New parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and some explicit formulas for its logarithm. (English) Zbl 1489.22014
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.
Reviewer: Salah Mehdi (Metz)
MSC:
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
parameterization of \(\mathrm{SL}(2,\mathbb{R})\)-matrices; logarithm of \(\mathrm{SL}(2,\mathbb{R})\)-matrices; field of valuesReferences:
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