Quantum $D = 3$ Euclidean and Poincar\'{e} symmetries from contraction limits

Following the recently obtained complete classification of quantum-deformed $\mathfrak{o}(4)$, $\mathfrak{o}(3,1)$ and $\mathfrak{o}(2,2)$ algebras, characterized by classical $r$-matrices, we study their inhomogeneous $D = 3$ quantum IW contractions (i.e. the limit of vanishing cosmological constant), with Euclidean or Lorentzian signature. Subsequently, we compare our results with the complete list of $D = 3$ inhomogeneous Euclidean and $D = 3$ Poincar\'{e} quantum deformations obtained by P. Stachura. It turns out that the IW contractions allow us to recover all Stachura deformations. We further discuss the applicability of our results in the models of (2+1)d quantum gravity in the Chern-Simons formulation (both with and without the cosmological constant), where it is known that the relevant quantum deformations should satisfy the Fock-Rosly conditions. The latter deformations in part of the cases are associated with the Drinfeld double structures, which also have been recently investigated in detail.


I. INTRODUCTION
The emergence of non-commutative quantum spacetimes at ultra-short distances, comparable with the Planck length λ P ≈ 10 −35 m, is a recurring theme, constantly re-surfacing during decades of efforts to construct quantum gravity models (see e.g. [1][2][3][4][5]). There are also good reasons to believe that a direct consequence of the quantization of gravity is a necessity to replace the classical symmetries with their quantum counterparts, where the quantum versions of Lie algebras and groups are given by the algebraic groups having the structure of Hopf algebras [6]. It was realized in late 1980s and 1990s that of particular importance in such a context are the studies and classification of quantum spacetime symmetries, described in the language of Hopf algebras, especially the quantum Poincaré algebras and quantum Poincaré groups, as well as quantum versions of the (anti-)de Sitter and conformal symmetries.
It has also been known since the 1980s [6,7] that the quantum deformations of symmetries can be characterized by the classical r-matrices, which are solutions of the classical Yang-Baxter equation (CYBE). A classical r-matrix r is linear in the deformation parameters q i and determines the coboundary Lie bialgebra structure of the algebra g, with the coproduct of algebra elements g ∈ g given by the perturbative formula ∆(g; q i ) = ∆ 0 (g) + [r, ∆ 0 (g)] + O(q 2 i ) , where ∆ 0 denotes the primitive (i.e. undeformed) coproduct and the second term provides the cobracket. Such an expansion leads to the quantization of Poisson-Lie structure and emergence of the corresponding quantum-deformed Hopf algebra. The procedure of generating associative and coassociative quantum Hopf algebras from classical r-matrices is called the quantization of bialgebras [8]. If a bialgebra is coboundary, then there is a one-to-one correspondence between r-matrices and possible Hopf algebraic structures describing quantum symmetries.
If t = 0, the Yang-Baxter equation (2) is homogeneous and describes the twist quantizations of algebra g, while its solutions are called the triangular r-matrices; if t = 0, the Yang-Baxter equation is inhomogeneous or modified, and its solutions are known as the quasitriangular r-matrices.
For semisimple Lie algebras, e.g. complex orthogonal algebras o(n; ) and their real forms, all bialgebras are coboundary and therefore the classification of classical r-matrices completes the task of classifying quantum deformations. In the case of D = 4 spacetime symmetries, all deformations of the Lorentz algebra have been described using four multiparameter r-matrices more than twenty years ago [9]. Recently in a similar fashion, by finding all classical r-matrices, all deformations of o(4; ) and of its real forms -the Euclidean o(4), Lorentzian o(3, 1), Kleinian o(2, 2) and quaternionic o * (4; ) ∼ = o(2; À) -were classified [10]. On the other hand, the complete classification of all deformations of the D = 4 Poincaré algebra (at the Poisson structure level) has not been obtained so far [11]. However, the complete classification of coboundary deformations for D = 3 inhomogeneous Euclidean and D = 3 Poincaré algebras has been accomplished in [12].
In what follows we denote by o(D + 1) the D-dimensional Euclidean de Sitter algebra and by o(D − 1, 2) the D-dimensional Lorentzian anti-de Sitter algebra, while we regard o(D, 1) either as the D-dimensional Lorentzian de Sitter algebra or D-dimensional Euclidean anti-de Sitter algebra 1 . For D = 4, all of these real rotation algebras are described by the (pseudo-)orthogonal real forms of the complex algebra o(4; ). The main task considered in this paper is to obtain all quantum inhomogeneous D = 3 Euclidean and Lorentzian (i.e. Poincaré) algebras that can be derived via the quantum versions ofİnönü-Wigner (IW) contraction, applied to the D = 4 quantum rotation algebras.
A standard, classical IW contraction is applied to Lie algebras that can be decomposed into g = h ⊕ n, where Rescaling n → Rñ and taking the limit of the contraction parameter R → ∞, we obtain a semidirect product g = h ⊲ <ñ, with [ñ,ñ] = 0 and h unmodified in the contraction limit. The quantum deformation parameters q i enter linearly into the definition of a classical r-matrix In the quantum IW contraction procedure, q i will usually depend on the powers of R, in a way that permits to obtain the finite contraction limit of a classical r-matrix as well as the finite form of the classical Yang-Baxter equation (2). In particular, if we appropriately rescale the g-invariant three-form Ω, we can answer which contracted classical r-matrices satisfy the modified Yang-Baxter equation.
As we already mentioned, classical r-matrices describe the infinitesimal version of quantum groups, which take the form of noncommutative Hopf algebras. The quantum IW contraction procedure applied to such a quantum Hopf algebra of g leads to quantum groups associated with the inhomogeneous algebrag.
The aim of the current paper is actually three-fold.
• Using the complete classification of classical r-matrices for the o(4; ) algebra and hence for the o(4), o (3,1) and o(2, 2) algebras [10], we perform all possible quantum IW contractions that lead to the inequivalent quantum deformations of D = 3 inhomogeneous Euclidean or D = 3 Poincaré algebra. Subsequently, we compare our results with the complete list of D = 3 quantum inhomogeneous algebras presented in [12]. In particular, we show that all such quantum algebras can be obtained as some quantum IW contraction of a D = 4 quantum (pseudo-)orthogonal algebra.
