we obtain the complete classifications of the non-equivalent (nonisomorphic) classical r -matrices for complex Lie algebra sl(2; C) and its real forms su(2), su(1, 1) and sl(2; R) with the help of explicit formulas for the automorphisms of these Lie algebras in terms of the Cartan?Weyl bases. In the case of sl(2; C) there are two non-equivalent classical r -matrices ? standard and Jordanian ones. For su(2) algebra there is only the standard non-equivalent r -matrix. These results are well known. For the su(1, 1) case we obtained three non-equivalent r -matrices ? standard, quasi-standard and quasi-Jordanian ones. In the case of sl(2; R) we find also three non-equivalent r -matrices ? standard, quasi-standard and Jordanian ones. Then using isomorphisms o(2, 1) su(1, 1) sl(2; R) we express these r -matrices in terms of the Cartesian basis of the D = 3 Lorentz algebra o(2, 1) and we see that two systems with three r -matrices for su(1, 1) and sl(2; R) algebras coincides. Thus we obtain that the isomorphic Lie algebras su(1, 1) and sl(2; R) have the isomorphic systems of their quasitriangle Lie bialgebras. In the case of o(2, 1) we see that the D = 3 Lorentz algebra has two standard q-deformations and one Jordanian. These Hopf deformations are presented in explicit form in terms of the quantum Cartan?Weyl generators for the quantized universal enveloping algebras of su(1, 1) and sl(2; R) and also in the terms of the quantum Cartesian generators. It should be noted that the full list of the non-equivalent classical r -matrices for sl(2; R) and o(2, 1) Lie algebras has been obtained early by different methods [9,10] (see also [11?13]), however, the complete list of the non-equivalent Hopf quantizations for these Lie algebras has not been presented in the literature. Furthermore, there was put forward the incorrect hypothesis that the isomorphic Lie algebra su(1, 1) and sl(2; R) do not have any isomorphic quasitriangular Lie bialgebras (see [14]). The isomorphic Lie algebras o(2, 1), sl(2; R), su(1, 1) and their quantum deformations play very important role in physics as well as in mathematical considerations, so the structure of these deformations should be understood with full clarity. The o(2, 1) Lie algebra has been used as D = 1 conformal algebra describing basic symmetries in conformal classical and quantum mechanics [15]; in such a case the o(2, 1) algebra is realized as a nonlinear realization on the one-dimensional time axis [16,17] and can be extended to osp(1|2) describing D = 1 N = 2 supersymmetric conformal algebra [18]. In field-theoretic framework the o(2, 1) Lie algebra describes Lorentz symmetries of three-dimensional relativistic systems with planar d = 2 space sector, which are often discussed as simplified version of the four-dimensional relativistic case. Due to the isomorphism o(2, 2) o(2, 1) ? o(2, 1) our results can be also applied to the description of D = 3 AdS symmetries [19]. We recall that o(2, 2) symmetry has been employed in Chern?Simons formulation of D = 3 gravity [20?22], with Lorentzian signature and nonvanishing negative cosmological constant. Subsequently, the quantum deformations of D = 3 Chern?Simons theory have been used for the description of D = 3 quantum gravity as deformed D = 3 topological QFT [23,24]. Three-dimensional deformed space-time geometry is also a basis of historical Ponzano?Regge formulation of D = 3 quantum gravity [25], which was further developed into spin foam [26] and causal triangulation [27] approaches. In mathematics and mathematical physics the importance of o(2, 1) and its deformations follows also from the unique role of the o(2, 1) algebra as the lowest-dimensional rank one noncompact simple Lie algebra, endowed only with unitary infinite-dimensional representations. One can point out that the program of deformations of infinite-dimensional modules of quantum-deformed U (su(1, 1)) algebra has been initiated already more than 20 years ago (see e.g. [28,29]). The (2 + 1)-dimensional models are also important in the theory of classical and quantum integrable systems [30,31] with their symmetries described by Poisson?Lie groups in classical case and after quantization by quantum groups. In particular recently, using a sigma model formulation of (super)string actions (see e.g. [32]), the integrable deformations of string target (super)spaces were obtained by Yang?Baxter deformations [33?37] of the principal as well as coset sigma models