Möbius groups over general fields using Clifford algebras associated with spheres
The space H(2,\({\mathbb{C}})\) of 2-by-2 Hermitian matrices is isometric to Minskowski space \({\mathbb{R}}^{1,3}\). This is commonly used to exhibit the group SL(2,\({\mathbb{C}})\) as a twofold cover of the identity component of the Lorentz group O(1,3). That the Hermitian matrices of H(2,\({\mathbb{C}})\) also represent equations of circles in the Euclidean plane \(E^ 2\) leads to the group PSL(2,\({\mathbb{C}})\) as the Möbius group - a group of complex point transformations that send ``circles'' to ``circles'' - of \(E^ 2\). Clifford algebras naturally arise in the construction of covers of the orthogonal group of a vector space V by spin groups. The authors here extend the classical case \(V={\mathbb{C}}\)- using a new approach which is quite different from that of Vahlen as revived recently by Ahlfors (1986) and others - to the case of a finite-dimensional vector space V over an arbitrary commutative field of characteristic \(\neq 2\) as follows: the equation of a ``sphere'' is determined by a nondegenerate quadratic form of V (instead of by the Hermitian form associated with \({\mathbb{C}})\); the cover of the full orthogonal group of V is constructed by using the whole Clifford algebra of the space of equations of spheres.
Möbius group of a finite- dimensional vector space over a general commutative field
scientific article; zbMATH DE number 4151235
scientific article