In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that lies in a subalgebra that has the norm composition property and that is either associative (Clifford algebra case) or at least alternative (octonionic case). The orthonormal frames are elements of the spin group Spin\((n+1)\). We round off by relating the nature of the orthonormal frames to the associated Möbius transformation which are related to SO(9, 1) in the octonionic case and to the Ahlfors–Vahlen group in the case of a Clifford algebra.

中文翻译：

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在任意维数的Octonion和Clifford代数中重新讨论了共形映射

在本文中，我们在正八调和Clifford代数的上下文中重新审视了高斯意义上的共形概念。我们使用偏态方程组和偏微分方程组，使用特殊的正交函数在以前的特殊四元数设定中扩展了连续函数的特殊正交集，扩展了共形性的表征。目的是描述该特征可以精确地扩展到哪个高维代数以及在什么情况下。事实证明至关重要的是，这种表征需要定义域位于具有范数组成性质的子代数中，并且该子代数是关联的（Clifford代数的情况）或至少是替代的（张调的情况）。正交框架是旋转组Spin \（（n + 1）\）的元素。我们通过将正交框架的性质与相关的Möbius变换相关联来完成，在莫托乌斯变换中，该Obius变换与八种正离子情况下的SO（9，1）有关，而对于Clifford代数而言，它们与Ahlfors-Vahlen组有关。