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 - <p style="text-align: justify;">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.</p>