Vahlen's group of Clifford matrices and spin-groups
It is well known that the groups of orientation preserving isometries of the upper half-plane and half-space models of the hyperbolic plane and of the three-dimensional hyperbolic space can be identified with SL(2,\({\mathbb{R}})\) and SL(2,\({\mathbb{C}})\), respectively. \textit{K. T. Vahlen} [Math. Ann. 55, 585-593 (1902)] extended this result to arbitrary dimensions and described the orientation preserving isometries of the upper half-space model of n-dimensional hyperbolic space in terms of suitable two-by-two matrices over a Clifford algebra over \({\mathbb{R}}\). We modify Vahlen's construction appropriately and introduce the Vahlen group \(SV_ n(K,Q)\) for a field K of characteristic \(\neq 2\) and the Clifford algebra associated with a nondegenerate quadratic form Q over the n- dimensional K-vector space E. Then the Vahlen group turns out to be isomorphic to a certain spin group. In the special case \(K={\mathbb{R}}\) and Q negative definite we investigate the relation of the Vahlen group with various groups known from hyperbolic geometry. One main aim of this work is the following one. As is well known from the work of \textit{J. Dieudonné} [Acta Math. 87, 175-242 (1952; Zbl 0049.025)] there exist many ''exceptional'' isomorphisms between classical groups in low dimensions. We prove here that most isomorphisms from Dieudonné's list can easily be obtained from the isomorphism of the Vahlen group with the associated spin-group. This simplifies and unifies the proofs of the known results on exceptional isomorphisms between classical groups.
groups of orientation preserving isometries
upper half-space model of n- dimensional hyperbolic space
exceptional isomorphisms between classical groups
scientific article; zbMATH DE number 3987561
scientific article