New parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and some explicit formulas for its logarithm (Q1980478)

The exponential map for Lie groups has been extensively studied and it is neither injective nor surjective in general, as it is already the case for \(\mathrm{SL}(2,\mathbb R)\), the Lie group of invertible \(2\times 2\) real matrices. In the paper under review, the authors give a new parametrization of \(\mathrm{SL}(2,\mathbb R)\) and derive explicit formulas for the inverse of the exponential map. Moreover a geometric description of the field of values of matrices in \(\mathrm{SL}(2,\mathbb R)\) is given in terms of ellipses.