Vector-Parameter Forms of SU(1,1), SL(2,R) and Their Connection to SO(2,1)

Abstract

The Cayley maps for the Lie algebras su(1, 1) and so(2, 1) converting them into the corresponding Lie groups SU(1, 1) and SO(2, 1) along their natural vector-parameterizations are examined. Using the isomorphism between SU(1, 1) and SL(2, R), the vector-parameterization of the latter is also established. The explicit form of the covering map SU(1, 1) → SO(2, 1) and its sections are presented. Using the so developed vector-parameter formalism, the composition law of SO(2, 1) in vector-parameter form is extended so that it covers compositions of all kinds of elements including also those that can not be parameterized properly by regular SO(2, 1) vectorparameters. The latter are characterized and it is shown that they can be represented by SU(1, 1) vector parameters with pseudo length equal to minus four. In all cases of compositions inside SO(2, 1), criteria for determination of their type (elliptic, parabolic, hyperbolic) have been presented. On the base of the vector-parameter formalism the problem of taking a square root in SO(2,1) is solved explicitly. Also, an analogue of Cartan's theorem about the decomposition of orthogonal matrix of order n into product of at most n reflections is formulated and proved for the subset of hyperbolic elements of the group of pseudo-orthogonal matrices from SO(2, 1).

A Review in Ermakov Systems and Their Symmetries

“…The invariants eliminate one of the angular variables and substituting it with an Ermakov invariant. This is due to the symmetry SO(2,1) given by (8) although one should firstly to accommodate the representation of this Lie Algebra [22] to the variables of the real physical system. The important property of the SO(2,1) invariance must be always present regardless of the dimension.…”