with symmetries, which may contain Ad S2 o(2; 1) and Ad S3 o(2, 2) factors [38?40]. The plan of our paper is the following. In Sect. 2 we consider the complex Lie algebra o(3; C) and its all real forms: o(3) su(2) and o(2, 1) su(1, 1) sl(2; R). In Sect. 3 we classify all classical r -matrices for these real forms and in Sect. 4 we provide the explicit isomorphisms between the real su(1, 1), sl(2; R) and o(2, 1) bialgebras. In Sect. 5 all three Hopf-algebraic quantizations (explicit quantum deformations) of the real D = 3 Lorentz symmetry are presented in detail: quantized bases, coproducts and universal R-matrices are given. In Sect. 6 we present short summary and outlook. 2 Complex D = 3 Euclidean Lie algebra o(3; C) and its real forms We first recall different most popular bases of the complex D = 3 Euclidean Lie algebra o(3; C): metric, Cartesian and Cartan?Weyl bases (see [8]). The metric basis contains in its commutation relations an explicit metric, namely, the complex D = 3 Euclidean Lie algebra o(3; C) is generated by three Euclidean basis elements Li j = ?L ji ? o(3; C) (i, j = 1, 2, 3) satisfying the relations [Li j , Lkl ] = g jk Lil ? g jl Lik + gil L jk ? gik L jl , where gi j is the Euclidean metric: gi j = diag (1, 1, 1). The Euclidean algebra o(3; C), as a linear space, is a linear envelope of the basis {Li j } over C. The Cartesian (or physical) basis of o(3; C) is related with the generators Li j as follows: From (2.1) and (2.2) we get If we consider a Lie algebra over R with the commutation relations (2.3) then we get the compact real form o(3) := o(3; R) with the anti-Hermitian basis Ii? = ?Ii (i = 1, 2, 3) for o(3). The real form o(2, 1) is given by the formulas Ii? = (?1)i?1 Ii (i = 1, 2, 3) for o(2, 1). For the description of quantum deformations and in particular for the classification of classical r -matrices of the complex Euclidean algebra o(3; C) and its real forms o(3) and o(2, 1) it is convenient to use the Cartan?Weyl (CW) basis of the isomorphic complex Lie algebra sl(2; C) and its real forms su(2), sl(1, 1) and sl(2, R). In the case of o(3) the su(2) Cartan?Weyl basis can be chosen as follows: H := ? I3, E? := ? I1 ? I2, E , E?] = 2H, [H, E?] = ?E?, [ + H ? = H, E ?? = E?, where the conjugation (?) is the same as in (2.4).1 For the real form o(2, 1) we will use two CW bases of sl(2; C) real forms: sl(1, 1) and sl(2, R). Such bases are given by H := ? I2, E? := ? I1 ? I3, [H, E?] = ?E?, [E+, E?] = 2H H := ? I3, E? := ? I1 ? I2, E , E?] = 2H [H , E?] = ?E?, [ + for su(1, 1), (2.7) for sl(2, R). (2.8) The two bases {E?, H } and {E?, H } have the same commutation relations but they have different reality properties, namely H ? = H, E ?? = ?E? H ? = ?H , E?? = ?E? for su(1, 1), 1 The basis elements E?, H over C with the defining relations in the second line of (2.6) generate the complex Lie algebra sl(2; C). The relations in the first line of (2.6) reproduce the isomorphism between o(3; C) and sl(2; C). where the conjugation (?) is the same as in (2.5).2 The relations between the su(1, 1) and su(2, R) bases look as follows: ? H = ? 2 (E+ ? E?), 3 Classical r-matrices of sl(2; C) and its real forms: su(2), su(1, 1) and sl(2; R) By definition any classical r -matrix of arbitrary complex or real Lie algebra g, r ? g?g, satisfy the classical Yang?Baxter equation (CYBE): [[r, r ]] = ? . Here [[?, ?]] is the Schouten bracket which for any monomial skew-symmetric two-tensors r1 = x ? y and r2 = u ? v (x , y, u, v ? g) is given by3 [[x ? y, u ? v]] := x ? ([y, u] ? v + u ? [y, v]) ? y ? ([x , u] ? v + u ? [x , v]) = [[u ? v, x ? y]], and ? is the g-invariant element which in the case of g := sl(2; C) looks as follows: (sl(2; C)) = ? (4E? ? H ? E+) where ? ? C, and E?, H is the CW basis of sl(2; C) with the defining relations on the second line of (2.6). Firstly we show that any two-tensor of sl(2; C) ? sl(2; C) is a classical sl(2; C) r -matrix. Indeed, let be an arbitrary element of sl(2; C) ? sl(2; C), where r+ := E+ ? H, r0 := E+ ? E?, r? := H ? E?, (3.5) are the basis elements of sl(2; C)?sl(2; C). Because all terms (3.5) are classical r -matrices, moreover, [[r?, r?]] = 0, as well as the Schouten brackets of the elements r? with r0 are also equal to zero, [[r?, r0]] = 0, we have [[r, r ]] = 2?+??[[r+, r?]] + ?02[[r0, r0]] = (?02 + ?+??) (4E? ? H ? E+) ? ? . Thus an arbitrary element (3.4) is a classical r -matrix, and if its coefficients ??, ?0 satisfy the condition ? := ?02 + + ? = 0 then it satisfies the homogeneous CYBE; if ? := ? ? 