Gedichte Musik Bilder Quellcodebibliothek Diashow Normaldarstellung
Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/nock/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 30.2.2022 mit Größe 7 kB image not shown  

Quelle  chap2.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/nock/doc/chap2.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (NoCK) - Chapter 2: Obstruction for the existence of compact Clifford-Klein form</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap2"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap1.html">[Previous Chapter]</a>    <a href="chap3.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p>
<p><a id="X7DD2840E797415E3" name="X7DD2840E797415E3"></a></p>
<div class="ChapSects"><a href="chap2.html#X7DD2840E797415E3">2 <span class="Heading">Obstruction for the existence of compact Clifford-Klein form</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X805C42557996F58A">2.1 <span class="Heading">Technical functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X832D9E887973AABE">2.1-1 NonCompactDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X868FC1B77B4B4D4C">2.1-2 PCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X827DC41787C8BC7B">2.1-3 PCalculate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X87164CA87A56E5B8">2.1-4 AllZeroDH</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Obstruction for the existence of compact Clifford-Klein form</span></h3>

<p>In this chapter we describe functions for algorithm from <a href="chapBib.html#biBour">[BJS+]</a>.</p>

<p><a id="X805C42557996F58A" name="X805C42557996F58A"></a></p>

<h4>2.1 <span class="Heading">Technical functions</span></h4>

<p><a id="X832D9E887973AABE" name="X832D9E887973AABE"></a></p>

<h5>2.1-1 NonCompactDimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonCompactDimension</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a real Lie algebra <span class="SimpleMath">G</span> constructed by the function <var class="Arg">RealFormById</var> (from <a href="chapBib.html#biBCoReLG">[DFdG14]</a>), this function returns the non-compact dimension of <span class="SimpleMath">G</span> (dimension of a non-compact part in Cartan decomposition of <span class="SimpleMath">G</span>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=RealFormById("E",6,2); # E6(6)</span>
<Lie algebra of dimension 78 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">dG:=NonCompactDimension(G);</span>
42
</pre></div>

<p><a id="X868FC1B77B4B4D4C" name="X868FC1B77B4B4D4C"></a></p>

<h5>2.1-2 PCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PCoefficients</code>( <var class="Arg">type</var>, <var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">G</span> be a compact connected Lie group of the type <var class="Arg">type</var> and the rank <var class="Arg">rank</var>. Let <span class="SimpleMath">ΛP_G=Λ (y_1,...,y_l)</span> be the exterior algebra over the spaces <span class="SimpleMath">P_G</span> of the primitive elements in <span class="SimpleMath">H^*(G)</span>. Denote the degrees as follows <span class="SimpleMath">|y_j|=2p_j-1,j=1,...,l</span>. This function returns coefficients <span class="SimpleMath">p_1,...,p_l</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCoefficients("D",5);</span>
[ 2, 4, 6, 8, 5 ]
</pre></div>

<p><a id="X827DC41787C8BC7B" name="X827DC41787C8BC7B"></a></p>

<h5>2.1-3 PCalculate</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PCalculate</code>( <var class="Arg">pi</var>, <var class="Arg">qi</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Here <span class="SimpleMath">pi={ p_1,...,p_l}</span> and <span class="SimpleMath">qi={ q_1,...,q_m}</span> are sets of coefficients (<span class="SimpleMath">l≥ m</span>). This function returns the polynomial: <span class="SimpleMath">P(t)=∏_j=m+1^l(1+t^2p_j-1)∏_i=1^m(1-t^2p_i)/(1-t^2q_i)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PCalculate([4,2,3],[2,2]);   </span>
t^9+t^5+t^4+1
</pre></div>

<p><a id="X87164CA87A56E5B8" name="X87164CA87A56E5B8"></a></p>

<h5>2.1-4 AllZeroDH</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllZeroDH</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">G^C</span> be a complex Lie algebra of the type <var class="Arg">type</var> and the rank <var class="Arg">rank</var>. Let <span class="SimpleMath">G</span> be a real form of <span class="SimpleMath">G^C</span> with the index <var class="Arg">id</var> (see <var class="Arg">RealFormsInformation</var>,<a href="chapBib.html#biBCoReLG">[DFdG14]</a>). This function returns the set of degrees of <span class="SimpleMath">P(t)</span> that have zero coefficients over all permutation (see Section 7 in <a href="chapBib.html#biBour">[BJS+]</a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllZeroDH("F",4,2); </span>
[ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27 ]
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap1.html">[Previous Chapter]</a>    <a href="chap3.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

100%


¤ Dauer der Verarbeitung: 0.15 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.