• An arbitrary classical r-matrix for the inhomogeneous (Euclidean or Lorentzian) Lie algebra L ⊲ < T can be decomposed as follows where L denotes the homogeneous subalgebra and T is the momentum subalgebra, generating Abelian translations. A classical r-matrix with c = 0 but b = 0 is usually said to describe a generalized κ-deformation (the best known example is the standard κ-deformation, introduced in [13,14]). We show that almost all non-Abelian contracted r-matrices belong to the D = 3 generalized κ-deformations (with the exception of r-matrices given only by c before the contraction). A distinguishing feature of such deformations in the physical context is their characterization by the dimensionful deformation parameter, which allows for the geometrization of the Planck mass m P = c −1 /λ P . This suggests the potential applicability of generalized κ-deformations in the context of quantum gravity.
• On the other hand, in the context of Chern-Simons formulation of 3D gravity it has been shown [15] that 3D quantum gravity models can only be constructed with quantum symmetries satisfying the Fock-Rosly compatibility conditions (see Sec. VI). The Fock-Rosly compatible r-matrices are supposed to be completely classified in [16] but our results for such r-matrices appear to contain less constrained sets of deformation parameters. Moreover, some authors argue that the relevant class of symmetries is actually smaller and given by the so-called Drinfeld doubles. It is known that there exist four r-matrices associated with Drinfeld doubles on the D = 3 de Sitter algebra and three on the D = 3 anti-de Sitter algebra [17], while there are eight for the D = 3 Poincaré algebra [18]. We point out where and how all of them arise in our calculations.
The plan of our paper is the following. In the next Sec. II we recall all complex o(4; ) r-matrices and perform their contractions leading to quantum-deformed inhomogeneous o(3; ) algebras, i.e. quantum o(3; ) ⊲ < T 3 ( ) algebras. Sec. III is devoted to the description of real forms of the complex algebras o(3; ), o(3; ) ⊲ < T 3 ( ) and o(4; ). The focal point of our paper is Sec. IV, where we analyze all possible quantum IW contractions of the deformed real forms of o(4; ): in Subsec. IV A we obtain r-matrices of the quantum D = 3 inhomogeneous Euclidean algebra from contractions of the o(4) r-matrices; then we study contractions of the o(3, 1) r-matrices, leading to D = 3 inhomogeneous Euclidean (in Subsec. IV B) or D = 3 Poincaré (in Subsec. IV C) quantum algebras; in Subsec. IV D we consider contractions of r-matrices of three different o(2, 2) real forms, which provide another set of quantum D = 3 Poincaré algebras. Subsequently, in Sec. V we recall the complete list of D = 3 inhomogeneous Euclidean and D = 3 Poincaré r-matrices given in [12] and compare them with the ones derived by us via the quantum IW contractions. Finally, Sec. VI is mainly devoted to a discussion of relevance of quantum Poincaré algebras (at some points we also refer to quantum (anti-)de Sitter algebras) in 3D quantum gravity models. We conclude the paper with a summary and a short list of open problems in Sec. VII.

II. DEFORMED o(4; ) ALGEBRAS AND THEIR INHOMOGENEOUS o(3; ) CONTRACTIONS
As we mentioned in the Introduction, the o(4; ) algebra provides a unified setting that we can use to describe all (deformed) rotational symmetries of Euclidean, Lorentzian or Kleinian space(time) in D = 4. Therefore, we will first consider quantum deformations of o(4; ), described by five families of classical r-matrices, and study the existence of their inhomogeneous o(3; ) quantum IW contraction limits. We subsequently pass to the o(4; ) real forms, whose quantum IW contraction limits lead to the deformed D = 3 inhomogeneous Euclidean and D = 3 Poincaré algebras. A convenient framework for the description of the corresponding r-matrices is obtained by expressing the o(4; ) algebra in the chiral Cartan-Weyl basis (left {H, E ± } and right {H,Ē ± }), where the algebra generators satisfy 2 we arrive at another chiral basis (left {X i } and right {X i }, i = 1, 2, 3), such that For our aim of studying the contraction limits we introduce the following orthogonal basis which can be obtained by taking The basis (11) provides a three-dimensional decomposition of the standard orthogonal basis where M AB = −M BA , A, B = 1, . . . , 4 and in this Section we choose the transformation (cf. (33)) K i are the generators of curved translations on the coset group O(4; )/O(3; ). Therefore, in order to examine the existence of inhomogeneous o(3; ) quantum contraction limits, we will perform the rescaling of K i tõ The contraction parameter R, which is chosen to be real and positive, for the real forms o(4), o(3, 1) and o(2, 2) (more precisely, as we will describe in Sec. III, there are three different o(2, 2) real forms) in the context of 3d gravity will be interpreted as the de Sitter (or respectively anti-de Sitter) radius R, related to the cosmological constant Λ via the formula R 2 = |Λ| −1 . In the present complex case, in the contraction limit R → ∞ we obtain and hence (11) reduces then to the inhomogeneous o(3; ) algebra. The generators P i ≡ lim R→∞Ki can be interpreted as the complexified Abelian three-momenta, generating commutative complex translations. There exist five families of r-matrices, each depending on some parameters, that determine all possible deformations of the o(4; ) algebra. We will denote them as r I , r II , r III , r IV and r V (see [10], eqs. (4.5-4.9)). Let us first consider the cases of r I , r II and r V . In the chiral Cartan-Weyl basis (8) they are given by We transform them to the Cartesian basis using (9) and (12) and subsequently rescale the generators K i according to (15). This leads to In order to get the finite result in the R → ∞ limit of these r-matrices, we should rescale the deformation parameters in the following wayχ Then in the contraction limit we ultimately obtaiñ As one can notice, the expression forr V in (20) vanishes if the original parameters satisfy the relations ρ =χ and γ = 0. On the other hand, for such values of γ,χ, ρ in (18) we may perform the alternative rescaling of the remaining free parameter toχ ≡ Rχ and in the R → ∞ limit it giveŝ We now turn to the r-matrix r IV , which has the form Changing the basis via (9) and (12), and subsequently performing the rescaling (15), we arrive at Again, the finite R → ∞ limit can be obtained after the rescaling of parameters This leads tor The above expression vanishes if the original parameter ς = −2γ. However, for (23) with such a fixed ς, the alternative rescalingγ ≡ R γ allows us to obtain in the contraction limit which is actually the r-matrix describing the κ-deformation of o(3; ) ⊲ < T 3 ( ). Finally, we analyze the r-matrix r III , given by This is the only r-matrix compatible with all possible real forms of o(4; ) (see Table I). Performing the same basis transformation as in (20) and (23), we obtain which has the finite R → ∞ limit if we supplement it by the rescaling of parameters tõ The contraction limit has the formr Similarly tor IV andr V , the above expression vanishes if the original parameterγ = γ but applying the alternative rescalingγ in (28) leads to another contraction limit Furthermore, withγ = γ and after the rescaling (31), the r-matrix (28) actually does not depend on R.
