Gedichte Musik Bilder Quellcodebibliothek Diashow Normaldarstellung
Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/hap/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 19.6.2025 mit Größe 12 kB image not shown  

Quelle  chap16_mj.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/hap/doc/chap16_mj.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>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (HAP commands) - Chapter 16:  Lie commutators and nonabelian Lie tensors</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="chap16"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chap14_mj.html">14</a>  <a href="chap15_mj.html">15</a>  <a href="chap16_mj.html">16</a>  <a href="chap17_mj.html">17</a>  <a href="chap18_mj.html">18</a>  <a href="chap19_mj.html">19</a>  <a href="chap20_mj.html">20</a>  <a href="chap21_mj.html">21</a>  <a href="chap22_mj.html">22</a>  <a href="chap23_mj.html">23</a>  <a href="chap24_mj.html">24</a>  <a href="chap25_mj.html">25</a>  <a href="chap26_mj.html">26</a>  <a href="chap27_mj.html">27</a>  <a href="chap28_mj.html">28</a>  <a href="chap29_mj.html">29</a>  <a href="chap30_mj.html">30</a>  <a href="chap31_mj.html">31</a>  <a href="chap32_mj.html">32</a>  <a href="chap33_mj.html">33</a>  <a href="chap34_mj.html">34</a>  <a href="chap35_mj.html">35</a>  <a href="chap36_mj.html">36</a>  <a href="chap37_mj.html">37</a>  <a href="chap38_mj.html">38</a>  <a href="chap39_mj.html">39</a>  <a href="chap40_mj.html">40</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap15_mj.html">[Previous Chapter]</a>    <a href="chap17_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap16.html">[MathJax off]</a></p>
<p><a id="X7A3DC9327EE1BE6C" name="X7A3DC9327EE1BE6C"></a></p>
<div class="ChapSects"><a href="chap16_mj.html#X7A3DC9327EE1BE6C">16 <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16_mj.html#X7CFDEEC07F15CF82">16.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X80BBA6247ED4DCCF">16.1-1 LieCoveringHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X7BEEE3D380CF22F1">16.1-2 LeibnizQuasiCoveringHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X7B384F9486A7C92B">16.1-3 LieEpiCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X849324D680C0EE5E">16.1-4 LieExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X809B166C835516EB">16.1-5 LieTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16_mj.html#X802F2E417D872042">16.1-6 LieTensorCentre</a></span>
</div></div>
</div>

<h3>16 <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></h3>

<p>Functions on this page are joint work with <strong class="button">Hamid Mohammadzadeh</strong>, and implemented by him.</p>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>16.1 <span class="Heading">  </span></h4>

<p><a id="X80BBA6247ED4DCCF" name="X80BBA6247ED4DCCF"></a></p>

<h5>16.1-1 LieCoveringHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieCoveringHomomorphism</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field, and returns a surjective Lie homomorphism <span class="SimpleMath">\(phi : C\rightarrow L\)</span> where:</p>


<ul>
<li><p>the kernel of <span class="SimpleMath">\(phi\)</span> lies in both the centre of <span class="SimpleMath">\(C\)</span> and the derived subalgebra of <span class="SimpleMath">\(C\)</span>,</p>

</li>
<li><p>the kernel of <span class="SimpleMath">\(phi\)</span> is a vector space of rank equal to the rank of the second Chevalley-Eilenberg homology of <span class="SimpleMath">\(L\)</span>.</p>

</li>
</ul>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">2</a></span> </p>

<p><a id="X7BEEE3D380CF22F1" name="X7BEEE3D380CF22F1"></a></p>

<h5>16.1-2 LeibnizQuasiCoveringHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeibnizQuasiCoveringHomomorphism</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field, and returns a surjective homomorphism <span class="SimpleMath">\(phi : C\rightarrow L\)</span> of Leibniz algebras where:</p>


<ul>
<li><p>the kernel of <span class="SimpleMath">\(phi\)</span> lies in both the centre of <span class="SimpleMath">\(C\)</span> and the derived subalgebra of <span class="SimpleMath">\(C\)</span>,</p>

</li>
<li><p>the kernel of <span class="SimpleMath">\(phi\)</span> is a vector space of rank equal to the rank of the kernel <span class="SimpleMath">\(J\)</span> of the homomorphism <span class="SimpleMath">\(L \otimes L \rightarrow L\)</span> from the tensor square to <span class="SimpleMath">\(L\)</span>. (We note that, in general, <span class="SimpleMath">\(J\)</span> is NOT equal to the second Leibniz homology of <span class="SimpleMath">\(L\)</span>.)</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X7B384F9486A7C92B" name="X7B384F9486A7C92B"></a></p>

