lie algebras - s. sternberg

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1、LiealgebrasShlomoSternbergApril23,20042Contents1TheCampbellBakerHausdorffFormula71.1Theproblem..............................71.2ThegeometricversionoftheCBHformula.............81.3TheMaurer-Cartanequations....................111.4ProofofCBHfromMaurer-Cartan......

2、............141.5Thedifferentialoftheexponentialanditsinverse.........151.6Theaveragingmethod.........................161.7TheEulerMacLaurinFormula....................181.8Theuniversalenvelopingalgebra...................191.8.1Tensorproductofvectorspaces........

3、.......201.8.2Thetensorproductoftwoalgebras.............211.8.3Thetensoralgebraofavectorspace.............211.8.4Constructionoftheuniversalenvelopingalgebra......221.8.5ExtensionofaLiealgebrahomomorphismtoitsuniversalenvelopingalgebra.......................221.

4、8.6Universalenvelopingalgebraofadirectsum........221.8.7Bialgebrastructure......................231.9ThePoincar´e-Birkhoff-WittTheorem................241.10Primitives................................281.11FreeLiealgebras...........................291.11.1Magmasan

5、dfreemagmasonaset.............291.11.2TheFreeLieAlgebraLX...................301.11.3ThefreeassociativealgebraAss(X).............311.12AlgebraicproofofCBHandexplicitformulas............321.12.1AbstractversionofCBHanditsalgebraicproof......321.12.2Explicitformula

6、forCBH...................322sl(2)anditsRepresentations.352.1LowdimensionalLiealgebras.....................352.2sl(2)anditsirreduciblerepresentations...............362.3TheCasimirelement.........................392.4sl(2)issimple.............................402.

7、5Completereducibility.........................412.6TheWeylgroup............................4234CONTENTS3Theclassicalsimplealgebras.453.1Gradedsimplicity...........................453.2sl(n+1)................................473.3Theorthogonalalgebras............

8、............483.4Thesymplecticalgebras........................503.5Therootstructures..........................523.5.1An=sl(n+1).........................523.5.2Cn=sp(2n),n≥2.