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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 >div class="chlinkprevnexttop" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0_mj.html" >[MathJax on]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >SLA</h1 >1 .6 .2 </p>12 July 2024 div >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 >"X7AA6C5737B711C89" name="X7AA6C5737B711C89" ></a></p>e algebras in characteristic zero.</p>"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>2013 -2018 Willem de Graaf</p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"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 >br />div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap1.html" >[Next Chapter]</a> 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