The result of the IW contraction o(4; ) → o(3; ) ⊲ < T 3 ( ) does not depend on a coordinate axis in 4 along which the rescaling and contraction is performed (this is no longer true for the pseudo-orthogonal real forms of o(4; ), cf. Subsec. III C). In (15) we picked the fourth axis (i.e. A = 4) but this is only a matter of convention. However, since r-matrices are not symmetric under O(4) rotations, their quantum IW contraction limits obtained with the rescaling of different axes are not always equivalent under automorphisms of o(3; ) ⊲ < T 3 ( ). To see this explicitly, let us pick the first axis, associated with the corresponding orthogonal basis: where now p, q, r = 2, 3, 4 (cf. (13)). The generators J ′ p and K ′ p satisfy the same commutation relations as (11), modulo the substitution 1 → 4 (due to the Euclidean metric we have ǫ 234 = 1). The quantum IW contraction along the first axis is performed by taking the rescalingK ′ p ≡ R −1 K ′ p (34) and subsequently the limit R → ∞,K ′ p → P ′ p . In order to pass from the Cartan-Weyl basis (8) to the basis {J ′ p ,K ′ p }, one has to substitute the formulae (12) into (9), with J i and K i expressed in terms of J ′ p andK ′ p . The obtained transformation between the bases has the form Applying (35) to the r-matrices (17), (22) and (27), one can express them in the orthogonal basis rescaled along the third axis. Let us consider the r-matrices r I , r II and r IV as examples 3 . The procedure described above gives which are respectively equivalent to (18) and (23) However, rescaling the deformation parameters to (under the condition we find the contraction limits that are inequivalent to (20) and (25)(26): This dependence on a contraction axis will be discussed in detail for deformations of the o(4; ) real forms in Sec. IV.

A. Real algebras, bialgebras and -Hopf algebras
A real Lie algebra structure (a real form) (g, ) is introduced on a complex Lie algebra g by defining an involutive antilinear antiautomorphism (called a -conjugation or -operation) : g → g. Then one can find such a basis of the algebra that its structure constants are real and the -conjugation is anti-Hermitian (i.e. ∀g ∈ g : g = −g) 4 . However, in an arbitrary basis, a given real form is characterized by certain nontrivial reality conditions that have to be satisfied by the algebra generators under the action of the -conjugation.
A real coboundary Lie algebra (see (1)) is introduced as a triple (g, , r), with the classical r-matrix r assumed to be anti-Hermitian, namely where τ is the flip map, τ : a ⊗ b → b ⊗ a. This leads to the appropriate reality conditions for the deformation parameters, which is especially important for the physical description of quantum deformed symmetries. Furthermore, due to the antiautomorphism property (i.e. ∀g, h ∈ g : (gh) = h g ), the -conjugation extends to the universal enveloping algebra U (g) of a Lie algebra g, as well as its quantum deformations U q (g), making each of them an associative -algebra. U q (g) is defined as a Hopf algebra (U q (g), ε, ∆, S), where ǫ denotes a counit, ∆ a coproduct and S an antipode [6]. The -conjugation has to preserve the Hopf-algebraic structure of U q (g) by satisfying the following compatibility conditions 5 (here and elsewhere * denotes the complex conjugation) where the -conjugation is assumed to act on tensor products as Let us note that sometimes the alternative rule (a ⊗ b) = b ⊗ a is used instead. The finite quantum counterpart of a classical r-matrix r is the universal R-matrix R, determining the Hopf algebra structure U q (g) of quantum deformations of U (g). An invertible element R ∈ U q (g) ∧ U q (g) provides the flip of the coproduct ∆ τ = R∆R −1 . U q (g) together with R satisfying certain additional conditions is called a quasitriangular Hopf algebra. In such a case, R allows us to define a quasitriangular -Hopf algebra if [6] it is either real, i.e. R ⊗ = R τ , or antireal, i.e. R ⊗ = R −1 and the corresponding quantum R-matrix is -unitary. For the triangular R-matrix, i.e. R τ = R −1 , the above two reality conditions are identical. In the non-triangular case, there exist two universal R-matrices and the second of them (R τ ) −1 satisfies the same reality conditions as R. Any R can be expanded as R = ½ ⊗ ½ + r + . . . and there are two possibilities for the element r. Firstly, if r is skew-symmetric, it is a classical r-matrix, which should also be anti-Hermitian, i.e. satisfy the relation (39). Secondly, if the element r is not skew-symmetric, we have a non-triangular case and then r ⊗ = r τ for the real R, while r ⊗ = −r for the antireal R.  (2) , where su(2) ∼ = o(3; Ê) and sl(2; Ê) ∼ = su(1, 1) ∼ = o(2, 1; Ê). The isomorphism between the su(1, 1) and sl(2; Ê) algebras can be realized as an automorphism of o(3; ) given by (here we denote the Cartan-Weyl basis in the su(1, 1) case as In the Cartesian basis, corresponding to the {X i } sector of (10), we have only two real forms, with the reality conditions 5 I.e. a real Hopf algebra is defined as a -Hopf algebra, which is a complex Hopf algebra equipped with a -conjugation. The presence of this conjugation turns the algebraic sector into a -algebra, while the coalgebraic sector becomes a -coalgebra, satisfying (40).
