Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic $\mathfrak{o}^{\star}(4)$ symmetries

We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong \mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$. For $\mathfrak{o}(3,1)$ we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra $\mathfrak{o}(4;\mathbb{C})$ we present new results.


Introduction
In recent years due to the efforts to construct models of quantum gravity characterized by non-commutative spacetime structures at Planckian distances [1]- [3], the ways in which one can deform the algebras of spacetime coordinates and space-time symmetries became important. In noncommutative description of spacetime the numerical coordinates are replaced by noncommutative algebra, which is consistent with new type of uncertainty relations between the pairs of operator-valued coordinates [2] called further DFR uncertainty relation relation. This extension of noncommutativity into the spacetime sector describes the limitations on spcetime localization measurements if the quantum gravitational background is present. It appears that during such procedure the high density of energy added by measurement leads to the creation of mini black holes, and one can show that below Planck distance λ P = 10 −33 m operationally the classical spacetime is not longer applicable. The quantum spacetime is effectively atomized, with lattice structure, and following the derivation in QM of Heisenberg algebra from Heisenberg uncertainty relations, one can deduce from DFR uncertainty relation the noncommutativity of quantum spacetime. Such noncommutative and/or discrete nature of quantum spacetime follows as well from loop quantum gravity (LQG) approach [4,5,6], where the discretization is dynamical 1 , based on the existence in LQG framework of minimal lengths, minimal area surfaces or minimal volume quanta. In particular recently by applying LQG techniques to the quantum deformation of D = 3 gravity with positive cosmological constant Λ it has been shown [7] that one gets the quantum symmetry of U q (o(4)), where ln q ∼ Λ, in analogy with earlier results of [8] and [9] for Lorentz signature.
In this paper we shall consider the noncommutative structures as linked with the quantum groups, which are described as non-cocommutative Hopf algebras [10,11,12,13,14,15,16]. The deformed spacetime algebra is described as the irreducible representation (noncommutative Hopf algebra module) of quantum rotations algebra, with semidirect (smash) product structure and build-in covariant action of quantum-deformed symmetry algebra on quantum noncommutative spacetime.
The aim of this paper is to provide the quantum Hopf algebras and universal R-matrices which are obtained by quantization of classical r−matrices classified recently in [17,18,19]. In this paper we shall describe the Hopf-algebraic deformations of any real four-dimensional rotational algebra given by the real form of o(4; C). Fortunately, all classical r−matrices presented in [17,19] can be quantized by providing explicit formulae for coproducts, antipodes and universal R-matrices. 2 We see therefore that the present paper provides the completion of the research program which we started in ref. [17,18,19].
The plan of our paper is the following: In Sect.2 we shall present some generalities on quantization of 'infinitesimal' versions of quantum deformations described by complex and real classical r-matrices, which provides the triangular and nontriangular cases and further we present reality conditions for the universal R-matrices. Further, in Sect. 3, we shall illustrate the quantization of classical r-matrices by the explicit presentation, for sl(2; C) case, of Jordanian and standard deformations.
In Sect. 4 we shall recall D = 4 complex Lie algebra o(4; C) and its all four real forms. In particular, besides three real forms o(4 − k, k) (k = 0, 1, 2) differing by the choice of signature, it should be added fourth quaternionic real form o(2; H) ≡ o(2, 1) ⊕ o(3) ≡ o * (4). We shall work mostly with the generators of four-dimensional complex rotations in Cartan-Weyl bases.
In Sect. 5 we quantize the full list of five classical r-matrices from [17,19], i.e. provide the complete list of all Hopf-algebraic deformations of U q (o(4; C)): two of them triangular, and remaining three quasitriangular. 3 In order to present the results in detail we shall calculate the coproducts, antipods and universal quantum R-matrices. Further we specify all real forms of the quantum deformations of o(4; C) described by ⋆-Hopf algebras. Following the standard recipe (see e.g. [22,23,24]) we assume that the ⋆-operation, defining respective real form, acts on tensor product (coproduct) in unflipped way (a ⊗ b) ⋆ = a ⋆ ⊗ b ⋆ . We add that all quantum deformations of Lorentz algebra were already obtained earlier by the present authors [25,26] and five out of eight Kleinian D = 4 real deformations can be obtained from the complex o(4; C) deformations listed in Sect.4 simply by replacing the complex sl(2; C) generators by the real ones describing sl(2; R) algebra. The Hopf-algebraic deformation of Euclidean o(4) algebra some o(2, 2) deformations and quaternionic o(2; H) case are the most important because the obtained results are new.
