Quantizations of D=3 Lorentz symmetry

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 deformations. 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 $\mathfrak{o}(2,1)$. Finaly, some applications of the deformed $D=3$ Lorentz symmetry are mentioned.


Introduction
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. [1,2]) 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, Poincaré) are modified into quantum symmetries, described by noncocommutative Hopf algebras, named after Drinfeld quantum deformations or quantum group [3]. We recall that in relativistic theories the basic role is played by Lorentz symmetries and Lorentz algebra, i.e. all aspects of their quantum deformations should be studied in 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, δ) play an essential role (see e.g. [3,4] and [5,6]). Here the cobracket δ is a linear skew-symmetric map g → g ∧ g with the relations consisted with the Lie bracket in g: δ([x, y]) = [x ⊗ 1 + 1 ⊗ x, δ(y)] − [y ⊗ 1 + 1 ⊗ y, δ(x)], (δ ⊗ id)δ(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 [3,6]). The Lie bialgebra (g, δ) is a correct infinitesimalisation of the quantum Hopf deformation of U(g) and the operation δ is a an infinitesimal part of difference between a coproduct ∆ and an oposite coproduct∆ in the Hopf algebra, δ(x) = h −1 (∆ −∆) mod h where h is a deformation parameter. Any two Lie bialgebras (g, δ) and (g, δ ′ ) are isomorphic (equivalent) if they are connected by a g-automorphism ϕ satisfying the condition for any x ∈ g. Of special our interest here are the quasitriangle Lie bialgebras (g, δ (r) ):=(g, δ, r), where the cobracket δ (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, δ (r) ) and (g, δ (r ′ ) ) are isomorphic iff the classical r-matrices r and r ′ are isomorphic, i.e. (ϕ ⊗ ϕ)r ′ = r. Therefore for a classification of all nonequivalent quasitriangular Lie bialgebras (g, δ (r) ) of the given Lie algebra g we need find all nonequivalent (nonisomorphic) classical r-matrices. Because nonequivalent quasitriangular Lie bialgebras uniquely determine non-equivalent quasitriangular quantum deformations (Hopf algebras) of U(g) (see [3,4]) therefore the classification of all nonequivalent quasitriangular Hopf algebras is reduced to the classification of the nonequivalent classical r-matrices.
In this paper we investigate the quantum deformations of D = 3 Lorentz symmetry. Firstly, following [7], we obtain the complete classifications of the nonequivalent (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 explicite 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 noniquivalent classical r-matrices -standard and Jordanian ones. For su (2) algebra there is only the standard nonequivalent r-matrix. These results are well known. For the su(1, 1) case we obtained three noneqvivalent r-matrices -standard, quasi-standard and quasi-Jordanian ones. In the case of sl(2; R) we find also three noneqvivalent 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 bealgebras. In the case of o(2, 1) we obtain that the D = 3 Lorentz algebra has two standard q-deformations and one Jordanian. These Hopf deformations are presented in explicite 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 noniquivalent classical r-matrices for sl(2; R) and o(2, 1) Lie algebras has been obtained early by different methods [8,9] (see also [10,11,12]), however the complete list of the nonequivalent Hopf quantisations for these Lie algebras has not been presented in the literature. Furthermore, there was put forward an incorrect hypothesis that the isomorphic Lie algebra su(1, 1) and sl(2; R) do not have any isomorphic quasitriangular Lie bialgebras (see [13]).
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 [14]; in such a case o(2, 1) algebra is realized as a nonlinear realization on the one-dimensional time axis [15,16] and can be extended to osp(1|2) describing D = 1 N = 2 supersymmetric conformal algebra [17]. 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 [18]. We recall that o(2, 2) symmetry has been employed in Chern-Simons formulation of D = 3 gravity [19,20,21], 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 [22,23]. Three-dimensional deformed space-time geometry is also a basis of historical Ponzano-Regge formulation of D = 3 quantum gravity [24], which was further developed into spin foam [25] and causal triangulation [26] 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 programm of deformations of infinite-dimensional modules of quantum-deformed U(su(1, 1)) algebra has been initiated already more than twenty years ago (see e.g. [27]). The (2 + 1)-dimensional models are also important in the theory of classical and quantum integrable systems [28,29] with their symmetries described by Poisson-Lie groups in classical case and after quantization by quantum groups. In particular recently, using sigma model formulation of (super)string actions (see e.g. [30]), there were introduced the integrable deformations of string target (super)spaces obtained by Yang-Baxter deformations [31]- [34] of the principal as well as coset sigma models with symmetries, which may contain AdS 2 ≃ o(2; 1) and AdS 3 ≃ o(2, 2) factors [35]- [37].
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 explicite isomorphisms between the real su(1, 1), sl(2; R) and o(2, 1) bialgebras. In Sect. 5 all three Hopfalgebraic quantizations (explicite 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 remind different most popular bases of the complex D = 3 Euclidean Lie algebra o(3; C): metric, Cartesian and Cartan-Weyl bases (see [7]). The metric basis contains in its commutation relations an explicite metric, namely, the complex D = 3 Euclidean Lie algebra o(3; C) is generated by three Euclidean basis elements where g ij is the Euclidean metric: g ij = diag (1, 1, 1). The Euclidean algebra o(3; C), as a linear space, is a linear envelope of the basis {L ij } over C.
The Cartesian (or physical) basis of o(3; C) is related with the generators L ij as follows From (2.1) and (2.2) we get 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 (2.11) 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): Here [[·, ·]] is the Schouten bracket which for any monomial skew-symmetric two-tensors r 1 = x ∧ y and andΩ is the g-invariant element which in the case of g := sl(2; C) looks as follows: where γ ∈ 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 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 elemets r ± with r 0 are also equal to zero, [[r ± , r 0 ]] = 0, and we have Thus an arbitrary element (3.4) is a classical r-matrix, and if its coefficients β ± , β 0 satisfy the condition γ := β 2 0 + β + β − = 0 then it satisfies the homogeneous CYBE, if γ := β 2 0 + β + β − = 0 it satisfies the non-homogeneous CYBE.
We shall call the parameter γ = β 2 0 + β + β − in (3.6) the γ-characteristic of the classical r-matrix (3.4). It is evident that the γ-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 γ-characteristic, γ = γ ′ . We can show also that any two sl(2; C) r-matrices r and r ′ with the same γ-characteristic can be connected by a sl(2; C)automorphism.
There are two types of explicite sl(2; C)-automorphisms which were presented in [7]. First type connecting the classical r-matrices with zero γ-characteristic is given by the formulas (see (3.15) in [7]) 4 : where χ is a non-zero rescaling parameter (including χ = 1), κ takes two values +1 or −1, and the parametersβ i (i = +, 0, −) satisfy the conditions: Let us consider two general r-matrices with zero γ-characteristics: where β 2 0 + β + β − = 0 and β Moreover, we suppose that the parameters β ± and β ′ ± satisfy the additional relations: where the parameters κ and χ are the same as in (3.7). One can check that the following formula is valid: where ϕ 0 is the sl(2; C)-automorphism (3.7) with the following parameters: It is easy to check that as expected the formulas (3.12) satisfy the conditions (3.8).
Let us assume in (3.9), (3.11) and (3.12) that the parameters β ′ 0 and β ′ − 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 authomorphism (3.7) with the parameters: Second type of sl(2; C)-automorphism connecting the classical r-matrices with non-zero γ-characteristic is given as follows 5 where χ is a non-zero rescaling parameter, andβ 2 0 +β +β− = 1. Let us consider two general r-matrices with non-zero γ-characteristics: where the parameters β ± , β 0 and β ′ ± , β ′ 0 can be equal to zero provided that γ = β 2 both r-matrices r and r ′ have the same non-zero γ-characteristic γ = γ ′ = 0. One can check the following relation: where ϕ 1 is the sl(2; C)-automorphism (3.14) with the parameters: .

