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<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="chap4.html">4</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>CoReLG</h1>


<h2>Computing with real Lie algebras</h2>

<p>
    Version 1.57</p>

<p>
    7 July 2024
  </p>

</div>
<p><b>
    Heiko Dietrich




  </b>
<br />Email: <span class="URL"><a href="mailto:heiko.dietrich@monash.edu">heiko.dietrich@monash.edu</a></span>
<br />Homepage: <span class="URL"><a href="http://users.monash.edu.au/~heikod/">http://users.monash.edu.au/~heikod/</a></span>
<br />Address: <br />School of Mathematics<br /> Monash University<br /> Wellington Road 1<br /> VIC 3800, Melbourne, Australia<br />
</p><p><b>
    Paolo Faccin



  </b>
<br />Email: <span class="URL"><a href="mailto:paolofaccin86@gmail.com">paolofaccin86@gmail.com</a></span>
<br />Address: <br />Dipartimento di Matematica<br /> Via Sommarive 14<br /> I-38050 Povo (Trento), Italy<br /> <br />
</p><p><b>
    Willem de Graaf




  </b>
<br />Email: <span class="URL"><a href="mailto:degraaf@science.unitn.it">degraaf@science.unitn.it</a></span>
<br />Homepage: <span class="URL"><a href="https://www.science.unitn.it/~degraaf">https://www.science.unitn.it/~degraaf</a></span>
<br />Address: <br />Dipartimento di Matematica<br /> Via Sommarive 14<br /> I-38050 Povo (Trento), Italy<br /> <br />
</p>

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<h3>Abstract</h3>
<p>This package provides functions for computing with various aspects of the theory of real simple Lie algebras.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2013-2019 Heiko Dietrich, Paolo Faccin, and Willem de Graaf</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The research leading to this package has received funding from the European Union's Seventh Framework Program FP7/2007-2013 under grant agreement no 271712, and from the Australian Research Council, grantor code DE140100088 and DP190100317.



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

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7A0F3100829CD1E1">1.1 <span class="Heading">The simple real Lie algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X80030DB07E5F4FBF">1.2 <span class="Heading">Cartan subalgebras and more</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8295733081A2BFF8">1.3 <span class="Heading">Nilpotent orbits</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7C0A369A841F5BC9">1.4 <span class="Heading">On base fields</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X83DD4ACD87694138">2 <span class="Heading">The field <var class="Arg">SqrtField</var></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80E89FFF7F52BE64">2.1 <span class="Heading"> Definition of the field </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E924375789E5F98">2.1-1 SqrtFieldIsGaussRat</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X850FE9D385B653D9">2.2 <span class="Heading"> Construction of elements </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84E7D3D787000313">2.2-1 Sqroot</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86C3EA257D7CF10C">2.2-2 CoefficientsOfSqrtFieldElt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B0063817B03422F">2.2-3 SqrtFieldEltByCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84E90EC582E8A921">2.2-4 SqrtFieldEltToCyclotomic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7EBF6AAC7A4189CC">2.2-5 SqrtFieldEltByCyclotomic</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X82EB5BE77F9F686A">2.3 <span class="Heading"> Basic operations </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X873983AD867AC476">2.3-1 SqrtFieldMakeRational</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X860A58627B6D5999">2.3-2 SqrtFieldPolynomialToRationalPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79C882567BC98D65">2.3-3 SqrtFieldRationalPolynomialToSqrtFieldPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82D6EDC685D12AE2">2.3-4 Factors</a></span>
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<div class="ContChap"><a href="chap3.html#X81152A5D7B4BF910">3 <span class="Heading">Real Lie Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X86598A16853C825D">3.1 <span class="Heading"> Construction of simple real Lie algebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BB53454857133FF">3.1-1 RealFormsInformation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78143E4187893A79">3.1-2 NumberRealForms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8443E03C868CA7D3">3.1-3 RealFormById</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E23043A7BBE7DF2">3.1-4 IdRealForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E8EA8457A5F01FC">3.1-5 NameRealForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85C3549A8537FBF6">3.1-6 AllRealForms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CA76CD087DBABF4">3.1-7 RealFormParameters</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8266EA5E7D3B4DD5">3.1-8 IsRealFormOfInnerType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79A0991B809A4D6C">3.1-9 IsRealification</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81E1C65282CE3130">3.1-10 CartanDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8318965D8692FC43">3.1-11 RealStructure</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8706BCC5858C3551">3.2 <span class="Heading">Maximal reductive subalgebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82A3668382971658">3.2-1 MaximalReductiveSubalgebras</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7D702EA087C1C5EF">3.3 <span class="Heading"> Isomorphisms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F84150B84B62412">3.3-1 IsomorphismOfRealSemisimpleLieAlgebras</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X82EAE07A8557719A">3.4 <span class="Heading">Cartan subalgebras and root systems</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D7B755F7E6B8471">3.4-1 MaximallyCompactCartanSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D593D72871F56B1">3.4-2 MaximallyNonCompactCartanSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F5A10E0782A4DBC">3.4-3 CompactDimensionOfCartanSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83BFAD338107D9FF">3.4-4 CartanSubalgebrasOfRealForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A8D86667BC7C033">3.4-5 CartanSubspace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F9943407A2F367E">3.4-6 RootsystemOfCartanSubalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82EBF10A7B3B6F6E">3.4-7 ChevalleyBasis</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X78932FB48237B18F">3.5 <span class="Heading">Diagrams</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7AE4B8A479E73F6D">3.5-1 VoganDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84042AAE7CF12E38">3.5-2 SatakeDiagram</a></span>
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<div class="ContChap"><a href="chap4.html#X845E3A7E87C93239">4 <span class="Heading">Real nilpotent orbits</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7A6BB7967FE7ABA4">4.1 <span class="Heading">Nilpotent orbits in real Lie algebras</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8424BB44791EAA48">4.1-1 NilpotentOrbitsOfRealForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A05B2957A625D85">4.1-2 RealCayleyTriple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X804830757E5971E9">4.1-3 WeightedDynkinDiagram</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7BE2BD6B79367FC8">4.2 <span class="Heading">The real Weyl group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8196CAF57F4CD8C7">4.2-1 RealWeylGroup</a></span>
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</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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