Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic $\mathfrak{o}^{\star}(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum

In our previous paper we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex $D=4$ Euclidean Lie symmetry $\mathfrak{o}(4;\mathbb{C})$. The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical $r$-matrices which provide 'directions' of the nonequivalent quantizations of $\mathfrak{o}(4;\mathbb{C})$. Applying reality conditions to the complex $\mathfrak{o}(4;\mathbb{C})$ $r$-matrices we obtained the nonisomorphic classical $r$-matrices for all possible real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$ Lie algebras. In the case of $\mathfrak{o}(4)$ and $\mathfrak{o}(3,1)$ real symmetries these $r$-matrices give the full classifications of the inequivalent quasitriangular quantum deformations, however for $\mathfrak{o}(2,2)$ and $\mathfrak{o}^{\star}(4)$ the classifications are not full. In this paper we complete these classifications by adding three new three-parameter $\mathfrak{o}(2,2)$-real $r$-matrices and one new three-parameter $\mathfrak{o}^{\star}(4)$-real $r$-matrix. All nonisomorphic classical $r$-matrices for all real forms of $\mathfrak{o}(4;\mathbb{C})$ are presented in the explicite form what is convenient for providing the quantizations. We will mention also some applications of our results to the deformations of space-time symmetries and string $\sigma$-models.


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 indicate that the presence of quantum gravity effects generates noncommutativity of D = 4 space-time coordinates, and as well the Lie-algebraic space-time symmetries (e.g. Euclidean, Lorentz, Kleinian, quaternionic and their ingomogeneous versions) are modified into respective quantum symmetries, described by noncocommutative Hopf algebras, named quantum deformations [2]. Therefore, studing all aspects of the quantum deformations in details is an important issue in the search of quantum gravity models.
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. [2]- [5]). 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]) = [δ(x), ∆ 0 (y)] + [∆ 0 (x), δ(y)], (δ ⊗ id)δ(x) + cycle = 0, (1.1) where ∆ 0 (·) is a trivial (non-deformed) coproduct 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 [2,5]). The Lie bialgebra (g, δ) is a correct infinitesimalization of the quantum Hopf deformation of U(g) and the operation δ is an infinitesimal part of difference between a coproduct ∆ and an oposite coproduct∆ in the Hopf algebra, 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 our special 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 to 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 [2,3]) therefore the classification of all nonequivalent quasitriangular Hopf algebras is reduced to the classification of all nonequivalent classical r-matrices. Let g * := (g, * ) be a real form of a semisimple complex Lie algebra g, where * is an antilinear involutive antiautomorphism of g, then the bialgebra (g * , δ (r) ) is real iff the classical r-matrix r is * -anti-real ( * -anti-Hermitian). 1 Indeed, the condition of * -reality for the bialgebra (g * , δ) means that δ(x) * ⊗ * = δ(x * ). (1.5) Applying this condition to the relations (1.4) we abtain that i.e. the r-matrix r is * -anti-Hermitian.
In the previous paper [1] we obtained a full classification of nonequivalent quasitriangular quantum deformations for the complex D = 4 Euclidean Lie symmetry o(4; C). The result was presented in the form of a list consisting of three three-parameter, one two-parameter and one one-parameter nonisomorphic classical r-matrices which provide directions of the nonequivalent quantizations of o(4; C). Applying reality conditions in [1] we obtained the non-