2 ?0 + ?+?? = 0 it satisfies the non-homogeneous CYBE. 2 It should be noted that in the case of su(1, 1) the Cartan generator H is compact, while for the case su(2, R) the generator H is noncompact. 3 For general polynomial (a sum of monomials) two-tensors r1 and r2 one can use the bilinearity of the Schouten bracket. We shall call the parameter ? = ?02 +?+?? in (3.6) the ? characteristic of the classical r -matrix (3.4). It is evident that the ? -characteristic of the classical r -matrix r is invariant under the sl(2; C)-automorphisms, i.e. any two r -matrices r and r , which are connected by a sl(2; C)-automorphism, have the same ? -characteristic, ? = ? . We can show also that any two sl(2; C) r -matrices r and r with the same ? characteristic can be connected by a sl(2; C)-automorphism. There are two types of explicit sl(2; C)-automorphisms which were presented in [8]. First type connecting the classical r -matrices with zero ? -characteristic is given by the formulas (see (3.15) in [8])4: ?0(E?) = ? ?1(??? E+ ? 2???0 H + ??+ E?), ?0(H ) = ??0 E+ + (???+ + ???) H + ???0 E?, where ? is a non-zero rescaling parameter (including ? = 1), ? takes two values +1 or ?1, and the parameters ??i (i = +, 0, ?) satisfy the conditions r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, where ?02 + ?+?? = 0 and ?02 + ?+?? = 0. Moreover, we suppose that the parameters ?? and ?? satisfy the additional relations: ??+ ? ?? = ??+ ? ? ?1??? = 0, where the parameters ? and ? are the same as in (3.7). One can check that the following formula is valid: ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E? = ?+?0(E+) ? ?0(H ) + ?0?0(E+) ? ?0(E?) where ?0 is the sl(2; C)-automorphism (3.7) with the following parameters: + = ? = ?0(??+ + ? ?1???) ? ?0(??+ + ??) , (??+ ? ??)(??+ ? ? ?1???) ?(??+ + ??)(??+ + ? ?1???) + 4?0?0 2(??+ ? ??)(??+ ? ? ?1???) (??+ + ??)(??+ + ? ?1???) + 4??0?0 2(??+ ? ??)(??+ ? ? ?1???) It is easy to check that as expected Eqs. (3.12) satisfy the conditions (3.8). Let us assume in (3.9), (3.11) and (3.12) that the parameters ?0 and ?? are equal to zero. Then the general classical r -matrix r in (3.9), satisfying the homogeneous CYBE, is reduced to usual Jordanian form by the automorphism (3.7) with the parameters: ??0 = ??+??0 ?? , ??? = ??+??? ?? . (3.13) Second type of sl(2; C)-automorphism connecting the classical r -matrices with non-zero ? -characteristic is given as follows:5 ?1(E?) = (??0 + 1) E+ + 2??? H ? ??0??+?2 1 E? , Let us consider two general r -matrices with non-zero ? characteristics: r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, r := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, where the parameters ??, ?0 and ??, ?0 can be equal to zero provided that ? = ?02 + ?+?? = ? = (?0)2 + ?+?? = 0, i.e. the two r -matrices r and r have the same non-zero ? characteristic ? = ? = 0. One can check the following relation: ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E? = ?+?1(E+) ? ?1(H ) + ?0?1(E+) ? ?0(E?) (?0 + ?0)2 ? (?+ ? ??+)(?? ? ? ?1??) , ??0 = (?0 + ?0)2 + (?+ ? ??+)(?? ? ? ?1??) ?? 2(?0 + ?0)(?? ? ? ?1??) . ? = (?0 + ?0)2 + (?+ ? ??+)(?? ? ? ?1??) It is easy to check that Eqs. (3.17) satisfy the condition ??02 + ??+??? = 1. If we assume in (3.15)?(3.17) that the parameters ?? are equal to zero then the general classical r -matrix r in (3.15), 4 Equations (3.7) are obtained from (3.15) in [8] by the substitution: ?0/(k?+ ? ??) = ?2??0, ??/(k?+ ? ??) = ???. 5 Equations (3.14) are obtained from (3.14) in [8] by the substitution ?0 = 2??0, ?? = ????, D = 4. satisfying the non-homogeneous CYBE, is reduced to the usual standard form by the automorphism (3.14) with the following parameters: where ?0 and ?0 are real numbers and we use the conditions (3.22). The r -matrices ri (i = 1, 2, 3) satisfy the nonhomogeneous CYBE Finally for sl(2, C) we obtain the following well-known result: For the complex Lie algebra sl(2, C) there exist up to sl(2, C) automorphisms two solutions of CYBE, namely Jordanian r J and standard rst : r J = ? E+ ? H, [[r J , r J ]] = 0, rst = ? E+ ? E?, [[rst , rst ]] = ? where the complex parameter ? in (3.19) can be removed by the rescaling automorphism: ?(E+) = ??1 E+, ?(E?) = ? E?, ?(H ) = H ; in (3.20) the parameter ? = e?? |? | for |?