However, the relation between the Cartan-Weyl and Cartesian bases does not look the same in the case of su(1, 1) as in sl(2; Ê). From (42) one can observe that H is compact for su(1, 1), while noncompact for sl(2; Ê). In general, the relations between the above mentioned bases can be chosen as 6 for su(2) or sl(2; Ê) , It means that if we consider e.g. the Drinfeld-Jimbo deformation of o(2, 1) in the Cartan-Weyl basis with the first or second set of relations (45), in the Cartesian basis we obtain two different types of nonlinearities (see [21], Sec. 5 for more details).
In order to enlarge the o(3; ) algebra to the inhomogeneous algebra o(3; ) ⊲ < T 3 ( ), one should extend the Cartan-Weyl basis by the generators P i ∈ T 3 ( ), i = 1, 2, 3 (commuting complex momenta). Denoting P ± ≡ P 1 ± iP 2 , we then write the cross brackets of o(3; In terms of the Cartesian basis they take the familiar simple form ) algebra can be obtained by imposing the appropriate conditions (42) together with the consistent proper reality conditions for P i ∈ T 3 or P i ∈ T 2,1 , respectively. The latter conditions define real momenta It is easy to see that the brackets (46) are invariant under either of the conjugations (48). On the other hand, the Poincaré algebra can also be introduced as the real form of (46) invariant under the su(1, 1) conjugation (i.e. su(1, 1) ⊲ < T 2,1 ), obtained by extending the reality conditions from the last line of (42) by which is equivalent to the second line of (48) with the generators P 2 and P 3 exchanged.
The completeness of the description of quantum deformed o(2, 1) algebras expressed in either sl(2; Ê) or su(1, 1) Cartan-Weyl basis has recently been proven in [21], where it has been explicitly demonstrated that the sl(2; Ê) and su(1, 1) bialgebras, determined by the respective classical r-matrices, are isomorphic. It turns out that one can conveniently choose the following three non-equivalent basic r-matrices for the D = 3 Lorentz algebra o(2, 1) [21]: where the parameter α ∈ Ê + (and the prime again denotes generators from the su(1, 1) basis (43)). The first two r-matrices describe the q-analogs of the sl(2; Ê) and su(1, 1) algebras, satisfying the following modified YB equation where the plus sign corresponds to r = r ′ st (the standard r-matrix of su(1, 1)) and the minus to r = r st (the standard r-matrix of sl(2; Ê)). The remaining r-matrix r J provides the Jordanian deformation of sl(2; Ê). We should also mention that the choice to employ two particular r-matrices in the sl(2; Ê) Cartan-Weyl basis and one in the su(1, 1) basis is inspired by the presence of explicit quantization procedure [6].
In terms of the chiral Cartesian basis {X i ,X i } introduced in (9), the reality conditions (52-56) simplify to (in particular, the above conditions are identical for different decompositions (54-56) of o(2, 2)). Similarly, one finds that in terms the orthogonal basis {J i , K i } introduced in (12) the reality conditions (57) become A natural basis for a real Lie algebra is either purely Hermitian or purely anti-Hermitian. Since choosing the Hermitian convention leads to the appearance of imaginary units in the algebra brackets, we will use the anti-Hermitian bases.
Finally, let us express the reality conditions (58) in terms of the standard orthogonal basis {M AB }, introduced in (14). We obtain For both o(4) and o(3, 1) it shows that J i = ǫ ijk M jk generate the o(3) algebra, corresponding to the IW contraction chosen along the fourth axis and the spacetime metric (1, 1, 1, −1) in the latter case.
As it was mentioned at the end of the previous Section, the IW contractions performed along other axes do not always lead to the equivalent result. In the case of the o(3, 1) real form adjusted to the IW contraction along the third axis (and analogously for the first or second one) we have J ′ p = 1 2 ǫ pqr M qr , K ′ p = M p3 , p, q, r = 1, 2, 4 and therefore (62) (with the same metric as above) gives the reality conditions   17), (27) and (22)), as well as completely exclude certain r-matrices. We collect all of this information in Table I. Looking at the o(4; ) rmatrices we may also observe that each of them is composed of two types of terms: the ones that describe independent deformations of the o(3; ) andō(3; ) subalgebras (for the o(2, 1) real form such r-matrices are given by (50)); and the terms that mix o(3; ) andō(3; ) deformations. It is the latter ones that introduce novel features to the results for o(4; ) (and its real forms) with respect to the already discussed o(3; ) deformations. When we introduce the cosmological constant, the o(4 − k, k), k = 0, 1, 2 algebras can appropriately be treated as the D = 3 Euclidean and Lorentzian (anti-)de Sitter algebras. Therefore, in such a way we obtain from the results of [10] the complete classification of Hopf-algebraic deformations of the above relativistic symmetry algebras. We will further introduce the unified notation {J i , L i }, i = 1, 2, 3 for the inhomogeneous algebra o(3) with curved translations forming o(4) or o(3, 1), and {J µ , L µ }, µ = 0, 1, 2 for the inhomogeneous algebra o(2, 1) with curved translations forming o(3, 1) or o(2, 2). We also need to mention that in the remaining part of this paper we use the natural system of physical units c = = 1.
We begin with the real form o(4), which is treated as the D = 3 Euclidean de Sitter algebra, with Λ = R −2 > 0 (after the IW rescaling analogous to (15)). In the chiral Cartan-Weyl basis it is characterized by the reality conditions (52). As shown in Table I, there is only one allowed family of Hopf-algebraic deformations of o(4), associated with the r-matrix (27). Quantum IW contractions of this r-matrix lead to quantum D = 3 inhomogeneous Euclidean algebras.
