We will prove that there is a direct relationship between the Poincare subgroup of translations, and the group of tetrad transformations LB1 introduced in a previous paper. LB1 is the group composed of SO(1, 1) plus two kinds of discrete transformations. Translations have been extensively studied under the scope of gauge theories. By using the geometric structures built to prove this elementary result we will generalize it to the case of what we might call local translations. A special case of the latter is the Bondi–Metzner–Sachs subgroup of supertranslations. In order to accomplish this goal and since the group of translations is four-dimensional we will prove first that it is isomorphic to (⊗ LB1) 4 . In order to prove this claim we will introduce a system of differential equations involving several kinds of fields—Abelian, non-Abelian, spinor, gravitational. These fields will constitute the structure needed to build local tetrads of a new kind that allow for the proof to be carried out with simplicity. Results already obtained involving similar but not equal tetrads will be useful in our constructions and demonstrations. Translations and generalized translations isomorphic to tensor products of LB1 groups are not trivial results, because the LB1 group is composed of SO(1, 1) and two discrete transformations

中文翻译：

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局部庞加莱广义平移群与时空变换群之间的同构 (⊗ LB1)4

我们将证明庞加莱翻译子群与前一篇论文中介绍的四分体变换群 LB1 之间存在直接关系。LB1 是由 SO(1, 1) 加上两种离散变换组成的群。翻译在规范理论的范围内得到了广泛的研究。通过使用构建的几何结构来证明这个基本结果，我们将把它推广到我们可能称之为本地翻译的情况。后者的一个特例是超翻译的 Bondi-Metzner-Sachs 子群。为了实现这个目标，并且由于翻译组是四维的，我们将首先证明它与 (⊗ LB1) 4 同构。为了证明这一说法，我们将引入一个涉及多种场的微分方程组——阿贝尔、非阿贝尔、旋量、引力。这些字段将构成构建新型局部四分体所需的结构，从而可以简单地进行证明。已经获得的涉及相似但不相等的四分体的结果将在我们的构建和演示中有用。与 LB1 群的张量积同构的平移和广义平移不是无关紧要的结果，因为 LB1 群由 SO(1, 1) 和两个离散变换组成