<h5>16.1-3 LieEpiCentre</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieEpiCentre</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field, and returns an ideal <span class="SimpleMath">\(Z^\ast(L)\)</span> of the centre of <span class="SimpleMath">\(L\)</span>. The ideal <span class="SimpleMath">\(Z^\ast(L)\)</span> is trivial if and only if <span class="SimpleMath">\(L\)</span> is isomorphic to a quotient <span class="SimpleMath">\(L=E/Z(E)\)</span> of some Lie algebra <span class="SimpleMath">\(E\)</span> by the centre of <span class="SimpleMath">\(E\)</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">2</a></span> </p>

<p><a id="X849324D680C0EE5E" name="X849324D680C0EE5E"></a></p>

<h5>16.1-4 LieExteriorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieExteriorSquare</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field. It returns a record <span class="SimpleMath">\(E\)</span> with the following components.</p>


<ul>
<li><p><span class="SimpleMath">\(E.homomorphism\)</span> is a Lie homomorphism <span class="SimpleMath">\(µ : (L \wedge L) \longrightarrow L\)</span> from the nonabelian exterior square <span class="SimpleMath">\((L \wedge L)\)</span> to <span class="SimpleMath">\(L\)</span>. The kernel of <span class="SimpleMath">\(µ\)</span> is the Lie multiplier.</p>

</li>
<li><p><span class="SimpleMath">\(E.pairing(x,y)\)</span> is a function which inputs elements <span class="SimpleMath">\(x, y\)</span> in <span class="SimpleMath">\(L\)</span> and returns <span class="SimpleMath">\((x \wedge y)\)</span> in the exterior square <span class="SimpleMath">\((L \wedge L)\)</span> .</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X809B166C835516EB" name="X809B166C835516EB"></a></p>

<h5>16.1-5 LieTensorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieTensorSquare</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field and returns a record <span class="SimpleMath">\(T\)</span> with the following components.</p>


<ul>
<li><p><span class="SimpleMath">\(T.homomorphism\)</span> is a Lie homomorphism <span class="SimpleMath">\(µ : (L \otimes L) \longrightarrow L\)</span> from the nonabelian tensor square of <span class="SimpleMath">\(L\)</span> to <span class="SimpleMath">\(L\)</span>.</p>

</li>
<li><p><span class="SimpleMath">\(T.pairing(x,y)\)</span> is a function which inputs two elements <span class="SimpleMath">\(x, y\)</span> in <span class="SimpleMath">\(L\)</span> and returns the tensor <span class="SimpleMath">\((x \otimes y)\)</span> in the tensor square <span class="SimpleMath">\((L \otimes L)\)</span> .</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X802F2E417D872042" name="X802F2E417D872042"></a></p>

<h5>16.1-6 LieTensorCentre</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieTensorCentre</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">\(L\)</span> over a field and returns the largest ideal <span class="SimpleMath">\(N\)</span> such that the induced homomorphism of nonabelian tensor squares <span class="SimpleMath">\((L \otimes L) \longrightarrow (L/N \otimes L/N)\)</span> is an isomorphism.</p>

<p><strong class="button">Examples:</strong></p>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap15_mj.html">[Previous Chapter]</a>    <a href="chap17_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chap14_mj.html">14</a>  <a href="chap15_mj.html">15</a>  <a href="chap16_mj.html">16</a>  <a href="chap17_mj.html">17</a>  <a href="chap18_mj.html">18</a>  <a href="chap19_mj.html">19</a>  <a href="chap20_mj.html">20</a>  <a href="chap21_mj.html">21</a>  <a href="chap22_mj.html">22</a>  <a href="chap23_mj.html">23</a>  <a href="chap24_mj.html">24</a>  <a href="chap25_mj.html">25</a>  <a href="chap26_mj.html">26</a>  <a href="chap27_mj.html">27</a>  <a href="chap28_mj.html">28</a>  <a href="chap29_mj.html">29</a>  <a href="chap30_mj.html">30</a>  <a href="chap31_mj.html">31</a>  <a href="chap32_mj.html">32</a>  <a href="chap33_mj.html">33</a>  <a href="chap34_mj.html">34</a>  <a href="chap35_mj.html">35</a>  <a href="chap36_mj.html">36</a>  <a href="chap37_mj.html">37</a>  <a href="chap38_mj.html">38</a>  <a href="chap39_mj.html">39</a>  <a href="chap40_mj.html">40</a>  <a href="chapInd_mj.html">Ind</a>  </div>

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

100%


¤ Dauer der Verarbeitung: 0.16 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.