In Sect. 6 we shall present a brief outlook; the paper contains also two appendices. The quantum deformations of four-dimensional rotational symmetries presented in this paper can be applied at least in the following contexts: i) The deformed D = 4 rotation groups with the signature (+, +, −, −) (Kleinian case) describe the deformed D = 3 AdS symmetry and for Lorentz signature (+, −, −, −) the D = 3 dS quantum symmetries. If we introduce (A)dS radius Λ and the re-scaling of three rotations M 1k →M 1k = ΛP k (P k describes curved (A)dS momenta, k = 1, 2, 3), by suitable quantum Wigner-Inönü contraction [27,8] one can get various κ-deformed D = 3 Poincaré algebras.
ii) The knowledge of classical r−matrices permits to introduce explicitly the action of deformed (super)string models, described by so-called YB (Yang-Baxter) sigma models [28]- [36]. The quantum deformations presented in this paper can be applied to the description of the YB deformation of principal o(4 − k, k) σ−models (k = 0, 1, 2) as well as to the coset sigma models with noncommutative target space, described by the deformed cosets o(4−k,k) o(3−k,k) . It appears that such deformed o(4 − k, k) group or their coset manifolds can appear as parts of internal symmetry target spaces obtained by the reduction to D = 4 of deformed D = 10 Green-Schwarz superstrings.
iii) The classical r-matrices and their quantizations provide a powerful algebraic tool in description of integrable models and provide effective methods for studying their multihamiltonian systems [37,38,39]. In particular, the methods of noncommutative geometry permits to consider as well the Hamiltonian theories over the noncommutative rings [40,41] and their integrability conditions. iv) Eight quantum deformations of o(2, 2) presented in the paper provide the set of finite D = 2 quantum conformal algebras, with six generators, which in general case cannot be factorized into a sum of "'left"' and "'right"' (X ± = X 1 ± X 0 ) D = 1 quantum deformed conformal algebras. It is interesting to study which o(2, 2) deformations presented in the paper can be consistently extended to infinite-dimensional quantum groups, describing new classes of deformed D = 2 infinite-dimensional conformal Virasoro algebras. v) For various real forms of quantum-deformed o(4; C) groups one can obtain corresponding four-dimensional spacetime with different signatures (see e.g. [42]). With all Hopfalgebraic deformations of o(4; C) which will be presented in this paper one can obtain the complete list of quantum spacetimes with signatures (4, 0), (3,1) and (2,2).
Further remarks related with the applications of quantum deformations considered in this paper we shall present also in Sect. 6.
2 Quantizations of complex and real Lie algebras: general remarks 2.1 From classical r-matrices to quantum universal R-matrices It is known that formulated by Drinfeld [10] the quantization problem of Lie bialgebras has been answered by Etingof and Kazhdan [11]: to each Lie bialgebra one can associate a quantized enveloping algebra supplemented with Hopf algebra structure. Unfortunately, their proof is not constructive and the methods of explicit quantizations are known only in specific situations, as e.g. Drinfeld-Jimbo quantization of semi-simple Lie algebras and twist quantization in the triangular case (when twist tensor can be constructed explicitly). We shall show however that the known quantization techniques are sufficient for finding all explicit non-isomorphic quantizations of the enveloping algebra o(4; C) and their real form. Principal tool for the classification of quantum deformations is provided by the classical r-matrices [20,21,44] which determine coboundary Lie bialgebra 4 structures. Quantization procedure od bialgebras leads to the construction of quantum-deformed associative and coassociative Hopf-algebras [45] and determine the corresponding universal (quantum) R−matrices [13,14,22].
For semi-simple Lie algebras, due to the classical Whithead lemma, all bialgebras are coboundary. In such a case there is one-to-one correspondence between the Lie bialgebra structure and the corresponding classical r-matrix given as the skewsymmetric element r ∈ g ∧ g satisfying the classical (homogenous or inhomogenous) YB equation: (2. where r 12 = r (1) ⊗ r (2) ⊗ 1 ∈ g ⊗ g ⊗ g etc. For skew-symmetric 2-tensor monomials x ∧ y = x ⊗ y − y ⊗ x and u ∧ v (x, y, u, v ∈ g) the explicit formula for Schouten brackets reads 5 where the three-form Ω is the g-invariant element in g ∧ g ∧ g, i.e.
The complex Lie bialgebra is described by a pair g ≡ (g, r) consisting of complex Lie algebra g and skew-symmetric classical r-matrix r satisfying the equation (2.1). One can distinguish two cases (cf. e.g. [13,14,16]): A) If in (2.1) t = 0, one gets the so-called triangular or non-standard case with vanishing Schouten brackets describing homogenous classical Yang-Baxter equation (denoted as CYBE). In such a case the r-matrix can be rescaled arbitrarily without changing the corresponding bialgebra structure. The triangularity is preserved by Lie algebra homomorphisms and can be reduced to non-degenerate case on the Borel subalgebra. B) If t = 0, eq. (2.1) describes so-called non-triangular (quasitriangular) classical r-matrix, satisfying inhomogenous or modified classical Yang-Baxter equation (mCYBE). In such case one can introduce r BD ∈ g ⊗ g called Belavin-Drinfeld form of the r-matrix satisfying CYBE, such that r = r BD − r τ BD ((x ⊗ y) τ = y ⊗ x is the flip operation) and the symmetric element r s BD ≡ r BD + r τ BD which is ad-invariant. 6 In general, the initial skew-symmetric r-matrix is not scale invariant nor preserved by a Lie algebra homomorphisms. It is remarkable that Belavin-Drinfeld r-matrices for simple Lie algebras has been fully classified by means of socalled Belavin-Drinfel triples in [21].
