Euler angle parametrization of the complex rotation group and its subgroups. I

Abstract

A parametrization of the n-dimensional complex rotation group O+(n, C) by a set of (1/2)n(n−1) complex Euler angles is obtained in exactly the same way as that for the n-dimensional real rotation group O+(n, R). Certain subgroups L̂(n, r) of O+(n, C) are then considered and found isomorphic with the ‘‘generalized Lorentz groups’’ L(n, r), of which the special case L(4, 1) is the usual Lorentz group L of special relativity. Distinguishing features, in the form of reality nature, of Euler angles of L̂(n, r) are obtained for r=1, 2, and it is proved that these lead naturally to the definition of ‘‘real’’ Euler angles of the corresponding L(n, r), which, of course, include the Lorentz group L.