We already showed at the end of Sec. II that it is enough to consider such contractions along the fourth axis of Ê 4 (corresponding to A = 4 in (13)), since the results for other axes would differ only by automorphisms. The considered basis {J i ,K i }, i = 1, 2, 3 is introduced via the transformation This rescaled orthogonal basis (as well as the analogous bases in Subsec. IV B and IV D) will be called the physical basis. In terms of J i and L i ≡K i , the undeformed brackets of the o(4) algebra have the form The r-matrix (27) for the considered real form is given by and in this case the parameters γ,γ ∈ Ê, η ∈ iÊ. As follows from the analysis in Sec. II, (66) has two inequivalent inhomogeneous contraction limits (if contraction is performed along the first or second axis, obtainingr III requires γ = −γ instead ofγ = γ)r where P i denote the Euclidean 3-momenta. The comparison with the Stachura classification of r-matrices for the D = 3 inhomogeneous Euclidean algebra will be presented in Subsec. V A.
The second considered real form of o(4; ) is o(3, 1), characterized by the reality conditions (53), which are in agreement with the spacetime metric (1, 1, 1, −1) (the convention chosen by us in Subsec. III C). The algebra o(3, 1) in D = 3 can be treated as either the (Lorentzian) de Sitter algebra (with Λ = R −2 > 0) or Euclidean anti-de Sitter algebra (with Λ = −R −2 < 0). This corresponds to taking either a spacelike or timelike direction as the axis associated with the generators rescaled by R −1 , along which the IW contraction (in the limit R → ∞) is later performed.
Let us first pick the fourth (i.e. timelike) axis and consider o(3, 1) as the Euclidean anti-de Sitter algebra. For the latter algebra there is no possibility to choose another axis. Consequently, the relation between the chiral Cartan-Weyl basis and the basis {J i ,K i } has the same form as (64), to wit Due to the reality conditions (53) the rotation generators J i are anti-Hermitian, whileK i 's (which are the D = 4 boost generators since K i = M i4 ) become Hermitian,K i =K i . As it was described in the previous Section, we need to move to the basis where all generators are anti-Hermitian, using the transformation The undeformed brackets of the o(3, 1) algebra in this physical basis have exactly the same form as (65) but with Λ > 0 being replaced by Λ < 0. It is known [9,20] that one may define four different families of Hopf-algebraic deformations of the real form o(3, 1), determined by the r-matrices r I , r II , r III and r IV [10] (see Table I). The first two r-matrices in the physical basis can be written as where χ ∈ iÊ and ς ∈ Ê, whileχ is eliminated by the relationχ = χ. Next, r III is given by where γ ∈ , η ∈ Ê and we had to impose the relationγ = −γ * . The remaining r IV has the form where γ, ς ∈ Ê.
The possible quantum IW contractions of these real r-matrices can be deduced from Sec. II and (for the considered real form and physical basis) they lead to the corresponding quantum inhomogeneous Euclidean algebras. Namely, r I and r II have the following inhomogeneous contraction limits r III has two inequivalent contraction limits (it is easy to notice thatγ ∈ Ê andγ ∈ iÊ) and for r IV we similarly havẽ We again refer the reader to Subsec. V A for a comparison with the known results of Stachura.
We will now consider the real form o(3, 1) as the D = 3 de Sitter algebra. To this end we choose the third spatial axis as the direction associated with the rescaled generators. The corresponding basis {J ′ p ,K ′ p }, p = 1, 2, 4 is introduced via the following transformation of the chiral Cartan-Weyl basis From (63) we can write down familiar brackets of the o(3, 1) algebra: (assuming the convention ǫ 012 = 1 and rising indices with the Lorentzian metric). When we consider o(3, 1) as the D = 3 de Sitter algebra, the list of allowed r-matrices naturally remains the same as in (70-72) but they are expressed in a different physical basis, namely (77). r I and r II are now written as where χ ∈ iÊ and ς ∈ Ê. r III acquires the form where γ ∈ , η ∈ Ê and we have the relationγ = −γ * . Finally, r IV is given by where γ, ς ∈ Ê.
Quantum IW contractions of these r-matrices lead to the quantum D = 3 Poincaré algebras. Such inhomogeneous contraction limits of r I and r II arer where P µ denote the Lorentzian 3-momenta; r III has the following contraction limits (withγ ∈ Ê andγ ∈ iÊ) and for r IV we havẽ wherer IV is obtained for ς = 2γ instead of ς = −2γ, as it was the case in (26). Furthermore, it turns out that the quantum IW contraction of o(3, 1) r-matrices along the first or second spatial axis leads to the set of r-matrices different than the one obtained for the third axis. This is because the considered r-matrices are not invariant under the rescaled o(3, 1) automorphisms. However, we may almost completely restrict ourselves to the contractions along the first axis (results for the second axis differ only by automorphisms, except in the cases of r I and r II , as we will mention), introducing another anti-Hermitian physical basis in which the o(3, 1) brackets preserve their form (78). Then instead of (79-81) we obtain the expressions which are equivalent to (79-81) under the o(3, 1) automorphism (J 0/2 → ∓R L 2/0 , J 1 → J 1 , L 0/2 → ±R −1 J 2/0 , L 1 → −L 1 ). However, taking the R → ∞ limit leads to results inequivalent to (82-84).
Performing the rescaling of deformation parameters as it was done forr I -r IV in (82-84), we find that (86) have only one contraction limit (equivalent tor III in (83)) r a III (γ) = iImγ P 0 ∧ P 1 .
On the other hand, the alternative rescaling (under the condition Imγ = 0 in the case of r III ) leads to the following set of new contraction limitŝ . The quantum IW contraction along the second axis can be performed in the analogous way but r I will now be expressed only in terms of the J µ generators and therefore both before and after the contraction (without the necessity of rescaling χ!) we have Moreover, in this caser a II does not exist (since r a II withχ = −χ is forbidden for o(3, 1)). All obtained contraction limits will be compared with the Stachura classification of r-matrices for the D = 3 Poincaré algebra in Subsec. V B.
