Transformations of the symplectic form under matrix multiplication¶

Exercise I.4.20.iii [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}} $ The symplectic form on $\Space[\sigma]{R}{2}$ is invariant if vectors are multiplied by a $2\times 2$ real matrix with the unit determinant.

Solution. Define the symplectic form

from init_cycle import *
def sf(P,P1):
    return P.op(0)*P1.op(1)-P.op(1)*P1.op(0)

            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

Two generic initial vectors

Z=matrix([[u],[v]])
Z1=matrix([[u1],[v1]])

Now we define the generic matrix and transformed vectors

M=matrix([[a,b],[c,d]])
Y=(M*Z).evalm()
Y1=(M*Z1).evalm()

The result check

"Symplectic form is invariant up to the factor det(M): %s" %\
(sf(Y,Y1)-M.determinant()*sf(Z,Z1)).normal().is_zero()

'Symplectic form is invariant up to the factor det(M): True'

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

References¶

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.
Vladimir V. Kisil, MoebInv notebooks, 2019.

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