| ? ?2 is effective. The general non-reduced expression (3.4) is convenient for the application of reality conditions ? E? where is the conjugation associated with corresponding real form ( = ?, ?), and ?i? (i = +, 0, ?) means the complex conjugation of the number ?i . It should be noted that for any classical r -matrix r , r is again a classical r -matrix. Moreover, if r -matrix is anti-real (anti-Hermitian),6 i.e. it satisfies the condition (3.21), then its ? -characteristic is real. Indeed, applying the conjugation to the relation (3.6) we have for the left-side [[r, r ]] = ?[[r , r ]] = ?[[r, r ]] and for the right-side: (? ) = ?? ? for all real forms su(2), su(1, 1), su(2; R). It follows that the parameter ? is real, ? ? = ? . I. The compact real form su(2) (H ? = H , E ?? = E?). In this case it follows from (3.21) that 2 If in (3.4) ? = ?0 +?+?? = 0 then ?0?0? +?+?+? = 0 and it follows that ?0 = ?+ = ?? = 0, i.e. any classical r -matrix, which satisfies the homogeneous CYBE and the su(2) reality condition, is equal zero. If in (3.4) ? = ?02 + ?+?? = 0 we have three possible su(2) real classical r -matrices: 6 The anti-real property r = ?r is a direct consequence of the reality condition for the co-bracket ?(x) := [x ?1+1?x, r ], namely ?(x) = ?(x ) for ?x ? g (= {su(2), su(1, 1), sl(2, R)}). where all ?i (i = 1, 2, 3) are positive: ?1 = ?02 > 0, ?2 = ?+?+? > 0, ?3 = ?02 + ?+?+? > 0. Let the classical r -matrices (3.15) be su(2)-anti-real, i.e. their parameters satisfy the reality conditions (3.22). It follows that the functions (3.17) for ? = e?? have the same conjugation properties, i.e. ??0? = ??0, ???? = ???, and we see that the automorphism (3.14) with such parameters is su(2)real, i.e.: ?1(E?)? = ?1(E ??) = ?1(E?), ?1(H )? = ?1(H ?) = ?1(H ). We see that the r -matrices r2 and r3 in (3.23) can be reduced to the standard r -matrix rst := r1 using Eq. (3.16). It is easy to see that the standard r -matrix rst = r1 in (3.23) effectively depends only on positive values of the parameter ? := ?0. Indeed, we see that where ? is the simple su(2) automorphism: ?(E?) = E?, ?(H ) = ?H , i.e. any negative value of parameter ? in rst can be replaced by the positive one. We obtain the following result: For the compact real form su(2) there exists up to the su(2) automorphisms only one solution of CYBE and this solution is the usual standard classical r -matrix rst : rst := ? E+ ? E?, [[rst , rst ]] = ? , II. The non-compact real form su(1, 1) (H ? = H , E ?? = I?f??02=+??e+?????|?=0|, 0anidn w(3e.4h)atvheenth?e0f?o0?llo?w?in+g??+?-f=am0il,yi.oef. su(1, 1) homogeneous CYBE solutions: r? := ?0 ei? |??00| E+ ? H + E+ ? E? ? e?i? |??00| H ? E? , where ?0 is real. By using the su(1, 1)-real rescaling automorphism ?(E?) = (?? ei? |?0| )?1 E?, ?(H ) = H we can ?0 reduce the ?-family (3.29) to rq J := ?0(? E+ ? H + E+ ? E? + ? H ? E?): r? = ?0 e?? |??00| E+ ? H + E+ ? E? ? e??? |??00| H ? E? = ?0(? (?(E+) ? ?(E?)) ? ?(H ) + ?(E+) ? ?(E?)). (3.30) We shall call a su(1, 1)-real r -matrix ?quasi-Jordanian? if it cannot be reduced to Jordanian form by a su(1, 1)-real automorphism, but after complexification of su(1, 1) it can be reduced to Jordanian form by an appropriate complex sl(2, C)-automorphism. Thus all r -matrices in the ?-family (3.29) are quasi-Jordanian and they are connected with each other by the su(1, 1)-real rescaling automorphism. We take rq J as an representative of the ?-family. It is easy to see that the quasi-Jordanian r -matrix rq J effectively depends only on positive values of the parameter ?0, indeed, rq J = ?0(i E+ ? H + E+ ? E? + i H ? E?) = ??0(i ?(E+) ? ?(H ) + i ?(E+) ? ?(E?) + ?(H ) ? ?(E?)), where ? is the simple su(1, 1) automorphism ?(E?) = ?E?, ?(H ) = ?H , i.e. any negative value of parameter ?0 in rq J can be changed into a positive one. In the case ?02 + ?+?? = 0 in (3.4) we have four versions of su(1, 1)-real classical r -matrices. Two of them are characterized by positive value of ?i , (i = 1, 2): where ?0 and ?0 are real (see (3.28)), and ?1 = ?02 > 0, ?2 = ?0?0? ? ?+?+? > 0. The remaining two are with negative values of ?i , (i = 3, 4): where ?0 is real (see (3.28)), and ?3 = ??+?+? < 0, ?4 = ?0 ?0 ? ? ?+ ?+ ? < 0. Let the classical r -matrices (3.15) be su(1, 1)-anti-real, i.e. their parameters satisfy the reality conditions (3.28). In such case the functions (3.17) for ? = e?? have the same conjugation properties, i.e. ??0? = ??0, ???? = ????, and we see that the automorphism (3.14) with these parameters is su(1, 1)-real, i.e.: ?1(E?)