The Lorentz–Möbius group PSO(1,d)

Abstract

The Lorentz group PSO(1, d) is the linear isometry group of Minkowski space–time, isomorphic to the Möbius group of direct hyperbolic isometries. The chapter briefly introduces Lie algebras and associated notions in the context of matrices: Lie groups, adjoint representations, the Killing form and Lie derivatives. The chapter starts with subalgebras of the Lie algebra of square matrices and use the exponential of matrices to define the associated Lie group. Then Minkowski space and its pseudo-metric are introduced, with the Lorentz frames on which PSO(1, d) acts both on the right and on the left. The chapter then introduces the affine subgroup A d , and determine the conjugacy classes of PSO(1, d). The hyperbolic space H d is the unit pseudo-sphere of the Minkowski space R1,d . The Iwasawa decomposition of PSO(1, d) yields Poincaré coordinates in H d , while the hyperbolic metric is related to the Cartan decomposition.