None ex.I.3.21.i
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Derived action of subgroups A' and N'

Derived action of the subgroup K was calculated in Ex. I.3.13.i aka ex-k-action-der.ipynb.

Exercise I.3.21.i [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\alli}{\iota} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmd}{\mathrm{d}} \newcommand{\rmh}{\mathrm{j}} \newcommand{\rmp}{\varepsilon} \newcommand{\modulus}[2][]{\left| #2 \right|_{#1}} \newcommand{\sperp}{⋋} $ Using the parameter $\tau=-1$, $0$, $1$ for the subgroups $K$, $N'$ and $A'$ respectively, check the following properties of the actions of the subgroups $K$, $N'$ and $A'$: Vector fields of the respective actions are \begin{equation} (u^2+\sigma v^2-\tau)\partial_u+ 2uv\partial_v, \end{equation} where $\sigma=\iota^2$ represent the metric of the space .

Solution. Define the Möbius transformation with the K subgroup

In [1]:
from init_cycle import *
[U,V]=clifford_moebius_map(sl2_clifford([cos(t),sin(t),-sin(t),cos(t)],e),[u,v],e)

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

Using vector formalism and idx

Derived action is:

In [2]:
Latex("Derived action of the K subgroup: $(%s, %s)$" % \
(U.diff(t).subs({t : 0}), V.diff(t).subs({t : 0})))
Out[2]:
Derived action of the K subgroup: $(1+u^{2}+ \sigma v^{2}, 2 u v)$

We define the Moebius transformation with the N' subgroup

In [3]:
[U,V]=clifford_moebius_map(sl2_clifford([1, 0, -t, 1], e), [u, v], e)

Derived action is:

In [4]:
Latex("Derived action of the N' subgroup: $(%s, %s)$" % \
(U.diff(t).subs({t : 0}), V.diff(t).subs({t : 0})))
Out[4]:
Derived action of the N' subgroup: $(u^{2}+ \sigma v^{2}, 2 u v)$

If the below method fails, uncommed this next part The failure is due to absence of conjucate() method for sinh/cosh We define the Moebius transformation with the A' subgroup

In [5]:
[U,V]=clifford_moebius_map(sl2_clifford([a, b, b, a], e), [u, v], e)

Derived action is:

In [6]:
Latex("Derived action of the A' subgroup: $(%s, %s)$" % \
(U.subs({a : cosh(t), b : sinh(t)}).diff(t).subs({t : 0}), \
 V.subs({a : cosh(t), b : sinh(t)}).diff(t).subs({t : 0})))
Out[6]:
Derived action of the A' subgroup: $(1-u^{2}- \sigma v^{2}, -2 u v)$

We define the Moebius transformation with the A' subgroup

In [7]:
[U,V]=clifford_moebius_map(sl2_clifford([cosh(t),-sinh(t),-sinh(t),cosh(t)],e),[u,v],e)

Derived action is:

In [8]:
Latex("Derived action of the A' subgroup: $(%s, %s)$" % \
(U.diff(t).subs({t : 0}), V.diff(t).subs({t : 0})))
Out[8]:
Derived action of the A' subgroup: $(-1+u^{2}+ \sigma v^{2}, 2 u v)$

This notebook is a part of the MoebInv notebooks project [2] .

References

  1. Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.

  2. Vladimir V. Kisil, MoebInv notebooks, 2019.

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