group theory 2000

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1、printedApril14,2000CLASSICSILLUSTRATEDGROUPTHEORYExceptionalLiegroupsasinvariancegroupsPredragCvitanovi´cAbstractWeoffertheultimatebirdtrackerguidetoexceptionalLiegroups.Keywords:exceptionalLiegroups,invarianttheory,Titsmagicsquare—————————————————————-PRELIMINARYversionof30March

2、2000availableon:www.nbi.dk/GroupTheory/p-cvitanovic@nwu.eduiiContents1Introduction12Apreview52.1Basicconcepts.............................52.2Firstexample:SU(n).........................92.3Secondexample:E6family......................123Invariantsandreducibility153.1Preliminaries

3、..............................153.1.1Groups.............................153.1.2Vectorspaces..........................163.1.3Algebra.............................173.1.4Definingspace,tensors,representations...........183.2Invariants................................203.2.1Algebraofin

4、variants......................223.3Invariancegroups............................233.4Projectionoperators..........................243.5Furtherinvariants...........................253.6Birdtracks................................273.7Clebsch-Gordancoefficients......................293

5、.8Zero-andone-dimensionalsubspaces.................313.9Infinitesimaltransformations.....................323.10Liealgebra...............................363.11OtherformsofLiealgebracommutators...............383.12Irrelevancyofclebsches........................384Recouplings414.1Cou

6、plingsandrecouplings.......................414.2Wigner3n−jcoefficients.......................444.3Wigner-Eckarttheorem........................455Permutations495.1Permutationsinbirdtracks......................495.2Symmetrization.............................50iiiivCONTENTS5.3Antisym

7、metrization..........................525.4Levi-Civitatensor...........................535.5Determinants..............................555.6Characteristicequations........................575.7Fully(anti)symmetrictensors.....................575.8Youngtableaux,Dynkinlabels..........

8、..........586Casimiroperators596.1CasimirsandLi