Conformal mappings revisited in the octonions and Clifford algebras of arbitrary dimension

TL;DR

This paper extends the concept of conformality to higher-dimensional algebras like octonions and Clifford algebras, characterizing it through PDEs and differential forms, and relating it to spin groups and M"obius transformations.

Contribution

It generalizes conformality characterization to non-associative and higher-dimensional algebras, identifying conditions for its validity.

Findings

Conformal mappings can be characterized via PDEs in certain higher algebras.

The characterization depends on the domain lying in subalgebras with norm composition.

Connections between orthonormal frames, spin groups, and M"obius transformations are established.

Abstract

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 M\"obius 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.