Volume 10, Issue 2
A Three-Dimensional Model of $SL(2, \mathbb{R})$ and the Hyperbolic Pattern of $SL(2, \mathbb{Z})$

Yoichi Maeda

Research Journal of Mathematics & Technology., 10 (2021), pp. 74-81.

Published online: 2022-03

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  • Abstract

The special linear group $SL(2,\mathbb{R})$, the group of $2 × 2$ real matrices with determinant one, is one of the most important and fundamental mathematical objects not only in mathematics but also in physics. In this paper, we propose a three-dimensional model of $SL(2, \mathbb{R})$ in $\mathbb{R}^3,$ which is realized by embedding $SL(2,\mathbb{R})$ into the unit $3$-sphere. In this model, the set of symmetric matrices of $SL(2,\mathbb{Z})$ forms a hyperbolic pattern on the unit disk, like the islands floating on the sea named $SL(2,\mathbb{R}).$ The structure of this hyperbolic pattern is described in the upper half-plane $H.$ The upper half-plane $H$ also enables us to generate symmetric matrices of $SL(2,\mathbb{R})$ with three circles. Furthermore, the well-known fact $H = SL(2, \mathbb{R})/SO(2)$ is visualized as $S^1$ fibers of Hopf fibration in the unit $3$-sphere. With this three-dimensional model in $\mathbb{R}^3,$ we can have a concrete image of $SL(2,\mathbb{R})$ and its noncommutative group structure. This kind of visualization might bring great benefits for the readers who have background not only in mathematics, but also in all areas of science.

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@Article{RJMT-10-74, author = {Yoichi and Maeda and and 22767 and and Yoichi Maeda}, title = {A Three-Dimensional Model of $SL(2, \mathbb{R})$ and the Hyperbolic Pattern of $SL(2, \mathbb{Z})$}, journal = {Research Journal of Mathematics & Technology}, year = {2022}, volume = {10}, number = {2}, pages = {74--81}, abstract = {

The special linear group $SL(2,\mathbb{R})$, the group of $2 × 2$ real matrices with determinant one, is one of the most important and fundamental mathematical objects not only in mathematics but also in physics. In this paper, we propose a three-dimensional model of $SL(2, \mathbb{R})$ in $\mathbb{R}^3,$ which is realized by embedding $SL(2,\mathbb{R})$ into the unit $3$-sphere. In this model, the set of symmetric matrices of $SL(2,\mathbb{Z})$ forms a hyperbolic pattern on the unit disk, like the islands floating on the sea named $SL(2,\mathbb{R}).$ The structure of this hyperbolic pattern is described in the upper half-plane $H.$ The upper half-plane $H$ also enables us to generate symmetric matrices of $SL(2,\mathbb{R})$ with three circles. Furthermore, the well-known fact $H = SL(2, \mathbb{R})/SO(2)$ is visualized as $S^1$ fibers of Hopf fibration in the unit $3$-sphere. With this three-dimensional model in $\mathbb{R}^3,$ we can have a concrete image of $SL(2,\mathbb{R})$ and its noncommutative group structure. This kind of visualization might bring great benefits for the readers who have background not only in mathematics, but also in all areas of science.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/rjmt/20368.html} }
TY - JOUR T1 - A Three-Dimensional Model of $SL(2, \mathbb{R})$ and the Hyperbolic Pattern of $SL(2, \mathbb{Z})$ AU - Maeda , Yoichi JO - Research Journal of Mathematics & Technology VL - 2 SP - 74 EP - 81 PY - 2022 DA - 2022/03 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/rjmt/20368.html KW - AB -

The special linear group $SL(2,\mathbb{R})$, the group of $2 × 2$ real matrices with determinant one, is one of the most important and fundamental mathematical objects not only in mathematics but also in physics. In this paper, we propose a three-dimensional model of $SL(2, \mathbb{R})$ in $\mathbb{R}^3,$ which is realized by embedding $SL(2,\mathbb{R})$ into the unit $3$-sphere. In this model, the set of symmetric matrices of $SL(2,\mathbb{Z})$ forms a hyperbolic pattern on the unit disk, like the islands floating on the sea named $SL(2,\mathbb{R}).$ The structure of this hyperbolic pattern is described in the upper half-plane $H.$ The upper half-plane $H$ also enables us to generate symmetric matrices of $SL(2,\mathbb{R})$ with three circles. Furthermore, the well-known fact $H = SL(2, \mathbb{R})/SO(2)$ is visualized as $S^1$ fibers of Hopf fibration in the unit $3$-sphere. With this three-dimensional model in $\mathbb{R}^3,$ we can have a concrete image of $SL(2,\mathbb{R})$ and its noncommutative group structure. This kind of visualization might bring great benefits for the readers who have background not only in mathematics, but also in all areas of science.

Yoichi Maeda. (2022). A Three-Dimensional Model of $SL(2, \mathbb{R})$ and the Hyperbolic Pattern of $SL(2, \mathbb{Z})$. Research Journal of Mathematics & Technology. 10 (2). 74-81. doi:
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