Finally, in this Subsection we investigate the (Kleinian) rotation algebra o(2, 2), i.e. the D = 3 anti-de Sitter algebra, with Λ = −R −2 < 0. The corresponding reality conditions (62) are in agreement with the spacetime metric (1, −1, 1, −1) (as discussed in Subsec. III C). The IW contraction of o(2, 2) leads to the Poincaré algebra but in principle there are two distinct possibilities: the contraction can be performed either along a timelike (e.g. the fourth) axis or spacelike (e.g. the third) axis, giving us the o(2, 1) or o(1, 2) algebra, respectively. In the absence of a deformation they differ only by a trivial change of the metric signature. As we will discuss, for deformed algebras it is actually sufficient to consider quantum IW contractions along the fourth and second axis, which lead to deformed Poincaré algebras with the metric (−1, 1, 1), as it is also the case in our convention for o(3, 1) contracted along a spacelike axis.
In the Cartan-Weyl basis the o(2, 2) algebra can arise as one of three different real forms of o(4; ), which were presented in (54-56). Let us first consider the set of reality conditions (54) (of the real form denoted as o ′′ (2, 2) in [10] but in this paper asô(2, 2)). We first calculate quantum IW contractions along the fourth axis. For the third axis the results differ only by automorphisms and a change of the metric signature (corresponding to the multiplication of all generators by −1). The relation between the chiral Cartan-Weyl basis and the basis {J i ,K i } has again the form (64), namely The reality conditions (54) determine that J 2 andK 2 are anti-Hermitian generators, while J 1/3 ,K 1/3 are Hermitian. The physical basis, in which all generators are anti-Hermitian, is now defined via the transformation 9 9 For the IW rescaling along the third axis it would be instead , where ± in the formulae for J 1/2 and L 1/2 allows to recover the (1, −1, −1) signature.
It has been shown (see Table I) thatô (2,2) is the only real form of o(4; ) that inherits all possible Hopf-algebraic deformations of the latter, given by the r-matrices r I , r II , r III , r IV and r V . Furthermore, in this case all deformation parameters are imaginary, χ,χ, ς, γ,γ, η, ρ ∈ iÊ. The first two r-matrices in the physical basis (92) become and the remaining three are Sec. II once again shows us what are the possible quantum IW contractions of these r-matrices. Namely, r I and r II have the following inhomogeneous contraction limits Each of the remaining r-matrices has two independent contraction limits. r III leads tõ and r V tor See Subsec. V B for a comparison of these results, as well as the cases of o ′′ (2, 2) and o ′ (2, 2) discussed below, with the known classification of r-matrices. The only subtlety for contractions along the third axis is thatr IV is then obtained under the condition ς = 2γ. Similarly as it was the case for o(3, 1), the quantum IW contraction ofô(2, 2) r-matrices along the first or second spatial axis leads to different set of r-matrices than above. We restrict to the contraction along the second axis (results for the first axis differ only by automorphisms and a change of the metric signature, except the case of r I , as we will show), introducing the anti-Hermitian physical basis where γ,γ ∈ Ê and η ∈ iÊ. Sec. II shows us that two D = 3 Poincaré r-matrices obtained via the quantum IW contraction of (106) arer Furthermore, if we choose instead the second axis, using the anti-Hermitian basis (98), we still obtain the same contraction limits as (107) (up to some signs, which can be changed via automorphisms), i.e.
The above results are also equivalent to what is obtained for the third or first axis, although in both these cases the r-matrixr III /r a III requires the relationγ = −γ instead ofγ = γ. The last pseudo-orthogonal real form of o(4; ) is o ′ (2, 2) (as denoted in [10]), characterized by the reality conditions (56). The transformation from the chiral Cartan-Weyl basis is introduced by (cf. (61)) o ′ (2, 2) has two possible Hopf-algebraic deformations, given by the r-matrices r III and r V . In the physical basis (92) they acquire the following form where γ, η, ρ ∈ Ê andγ,χ ∈ iÊ. For this particular real form there is an essential difference with respect to the contractions of o(4; ) r-matrices along the fourth axis discussed in Sec. II. Namely, both r III and r V from (110) have only one contraction limit As one can notice, we obtain no r-matrices that depend on the J i generators. Furthermore, if we perform the contraction along the second axis, using the anti-Hermitian basis (98), it leads to identical contraction limits as (111) (up to some irrelevant sign changes), i.e.
r a III (γ,γ,η) = − iγ 2 P 1 ∧ P 2 +γ 2 The situation is the same for contractions along the third or first axis. In the case of D = 3 inhomogeneous Euclidean algebra o(3) ⊲ < T 3 , we identify the following relation between the notation of [12] and ours: as well as we replace the names of parameters α and ρ by β and ̺, respectively. Then we rewrite the complete classification (up to an algebra automorphism) of Hermitian r-matrices for the D = 3 inhomogeneous Euclidean algebra from Subsec. 3.2 of [12], which includes where β ∈ {0, 1}, ̺ ≥ 0, β = 0 ⇔ ̺ = 0 and θ ij = −θ ji ∈ Ê, i, j = 1, 2, 3 (the latter can be further restricted via automorphisms). Let us note that the part c ∈ L ∧ L of a classical r-matrix of a Lie alebra L ⊲ < T vanishes for all r-matrices (114).  A comparison with the results of Subsec. IV A and IV B shows (cf. Table II): • the r-matricesr I ,r II ,r III andr IV in (73-75), as well asr III in (67) depend only on the translation generators and therefore they belong to the type r 3 above; •r IV in (75) is proportional to r 2 ; •r III in (67) and (74) are equivalent to r 1 (with θ ij = 0 and up to the automorphism (J 1 → −J 1 , J 3 → −J 3 , P 2 → −P 2 ) or (J 2 → −J 2 , J 3 → −J 3 , P 1 → −P 1 )).
