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/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 >div class="chlinkprevnexttop" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >"mathjaxlink" class="pcenter" ><a href="chap0.html" >[MathJax off]</a></p>"X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5" ></a></p>div class="pcenter" >h1 >QuaGroup</h1 >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 >"X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>"X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>div class="contents" >"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 >div ></div >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>span >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 >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 >br />div >div class="chlinkprevnextbot" > <a href="chap0_mj.html" >[Top of Book]</a> <a href="chap0_mj.html#contents" >[Contents]</a> <a href="chap1_mj.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" 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