三维引力场的四元数结构和Poisson-Lie结构：以宇宙学常数为形变参数
Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
摘要：
三维引力中产生的每一个局部等距群都可以看作是环上的单位（分裂）四元数群，它依赖于宇宙学常数。在本文中，我们解释和证明这一说法，并使用它作为一个统一的框架，研究泊松结构与局部等距群。我们证明了，除了具有正宇宙学常数的欧几里德签名，在所有的情况下，局部等距群都具有经典对偶的Poisson-Lie结构。我们计算了因子群对彼此的修饰作用，并找到了SU（2）和SL（2，R）的辛叶的一个简单而统一的描述。我们还计算泊松结构的对偶泊松李群的局部等距群和海森堡双打;一起，他们确定的泊松结构的相空间的三维重力在所谓的组合描述。
英文摘要：
Each of the local isometry groups arising in 3d gravity can be viewed as the group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement, and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson-Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, amongst others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,R). We also compute the Poisson structure on the dual Poisson-Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.
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