<h4>4.1 <span class="Heading">Nilpotent orbits in real Lie algebras</span></h4>
<p><strong class="pkg">CoReLG</strong> has a database of the nilpotent orbits of the real forms of the simple Lie algebras of ranks up to 8. When called the first time in a GAP session, <strong class="pkg">CoReLG</strong> will first read the database of nilpotent orbits.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentOrbitsOfRealForm</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real form of a complex simple Lie algebra of rank up to 8. This function returns the list of nilpotent orbits (under the action of the adjoint group) of <var class="Arg">L</var>. For this function to work, <var class="Arg">L</var> must be defined over <var class="Arg">SqrtField</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= RealFormById( "F", 4, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">no:= NilpotentOrbitsOfRealForm( L );;</span>
#I CoReLG: read database of real triples ... done
<span class="GAPprompt">gap></span> <span class="GAPinput">no[1];</span>
<nilpotent orbit in Lie algebra>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealCayleyTriple</code>( <var class="Arg">o</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">o</var> is a nilpotent orbit constructed by <code class="func">NilpotentOrbitsOfRealForm</code> (<a href="chap4.html#X8424BB44791EAA48"><span class="RefLink">4.1-1</span></a>) of a simple real Lie algebra. This function returns a real Cayley triple <var class="Arg">[ f, h, e ]</var> corresponding to the orbit <var class="Arg">o</var>. The third element <var class="Arg">e</var> is a representative of the orbit.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WeightedDynkinDiagram</code>( <var class="Arg">o</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">o</var> is a nilpotent orbit constructed by <code class="func">NilpotentOrbitsOfRealForm</code> (<a href="chap4.html#X8424BB44791EAA48"><span class="RefLink">4.1-1</span></a>) of a simple real Lie algebra. This function returns the weighted Dynkin diagram of the orbit, which identifies its orbit in the complexification of the real Lie algebra in which <var class="Arg">o</var> lies.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealWeylGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealWeylGroup</code>( <var class="Arg">L</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra with Cartan subalgebra <var class="Arg">H</var>. (If <var class="Arg">H</var> is not given, then <var class="Arg">CartanSubalgebra(L)</var> will be taken.) This function returns the real Weyl group <span class="SimpleMath">N_G(H)/C_G(H)</span> associated with <var class="Arg">H</var>, where <span class="SimpleMath">G</span> is the connected component of the group of real points of the complex adjoint group of <var class="Arg">L</var>. The real Weyl group will be stored in the Cartan subalgebra, so that a new call to this function, with the same input, will return the real Weyl group immediately.</p>
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