? = ?1(E ??) = ??1(E?), ?1(H )? = ?1(H ?) = ?1(H ). It allows to reduce the r -matrix r2 to the standard r -matrix rst := r1 for ?1 = ?2 > 0 and the r -matrix r4 to the r matrix r3 for ?3 = ?4 < 0 by use of Eq. (3.16). By analogy to the notation of quasi-Jordanian r -matrix we shall call the r -matrices r3 and r4 as quasi-standard ones and take rqst := ?(E+ + E?) ? H as their representative.7 Finally for su(1, 1) we obtain the following results: For the non-compact real form su(1, 1) there exist up to su(1.1) automorphisms three solutions of CYBE, namely quasi-Jordanian rq J , standard rst and quasi-standard rqst : ? rq J = 2 (? (E+ ? E?) ? H + E+ ? E?), [[rq J , rq J ]] = 0, rst = ? E+ ? E?, [[rst , rst ]] = ?2 , rqst = ?(E+ + E?) ? H, [[rqst , rqst ]] = ??2 , where ? effectively is a positive number. III. The non-compact real form sl(2; R) (H ? = ?H , E?? = ?E?). In this case from (3.21) we obtain i.e. all parameters ?i (i = +, 0, ?) are purely imaginary. Consider the case ?02 + ?+?? = 0 in (3.4). We have three su(2; R) solutions of the homogeneous CYBE: r1 = ?+ E+ ? H , r2 = ?? H ? E?, r3 = ?+, E+ ? H + ?0 E+ ? E? + ?? H ? E?, (3.39) where all parameters ?i (i = +, ?), ?i (i = +, 0, ?) are purely imaginary, and ?02 + ?+?? = 0. If the classical r -matrices (3.9), where all generators H , E? are replaced by H , E?, are sl(2; R)-anti-real, i.e. their parameters satisfy the reality conditions (3.38), then for the real parameter ? all functions (3.12) are real, i.e. ??0? = ??0, ???? = ???. We see that the automorphism of the type (3.7) with such parameters is sl(2; R)-real, i.e.: It allows to reduce the r -matrices r2 and r3 in (3.39) to the Jordanian r -matrix r J := r1 by using Eq. (3.11). In the case ?02 +?+?? = 0 in (3.4) we have seven versions of sl(2; R)-real classical r -matrices. Five of them are with negative values of ?i , (i = 1, 2, . . . , 5): 7 The r -matrix rqst is connected with r3 (3.33) in the following way. Substituting ?+ = |?+|e?? in r3 (3.33) and using the su(1, 1)-real rescaling automorphism ?(E?) = e?i? E?, ?(H ) = H we obtain rqst with ? = |?+|. where all parameters ? are purely imaginary, and ?1 = ?2 = ?3 = ?02 < 0, ?4 = ?+?? < 0, ?5 = ?0 + ?+?? < 0; is the sl(2; R)-invariant element8: = (4E? ? H ? E+. The remaining two r -matrices ri (i = 6, 7) have positive values of ?i : r6 := ?+ E+ ? H + ?? H ? E?, r7 := ?+ E+ ? H + ?0 E+ ? E? + ?? H ? E?, (3.42) [[ri , ri ]] := ?i (i = 6, 7), where ?6 = ?+ ?? > 0 and ?7 = ?0 + ?+ ?? > 0. Let the classical r -matrices (3.15) be sl(2; R)-anti-real, i.e. with their parameters satisfying the reality conditions (3.38). In such a way the functions (3.17) for real ? are real, i.e. ??? = ??0, ???? = ???, and we see that the automorphism 0 (3.14) with such parameters is sl(2; R)-real. We can conclude that for the case of the negative ? -characteristics ?i < 0 (i = 1, . . . , 5) all r -matrices ri (i = 2, . . . , 5) in (3.41) are reduced to the standard formula rst := r1 and in the case of the positive ? -characteristics ?i > 0 (i = 6, 7) the classical r -matrix r7 in (3.42) is reduced to the quasi-standard r -matrix rqst := r6. Let us show that the r -matrix rqst effectively depends only on one positive parameter. Indeed, it is easy to see that rqst = E+ ? H + H ? E?? where ? is the sl(2, R)-real automorphism: ?(E?) = ?+?? is positive. ???? E?, ?(H ) = H , and ? = ?+ ?? Finally for sl(2, R) we obtain the following result: For the non-compact real form sl(2, R) there exist up to sl(2, R) automorphisms three solutions of CYBE, namely Jordanian r J , standard rst and quasi-standard rqst : r J = ? ? E+ ? H , [[r J , r J ]] = 0, rst = ? ? E+ ? E?, [[rst , rst ]] = ??2 , rqst = ? ? (E+ + E?) ? H , [[rqst , rqst ]] = ?2 4 Explicit isomorphism between su(1, 1) and sl(2; R) bialgebras and its application to o(2, 1) quantizations Using Eqs. (2.7) and (2.8) we express the triplets of the classical su(1, 1) and sl(2; R) r -matrices in terms of the o(2, 1) basis (2.3), (2.5). We get the following results. 8 Using (2.11) it is easy to check that (i) The su(1, 1) case: = ??(? I1 ? I2) ? I3, [[rq J , rq J ]] = 0, where the o(2, 1)-invariant element expressed in terms of the Cartesian basis (2.3) satisfying the reality condition (2.5) looks as follows: = ?8I1 ? I2 ? I3. (ii) The su(2; R) case: We see that the quasi-Jordanian r -matrix rq J in the su(1, 1) basis is the same as the Jordanian r -matrix r J in the sl(2; R) basis, and the standard r -matrix rst in the su(1, 1) basis becomes the quasi-standard r -matrix rqst in the sl(2; R) basis. Further, the quasi-standard r -matrix rqst in the su(1, 1) basis is the same as the standard r -matrix rst in the sl(2; R) basis. Equations (4.8)?