Our (parametrized families of) r-matrices can be obtained by multiplying an appropriate expression from (114) by an imaginary parameter and using the automorphism in the r 1 case.
As in the previous Subsection, our (parametrized families of) r-matrices are constructed by multiplying an appropriate expression from (117-119) by an imaginary parameter and acting with the automorphisms described above.
VI. D = 3 POINCARÉ r-MATRICES AND 3D GRAVITY The major reason why the (pseudo-)orthogonal groups considered in this paper are of physical interest is that they play the role of gauge groups in 3D gravity. Namely, in the Chern-Simons formulation of gravity in 2+1 dimensions, the local gauge group describes local isometries of spacetime and is given by the D = 3 Poincaré or (anti-)de Sitter group for vanishing, negative or positive cosmological constant, respectively. The above formalism extends also to the Euclidean version of the theory, where the corresponding gauge groups are the inhomogeneous Euclidean and Euclidean (anti-)de Sitter groups [22,23]. In this Section we restrict ourselves mainly to the physically more important Lorentzian signature (as can be seen from Subsec. V A, the situation in the Euclidean case is much simpler).
In order to formulate the Chern-Simons theory of classical gravity, we first introduce the gauge field A that is a Cartan connection with values in the appropriate local isometry (Lie) algebra. In terms of the physical basis {J µ , L µ } used in the previous Sections (in the Poincaré case L µ become P µ ), the gauge field is constructed as follows where e µ and ω µ are the dreibein and spin connection one-forms, respectively. The Chern-Simons action 10 is equivalent to the Einstein-Hilbert action of general relativity in 2+1 dimensions 11 if the Ad-invariant bilinear form (., .) is defined in terms of the gauge algebra generators as 12 [23] ( The Chern-Simons theory is a topological one and thus gravity in 2+1 dimensions does not have any dynamical degrees of freedom. In the Hamiltonian picture, after singling out the time direction, the action (121) contains two terms: the kinematical one, defining the symplectic structure, which is directly related to the bilinear form (122), and the constraint taking the form of the requirement that the curvature of the connection A vanishes on constant time surfaces Σ, (even for non-zero Λ, since it is not the Riemannian curvature of Σ). It follows that the action (121) describes a theory of flat connections on a two-dimensional manifold (Riemann surface) Σ. The punctures on a Riemann surface are interpreted as point particles, each labeled by its mass and spin. If such punctures are present, the right hand side of (123) becomes the sum of delta functions at the positions of particles, each one multiplied by a gauge algebra element parametrized by the particle's mass and spin. Another way to introduce (topological) degrees of freedom is via the nontrivial topology of a Riemann surface, with some number of handles, which do not modify (123) but imply additional continuity conditions on A (see [24] and [25][26][27][28] for details). In order to quantize such a theory we need to know its Poisson (or symplectic) structure. The symplectic structure can be derived from (121) and has the form This symplectic structure is invariant under the gauge transformations and one has to compute the symplectic form not on the space of the gauge potentials but on the space of their gaugeequivalent classes. At this point the r-matrices associated with the gauge group become relevant. The celebrated Fock-Rosly construction provides the auxiliary Poisson structures in the case when spacetime has the topology of R × S, where the space S is an oriented, closed two-dimensional manifold (the spatial infinity can be added as an from a Drinfeld double of the D = 3 Lorentz algebra (what is explained below). Furthermore, the r-matrices (129) satisfy the following set of Yang-Baxter equations We see thatr 2 ,r 6 andr 7 are FR-compatible, with α =γ/ √ 3 in the formula (126), whiler 3 is FR-compatible only if α = iγ/ √ 3, as it has recently been considered in [32] (it remains to be verified whether choosing α to be imaginary leads to a physically meaningful theory). It turns out that the terms ofr 2 ,r 3 andr 6 proportional toη andς are not involved in generating inhomogeneity in the Yang-Baxter equations, necessary for the FR-compatibility. Let us also observe (see Table III) that quantum IW contractions of r I and r II , which satisfy the classical Yang-Baxter equation [10], lead to the r-matricesr 1 ,r 5 andr 4 or the Stachura class r 8 (see (119)), also satisfying the classical Yang-Baxter equation. The case of r V is more peculiar: it satisfies the modified Yang-Baxter equation, while its contractionsr 5 and r 8 satisfy the classical one. Our list of FR-compatible D = 3 Poincaré r-matrices agrees with the most complete previous classification, which was given in [16] (in the case of the generalized bilinear form (122) and for an arbitrary value of the cosmological constant); however, since we find that the values of parameters are not constrained for the FR r-matricesr 2 ,r 6 andr 7 , our results appear to be more general.
Let us now discuss a special class of the classical r-matrices, called the Drinfeld double r-matrices, which all satisfy the Fock-Rosly conditions [18]. The Drinfeld double algebra is a 2d-dimensional extension of a d-dimensional Lie algebra, with a basis (Y 1 , . . . , Y d ; y 1 , . . . , y d ) such that We note that to a given Lie algebra one can in principle attach many inequivalent Drinfeld double structures. For example, for all 6-dimensional real Lie algebras there exist 22 inequivalent Drinfeld doubles [33]. The structure of a Drinfeld double allows to define the non-degenerate, symmetric, Ad-invariant bilinear form hence the Lie algebra g * generated by {y i } becomes a dual of g generated by {Y i }. The unique canonical r-matrix associated with a Drinfeld double has the form where Ω is the quadratic split Casimir of the universal enveloping algebra of (131). It can be shown by an explicit calculation that (133) satisfies the Fock-Rosly conditions. The r-matrices associated with all possible Drinfeld double structures on the D = 3 Poincaré algebra were recently classified in [18,34], where it was found that there are eight inequivalent D = 3 Poincaré Drinfeld doubles. In terms of the classification from Subsec. V B, four of them lead to the r-matrices of the type r 6 with ̺ = ±1 (and appropriate θ µν ), two have the r-matrix of the type r 2 with ̺ = β = 1 (and appropriate θ µν ), and the remaining two Drinfeld double r-matrices are of the types r 1 with β = 1 and r 7 , respectively.