(3.17)
It is easy to check that the formulas (3.17) satisfy the conditionβ 2 0 +β +β− = 1. If we assume in (3.15)-(3.17) that the parameters β ′ ± are equal to zero then the general classical r-matrix r in (3.15), satisfying the non-homogeneous CYBE, is reduced to the usual standard form by the automorphism (3.14) with the following parameters: Finally for sl(2, C) we obtain the well-known result: For the complex Lie algebra sl(2, C) there exists up to sl(2, C) automorphisms two solutions of CYBE, namely Jordanian r J and standard r st : where the complex parameter β in (3.19) can be removed by the rescaling automorphism: The general non-reduced expression (3.4) is convenient for the application of reality conditions: where is the conjugation associated with corresponding real form ( = * , †), and β * i (i = +, 0, −) means the complex conjugation of the number β 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 γ-characteristic is real. Indeed, applying the conjugation to the relation (3.6) we have for the left-side: r]] and for the right-side: (γΩ) = −γ * Ω for all real forms su(2), su(1, 1), su(2; R). It follows that the parameter γ is real, γ * = γ.
It is easy to see that the standard r-matrix r st = r 1 in (3.23) effectively depends only on positive values of the parameter α := β 0 . Indeed, we see that where ϕ is the simple su(2) automorphism: ϕ(E ± ) = E ∓ , ϕ(H) = −H, i.e. any negative value of parameter α in r st 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 r st : where the effective parameter α is a positive number, and γ = α 2 .