(4.10) show that the su(1, 1) and sl(2; R) bialgebras are isomorphic. This result finally resolves the doubts about isomorphism of these two bialgebras (for example, see [14]). Using the isomorphisms of the su(1, 1) and sl(2; R) bialgebras we take as basic r -matrices for the D = 3 Lorentz algebra o(2, 1) the following ones: The first two r -matrices rst and rst with the effective positive parameter ? correspond to the q-analogs of su(1, 1) and sl(2; R) real algebras, the third r -matrix r J presents the Jordanian twist deformation of sl(2; R). In the next section we shall show how to quantize the r -matrices (4.11)?(4.13) in an explicit form. 5 Quantizations of the D = 3 Lorentz symmetry The q-analogs of the universal enveloping algebras U (g) for the real Lie algebras g = su(1, 1), sl(2; R) were already considered (see e.g. [7,28,29,41]) and they are given as follows. The quantum deformation (q-analog) of U (g) is an unital associative algebra Uq (g) with generators X?, q?X0 and the defining relations: q X0 q?X0 = q?X0 q X0 = 1, q ? q?1 with the reality conditions: (i) X ?? = ?X?, (q X0 )? = q X0 , q := e? for Uq (su(1, 1)), (ii) X ?? = ?X?, (q X0 )? = q X0 , q := e?? where ? is real in accordance with (4.11) and (4.12). A Hopf structure on Uq (g) (g = su(1, 1), sl(2; R)) is defined with the help of three additional operations: coproduct (comultiplication) q , antipode Sq and counit q : q (q?X0 ) = q?X0 ? q?X0 , q (X?) = X? ? q X0 + q?X0 ? X?, Sq (q?X0 ) = q?X0 , Sq (X?) = ?q?1 X?, q (q?X0 ) = 1, q (X?) = 0 with the reality conditions9: q? (X ) = q (X ?), Sq?(X ) = Sq?1(X ?), q?(X ) = q (X ?) for any X ? Uq (g). The quantum algebra Uq (g) is endowed also with the opposite Hopf structure: opposite coproduct ? q ,10 corresponding antipode S?q and counit ?q . 9 q? (X ) := ( q (X ))???. 10 The opposite (transformed) coproduct ? q (?) is a coproduct with permuted components, i.e. ? q (?) = ? ? q (?) where ? is the flip operator: ? ? X(1) ? X(2) = X(2) ? X(1). Rq q (X ) = ? q (X )Rq , ?X ? Uq (g), ( q ? id)Rq = Rq13 Rq23, (id ? Rq12 Rq13 R23 q q = Rq23 Rq13 R12 is called the universal R-matrix. Let Uq (b+) and Uq (b?) be quantum Borel subalgebras of Uq (g), generated by X+, q?X0 and X?, q?X0 , respectively. We denote by Tq (b+ ?b?) the Taylor extension of Uq (b+) ? Uq (b?).11 One can show (see [42,43]) that there exists a unique solution of equations (5.5) in the space Tq (b+ ? b?) and such a solution has the following form: Rq (g) := Rq = expq?2 ((q ? q?1)X+q?X0 ? q X0 X?)q2X0?X0 = q2X0?X0 expq?2 ((q ? q?1)X+q X0 ? q?X0 X?), where q = e? for Uq (su(1, 1)) and q = e?? for Uq (sl(2; R)). Here we use the standard definition of the q-exponential: expq (x ) := Analogously, there exists a unique solution of equations (5.5) in the space Tq (b? ? b+) = ? ? Tq (b+ ? b?) and such a solution is given by the formula = expq2 ((q?1 ? q)X?q?X0 ? q X0 X+)q?2X0?X0 = q?2X0?X0 expq2 ((q?1 ? q)X?q X0 ? q?X0 X+), where q satisfies the conditions (5.2). As formal Taylor series the solutions (5.7) and (5.9) are independent and they are related by It should be noted also that (Rq )?1 = Rq?1 , (Rq?)?1 = Rq??1 . 11 Tq (b+ ? b?) is an associative algebra generated by formal Taylor series of the monomials X +n ? X?m with coefficients which are rational functions of q?X0 , q?X0?X0 , provided that all values |n ? m| for each formal series are bounded, |n ? m| < N . From the explicit forms (5.7) and (5.9) we see that (Rq?)? = ? ? Rq? = (Rq )?1 for Uq (su(1, 1)), (Rq )? = (Rq )?1, (Rq?)? = (Rq?)?1 for Uq (sl(2; R)), i.e. in the case Uq (sl(2; R)) the two R-matrices Rq , Rq? are unitary and in the case Uq (su(1, 1)) they can be called ?flipHermitian? or ?? -Hermitian?. In the limit ? ? 0 (q ? 1) we obtain for the R-matrix (5.5) Here rB D is the classical Belavin?Drinfeld r -matrix: rB D = 2?