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 $\mathfrak{sl}(2)$ 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.
AdS Poisson homogeneous spaces and Drinfel'd doubles
在Poisson-Lie群$G$上的Poisson齐性空间与经典双$G$的Lagrangian李子代数之间的对应关系被重新讨论，并在${\mathfrak g}=D(\mathfrak a)$是经典双$G$的情况下进行了详细的探索.我们应用这些结果给出了群$\mathrm{SL}(2,R)\cong\mathrm{SO}(2,1)$上某些共迷向的2d Poisson齐性空间，即2d anti de Sitter空间，2d双曲空间和3d Minkowski空间中的光锥的一个显式描述.我们展示了如何获得这些空间的商相对于一个泊松子群的三个不等价的李双代数结构之一的${sl}(2,R)$和作为一个共各向同性的其他。然后，我们构建家庭的共向泊松齐次结构的三维反德西特空间$\mathrm{AdS}_3$，并表明，由泊松子群的从属性是由一个三参数家庭的经典$r$ -矩阵的${so}(2,2)$，而非泊松子群的情况下，要多得多。特别是，我们提出了两个泊松均匀结构$\mathrm{AdS}_3$上产生的两个Drinfel的双结构$\mathrm{SO}(2,2)$。第一个将$\mathrm{AdS}_3$实现为$\mathrm{SO}(2,2)$与泊松子群$\mathrm{SL}(2,R)$的商，而第二个，扭曲$κ$-AdS变形的非对易时空，将$\mathrm{AdS}_3$实现为各向同性泊松齐性…
