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<div class="ChapSects"><a href="chap3_mj.html#X7F57F15D7A4099A1">3 <span class="Heading">Algorithm example</span></a>
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<h3>3 <span class="Heading">Algorithm example</span></h3>

<p>In this chapter we use additionaly functions from the following packages: CoReLG <a href="chapBib_mj.html#biBCoReLG">[DFdG14]</a> and SLA <a href="chapBib_mj.html#biBSLA">[dG]</a>. We will show in detail the split case (for a non-split case you should use algoritm to generate regular subalgebras from <a href="chapBib_mj.html#biBDFG">[DFdG15]</a>). For example, we take <span class="SimpleMath">\(G=\mathfrak{e}_{6(6)}\)</span> (tuple "E",6,2 in CoReLG notation). We calculate <var class="Arg">AllZeroDH</var> on it.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllZeroDH("E",6,2);</span>
[ 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27, 
 28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 41 ]
</pre></div>

<p>We generate all regular subalgebras of complexification.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GC:=SimpleLieAlgebra("E",6,Rationals);;  </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">REG:=RegularSemisimpleSubalgebras(GC);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L0:=List( REG, SemiSimpleType );   </span>
[ "A1", "A1 A1", "A2 A1", "A4", "D5", "A4 A1", "A2 A1 A1", "A2 A1 A2", "A3 A1", 
 "A1 A1 A1", "A2", "A3", "A5", "A2 A2", "D4", "A5 A1", "A3 A1 A1", "A1 A1 A1 A1", 
 "A2 A2 A2" ]
</pre></div>

<p>For each subalgebras we take the split real form and calculate its non-compact dimension.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L0[4]; </span>
"A4"
<span class="GAPprompt">gap></span> <span class="GAPinput">RealFormsInformation( "A", 4 ); </span>

  There are 4 simple real forms with complexification A4
    1 is of type su(5), compact form
    2 - 3 are of type su(p,5-p) with 1 <= p <= 2
    4 is of type sl(5,R)
  Index '0' returns the realification of A4

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=RealFormById("A",4,4);;     </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NonCompactDimension( G );      </span>
14
</pre></div>

<p>Number 14 is in output of <var class="Arg">AllZeroDH</var> function, so for <span class="SimpleMath">\(\mathfrak{g}=e_{6(6)}\)</span> and <span class="SimpleMath">\(\mathfrak{h}=\mathfrak{sl}(5,\mathbb{R})\)</span> corresponding homogeneous spaces <span class="SimpleMath">\(G/H\)</span> do not have compact Clifford–Klein forms.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L0[5];                                                          </span>
"D5"
<span class="GAPprompt">gap></span> <span class="GAPinput">RealFormsInformation( "D", 5 ); </span>

  There are 7 simple real forms with complexification D5
    1 is of type so(10), compact form
    2 - 3 are of type so(2p,10-2p) with 1 <= p <= 2
    4 is of type so*(10)
    5 is of type so(9,1)
    6 - 7 are of type so(2p+1,10-2p-1) with 1 <= p <= 2
  Index '0' returns the realification of D5

<span class="GAPprompt">gap></span> <span class="GAPinput">G:=RealFormById("D",5,7);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NonCompactDimension( G );                                       </span>
25
</pre></div>

<p>Number 25 is not in output of <var class="Arg">AllZeroDH</var> function, so for <span class="SimpleMath">\(\mathfrak{g}=e_{6(6)}\)</span> and <span class="SimpleMath">\(\mathfrak{h}=\mathfrak{so}(5,5)\)</span> our algoritm does not provide a solution to the problem.</p>


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