(X+ ? X? + X0 ? X0), where ? = ?, X? = E?, X0 = H for the case g = su(1, 1), and ? = ? ?, X? = E?, X0 = H for the case g = sl(2; R). The r -matrix rB D is not skew-symmetric and it satisfies the standard CYBE [r B12D, r B13D + r B23D] + [r B13D, r B23D] = 0, which is obtained from QYBE (5.6) in the limit (5.13). The standard r -matrix (4.11) or (4.12) is the skew-symmetric part of rB D, namely where r?st = r B12D ? r B21D is the standard r -matrix (4.11) or (4.12) and C? 2 = 2?C2 = r B12D + r B21D where C2 is the split Casimir element of su(1, 1) or sl(2; R). We can introduce the quantum Cartesian generators by the formulas X? = ? J1 ? J3, q?X0 = q?? J2 .12 In terms of these generators the quantum algebra Uq (su(1, 1)), which will be denoted by U(rst )(o(2, 1)), can be reformulated as follows. The quantum deformation of U (o(2, 1)), corresponding to the classical r -matrix (4.11), is an unital associative algebra U(rst )(o(2, 1)) with the generators { J1, J3, q?? J2 } and the defining relations (k = 1, 3): q? J2 q?? J2 = q?? J2 q? J2 = 1, [ J1, J3] = ? (q2? J2 ? q?2? J2 ) 2(q ? q?1) with the reality condition J1? = J1, J3? = J3, (q?? J2 )? = q?? J2 , q? = q (q := e?, ? ? R). These relations are the q-analog of Eqs. (2.3) with the reality condition (2.5). The q ( Jk ) = Jk ? q? J2 + q?? J2 ? Jk , q (q?? J2 ) = q?? J2 ? q?? J2 , Sq (q?? J2 ) = q?? J2 , 1 ? Sq ( Jk ) = ? 2 (q + q?1) Jk + 2 (q ? q?1)?k2l Jl , q (q?? J2 ) = 1, q ( Jk ) = 0. Substituting in Eqs. (5.7) and (5.9) the expressions X? = ? J1 ? J3, q?X0 = q?? J2 we obtain the universal R-matrix in the terms of the quantum Cartesian generators Ji (i = 1, 2, 3) with the defining relations (5.17). We can also introduce other quantum Cartesian generators, by the formulas X? = ? J1 ? J2, q?X0 = q?? J3 .13 In terms of these generators the quantum algebra Uq (sl(2; R), which will be denoted by U(rst )(o(2, 1)), can be reformulated as follows. The quantum deformation of U (o(2, 1)), corresponding to the classical r -matrix (4.12), is an unital associative algebra U(rst )(o(2, 1)) with the generators { J1, J2, q?? J3 } and the defining relations (k = 1, 2): q? J3 q?? J3 = q?? J3 q? J3 = 1, [ J1, J2] = ? ? (q2? J3 ? q?2? J3 ) 2(q ? q?1) 1 ? q?? J3 Jk = 2 (q + q?1) Jk q?? J3 ? 2 (q ? q?1)?3kl Jl q?? J3 with the reality conditions J1? = J1, J2? = ? J2, (q? J3 )? = q? J3 , q? = q?1 (q := e??, ? ? R). The Hopf structure on U(rst )(o(2, 1)) are provided by the formulas (k = 1, 2): q ( Jk ) = Jk ? q? J3 + q?? J3 ? Jk , q (q?? J3 ) = q?? J3 ? q?? J3 , Sq (q?? J3 ) = q?? J3 , 1 ? Sq ( Jk ) = ? 2 (q + q?1) Jk + 2 (q ? q?1)?k3l Jl , q (q?? J3 ) = 1, q ( Jk ) = 0. Substituting in Eqs. (5.7) and (5.9) the expressions X? = ? J1 ? J2, q?X0 = q?? J3 we obtain the universal R-matrix in terms of the quantum physical generators Ji (i = 1, 2, 3) with the defining relations (5.19). The quantization of U (sl(2; R)) corresponding to the classical Jordanian r -matrix (4.13) has well been known for a long time [44?46] and it is defined by the twist F (see [45]): F = exp(H ? ? ), ? = ln(1 + ? ? E+). The two-tensor F satisfies the 2-cocycle condition ? id)(F ) = F 23(id ? 12 The generators Ji = (?1)i?1 Ji? (i = 1, 2, 3) are q-analogs of the Cartesian basis (2.3), (2.5) (limq?1 Ji ? Ii ). 13 The generators Ji (i = 1, 2, 3) are also the q-analog of the Cartesian basis given by (2.3), (2.5) (limq?1 Ji ? Ii ). and the ?unital? normalization ( ? id)(F ) = (id ? )(F ) = 1. It is evident that the twist (5.21) is unitary F ? = F ?1. The twisting element F defines a deformation of the universal enveloping algebra U (sl(2; R)) considered as a Hopf algebra. The new deformed coproduct and antipode are given as follows: (F)(X ) = F (X )F ?1, S(F)(X ) = u S(X )u?1 for any X ? U (sl(2; R)), where (X ) and S(X ) are the coproduct and the antipode before twisting: (X ) = X ? 1 + 1 ? X , S(X ) = ?X ; and u = m(id ? S)(F ) = exp(?? ? H E+). It is easy to see that we get the ?-Hopf algebra, i.e. ( (F)(X ))? = (F)(X ?), (S(F)(X ))? = S(F)(X ?) for any X ? U (sl(2; R)). One can calculate the following formulas for the deformed coproducts (F) (see [45]): Using (5.25) and (5.26) one gets the formulas for the deformed antipode S(F): S(F)(H ) = ?H e?? , S(F)(E+) = ?E+e?? , S(F)(E?) = ?E?e? + 2? ? H 2e? ? ?2 H (H ? 1)E+e? . (5.29) It is easy to see that the universal R-matrix R(F) for this twisted deformation looks as follows: R(F) = F? F ?1, (R(F))? = (R(F))?1. where r J is the classical Jordanian r -matrix (4.13). Using Eqs. (2.8) we can express all the formulas (5.28)?(5.30) in terms of the Cartesian basis (2.3) and (2.5). We add that the Jordanian deformation has been described as well in a deformed sl(2; R) algebra basis [47?49]. 