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Non-standard quantum so(3,2) and its contractions

A full (triangular) quantum deformation of so(3,2) is presented by considering this algebra as the conformal algebra of the 2+1 dimensional Minkowskian spacetime. Non-relativistic contractions are analysed and used to obtain quantum Hopf structures for the conformal algebras of the 2+1 Galilean and Carroll spacetimes. Relations between the latter and the null-plane quantum Poincar\'e algebra are studied.

The two non-relativistic limits of the Poincaré algebra P(2 + 1) are the Galilean G(2 + 1) and Carroll C(2 + 1) algebras which correspond, in this order, to a speedspace and a speed-time contraction of P(2 + 1) [1]. These contraction processes can be implemented at a conformal level in order to obtain the conformal algebras of the 2+1 Galiean and Carroll spacetimes [2], here denoted G 3 and C 3 , by applying the following mappings to the generators of M 3 : Once these transformations have been performed on the Lie brackets (1.1) we get after the limit ε → 0 the commutation relations of G 3 and C 3 . For the Galilean case, the non-vanishing commutators are: The conformal Galilean Lie algebra G 3 is isomorphic to t 6 (so(2) ⊕ so(2, 1)) (the structure of this type of algebras is described in [3,4]). We also have a sequence of subalgebras G(2 + 1) ⊂ G(2 + 1) ⊂ G 3 , where G(2 + 1) is the 2+1 Galilean algebra with dilation.
Non-standard quantum deformations for these conformal algebras haven been already obtained for the 1+1 case [5,6] being inspired in the well known nonstandard or Jordanian quantum sl(2, IR) algebra [7,8,9]. Their underlying Lie bialgebras come from classical r-matrices which satisfy the classical Yang-Baxter equation. The results so obtained show that non-standard deformations are naturally adapted to a conformal basis, although for the particular case of the quantum Poincaré algebra an alternative interpretation has been considered in a null-plane framework [10].
An analysis of non-standard conformal Lie bialgebras for higher dimensions can be found in [11] where the deformation parameters are interpreted as fundamental mass parameters. However, to our knowledge, no explicit non-standard quantum Hopf structure for the conformal algebra further the 1+1 case (so(2, 2)) has been given yet. In this letter we solve this problem for a precise non-standard quantum deformation of M 3 . To begin with we consider in section 2 the 2+1 conformal Lie bialgebra which generalizes that introduced in [5, 6], and we study its possible nonrelativistic Lie bialgebra contractions. It is shown that there is a unique possible (coboundary) contraction for each conformal bialgebra G 3 and C 3 . In section 3 the quantum Hopf structure of M 3 is introduced and those corresponding to G 3 and C 3 are obtained via contraction in section 4. All of them have as Hopf subalgebra the corresponding kinematical algebra together the dilation generator, but not the kinematical algebra itself (a feature already pointed out in [12]); hence only the Weyl subalgebra is promoted to a Hopf subalgebra. The quantum conformal Carroll algebra is related with the 3+1 null-plane quantum Poincaré algebra; this fact allows to get its universal R-matrix from the results given in [13].

Conformal Lie bialgebras
The classical r-matrices underlying the non-standard quantum deformations of sl(2, IR) ≡ P 0 , C 0 , D and so(2, 2) ≡ P 0 , P 1 , K 1 , C 0 , C 1 , D [5,6] can be written as where z is the deformation parameter. The generalization of these expressions for which fulfills the classical Yang-Baxter equation (the presence of the term J ∧ P 2 is essential for this purpose). The cocommutator of a generator X is obtained as In order to analyse the possible non-relativistic contractions of this conformal Lie bialgebra one has to consider the Lie algebra transformations (1.2) and (1.3) together a mapping on the deformation parameter: z → ε −n z where n is any real number [14]. The result is that there exists a unique minimal value n 0 of n for each contraction from M 3 to G 3 and C 3 in such way the classical r-matrix and the cocommutators do not present divergencies: For n > n 0 the contracted r-matrix and cocommutators go to zero and for n < n 0 they diverge.
The classical (non-standard) r-matrix and cocommutators of the conformal Galilean Lie bialgebra so obtained are given by: and for the Carroll case we get: It is worth remarking that in each of the above Lie bialgebras the corresponding Weyl subalgebra {J, P µ , K i , D} is preserved at a bialgebra level. Note also that the cocommutator of D coincides with the classical r-matrix.

Quantum conformal Hopf algebra
We proceed to introduce the Jordanian quantum deformation of the conformal Minkowskian bialgebra, U z (M 3 ), in two steps. Firstly we close the Hopf structure for the Weyl subalgebra, and secondly we complete the quantum deformation with the expressions involving the special conformal transformations. None direct procedure as the deformation embedding method (applied for instance to the nullplane quantum Poincaré algebra [10]) seems to be useful now, so that we are forced to deduce formerly the coproduct ∆ by solving the coassociativity condition and by taking into account that the cocommutators (2.3) are related with the first order of ∆ on z, ∆ (1) , by means of Afterwards, the deformed commutation rules follow by imposing the coproduct to be an algebra homomorphism, this is, In the sequel we write down the coproduct and the commutation relations for U z (M 3 ); the counit is trivial and the antipode can be easily derived from these results so we omit them. a) Weyl Hopf subalgebra U z (P(2 + 1)).
deformed commutation relations of the non-standard quantum conformal Galilean algebra U z (G 3 ) are the remaining commutators are non-deformed and given by (1.4). On the other hand, the element is a trivial solution of the quantum Yang-Baxter equation since the four generators involved commute. Furthermore, it is easy to check that the property Similarly, the contraction U z (M 3 ) → U z (C 3 ) is provided by the mappings (1.3) and (2.5) applied on the conformal Hopf algebra U z (M 3 ) together the limit ε → 0. The coproduct and non-vanishing commutation relations of the quantum conformal Carroll algebra U z (C 3 ) are given as follows: It is rather remarkable that U z (C 3 ) can be shown to be isomorphic to the nullplane quantum Poincaré algebra [10] in the basis used in [13] by means of where the primed generators and deformation parameter correspond to the nullplane quantum Poincaré algebra. As a straightforward consequence the universal R-matrix for U z (C 3 ) (satisfying the quantum Yang-Baxter equation and relation (4.4)) reads

Concluding remarks
Summarizing, we have obtained a new quantum deformation of so(3, 2) and we have related three non-standard quantum conformal algebras via contraction processes, all of them containing the corresponding Weyl subalgebra as a Hopf subalgebra: For the contracted quantum algebras the universal R-matrices have been given in a factorized form. The expression of the R-matrix associated to U z (M 3 ) remains as an open problem.
From a mathematical point of view, the corresponding quantum deformation is equivalent to the one just studied via a simple redefinition of the generators. However both deformations exhibit different physical features which are stronger when contractions are carried out. More explicitly, the quantum conformal Galilean and Carroll algebras coming from (5.3) are no longer equivalent to those above obtained. For both of them the transformation of z would be z → ε −2 z (n 0 = 2) leading to the following classical r-matrices: Therefore the latter could not be related to the null-plane quantum Poincaré algebra. The analysis of these and further possibilities will be presented elsewhere.