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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>PSp(4,R)</title>
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<meta name="keywords" content="mathematics, geometry, visualisation, mobius, symplectic, group, veronese, lagrangian">
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<h1>\(\mathrm{PSp}(4, \mathbb{R})\)</h1>
<p>
Let \(V\) be a vector space and let \(K\) be a field. A <i>symplectic bilinear form</i> on \(V\) is a bilinear map
\(\omega : V\times V \to K\) such that \(\omega\) is
</p>
<ol>
<li>Alternating: \(\omega(u,u) = 0\) for all \(u \in V\);</li>
<li>Non-degenerate: \(\omega(u,v) = 0\) for all \(v \in V\) implies that \(u=0\).</li>
</ol>
<p>
A <i>symplectic vector space</i> is a vector space equipped with a symplectic bilinear form. Also, by fixing a basis,
it's possible to represent the symplectic bilinear form \(\omega\) by a matrix \(\Omega\).
</p>
<p>
The <i>symplectic group</i>, denoted \(\mathrm{Sp}(2n, K)\), is the group of linear transformations of a symplectic \(K\)-vector space
of dimension \(2n\) preserving the symplectic bilinear form. In our case, we are interested in the group
\[\mathrm{Sp}(4, \mathbb{R}) = \{M \in M_{4}(\mathbb{R}) \mid M^{T}\Omega M = \Omega\},\]
and more precisely by the <i>projective symplectic group</i>
\[\mathrm{PSp}(4,\mathbb{R}) = \mathrm{Sp}(4,\mathbb{R})/\{-I,I\}.\]
It is a group of \(10\) dimensions.
</p>
<p>
Let's consider the group of Möbius transformations \(\mathcal{Möb}\) as a subgroup of \(\mathrm{PGL}(4,\mathbb{R})\). This group
is a subgroup of \(\mathrm{PSp}(4, \mathbb{R})\) of \(6\) dimensions. By parameterizing the \(4\) other dimensions with sliders, we obtain
the entirety of \(\mathrm{PSp}(4, \mathbb{R})\). The Möbius transformations are selected the same way that is described in their <a href="mobius.html">dedicated section</a>.
</p>
<p>
On the left, we visualise the action of \(\mathrm{PSp}(4, \mathbb{R})\) on isotropic flags (visualised as oriented circles with a point), and on the right, on the Veronese curve.
</p>
<div class="picture_box">
<a href="visualisation/isotropicflags.html"><img src="images/picture_flags.png"></a>
<a href="visualisation/veronese.html"><img src="images/picture_veronese.png"></a>
</div>
</body>
</html>