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LIE ALGEBRAS

 
Acknowledgements
 
Introduction
      Guide for the Reader
 
Constructors for Lie Algebras
 
Finitely Presented Lie Algebras
      Construction of the Free Lie Algebra
      Properties of the Free Lie Algebra
      Operations on Elements of the Free Lie Algebra
      Construction of a Finitely- Presented Lie Algebra
      Homomorphisms of the Free Lie Algebra
 
Lie Algebras Generated by Extremal Elements
      Constructing Lie Algebras Generated by Extremal Elements
      Properties of Lie Algebras Generated by Extremal Elements
      Instances of Lie Algebras Generated by Extremal Elements
      Studying the Parameter Space
 
Families of Lie Algebras
      Almost Reductive Lie Algebras
      Cartan-Type Lie Algebras
      Melikian Lie Algebras
 
Construction of Elements
      Construction of Elements of Structure Constant Algebras
      Construction of Matrix Elements
 
Construction of Subalgebras, Ideals and Quotients
 
Operations on Lie Algebras
      Basic Invariants
      Changing Base Rings
      Bases
      Operations for Semisimple and Reductive Lie Algebras
 
Operations on Subalgebras and Ideals
      Standard Ideals and Subalgebras
      Cartan and Toral Subalgebras
      Standard Series
      The Lie Algebra of Derivations
 
Properties of Lie Algebras and Ideals
 
Operations on Elements
      Indexing
 
The Natural Module
 
Operations for Matrix Lie Algebras
 
Homomorphisms
 
Automorphisms of Classical- type Reductive Algebras
 
Restrictable Lie Algebras
 
Universal Enveloping Algebras
      Background
            Universal Enveloping Algebras
            The Integral Form of a Universal Enveloping Algebra
      Construction of Universal Enveloping Algebras
      Related Structures
      Elements of Universal Enveloping Algebras
            Creation of Elements
            Operations on Elements
 
Solvable and Nilpotent Lie Algebras Classification
      The List of Solvable Lie Algebras
      Comments on the Classification over Finite Fields
      The List of Nilpotent Lie Algebras
      Intrinsics for Working with the Classifications
 
Semisimple Subalgebras of Simple Lie Algebras
 
Nilpotent Orbits in Simple Lie Algebras
 
Bibliography







DETAILS

 
Introduction

      Guide for the Reader

 
Constructors for Lie Algebras
      LieAlgebra<R, n | Q : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie
      LieAlgebra<R, n | T : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie
      LieAlgebra< t | T : parameters > : SeqEnum, SeqEnum -> AlgLie
      LieAlgebra(A) : AlgAss -> AlgLie, Map
      LieAlgebra(A) : AlgMat -> AlgLie
      AbelianLieAlgebra(R, n) : Rng, RngIntElt -> AlgLie
      Example AlgLie_LieAlgebraCons (H100E1)
      ChangeBasis(L, B) : AlgLie, [AlgLieElt] -> AlgLie
      MatrixLieAlgebra(R, n) : Rng, RngIntElt -> AlgMatLie
      MatrixLieAlgebra(A) : AlgMat -> AlgMatLie
      Algebra(M) : AlgMatLie -> AlgLie, Map
      Example AlgLie_MatrixLieAlgebra (H100E2)

 
Finitely Presented Lie Algebras

      Construction of the Free Lie Algebra
            FreeLieAlgebra(F, n) : Rng, RngIntElt -> AlgFPLie
            Example AlgLie_FreeLie (H100E3)

      Properties of the Free Lie Algebra
            Rank(L) : AlgFPLie -> RngIntElt
            CoefficientRing(L) : AlgFPLie -> Rng

      Operations on Elements of the Free Lie Algebra
            L ! 0 : AlgFPLie, RngIntElt -> AlgFPLieElt
            Example AlgLie_FreeLieAlgebra (H100E4)
            IsLeaf(m) : AlgFPLieElt -> BoolElt, AlgFPLieElt, AlgFPLieElt
            Example AlgLie_IsLeaf (H100E5)

