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<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="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

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<h1>QuaGroup</h1>


<h2>Computations with quantum groups</h2>

<p>
    Version 1.8.4</p>

<p>
    11 January 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>
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2002 Willem A. 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_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
</div>
<div class="ContChap"><a href="chap2_mj.html#X84AF2F1D7D4E7284">2 <span class="Heading">Background</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7AAC838B7CEB5E54">2.1 <span class="Heading">Gaussian Binomials</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X81394E207F6AA6CF">2.2 <span class="Heading">Quantized enveloping algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7DCC28DE7C626D33">2.3 <span class="Heading">Representations of <span class="SimpleMath">\(U_q(\mathfrak{g})\)</span>  </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83E7F39F7D16793B">2.4 <span class="Heading">PBW-type bases </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X798D979F7A53E05D">2.5 <span class="Heading">The <span class="SimpleMath">\({\mathbb Z}\)</span>-form of <span class="SimpleMath">\(U_q(\mathfrak{g})\)</span> </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X78BE3EB980F0A295">2.6 <span class="Heading">The canonical basis </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X83BBB99685FB9FB7">2.7 <span class="Heading"> The path model </span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7BB6B9777D54FA07">2.8 <span class="Heading"> Notes</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X80C9BD6584865157">3 <span class="Heading">QuaGroup</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C202C54806653DE">3.1 <span class="Heading">Global constants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8748A60A7AB09B0C">3.1-1 QuantumField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87FCF1537A81E11D">3.1-2 _q</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X83804A6F84BB2387">3.2 <span class="Heading">Gaussian integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7985B6A67A30FDA1">3.2-1 GaussNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EF34AC07BDFEF20">3.2-2 GaussianFactorial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F1A5A457926F3CF">3.2-3 GaussianBinomial</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X87064F787DEA9E21">3.3 <span class="Heading">Roots and root systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80D15C027BB8029B">3.3-1 RootSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79E8AE3A8739D11A">3.3-2 BilinearFormMatNF</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X853E620A79D796E7">3.3-3 PositiveRootsNF</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D2E78B678C73E69">3.3-4 SimpleSystemNF</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A5DFEAB7EC0D945">3.3-5 PositiveRootsInConvexOrder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A5BD2D77ED1BEDD">3.3-6 SimpleRootsAsWeights</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X798CE6777FC473DD">3.4 <span class="Heading">Weyl groups and their elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D738A0E85CD5C14">3.4-1 ApplyWeylElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X792A49A8808D1C64">3.4-2 LengthOfWeylWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83851E7D7ED84A48">3.4-3 LongestWeylWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81FC70FA84F1B6E9">3.4-4 ReducedWordIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8264626678D9F39A">3.4-5 ExchangeElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84138DD47BBEA4E0">3.4-6 GetBraidRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X800BB9837DCF1DBD">3.4-7 LongWords</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X81394E207F6AA6CF">3.5 <span class="Heading">Quantized enveloping algebras </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C263EA87C6E4F8A">3.5-1 QuantizedUEA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E3D93D47AFDE8D4">3.5-2 ObjByExtRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8542B32A8206118C">3.5-3 ExtRepOfObj</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7879B74F7C77E930">3.5-4 QuantumParameter</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82B42CD279854DFC">3.5-5 CanonicalMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87C302567A9B2AD3">3.5-6 WriteQEAToFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X874B8E4A83180063">3.5-7 ReadQEAFromFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X847A0459844B7A59">3.6 <span class="Heading"> Homomorphisms and automorphisms </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8185FE54820AA950">3.6-1 QEAHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81214E9B8532B06F">3.6-2 QEAAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79D886AB7B248F75">3.6-3 QEAAntiAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X851AE6987871E0E8">3.6-4 AutomorphismOmega</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E288BFE7FDF8BEA">3.6-5 AntiAutomorphismTau</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8576A5987AD999F0">3.6-6 BarAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8277D87D7D81AE75">3.6-7 AutomorphismTalpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DC17C2D8540FCED">3.6-8 DiagramAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7857704878577048"><code>3.6-9 \*</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X832B6D897BDBC8A3">3.7 <span class="Heading">Hopf algebra structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8245FA64874E085D">3.7-1 TensorPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783144818681D2E6">3.7-2 UseTwistedHopfStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X811565FF83FA4847">3.7-3 ComultiplicationMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7985B7EF7E31C4DF">3.7-4 AntipodeMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8361876D8770FC06">3.7-5 CounitMap</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8183A6857B0C3633">3.8 <span class="Heading">Modules</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7ED0FA2980330CE1">3.8-1 HighestWeightModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B00688F82AACDD0">3.8-2 IrreducibleQuotient</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B080C0078370E99">3.8-3 HWModuleByTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C03DC018016B93B">3.8-4 DIYModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X854FA09F7E07D2D0">3.8-5 TensorProductOfAlgebraModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F0AED7A85D603A2">3.8-6 HWModuleByGenerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8565D4CD82EB2503">3.8-7 InducedQEAModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8680D0D67BACAE16">3.8-8 GenericModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82B42CD279854DFC">3.8-9 CanonicalMapping</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DCD17BF7B76E7CA">3.8-10 U2Module</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E1B059780BB2E09">3.8-11 MinusculeModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79C8F1317C2E8C60">3.8-12 DualAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87EBE47D86292754">3.8-13 TrivialAlgebraModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7929063E7C428331">3.8-14 WeightsAndVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EEEBA347A816827">3.8-15 HighestWeightsAndVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799E7D5981B9F14A">3.8-16 RMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78E533D2813BDA9E">3.8-17 IsomorphismOfTensorModules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87FF182B86367CC4">3.8-18 WriteModuleToFile</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DD49D8282C516B9">3.8-19 ReadModuleFromFile</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X83BBB99685FB9FB7">3.9 <span class="Heading">The path model</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8025E1918067B03C">3.9-1 DominantLSPath</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82EDAFAB7E81BD1F">3.9-2 Falpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X855C478587112804">3.9-3 Ealpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X820413E57DEDB0D1">3.9-4 LSSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8573A4077DB4659F">3.9-5 WeylWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D06353282BE96C6">3.9-6 EndWeight</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8637218B80BDC906">3.9-7 CrystalGraph</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X827A177C8416C576">3.10 <span class="Heading"> Canonical bases </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8215C71E8361EE62">3.10-1 Falpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X814B1C7B7AF17B79">3.10-2 Ealpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C8EBFF5805F8C51">3.10-3 CanonicalBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8233212A79D722FB">3.10-4 PBWElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81631585816C4CCD">3.10-5 MonomialElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X782174177D696504">3.10-6 Strings</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X842A5CC07C3E5848">3.10-7 PrincipalMonomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C04F75683D0A512">3.10-8 StringMonomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E67BE5D856C77DC">3.10-9 Falpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X855EFC817CFCE2C7">3.10-10 Ealpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85A9229678572948">3.10-11 CrystalBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82CE52AB7E798902">3.10-12 CrystalVectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80A534637D8F3210">3.10-13 Falpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B9C76BF841FA70B">3.10-14 Ealpha</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D7607BB81ADF579">3.10-15 CrystalGraph</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7875070C85DD4E8E">3.11 <span class="Heading"> Universal enveloping algebras </span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E7B25307E6478CD">3.11-1 UEA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CA021E28527763E">3.11-2 UnderlyingLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8707EE2C8145701F">3.11-3 HighestWeightModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AD80E5C7E015859">3.11-4 QUEAToUEAMap</a></span>
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