Complex D = 4 Euclidean algebra and its real forms
In this section we remind a short necessary information about structure of D = 4 complex orthogonal Euclidean Lie algebra o(4; C) and its real forms.
The complex orthogonal Lie algebra o(4; C) has the chiral decomposition, i.e. it can be expressed as the following direct sum: One real form, the Lorentz algebra o(3, 1), which does not preserve the chiral decomposition, will be considered as well (see (2.11)). We shall use here two most popular bases of Lie algebra o(4; C) and its real forms: the Cartesian basis and Cartan-Weyl one.
Due to the decompositions (2.1)-(2.4) we shall consider these bases only for one factor o(3; C) in (2.1) and its real forms o(3) and o(2, 1). The Cartesian basis of o(3; C) is given by the generators I i (i = 1, 2, 3) with the defining relations: (2.5) If we consider a Lie algebra over R with the commutation relations (2.5) then we get the compact real form o(3) := o(3; R) with the anti-Hermitian basis (i = 1, 2, 3): The real form o(2, 1) is given by the formulas (i = 1, 2, 3): where the primed generators satisfy the same relations (2.5). For the right chiral Cartesian bases we shall use the notations with bar, i.e.Ī i ,Ī ′ i , (i = 1, 2, 3). In terms of the Cartesian bases the reality conditions for real forms (2.2)-(2.4) and for the Lorentz algebra o(3, 1) look as follows (i = 1, 2, 3): where the conjugation ( * ) is the same as in (2.6).
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 where the conjugation ( † ) is the same as in (2.7) 2 .
In the basis (2.12)-(2.17) all possible real forms of o(4; C) are described by the following reality conditions: 3 Classical r-matrices of the o(4; C) real forms In previous paper [1] we found a total list of nonequivalent (nonisomorphic -unrelated by automorphisms) classical r-matrices for the complex D = 4 Euclidean Lie symmetry o(4; C). This result are presented in the form of three three-parameter, one two-parameter and one one-parameter r-matrices as follows 3 : Here all parameters γ,γ, η, χ,χ, χ ′ ,χ ′ are arbitrary complex numbers. It should be noted that all three parameters in r 1 are effective, i.e. there does not exist any o(4; C) automorphisms which can reduce the number of parameters. In the case of r 2 only two parameters are effective because the parameterχ orχ ′ can be removed by the rescaling automorphism ϕ(Ē ± ) = λ ±1Ē ± , ϕ(H) =H. In r 3 only one parameters is effective because any two parameters can be removed by the rescaling automorphisms. Analogously, in r 4 and r 5 the parameters χ ′ and χ can be removed, i.e. they can be replaced by one. 4 In [1] we employed to (3.1)-(3.5) the reality conditions (2.18)-(2.21) without ones, which contain su(1, 1) conjugations. In such a way we obtained the classical r-matrices for all possible real forms of o(4; C): compact Euclidean o(4), noncompact quaternionic o ⋆ (4), noncompact Kleinian o(2, 2) and noncompact Lorentz o(3, 1) Lie symmetries. It follows from [7,8] that the obtained in [1] sets of the classical r-matrices for o(4) and o(3, 1) are complete, whereas for o ⋆ (4) and o(2, 2) these results are only partial. If we apply to (3.1)-(3.5) all conditions (2.18)-(2.21) including ones which contain su(1, 1) conjugations then we obtain the following results.
I. Classical r-matrices of the real Euclidean algebra o(4). If we employ the reality conditions (2.18) to the results (3.1)-(3.5) we obtain all nonequivalent classical rmatrices for the compact Euclidean symmetry o(4). These classical r-matrices are described by one * -anti-Hermitian three-parameter r-matrix: with three real parameters γ,γ, η, and . It should be noted that all parameters in (3.6) are effective, i.e. the number of parameters can not be reduced by any o(4) automorphism.
II. Classical r-matrices for the quaternionic algebra o ⋆ (4). Applying the realty conditions (2.19) to the complex formulas (3.1)-(3.5) we obtain all nonisomorphic classical r-matrices for the quaternionic algebra o ⋆ (4). These classical r-matrices are described by three three-parameter anti-Hermitian r-matrices: (3.7) where r 1 supplements the results obtained in [1]. All parameters γ,γ, η,χ,χ ′ are arbitrary real numbers, and 1, 2, 3). Moreover all parameters in r 1 and r 2 are effective, and in the case of r 3 only two parameters are effective because the parameterχ orχ ′ can be removed by the sl(2, R)-real rescaling automorphism: III. Classical r-matrices for the Kleinian algebra o(2, 2). If we employ the reality conditions (2.20) to the complex r-matrices (3.1)-(3.5) we obtain all nonisomorphic o(2, 2)real classical r-matrices for the Kleinian symmetry o(2, 2). These classical r-matrices are described by six three-parameter, one two-parameter and one one-parameter nonequivalent †-anti-Hermitian r-matrices: (3.10) (3.11) (3.13) (3.14) where r 1 , r 2 and r 3 supplement the results obtained in [1]. All parameters γ,γ, η, χ,χ, χ ′ , χ ′ are arbitrary real numbers, and ( . Moreover all parameters in r 1 , r 2 and r 4 are effective. In r 3 and r 5 only two parameters are effective, because the parameterχ orχ ′ can be removed by the sl(2, R)real rescaling automorphism: ϕ(Ē ′′ ± ) = λ ±1Ē′′ ± , ϕ(H ′′ ) =H ′′ , where λ is real. In the case r 6 only one parameter is effective because any two of them can be removed by the sl(2, R)-real rescaling automorphisms. Analogously for r 7 and r 8 the parameters χ ′ and χ can be removed in the same way.