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2021 New Parameterizations of $\mathrm{SL}(2,\mathbb{R})$ and Some Explicit Formulas for Its Logarithm
Tihomir I. Valchev, Clementina D. Mladenova, Ivaïlo M. Mladenov
J. Geom. Symmetry Phys. 60: 65-81 (2021). DOI: 10.7546/jgsp-60-2021-65-81

Abstract

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\mathbb{R}) ≅ \mathrm{SL}(2,\mathbb{R})$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\mathbb{R}) \rightarrow \mathrm{SL}(2,\mathbb{R})$, we have found a few explicit formulas for the logarithm of the matrices in these groups.

Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

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Tihomir I. Valchev. Clementina D. Mladenova. Ivaïlo M. Mladenov. "New Parameterizations of $\mathrm{SL}(2,\mathbb{R})$ and Some Explicit Formulas for Its Logarithm." J. Geom. Symmetry Phys. 60 65 - 81, 2021. https://doi.org/10.7546/jgsp-60-2021-65-81

Information

Published: 2021
First available in Project Euclid: 4 August 2021

Digital Object Identifier: 10.7546/jgsp-60-2021-65-81

Rights: Copyright © 2021 Institute of Mechanics, Bulgarian Academy of Sciences

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