Quantization of (complex) Lie bialgebra leads to quantum groups in Drinfeld sense [10] with the Hopf algebra structure supplementing the complex deformed universal enveloping algebra U(g) (in general one needs its topological ξ-adic extension U ξ (g) ≡ U(g) [[ξ]] formulated also for multiparameter deformation, i.e. ξ → (ξ 1 , . . . , ξ k ), see e.g. [10,13,14,15]). According to the cases A), B) indicated above, there are two ways of introducing quantum-enveloping algebra what will be described shortly below. Before we would like to focus our attention on the quantum universal R-matrix as an important byproduct of the quantization procedure. 7 The universal R-matrix is an invertible element of U ξ (g) ⊗ U ξ (g) which provides the flip τ of the noncocommutative coproduct τ : ∆ ξ → ∆ τ ξ given by the following similarity transfor- The universal R−matrix describes quantum group (see [45]) if it satisfies quasitriangularity conditions as well as the following normalization conditions where ǫ denotes a counit.
In fact, the same properties (2.5)-(2.8) are satisfied by another universal R-matrix, which is (R τ ) −1 . Therefore one can distinguish two case: i) the element Q R = RR τ = 1 is trivial ii) Q R = 1 is non-trivial It turns out that the first case corresponds to the triangular or twist quantization case while the second characterizes the non-triangular case. In order to describe their difference let us expand the R-matrix (2.5) in the powers of the deformation parameter ξ, entering linearly in the definition of classical r-matrix 8 From (2.9) and (2.7) it follows that the elementr ∈ g ⊗ g satisfies classical Yang-Baxter equation (CYBE) [13,14]. In triangular case one hasr +r τ = 0, i.e.r is skew-symmetric and can be identified with the classical r-matrix satisfying (2.1) with t = 0. In the second (non-triangular, see ii)) case,r is not skew-symmetric, satisfies CYBE and takes the Belavin-Drinfeld form of r-matrix, i.e.r = r BD . The classical r-matrices describe the infinitesimal version of quantum deformed Lie-algebraic symmetries; the quantum deformation parameterized by an arbitrary (formal) deformation parameter ξ determines Hopf-algebraic quantization and universal R−matrix .
In general case it is not known how to obtain the universal R-matrix from the solutions of (2.2); however for canonical Belavin-Drinfeld nontriangular r−matrices [20] the explicit formula for universal R−matrices is well-known (see e.g. [22]). It is worth noticing that in contrast to the triangular case, the non-triangular one provides two different quantum R-matrices: R(ξ) and R τ (ξ) −1 . The element Q(ξ) ≡ R τ (ξ)R(ξ) = 1 + (r + r τ ) + O(ξ 2 ) is called a quantum Killing form since its first order term, if not degenerate, defines a classical Cartan-Killing form on g. We would like to add that the skew-symmetric classical r-matrices are sufficient for classification 9 as well as for the description of correspondence with classical Lie-Poisson groups.

Reality conditions providing the quantizations of real bialgebras
We remind that a real Lie algebra structure (g, ) can be introduced by adding an antilinear involutive (Lie algebra) anti-automorphism : g → g ( -operation, conjugation) acting on the complex Lie algebra g. It relies on finding Lie algebra basis with real structure constants for which -operation is anti-Hermitian (i.e. x = −x) 10 . Subsequently, the real coboundary Lie bialgebra can be considered as a triple (g, ) ≡ (g, , r), where the skew-symmetric element r is assumed to be anti-Hermitian, i.e.
Such conditions lead to the suitable reality conditions for parameters, what is particularly important in the inhomogenous case (t = 0), due to the lack of scale invariance. The -operation extends, by the property (ab) = b a (i.e. as an antilinear antiautomorphism), to the enveloping algebra U(g), as well as to quantized enveloping algebra, making both of them associative -algebras. The real Hopf-algebraic structure represented on quantized enveloping algebra U q (g) by -involution is defined by the following conditions for coproducts and antipodes (see also [23,24]) where the -involution on the tensor product (2.11) acts as follows 11 One can get (2.5) and (2.11) compatible and consistently defined quasitriangular -Hopf algebras by imposing two distinct reality constraints on the universal R-matrix (see e.g. [14]): a) R ⊗ = R τ (R is called real); b) R ⊗ = R −1 and the corresponding quantum R-matrix is -unitary (R is called antireal).
Particularly, in the triangular case, due to the identity R τ = R −1 , the conditions a) and b) are the same. In non-triangular case (R τ = R −1 ), the second universal R-matrix (R τ ) −1 satisfies the same reality constraints 12 .
It should be noted that for any elementr ∈ g ⊗ g satisfying CYBE,r ⊗ satisfies again CYBE. Therefore, one can distinguish two cases: i) the classical r-matrix r =r corresponding to the universal R-matrix (cf. 2.9) is skewsymmetric; then r should be anti-Hermitian and satisfy the relation (2.10).
ii) if the elementr is not skew-symmetric this corresponds to the non-triangular case; theñ r ⊗ =r τ 13 for R real andr ⊗ = −r for R antireal. It is easy to check that in any case 10 In a case of Hilbert space realization this condition leads to operators with imaginary spectrum. For this reason some authors do prefer instead Hermitian generators and imaginary structure constants as representing real Lie algebras. 11 In other words a real Hopf algebra is identified with a ⋆-Hopf algebra which is a complex Hopf algebra equipped with an additional star operation making the algebraic sector into ⋆-algebra and coalgebraic sector satisfying (2.11). 12 It should be observed that the presence of these two universal R-matrices may help to obtain finite contraction limits (see e.g. [43]) 13 This condition can be rewritten asr τ ( ⊗ ) =r, where (a ⊗ b) τ ( ⊗ ) = b ⊗ a denotes so-called flipped conjugation (cf. [23,26] and formula (3.34) below). the skew-symmetric part ofr remains anti-Hermitian, i.e. (r −r τ ) ⊗ = −(r −r τ ).
