The Poincaré group as a Drinfel’d double

Abstract

The eight nonisomorphic Drinfel'd double (DD) structures for the Poincaré Lie group in (2+1) dimensions are explicitly constructed in the kinematical basis. Also, the two existing DD structures for a non-trivial central extension of the (1+1) Poincaré group are also identified and constructed, while in (3+1) dimensions no Poincaré DD structure does exist. Each of the DD structures here presented has an associated canonical quasitriangular Poincaré r-matrix whose properties are analysed. Some of these r-matrices give rise to coisotropic Poisson homogeneous spaces with respect to the Lorentz subgroup, and their associated Poisson Minkowski spacetimes are constructed. Two of these (2+1) noncommutative DD Minkowski spacetimes turn out to be quotients by a Lorentz Poisson subgroup: the first one corresponds to the double of sl(2) with trivial Lie bialgebra structure, and the second one gives rise to a quadratic noncommutative Poisson Minkowski spacetime. With these results, the explicit construction of DD structures for all Lorentzian kinematical groups in (1+1) and (2+1) dimensions is completed, and the connection between (anti-)de Sitter and Poincaré r-matrices through the vanishing cosmological constant limit is also analysed.

Drinfel’d double structures for Poincaré and Euclidean groups

“…Some of these Poisson Minkowski spacetimes have the extra property of being defined by an r-matrix that comes from a DD structure of the Poincaré Lie group and they are specially interesting due to their connections to (2+1) quantum gravity, as explained above. Here we sketch the main results given in [16], where a complete study of this matter was presented.…”

“…In this section we review the relevant outcomes of the recent work [16], in which the Poincaré case was thoroughly analyzed. From now on, we will work on a kinematical basis {J, K 1 , K 2 , P 0 , P 1 , P 2 } corresponding to the generators of rotations, boosts, time translation and space translations, respectively.…”

“…In Table 1 these DD r-matrices are presented together with the Poisson subgroup/coisotropy condition for each of them and the class of the classification [22] they belong to. [16] as well as the corresponding class in the Stachura classification. [16].…”

“…An interesting remark concerning DDs for the Poincaré group is that the (2+1)-dimensional situation is quite special, since in (3+1) and higher dimensions, the existence of any DD is precluded by the non-existence of a suitable non-degenerate bilinear form (2), both symmetric and associative. In (1+1)-dimensions there is obviously no DDs (given that the group has odd dimension), but it turns out that the non-trivially centrally extended Poincaré group in (1+1) dimensions has two non-isomorphic DD structures which were also explicitly constructed in [16].…”

Cayley--Klein Poisson Homogeneous Spaces

“…Nevertheless, the quantum deformation scheme is rather involved, not only because to raise the dimension requires cumbersome computations, but mainly due to the fact that in higher dimensions different possible non-equivalent deformations (and, therefore, PL structures and PHSs) can be considered. In this respect, see [4] and references therein for recent results on Lorentzian kinematical algebras.…”