In the set of r-matrices that we obtained via quantum IW contractions, r 1 is replaced byr 1 , which is not associated with the Drinfeld double (and also not FR-compatible, cf. (130)). Further, if we taker 2 withη =γ (or evenη = 0), as well asr 6 withς = ±2γ, it violates the Drinfeld double structure but not the FR-compatibility. The most peculiar case isr 3 , which can not be associated with a Drinfeld double for any set of values of its parameters but in principle [32] may be FR-compatible (i.e. satisfy the condition (128)), although in the literature there is no agreement about the latter claim [16].
Let us also comment, for the sake of completeness, on the relation between the known list [17] of Drinfeld double r-matrices in the case of D = 3 (anti-)de Sitter algebra and the list of all possible classical r-matrices associated with these algebras, which is provided by us in Subsec. IV C-IV D by rewriting the results of [10] in the corresponding physical bases. We again have to consider all our r-matrices as endowed with real-valued parameters. If we use the notation of [17], de Sitter r-matrices r ′ A and r ′ B are equivalent (under certain automorphisms that do not mix J i with L i ) to our r IV in (81) with ς = ±2γ = 1, while r ′ C and r ′ D are respectively equivalent to r a III in (89) with η = 2γ = 1 and with γ = µ 2 −1 2µ + i, η = µ 2 +1 µ , µ > 0. Similarly, the anti-de Sitter r-matrix r ′ E (in [17]) is equivalent (under certain automorphisms that do not mix J i with L i ) to our r IV in (94) with ς = −2γ = −1, while r ′ F and r ′ G are equivalent to r III in (94) (or r a III in (100)) with η = 2γ = 2γ = 1 and with 2γ = 1, 2γ = η 2 , −1 < η < 1, respectively.
In another approach to the quantization of 3D gravity [35], the algebra of symmetries of quantum 3D spacetime was derived with the help of the Loop Quantum Gravity quantization scheme. The algebra is introduced as a commutator algebra of the appropriately smeared quantum constraint operators, obtained from the classical Poisson algebra of Poincaré symmetries of flat spacetime. It turns out that such an algebra leads to a Hopf algebra, whose non-trivial coproducts are derived from the quantum R-matrix. The quantum R-matrix has been calculated within the particular regularization scheme, employed in the process of loop quantization, and it is the one corresponding to the classical r-matrix (66) forγ = −γ, η = 0 (describing the D = 3 Euclidean κ-de Sitter deformation). In this framework the presence of different r-matrices may lead to different Loop Quantum Gravity quantizations, associated with different regularization schemes for the non-commutative connections.
Finally, in [36,37] a different type of contraction of the D = 3 de Sitter symmetries has been considered, using a local decomposition of the gauge group O(3, 1) ∼ = SL(2; ) = SL(2; Ê) ⊲ ⊳ AN(2) and making the SL(2; Ê) generators Abelian via the IW contraction procedure. It could be interesting to apply such a type of contraction to the Hopfalgebraic deformations of symmetries.

VII. CONCLUSIONS AND OUTLOOK
The main aim of this paper was to check whether all classical r-matrices (up to an automorphism) for D = 3 inhomogeneous Euclidean and D = 3 Poincaré algebra, derived by Stachura as solutions of the homogeneous or modified Yang-Baxter equation [12], can be obtained as well via quantum IW contractions of classical r-matrices for the real (pseudo-)orthogonal algebras o(4 − k, k), k = 0, 1, 2. It turns out that this is indeed true -the contracted r-matricesr 1 -r 6 (cf. (129)) do not contain certain terms present in the Stachura classes r 1 -r 6 (cf. (116-118)) but these terms are actually equivalent to r 7 and r 8 . In Sec. V, we explicitly write down the o(3) ⊲ < T 3 and o(2, 1) ⊲ < T 2,1 automorphisms that are necessary in order to connect our formulae with the ones provided by Stachura. Let us stress that the success of this work is based on the recently obtained complete classification of r-matrices for o(4 − k, k) (k = 0, 1, 2), in the unified setting of the o(4; ) algebra [10]. We also recall that the IW contraction parameter R in the context of classical and quantum 3D gravity (with the Lorentzian or Euclidean metric signature) directly corresponds to the cosmological constant. As we discussed in detail in the previous Section, our classification of r-matrices for D = 3 Poincaré and (A)dS algebras (as well as for the D = 3 inhomogeneous Euclidean and Euclidean (A)dS algebras) contains an extension of earlier results [16,18] for r-matrices compatible with 3D gravity.
There are the following four problems that are worth to be explored in future investigations.
• The most obvious next step is to describe the quantizations of symmetries that are generated by all D = 3 classical r-matrices (129), i.e. to define the corresponding Hopf algebras. Since all o(4 − k, k) r-matrices have already been quantized in the Cartan-Weyl basis in [10], the deformed D = 3 inhomogeneous Euclidean and D = 3 Poincaré Hopf algebras can be obtained simply by introducing physical bases of the algebras and performing suitable quantum IW contractions of the Hopf-algebraic formulae presented in that paper.
• It is still not completely clear what is the actual role of different r-matrices in 3D gravity: whether do they lead to different physical theories or, effectively, reduce to the only one model when expressed in terms of suitably chosen gauge invariant observables.
• In order to study quantum symmetries of 4D quantum gravity models, one has to consider the classical rmatrices for o(5 − k, k) (k = 0, 1, 2) algebras as well as their quantum IW contractions to the o(4) ⊲ < T 4 and o(3, 1) ⊲ < T 3,1 r-matrices. Unfortunately, the complete classification of r-matrices describing deformed D = 5 rotations is still unknown (for some effort in this direction see [38]). As we already mentioned, for the D = 4 Poincaré algebra the most extensive list has been given a long time ago in [9] (see also [39]).
Concluding, we hope that the knowledge of different quantum-deformed D = 3 and D = 4 (super)symmetries will be helpful in the construction of consistent and physically reliable quantum 3D and 4D (super)gravity models.