It is easy to show that twisting of real form of quasitriangular Hopf algebra by unitary twist leads again to real quasitriangular Hopf algebra. More precisely, if (H, ∆, S, ǫ, R, ⋆) is a quasitriangular ⋆-Hopf algebra with R being real (resp. antireal) universal R-matrix, then for any unitary, normalized 2-cocycle twist F = (F −1 ) ⋆⊗⋆ ∈ H ⊗ H the quantized algebra (H, ∆ F , S F , ǫ, R F , ⋆) is a quasitriangular ⋆-Hopf algebra such that R F = F τ RF −1 is real (resp. antireal). This property will be used in the consideration of some cases of chain quantization of o(4; C) (see Sect. 5).
3 The basic sl(2; C) example: complex and real Lie bialgebra and their quantizations

Complex and real bialgebras
We recall that classical r-matrices, providing Lie bialgebra structure of a given Lie algebra as well as quantum deformations of the corresponding enveloping algebra, are classified up to the isomorphisms; in particular for real Lie algebras one should use the isomorphisms preserving reality condition. 14 Fixing the basis (structure constant) one deals with Lie algebra automorphisms. For simple Lie algebras these are (modulo discrete automorphisms) the internal automorphisms generated by the adjoint actions of the Lie algebra upon itself. The important example of such classification for real forms of o(4; C) has been investigated in [17,18,19]. It is well known that for the complex Lie algebra sl(2; C) ∼ = o(3; C) there exists up to sl(2, C) automorphisms two solutions of mCYBE 15 , namely Jordanian r J (triangular, called also non-standard) and the standard one r st (non-triangular): where we use the Cartan-Weyl (CW) basis In (3.13) the parmeter ξ can be replaced by ξ = 1 due to the scale invariance of CYBE. In the standard case (3.14) the non skew-symmetric counterpart of r st satisfies CYBE if it takes the Belavin-Drinfeld form Its symmetric part, described by sl(2; C) bilinear split Casimir is an invariant element in sl(2, C) ⊗ sl(2, C) and determines the Cartan-Killing form.
14 From now on all Lie algebras and bialgebras are real if not indicated otherwise. 15 There are only two orbit types under the action of o(3; C) in C 3 : null and non-null, see Appendix A.
In fact the formulae (3.25) can be used for the introduction of Cartesian basis in all classical o(2, 2) and o * (4) r-matrices containing the su(1, 1) and sl (2) sectors.
In the next subsection we shall describe explicitly the quantization of the complex bialgebras (3.13) and (3.14). In order to obtain the quantization of the bialgebras listed in (3.17) -(3.20) one should insert the generators (H, E ± ) satisfying the respective reality condition and impose the suitable restriction on the parameter γ. Standard deformation of simple Lie algebra is given by the explicit algorithm introduced firstly by Drinfeld and Jimbo. Non-standard quantum deformation of g ≡ (g, r), where r a skew symmetric solution of CYBE, is obtained by employing the 2-cocycle Drinfeld twist element F ∈ U(g) ⊗ U(g) which remains unchanged the algebra and modifies the coproduct ∆ and antipode S as follows (see e.g. [13,14]): where If classical enveloping Lie algebra U(g) is considered as a Hopf algebra H (0) = (U(g), m, ∆ (0) , S (0) , ǫ) then In order to get the coassociative coproduct one should postulate the normalized 2-cocycle condition for the invertible twist element F (see [10]) In H (0) one can introduce the universal (quantum) R-matrix by the formula which under the reality conditions (2.10) becomes unitary (at the same time real and antireal). More generally, twist deformation of quasitriangular Hopf algebra (H, R) give rise to quasitriangular Hopf algebra with new universal The simplest case one can deal is an Abelian twist where two primitive commuting elements [X, Y ] = 0 determines r A = χX ∧ Y the skewsymmetric solutions of CYBE. In fact, the same Abelian quantum R-matrix R A can be implemented by the one-parameter family of Abelian twists which are related with each other by a trivial (coboundary) twists . For example where W = exp( ξ 2 XY ). 18 Assuming X, Y real (antireal), i.e. X = ±X, Y = ±Y , the formal parameter χ has to be imaginary and all twist F A,s (χ) are unitary. Consequently, anyone can be used to deform equivalently -Hopf algebras. However, there is an advantage of using (3.31). In this case the element u (see (3.28)) reduces to the unit and the antipodes map remains unchanged.
