Vector-Parameter Forms of $\mathrm{SU}(1,1), \mathrm{SL}(2,\mathbb{R})$ and their Connection to $\mathrm{SO}(2,1)$

Abstract

The Cayley maps for the Lie algebras $\mathfrak{su}(1,1)$ and $\mathfrak{so}(2,1)$ converting them into the corresponding Lie groups $\mathrm{SU}(1,1)$ and $\mathrm{SO}(2,1)$ along their natural vector-parameterizations are examined. Using the isomorphism between $\mathrm{SU}(1,1)$ and $\mathrm{SL}(2,\mathbb{R})$, the vector-parameterization of the latter is also established. The explicit form of the covering map $\mathrm{SU}(1,1) \rightarrow \mathrm{SO}(2,1)$ and its sections are presented. Using the so developed vector-parameter formalism, the composition law of $\mathrm{SO}(2,1)$ 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 $\mathrm{SO}(2,1)$ vector-parameters. The latter are characterized and it is shown that they can be represented by $\mathrm{SU}(1,1)$ vector parameters with pseudo length equal to minus four. In all cases of compositions inside $\mathrm{SO}(2,1)$, 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 $\mathrm{SO}(2,1)$ 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 $\mathrm{SO}(2,1)$.