6 Short summary and outlook By using the three-fold isomorphism of classical Lie algebras o(2, 1) sl(2; R) su(1, 1) one can express the infinitesimal versions of the D = 3 Lorentz quantum deformations in terms of classical o(2, 1), sl(2; R) and su(1, 1) r -matrices. The first aim of our paper was to derive o(2, 1), su(1, 1) and sl(2; R) bialgebras using a representation-independent purely algebraic method (see Sect. 3) and further to provide the explicit maps which relate them (see Sect. 4). We start in Sect. 3 with the derivation of a well-known pair of inequivalent complex o(3; C) sl(2; C) r -matrices ? the Jordanian (nonstandard) one and the Drinfeld?Jimbo (standard) r -matrix. Passing from sl(2; C) to sl(2; R) we obtain three independent sl(2; R) r -matrices. The first two of them are the real forms of two basic complex sl(2; C) r -matrices, the third sl(2; R) r -matrix, which we called quasi-standard (see (3.46)), is the sum of two skew-symmetric 2-tensors. We do not know, however, how to obtain directly the universal R-matrix from the third r -matrix. We show that there is, however, a way out: the quasi-standard r -matrix (3.46)) (see also (3.7)) can be transformed by the map (2.11) into the standard r -matrix in su(1, 1) basis, with a well-known universal R-matrix (see e.g. [7]). In such a way we can derive the effective quantization of all three D = 3 Lorentz r -matrices; however, we recall that for such a purpose it is necessary to use both sl(2; R) and su(1, 1) bases. In the second part of the Introduction we mentioned main applications of D = 3 Lorentz symmetries and their deformations, but still more important for the description of noncommutative D = 3 space-time geometry and D = 3 quantum gravity are the quantum deformations of the D = 3 Poincar? algebra, with a noncommutative translation sector. These quantum deformations were classified (see e.g. [50]) in terms of classical r -matrices, but systematic studies of their Hopf quantizations still should be completed. Also the quantum deformations were considered of D = 3 de Sitter (d S) and anti-de Sitter ( Ad S) space-times, with nonvanishing cosmological constant . In the D = 3 d S case ( > 0) all Hopf-algebraic quantizations are known, because they were studied as the quantum deformations of D = 4 Lorentz algebra o(3, 1) [51]. In D = 3 Ad S case ( < 0) with o(2, 2) symmetry some Hopf-algebraic quantum deformations were also given, but recently a complete classification was presented of real o(2, 2) r -matrices.14 For physical applications it is very important to consider subsequently the quantum space-time deformations for 14 See [8] and the addendum (to be published); for earlier efforts to describe o(2, 2) quantum deformations see e.g. [52]. We recall that o(D, D) algebras describe the symmetries of double geometry [53?55], which were used recently e.g. in the description of self-dual models in so-called metastring theory [56,57]. D = 4, 5, 6. The deformations of physical D = 4 space-time and D = 4 Poincar? algebra have been extensively studied for more than a quarter of the century [58?62], but it should be observed that the complete list of D = 4 Poincar? r -matrices ( D = 4 Poincar? bialgebras) has not been obtained.15 The next task could be to describe all deformations of D = 4 space-times with constant curvature and arbitrary signature, which would classify all possible D = 4 quantum d S and Ad S algebras as well as the quantum-deformed D = 5 Euclidean o(5) symmetries. For such a purpose one can look for the extension of algebraic methods used to classify the deformations of o(4; C) and its real forms (see [8]) to the case of o(5; C) and the real forms o(5), o(4, 1) and o(3, 2). Finally the systematic study of deformations of o(6; C) is another important challenge, in particular because the deformations of its real form o(4, 2) su(2, 2) will provide the list of quantum D = 4 conformal algebras. Acknowledgements The authors would like to thank A. Borowiec for discussions. JL would like to acknowledge the financial support of NCN (Polish National Science Center) Grants 2013/09/B/ST2/02205, 2014/13/B/ST2/04043 and by European Project COST, Action MP1405 QSPACE. VNT is supported by RFBR Grant No. 16-01-00562-a. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. 1. L. Freidel , E.R. Livine , Ponzano-Regge model revisited III: Feynman diagrams and effective field theory . Class. Quantum Gravity 23 , 2021 ( 2006 ). arXiv:hep-th/0502106v2 2. L. Freidel , E.R. 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J. Lukierski, V. N. Tolstoy. Quantizations of \(D=3\) Lorentz symmetry, The European Physical Journal C, 2017, 226, DOI: 10.1140/epjc/s10052-017-4786-9