Alternatively, by introducing the (non-standard) flipped conjugation on the tensor product (see [23]) The quantum twist F J corresponding to the classical Jordanian r-matrix r J is well known since a long time [47] 19 (3.35) 18 We remind that a coboundary twist for a given Hopf algebra is constructed out of any invertible element W according to the following prescription and leads via twisting (3.27) to the isomorphic Hopf algebras (see e.g. [46] ). 19 Here χ is not an effective deformation parameter. It is a formal variable which enables to write the twist as a formal power series and an invertible element.
The twisted coproducts and antipodes are easy to derive

Standard quantization of sl(2; C)
The standard (non-triangular) quantum deformation is corresponding to the solution of CYBE given by (3.16). It is described by q-analog or Drinfeld-Jimbo quantum deformation, with algebraic and coalgebraic sectors given by the following formulae ǫ q (q ±h ) = 1 , ǫ q (e ± ) = 0 . where we denote by (q h , e ± ) the q-deformed or quantum CW basis. 20 The quantum universal R-matrix satisfying QYBE (2.7) as well as the conditions (2.5)-(2.6) is given by the formula: 20 Non-standard, e.g. Jordanian, deformation can be also expressed with the use of nonclassical quantum Lie algebra generators [48] obtained from twist (3.35) (see also [49]).
where we use the standard definition of q-exponential exp q −2 (cf. Appendix B) The alternative second version of the universal R-matrix has the form: and provides nontrivial element Q q = R q R τ q . This quantum R-matrices describe by their linear terms, in the limit γ → 0, q → 1, non-skewsymmetric classical r-matrices in the Belavin-Drinfeld form (3.16).
Three standard real forms (3.17)-(3.19) impose the following reality conditions on qdeformed generators (q ±h , e ± ):  .48)). The last two (non-compact) real forms coincide in the classical limit γ → 0. In the classical limit γ → 0 deformed and undeformed generators can be identified, i.e. h → H, e ± → E ± , q h → 1. For the first two real forms the corresponding universal R-matrix (3.42) is real, the last case (3.19) is antireal. We recall that however if the Jordanian Lie bialgebra has no effective deformation parameter its quantization requires the introduction of a (formal) parameter χ, which permits to construct the twist and the quantum R-matrix as a formal power series, elements of U(sl(2; C)) ⊗ U(sl(2; C)) [[χ]]. In contrast, Lie bialgebras corresponding to standard deformations are parametrized by numerical (complex or real) factor γ, describing effective deformation parameter.  Due to the fact that each sl(2; C) sector has two bialgebra structures (single Jordanian and standard one-parameter family) one can easily to identify three (up to the flip) types of 21 It is known that complex metric (symmetric, nondegenerate and bilinear form) has no signature 22 For the relation with other, physically more meaningful, Cartesian basis see e.g. (3.21) and [17]. with the classical r-matrices obtained by summing up the pair of chiral and antichiral contributions, e.g. r J,J = r J +r J = H ∧ E + +H ∧Ē + in (4.4), etc. The list (4.2)-(4.4) does not exhaust all possible bialgebra structures because it does not take into account the mixed terms belonging to sl(2; C) ∧sl(2; C), which can also contribute to the classical r-matrices. In [17] using purely algebraic methods we classified all o(4; C) bialgebras. We found five families of complex skewsymmetric r-matrices: three, each with three-parameters, one two-parameter and one with one parameter.The list of o(4; C) r-matrices looks as follows [17]: 23 Here all parameters γ,γ, η, χ,χ, ς, ρ,ρ are arbitrary complex numbers and they are independent in different r-matrices. The first two r-matrices r I (χ) and r II (χ,χ, ς), generate twist and they satisfy the homogeneous CYBE (2.2). Moreover the first r-matrix r I (χ) is pure Jordanian type and the second r-matrix r II (χ,χ, ς) is the sum of two Jordanian ones with third one describing Abelian twist: r II (χ,χ, ς) = r II (χ, 0, 0)+r II (0,χ, 0)+r II (0, 0, ς). The third r-matrix r III (γ,γ, η) is the sum of two standard r-matrices and one Abelian: r III (γ,γ, η) = r III (γ, 0, 0) + r III (0,γ, 0) + r III (0, 0, η). The fourth r-matrix r IV (γ, χ ′ ) is the sum of special choice of the third rmatrix and the Abelian r-matrix: r IV (γ, ς) := r III (γ, −γ, −2γ) + ςE + ∧Ē + . The last rmatrices r V (γ,χ, ρ) is the sum of standard, Jordanian and Abelian r-matrices: r V (γ,χ, ρ) = r V (γ, 0, 0) + r V (0,χ, 0) + r V (0, 0, ρ). The formulae for (r II , r III , r V ) are obtained by supplementing (4.2) -(4.4) with particular additional Abelian contributions belonging to sl(2; C) ∧ sl(2; C).