37)
where α effectively is a positive number.
Comparing the r-matrix expressions (4.1)-(4.3) with (4.5)-(4.7) we obtain that We see that the quasi-Jordanian r-matrix r qJ 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 r st in the su(1, 1) basis becomes the quasi-standard r-matrix r ′ qst in the sl(2; R) basis. Conversely, the quasi-standard r-matrix r qst in the su(1, 1) basis is the same as the standard r-matrix r ′ st in the sl(2; R) basis. The relations (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 [13]).
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 r st and r ′ st with the effective positive parameter α 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 explicite form.

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. [6,27,38]) and they are given as follows. The quantum deformation (q-analog) of U(g) is an unital associative algebra U q (g) with generators X ± , q ±X 0 and the defining relations: with the reality conditions: (1, 1)), (ii) X † ± = −X ± , (q X 0 ) † = q X 0 , q := e ıα for U q (sl(2; R)), where α is real in accordance with (4.11) and (4.12). A Hopf structure on U q (g) (g = su(1, 1), sl(2; R)) is defined with help of three additional operations: coproduct (comultiplication) ∆ q , antipode S q and counit ǫ q : with the reality conditions 9 : for any X ∈ U q (g). The quantum algebra U q (g) is endowed also with the opposite Hopf structure: opposite coproduct∆ q 10 , corresponding antipodeS q and counitǫ q . An invertible element R q := R q (g) which satisfies the relations: as well as, due to (5.5), the quantum Yang-Baxter equation (QYBE) is called the universal R-matrix. Let U q (b + ) and U q (b − ) be quantum Borel subalgebras of U q (g), generated by X + , q ±X 0 and X − , q ±X 0 respectively. We denote by 11 . One can show (see [39,40]) that there exists unique solution of equations (5.5) in the space T q (b + ⊗ b − ) and such solution has the following form where q = e α for U q (su(1, 1)) and q = e ıα for U q (sl(2; R)). Here we use the standard definition of the q-exponential: (n) q ! := (1) q (2) q . . . (n) q .
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 explicite forms (5.7) and (5.9) we see that 12) i.e. in the case U q (sl(2; R)) both R-matrices R ≻ q , R ≺ q are unitary and in the case U q (su(1, 1)) they can be called "flip-Hermitian" or "τ -Hermitian".
The quantization of U(sl(2; R)) corresponding to the classical Jordanian r-matrix (4.13) is well known for a long time [41,42,43] and it is defined by the twist F (see [42]): The two-tensor F satisfies the 2-cocycle condition (5.22) and the "unital" normalization It is evident that the twist (5.21) is unitary 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) = uS(X)u −1 (5.25) 13 The generators J ′ i (i = 1, 2, 3) are also the q-analoq of the Cartesian basis given by (2.3), (2.5) (lim q→1 J ′ i → I i ).

(5.29)
It is easy to see the universal R-matrix R (F ) for this twisted deformation looks as follows In the limit α → 0 we obtain for the R-matrix (5.23) where r J is the classical Jordanian r-matrix (4.13). Using the relations (2.6) 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 [44,45].

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 representation-independent purely algebraic method (see Sect. 3) and further to provide the explicite maps which relate them (see Sect. 4). We start in Sect. 3 with the derivation of 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. 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 (see Sect. 4): 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 known universal R-matrix (see e.g. [6]). 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 second part of 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 D = 3 Poincaré algebra, with noncommutative translations sector. These quantum deformations were classified (see e.g. [46]) in terms of classical r-matrices, but systematic studies of their Hopf quantizations still should be completed. There were considered also the quantum deformations of D = 3 de-Sitter (dS) and anti-de-Sitter (AdS) space-times, with nonvanishing cosmological constant Λ. In D = 3 dS case (Λ > 0) all Hopf-algebraic quantizations are known, because they were studied as the quantum deformations of D = 4 Lorentz algebra o(3, 1) [47]. In D = 3 AdS case (Λ < 0) with o(2, 2) symmetry some Hopf-algebraic quantum deformations were also given, but recently there was presented complete classification of real o(2, 2) r-matrices 14 .
For physical applications it is very important to consider subsequently the quantum spacetime deformations for D = 4, 5, 6. The deformations of physical D = 4 space-time and D = 4 Poincaré algebra were extensively studied for more than a quarter of the century [52]- [56], but it should be observed that the complete list of D = 4 Poincaré r-matrices (D = 4 Poincaré bialgebras) is still not complete 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 dS and AdS 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 [7]) 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 challege, 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.