标题： 时空来自对称性：从庞加莱-霍普夫代数得到的莫尔平面 显示英文标题

标题： Space-time from Symmetry: The Moyal Plane from the Poincare-Hopf Algebra

摘要： 我们展示如何从庞加莱群的余乘法的形变出发，使用德林费尔德扭变得到时空上函数的非交换乘积。 因此很容易看出，时空上的函数的交换代数（R^4）可以被识别为在洛伦兹群右作用下不变的庞加莱群上的函数集合，只要我们对庞加莱群使用标准的余乘法。 我们通过将这一结果推广到扭曲余乘法的情况，得到了非交换Moyal平面的结果。 这种扩展并不简单，涉及到上同调特征。 众所周知，时空代数决定了流形微分同胚群上的余乘法。 我们现在看到这种影响是相互的：它们紧密相连。 显示英文摘要

摘要： We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare' group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare' group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare' group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features. As is known, spacetime algebra fixes the coproduct on the dffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.