We shall calculate as well in next Section for all five quantizations generated by (4.5)-(4.9) the universal R-matrices. Using formula (2.9) one obtains in third, fourth and fifth cases the following Belavin-Drinfeld type of matrices which appear in the expansion (2.9): for o ⋆ (4), (4.12) for o(2, 2), (4.13) 14) The last real form (4.14) characterizing the Lorentz o(3, 1)-algebra, does not preserve the chiral decomposition. By imposing all real involutions in the list of classical complex r-matrices (4.5)-(4.9) we get complete set of real bialgebra structures on the Lie algebra o(4; C). The list of all real bialgebras for o(4; C), together with specified values for the corresponding parameters, is presented in the table below, where o ⋆ (4), o ′ ⋆ (4) denotes the bialgebras after imposing the reality conditions (4.12), and o ′′ (2, 2), o ′ (2, 2), o ′′ (2, 2) denotes three bialgebras obtained by applying three reality conditions (4.13). In the following Section we shall describe the Hopf-algebraic quantization of five complex o(4; C) r-matrices (4.5)-(4.9). Out of these five complex quantizations after imposing seven reality conditions we obtain sixteen real o(4; C) Hopf algebra structures: r III provides seven real forms, r V -three, and each of remaining three leads to two real quantizations.
where k ∈ {0, 1} ≡ Z 2 and in order to reduce the number of formulae we denoted 24 Similarly, the formulae for the antipodes look as follows The universal quantum R-matrix takes the form (R = F 21 F −1 ) This simple one-parameter deformation admits two real quantum group structures (cf. Table 2) as indicated below. Since the twist is Jordanian, the reality conditions (2.11) are valid if the deformation parameter χ is imaginary. The Lorentzian case requiring as well imaginary χ has been already studied in [26] with more details. Table 2: Real quantizations of r I (χ) = χ(E + +Ē + ) ∧ (H +H) 24 The same convention will be further used below in the paper.

Left and right Jordanian quantizations intertwined by Abelian twist (r-matrix r II )
We see that for ς = 0 the r-matrix (4.6) describes two complex Jordanian r-matrices, each one for chiral sectors sl(2, C) and sl(2; C). They do commute with each other and can be quantized as the product of two Ogievetsky twists (k = 1, 2) ( [47] see also (3.35)) where Σ = ln(1 + χE + ),Σ = ln(1 +χĒ + ). The next step is to consider the Abelian part of the classical r-matrix r II belonging to sl(2; C) ∧sl(2; C) intertwining two chiral coalgebra sectors which ceases to be independent. Because the generators (H, E ± ) and (H,Ē ± ) do commute the twist function corresponding to (4.6) is given by the following formula: where the Abelian twist F A takes the form 25 which follows from the property that elements Σ,Σ are primitive after performing Jordanian deformation. We would like to mention here that the form of the twist function given above by formula (5.6) was proposed firstly by Kulish and Mudrov [50]. If we use (3.27-3.28), and (5.6) we obtain the following formulae for the coproducts of sl(2; C) ⊕sl(2; C) generators 25 The normalization ς χχ in the deformation parameter is necessary in order to recover correct formula in the limit χ,χ → 0.
Using further the relations (3.27-3.28) one obtains the following formulae for the antipodes We notice that the Abelian twist (5.7) does not contribute to the antipodes (5.9). The quantum universal R-matrix R 2 ≡ R 2 (χ,χ, ς) takes the form (5.10) The formulae (5.8)-(5.10) present the general three-parameter deformation which can be studied in various two-parameter limits. For example, if χ → 0 one should take into account that lim χ →0 Σ χ = E + , lim χ →0 Λ = 0 and lim χ →0 Λ χ = −E + . In this case the left chiral sector will be deformed only by Abelian twist. The case ς = 0 provides obviously the product of two independent Jordanian deformations.
In real cases the independence of parameters may be not valid. Only for the real o(2, 2) deformation all three parameters are imaginary and independent. In the Lorentzian case 27 two Jordanian parameters (χ,χ) are replace by one as follows from the condition χ = (χ) * in the table below.  Table 3) and the reality conditions (2.11) are satisfied.
This implies that the starting point for further considerations is a pair of standard (Drinfeld-Jimbo) deformations in each chiral sector. They are described by nonlinear (quantum) generators q ±h k k , e k± (k = 0, 1) which satisfy the following defining relations The co-products ∆ 3 ′ and antipodes S 3 ′ are given by the formulas : The universal R-matrices R 3 ′ k for each chiral sector are well-known and using deformed CW generators (5.11) take the form (q k = exp 1 2 γ k ): Following the discussion of nontriangular case in Sect. 2.1, there exists alternative universal R-matrix in the form Therefore, the universal R-matrix R 3 ′ , which connects the coproducts ∆ 12 3 ′ := ∆ 3 ′ and the flipped one ∆ 21 3 ′ can be written in two equivalent forms: 28 Expanding (5.16) up to first order in deformation parameters (γ,γ) one gets where r 3 ′ BD is in Belavin-Drinfeld form 29 This r-matrix is not skew-symmetric and satisfies the condition where ω is the quadratic split Casimir of o(4; C) We recall that the Belavin-Drinfeld r-matrix r BD satisfies CYBE and the r-matrix r ′ 3 is a skew-symmetric part of it. Now we consider deformation of the quantum algebra U (γ,γ) (o(4; C)) ∼ = U γ (sl(2; C)) ⊗ Uγ(sl(2; C)) generated by the r-matrix r ′′ 3 = ηH ⊗H, (see (4.7)). Since the generators h and h have the primitive coproduct 21) 28 In fact, taking into account (5.15), there are four ways of describing universal R-matrix R 3 ′ . 29 In (5.18) and in other formulas describing classical r-matrices, the generators E ± ,Ē ± are not deformed.