      Construction of a Finitely- Presented Lie Algebra
            LieAlgebra(R) : [ AlgFPLieElt ] -> AlgLie, SeqEnum, SeqEnum, Map
            Example AlgLie_LieAlgebra (H100E6)
            Example AlgLie_LieRing (H100E7)
            quo<L | R> : AlgFPLie, [ AlgFPLieElt ] -> AlgLie, Map
            Example AlgLie_QuoAlgFPLie (H100E8)
            NilpotentQuotient(R, d) : [ AlgFPLieElt ], RngIntElt -> AlgLie, SeqEnum, SeqEnum, UserProgram
            Example AlgLie_NilpotentQuotient (H100E9)

      Homomorphisms of the Free Lie Algebra
            hom<L -> M | Q> : AlgFPLie, AlgFPLie, [ AlgFPLieElt ] -> Map
            Example AlgLie_AlgFPLHom (H100E10)

 
Lie Algebras Generated by Extremal Elements

      Constructing Lie Algebras Generated by Extremal Elements
            ExtremalLieAlgebra(K, n) : Rng, RngIntElt -> AlgLieExtr
            ExtremalLieAlgebra(K, G) : Rng, GrphUnd -> AlgLieExtr

      Properties of Lie Algebras Generated by Extremal Elements
            NumberOfGenerators(L) : AlgLieExtr -> RngIntElt
            CoefficientRing(L) : AlgLieExtr -> Rng
            CommutatorGraph(L) : AlgLieExtr -> GrphUnd
            Example AlgLie_AlgLieExtrConstr (H100E11)
            Basis(L) : AlgLieExtr -> SeqEnum[AlgFPLieElt], SeqEnum[UserProgram]
            ZBasis(L) : AlgLieExtr -> SeqEnum[AlgFPLieElt], SeqEnum[RngIntElt], SeqEnum[AlgFPLieElt]
            Dimension(L) : AlgLieExtr -> RngIntElt
            Example AlgLie_AlgLieExtrBasis (H100E12)
            MultiplicationTable(~L) : AlgLieExtr ->
            MultiplicationTable(L) : AlgLieExtr -> SeqEnum
            Example AlgLie_AlgLieExtrMultTable (H100E13)

      Instances of Lie Algebras Generated by Extremal Elements
            Instance(L) : AlgLieExtr -> AlgLie
            Instance(L, Q) : AlgLieExtr, SeqEnum -> AlgLie, Map
            Example AlgLie_AlgLieExtrMultInstance (H100E14)

      Studying the Parameter Space
            FreefValues(L) : AlgLieExtr -> SeqEnum, SeqEnum
            fValue(L, x, b) : AlgLieExtr, RngIntElt, RngIntElt -> RngElt
            fValueProof(L, x, b) : AlgLieExtr, RngIntElt, RngIntElt ->
            Example AlgLie_AlgLieExtrfVal (H100E15)
            DimensionsEstimate(L, g) : AlgLieExtr, UserProgram -> SeqEnum, SetMulti
            InstancesForDimensions(L, g, D) : AlgLieExtr, UserProgram, SetEnum[RngIntElt] -> Assoc
            Example AlgLie_AlgLieExtrVarietyDims (H100E16)

 
Families of Lie Algebras

      Almost Reductive Lie Algebras
            LieAlgebra(T, k) : MonStgElt, Rng -> AlgLie
            MatrixLieAlgebra(T, k) : MonStgElt, Rng -> AlgLie
            Example AlgLie_ReductiveLieAlgebra (H100E17)
            Example AlgLie_LieAlgebraIsogeny (H100E18)
            LieAlgebra(N, k, p) : MonStgElt, Rng, GrpPermElt -> AlgLie
            TwistedLieAlgebra(R, k) : RootDtm, Rng -> AlgLie
            Example AlgLie_TwistedLieAlgebra (H100E19)

