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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="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>SLA</h1>


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

<p>
    Version 1.6.2</p>

<p>
    12 July 2024
  </p>

</div>
<p><b>
    Willem Adriaan 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="http://www.science.unitn.it/~degraaf">http://www.science.unitn.it/~degraaf</a></span>
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<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>This package provides functions for computing with various aspects of the theory of simple Lie algebras in characteristic zero.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2013-2018 Willem de Graaf</p>

<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<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>
<div class="ContChap"><a href="chap2.html#X84EA8FA47E4A7BDF">2 <span class="Heading">Root Systems and Weyl Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7D22A7DF7EF96F24">2.1 <span class="Heading"> Root Systems </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8122C42F7CCFC6ED">2.1-1 ExtendedCartanMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7DABAF857F49C8EB">2.1-2 CartanType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7BBD0ECE8762DB3A">2.1-3 DisplayDynkinDiagram</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X808290B47CC22D62">2.2 <span class="Heading"> Weyl groups </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8789147A7A570A01">2.2-1 WeylTransversal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X78A375CD80A0F31E">2.2-2 SizeOfWeylGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X835F02DB7E5C1AF4">2.2-3 WeylGroupAsPermGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X824D038880907771">2.2-4 ApplyWeylPermToWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X79E8F3D5809EB6F8">2.2-5 WeylWordAsPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X821242C278AA7C5F">2.2-6 PermAsWeylWord</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X7944E8BD87DCAA24">3 <span class="Heading">Semisimple Lie Algebras and their Modules</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7844D90E7F25B423">3.1 <span class="Heading"> Semisimple Lie algebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FA167DB80B7D701">3.1-1 IsomorphismOfSemisimpleLieAlgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7BBD0ECE8762DB3A">3.1-2 DisplayDynkinDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A83F8227DFEAAB2">3.1-3 ApplyWeylPermToCartanElement</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X785D65257C93A7EF">3.2 <span class="Heading"> Representations of semisimple Lie algebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7866133B814EA2B2">3.2-1 AdmissibleLattice</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A4E769A87A9B2B3">3.2-2 DirectSumDecomposition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87A8E50683C5F286">3.2-3 IsIrreducibleHWModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EC5E998789DF177">3.2-4 HighestWeightVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8488EF397B8298D6">3.2-5 HighestWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8589BB6484842CBC">3.2-6 DisplayHighestWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B5CEF1E81AE0738">3.2-7 IsomorphismOfIrreducibleHWModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X79C8F1317C2E8C60">3.2-8 DualAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7889949D87615523">3.2-9 CharacteristicsOfStrata</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X8295733081A2BFF8">4 <span class="Heading">Nilpotent Orbits</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8173135A7D187358">4.1 <span class="Heading"> The functions </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A074A557A7347D2">4.1-1 NilpotentOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D5C0354810069A8">4.1-2 NilpotentOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X804830757E5971E9">4.1-3 WeightedDynkinDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X804830757E5971E9">4.1-4 WeightedDynkinDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X870F93A77E4F9CA7">4.1-5 DisplayWeightedDynkinDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X870F93A77E4F9CA7">4.1-6 DisplayWeightedDynkinDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F2B6308785707B9">4.1-7 AmbientLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8401CDC2859F8A85">4.1-8 SemiSimpleType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84E78DA17D8C7F74">4.1-9 SL2Triple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X832FB68587166C4F">4.1-10 RandomSL2Triple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8029297A7C3372E9">4.1-11 SL2Grading</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84E78DA17D8C7F74">4.1-12 SL2Triple</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E6926C6850E7C4E">4.1-13 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CF02C4785F0EAB5">4.1-14 IsRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X781CAF5D7FF46E66">4.1-15 RegularNilpotentOrbit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A9088098391EB5E">4.1-16 IsDistinguished</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80F0A7F07F78C06D">4.1-17 DistinguishedNilpotentOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CC92CF8796393CF">4.1-18 ComponentGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X830C432A838875A0">4.1-19 InducedNilpotentOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78795C607C2343C3">4.1-20 RigidNilpotentOrbits</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X856EEEB08169D020">4.1-21 RichardsonOrbits</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X80D10A5D7D73D871">5 <span class="Heading">Finite Order Automorphisms and <span class="SimpleMath">θ</span>-Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8173135A7D187358">5.1 <span class="Heading"> The functions </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X856FF71D8565C1E5">5.1-1 FiniteOrderInnerAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E8CCC4885E2A443">5.1-2 FiniteOrderOuterAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X84F59A2687C62763">5.1-3 Order</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8635ABCD7D5ACED8">5.1-4 KacDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DCA2568870A2D34">5.1-5 Grading</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D9BDD03811B3C98">5.1-6 NilpotentOrbitsOfThetaRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D4106C37DBD0943">5.1-7 ClosureDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8108B6487A3A363B">5.1-8 CarrierAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7A8D86667BC7C033">5.1-9 CartanSubspace</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X7FF8A8057E0BFAFD">6 <span class="Heading">Semisimple Subalgebras of Semisimple Lie Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X806713FE83F21540">6.1 <span class="Heading"> Branching </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8147807D7B92C613">6.1-1 ProjectionMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X806713FE83F21540">6.1-2 Branching</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7CC8010B7F31B486">6.2 <span class="Heading"> Constructing Semisimple Subalgebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7E95AD2C79B19D15">6.2-1 RegularSemisimpleSubalgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X857808A77C13E46C">6.2-2 SSSTypes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X832B17BA819FB057">6.2-3 LieAlgebraAndSubalgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82ECC8237E213AAB">6.2-4 InclusionsGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7C81C2B57D94EC2B">6.2-5 SubalgebrasInclusion</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X82458B807A8D77F6">6.2-6 DynkinIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7AAE2B317A4B7D9F">6.2-7 AreLinearlyEquivalentSubalgebras</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7B1A01FE86A7718F">6.2-8 MakeDatabaseEntry</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X85A6B70B83A36495">6.2-9 ClosedSubsets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A3EAA637F94D973">6.2-10 DecompositionOfClosedSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X798DF4D67DE56EB1">6.2-11 IsSpecialClosedSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7D31AA9780B054C1">6.2-12 LieAlgebraOfClosedSet</a></span>
</div></div>
</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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