the Abelian two-tensor (q = exp 1 4 η) satisfies the 2-cocycle condition (3.29). Thus the complete deformation generated by the rmatrix r 3 is the twist deformation of U (γ,γ) (o(4; C)); the resulting coproduct ∆ 3 is given as follows 23) and the antipode S 3 is not changed (S 3 = S 3 ′ ). Applying the twist (5.22) to the formulas (5.12) we obtain The universal R-matrix, R 3 (γ,γ, η), corresponding to the complete r-matrix r 3 , has the form In the linear limit we obtain (cf. (5.16), (5.17)) This deformation admits seven real forms which employ all four conjugations (cf. (4.11) -(4.14) ) . The list of real forms with corresponding restricted values of the deformation parameters γ,γ, η is presented in the Table 4, with real bialgebras denoted in the first column (cf. Table 1). Table 4: Real quantizations of r III (γ,γ, η) = γ E + ∧ E − +γĒ + ∧Ē − + η H ∧H The letters in the last column (R=real, A=antireal, H=hybrid) indicate the properties of the R-matrix under respective conjugation: R ⋆ = R τ for real, R ⋆ = R −1 for antireal cases. In the hybrid case the R-matrix decomposes into a product of three factors, with first real, second antireal and third is given by twist which satisfies both reality conditions. It shoul be mentioned that only the classical r-matrix r III provides the quantum deformations of real o(4) = su(2) ⊕ su)(2 algebra (see first line in Tab 4). Particular case, with η = 0 was derived as describing quantum symmetries of D = 3 LQG [7].

5.4
Twisting of o(4; C) Belavin-Drinfeld triple (r-matrix r IV ) Next, we describe quantum deformation corresponding to the classical r-matrix r IV (4.8).
It should be noted that the value ς = 0 in the Lorentzian case is due to the fact that twist (5.30), in contrast to the o ′′ (2, 2) case where |q| = 1, is not unitary for real q. In order to have formulae (5.31)-(5.34) compatible with the Lorentzian conjugation (4.14) it is helpful to introduce flipped conjugation (3.34) on the tensor product of quantized algebras (see [26]). Alternatively, one can keep the standard (non-flipped) conjugation and seek for the unitarizing coboundary twist -the quantum analog of (3.33) 30 . Examples of quantum coboundary twists can be found e.g. in [53]. The realization of this task is postponed to our future work.

Left q-analog and right Jordanian deformation intertwined by
Abelian twist (r-matrix r V ) In this case we start with the left sector as q-deformed with q = exp 1 2 γ Further we perform the subsequent quantization by using the quantized Abelian twist The explicit coproduct formuale are the following ∆ 5 (e ± ) = e ± ⊗ q hq±Σ + q −hq∓Σ ⊗ e ± . (5.41) The antipodes do not depend on the Abelian twist and look as follows: Quantum universal R-matrix generated from r V takes the following form Three real quantizations we describe in the Table 6 below. Table 6: Real quantizations of r V (γ, ρ,χ) = γ E + ∧ E − +χĒ + ∧H + ρH ∧Ē +

Concluding remarks and outlook
In this paper we presented the complete set of Hopf-algebraic quantum deformations generated by classical r-matrices for o(4; C) and its real forms given in [17,19]. The explicit formulae describing algebraic and coalgebraic sectors are provided as well as there are given the universal R-matrices which permits the tensoring of quantum modules (representations of quantum-deformed Hopf algebras). We recall that the universal R-matrices describe the braided structure of quantum-covariant tensor products of modules [56,14] what has been used in quantum-covariant NC field theory [57,58]. For quantum twist deformations of enveloping Lie algebras U(g) (g = o(4; C), o(4 − k, k) (k = 0, 1, 2), and o * (4) = 0(2; H)); for o(4; C) (see Sect.5.1, 5.2) the algebra of quantum modules, describing e.g. NC quantum fields, can be represented by the functions of classical (commutative) fields with twist-dependent nonlocal star product multiplication rule [59,60]. Basic role plays in relativistic physics D = 4 Minkowski space R 3,1 , with signature (+, +, +, −), and its Lorentz rotations o (3,1). In order to describe the Lie algebra generating relativistic group of motion, one adds four generators P µ ∈ T 3,1 of translations, i.e. one extends Lorentz algebra o(3, 1) to D = 4 Poincaré algebra o(3, 1) ⋉ T 3,1 . It is known that only two out of four quantum deformations of o(3, 1) can be extended to quantum deformations of D = 4 Poincaré algebra (see [61,62,63,26]). The studies providing the complete list of possible quantum deformations of inhomogeneous D = 4 Euclidean o(4) ⋉ T 4 algebra and of inhomogeneous D = 4 Kleinian o(2, 2) ⋉ T 2,2 algebra has not been presented 31 .We add that inhomogeneous extension of quaternionic real form o * (4) ≡ o(2; H) of o(4; C) contains four complex or two quaternionic translations and its applications to the description of physical symmetries are, according to our knowledge, not known.
