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<h1>Sophus</h1>


<h2>Computing in nilpotent Lie algebras</h2>

<p>
    1.27</p>

<p>
    9 August 2022
  </p>

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<p><b>
    Csaba Schneider




  </b>
<br />Email: <span class="URL"><a href="mailto:csaba@mat.ufmg.br">csaba@mat.ufmg.br</a></span>
<br />Homepage: <span class="URL"><a href="http://www.mat.ufmg.br/~csaba/">http://www.mat.ufmg.br/~csaba/</a></span>
<br />Address: <br />Departamento de Matemática<br /> Instituto de Ciências Exatas<br /> Universidade Federal de Minas Gerais (UFMG)<br /> Belo Horizonte, Brasil<br />
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<h3>Abstract</h3>
<p><strong class="pkg">Sophus</strong> is a GAP4 package to compute with nilpotent Lie algebras over finite prime fields. In particular, the package can be used to compute certain central extensions and the automorphism group of such Lie algebras. <strong class="pkg">Sophus</strong> also enables its user to test isomorphism between two nilpotent Lie algebras. The author of the package used it to construct all Lie algebras of dimension at most 9 over <span class="SimpleMath">\(\mathbb F_2\)</span>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2004, 2005 Csaba Schneider</p>

<p>The <strong class="pkg">Sophus</strong> package is free software; you can redistribute it and/or modify it under the terms of the <span class="URL"><a href="http://www.fsf.org/licenses/gpl.html">GNU General Public License</a></span> as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>

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<h3>Acknowledgements</h3>
<p>Most of the work on this package was carried out while I held a research position at the Technische Universität Braunschweig. I would like to express my gratitude to the staff and the students of the Institut für Geometrie for their interest in this work. Special thanks go to Bettina Eick for her rôle in completing this project.</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X818ED9677EDCB80E">1 <span class="Heading">The theory</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X864BE1A7820A35FC">2 <span class="Heading">A sample calculation with <strong class="pkg">Sophus</strong></span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X809610728132CED7">3 <span class="Heading"><strong class="pkg">Sophus</strong> functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E043B9C80BF5DDF">3.1 <span class="Heading">Some general functions to compute with Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7831B69779E3E5D6">3.1-1 SophusTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F110C75826C1A6D">3.1-2 IsLieNilpotentOverFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A9FAEC37ACA9285">3.1-3 MinimalGeneratorNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C7898C07C7711DB">3.1-4 AbelianLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A8E9A01835EDC3C">3.2 <span class="Heading">Functions to compute with nilpotent bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799C8C57797AE5F0">3.2-1 NilpotentBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E1BDBF08305FC7A">3.2-2 LieNBWeights</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80CF319778583B3C">3.2-3 LieNBDefinitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79D0807F784E9BEC">3.2-4 IsNilpotentBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X809032BB7ECF5F48">3.2-5 IsLieAlgebraWithNB</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7F149872830B45BA">3.3 <span class="Heading">The cover</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X830028407DC2D80A">3.3-1 LieCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F4D3B8B7E9C30F7">3.3-2 CoverHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783B60268536DD75">3.3-3 CoverOf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87BB5AEA80EB4E46">3.3-4 IsLieCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F9DFA097A920EEF">3.3-5 LieMultiplicator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84C5C1A37BCCB7B9">3.3-6 LieNucleus</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7AD387A67CDFF8A9">3.4 <span class="Heading">Automorphisms of nilpotent Lie algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86E8712281D7A532">3.4-1 NilpotentLieAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X856F9F0B87F8673B">3.4-2 IdentityNilpotentLieAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78778C3087CB7999">3.4-3 IsNilpotentLieAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X823B76A1836A4BD3">3.5 <span class="Heading">Automorphism group and isomorphism testing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87677B0787B4461A">3.5-1 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A9A3C7E7804704B">3.5-2 AutomorphismGroupNilpotentLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87B4D4C384E9B3DF">3.5-3 AreIsomorphicNilpotentLieAlgebras</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8396FA6279C8E439">3.6 <span class="Heading">Descendants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8396FA6279C8E439">3.6-1 Descendants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X825E625F84273954">3.6-2 DescendantsOfStep1OfAbelianLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X78DA04477DDD0ACE">3.7 <span class="Heading">Input and output</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B4AA5B97D4B53B9">3.7-1 WriteLieAlgebraToString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X841498AD80B13D34">3.7-2 ReadStringToNilpotentLieAlgebraOverFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D54F5838172718B">3.7-3 WriteLieAlgebraListToFile</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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