The real forms of considered quantum groups describe the quantum symmetries of D = 3 compact Euclidean (S 3 ), de Sitter (dS 3 ) or anti-de-Sitter (AdS 3 ) spacetimes, with finite nonvanishing constant curvature and curved D = 3 Euclidean, dS 3 or AdS 3 translations. In D = 4 rotation algebras o(3, 1) (o(2, 2)), the dS (AdS) radius R is introduced by suitable rescaling of the generators in the coset o(4−k,k) o(3−k,k) (k = 1, 2), with Λ = R −1 which can be treated as a deformation parameter. The quantum deformations of o(4 − k, k) (k = 1, 2) have been extensively studied as describing the NC geometry of 2 + 1-dimensional QG with cosmological constant Λ = 0 [66,67]. The classical action of D = 3 gravity can be introduced geometrically as gauge theory described by D = 3 Chern-Simons (CS) model. Following Fock-Rosly construction [68,69], in such framework we describe gravitational degrees of freedom as parameterizing the Poisson-Lie group manifold. If we search for quantum deformations of Fock-Rosly construction, it appears that only classical r-matrices obtained from Drinfeld double (DD) structures [10] are allowed [70,71]. The DD structures and corresponding classical r-matrices for o(3, 1) and o(2, 2) algebras were recently constructed and classified [67]. We see that such quantum deformation which are well adjusted to the description of quantum-deformed D = 3 gravity are generated by a subclass of classical r-matrices, listed in [17,19] and quantized in this paper. The next step in our program is to construct the complete list of classical r-matrices for the D = 4 complex inhomogeneous Euclidean algebra E(4; C) := io(4; C) := o(4; C) ⋉ T(4; C) (orthogonal rotations together with translations) and for its real forms, in particular o(4 − k, k) ⋉ T(4 − k, k; R) (k = 0, 1, 2). Until present time the most complete results were obtained for o(3, 1) ⋉ T(3, 1) by Zakrzewski [62], who provided almost complete list of 21 different, not related by Poincaré automorphism real D = 4 Poincaré r-matrices (see also [63,64]). It should be noticed that the complete classifications of r-matrices for both inhomogenous D = 3 Poincaré and D = 3 Euclidean algebras have been given by Stachura [72].
Recently in [73,74] the present authors complexified Zakrzewski results and then imposed 31 For partial results in D = 4 Euclidean case see e.g. [65] D = 4 Euclidean reality constraints. It appeared that 8 out of 21 complexified Zakrzewski r-matrices are consistent with the Euclidean conjugation in o(4; C) (see (4.11)). It can be shown, however, that the complexified Zakrzewski r-matrices do not describe all r-matrices for E(4; C) 32 . Using the constructive method analogous to the one proposed in this paper we intend to describe the complete classification of classical r-matrices for D = 4 complex inhomogeneous Lie algebra E(4; C) and for its all real forms. We add that in [73,74] we considered also the N = 1 superextension of Poincaré and Euclidean classical r-matrices. Recently we derived in analogous way as well new class of N = 2 Poincaré and Euclidean supersymmetric r-matrices (see [65]). We hope that our constructive method of providing the complete list of classical r-matrices for the complex E(4; C) case can be applied as well to N-extended Euclidean superalgebras E(4|N; C) for N = 1, 2, 4 and further classify and quantize the supersymmetric r-matrices for the corresponding real forms. orbits in R 3 . These are the 2-dimensional spheres represented by a radius ξ > 0 or by the vector ξ(1, 0, 0). Thus as a result of final classification one gets the following family of nontrivial o(3) r-matrices (the trivial r = 0 r-matrix corresponds to singular one-point orbit at (0, 0, 0)) r ξ = ξI 1 ∧ I 2 (A.3) Notice that the values of the real parameter ξ > 0 are effective and lead to nonequivalent Lie bialgebra structures. Similar analysis applied to the non-compact real form o(2, 1) provides qualitatively different results. Arbitrary o(2, 1) r-matrix satisfies the following YB equation Three canonical o(2, 1) r-matrices corresponding to three types of orbits take the form r 0 ξ = ξ(J 1 ∧ J 3 + J 1 ∧ J 2 ); r − ξ = ξJ 3 ∧ J 2 ; r + ξ = ξJ 1 ∧ J 3 (A. 6) where in the first case one gets the same deformation for any value of the parameter ξ = 0, while in the remaining two cases different values of ξ lead to different Lie bialgebras. In this setting we get one reality condition and three different types of r-matrices representing nonequivalent bialgebra structures.

B q-exponent and q-Hadamard formula
Our aim here is to introduce some formulas (mainly concerning a q-deformed Hadamard lemma), which were main tools for calculations presented in Sect. 5.4. Let A and B be two arbitrary elements of some quantum algebra and let exp q (A) be a formal q-exponential exp q (A) := n≥0 A n (n) q ! , (n) q ! := (1) q (2) q · · · (n) q , (n) q = 1 − q n 1 − q .
(B.1) of the element A. As the q-exponential exp q −1 (−A) is inverse to exp q (A), i.e. exp q (A) −1 = exp q −1 (−A) thus the q-analog of Hadamard formula can be obtained as follows (see [22]) (1 − q ′−1 q k )A n B = (q ′−1 ; q) n A n B .