      Cartan-Type Lie Algebras
            WittLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, Map
            Example AlgLie_witt-alg-ex (H100E20)
            SpecialLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie, Map, Map
            Example AlgLie_special-lie-alg-ex (H100E21)
            HamiltonianLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie
            Example AlgLie_hamilton-alg-ex (H100E22)
            ContactLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie
            Example AlgLie_contact-alg-ex (H100E23)

      Melikian Lie Algebras
            MelikianLieAlgebra(F, n1, n2) : Fld, RngIntElt, RngIntElt -> AlgLie, Map
            Example AlgLie_melikian-alg-ex (H100E24)

 
Construction of Elements
      Zero(L) : AlgLie -> AlgLieElt
      Random(L) : AlgLie -> AlgLieElt

      Construction of Elements of Structure Constant Algebras
            elt<L | r1, r2, ..., rn> : AlgLie, RngElt, RngElt, ..., RngElt -> AlgLieElt
            L ! Q : AlgLie, SeqEnum[RngElt] -> AlgLieElt
            BasisProduct(L, i, j) : AlgLie, RngIntElt, RngIntElt -> AlgLieElt
            BasisProducts(L) : AlgLie -> SeqEnum

      Construction of Matrix Elements
            elt<R | L> : AlgMatLie, [ RngElt ] -> AlgMatLieElt
            DiagonalMatrix(L, Q) : AlgMatLie, [RngElt] -> AlgMatLieElt
            ScalarMatrix(L, r) : AlgMatLie, RngElt -> AlgMatLieElt

 
Construction of Subalgebras, Ideals and Quotients
      sub<L | A> : AlgLie, List -> AlgLie, Map
      ideal<L | A> : AlgLie, List -> AlgLie, Map
      quo<L | A> : AlgLie, List -> AlgLie, Map
      L / S : AlgLie, AlgLie -> AlgLie
      Example AlgLie_LieAlgebraQuotient (H100E25)
      QuotientWithPullback(L, I) : AlgLie, AlgLie -> AlgLie, Map, UserProgram, UserProgram
      Example AlgLie_LieAlgebraQuotientPullback (H100E26)

 
Operations on Lie Algebras
      L eq K : AlgLie, AlgLie -> BoolElt
      L ne K : AlgLie, AlgLie -> BoolElt
      L subset K : AlgLie, AlgLie -> BoolElt
      L notsubset K : AlgLie, AlgLie -> BoolElt
      L meet M : AlgLie, AlgLie -> AlgLie
      L * M : AlgLie, AlgLie -> AlgLie
      L ^ n : AlgLie, RngIntElt -> AlgLie
      Morphism(L, M) : AlgLie, AlgLie -> Map
      IsIsomorphic(L, M) : AlgLie, AlgLie -> BoolElt, .
      IsKnownIsomorphic(L, M) : AlgLie, AlgLie -> BoolElt, BoolElt, .
      IsIsomorphism(m) : Map[AlgLie, AlgLie] -> BoolElt
      Example AlgLie_Isomorphic (H100E27)
      Example AlgLie_IsomorphicG2 (H100E28)

      Basic Invariants
            CoefficientRing(L) : AlgLie -> Rng
            Dimension(L) : AlgLie -> RngIntElt
            # L : AlgLie -> RngIntElt
            Moduli(L) : AlgLie -> SeqEnum
            Example AlgLie_ModuliOfLieAlgebra (H100E29)

      Changing Base Rings
            ChangeRing(L, S) : AlgLie, Rng -> AlgLie, Map
            ChangeRing(L, S, f) : AlgLie, Rng, Map -> AlgLie, Map

      Bases
            BasisElement(A, i) : AlgLie, RngIntElt -> AlgLieElt
            Basis(A) : AlgLie -> [ AlgLieElt ]
            IsIndependent(Q) : [ AlgLieElt ] -> BoolElt
            ExtendBasis(S, L) : AlgLie, AlgLie -> [ AlgElt ]

      Operations for Semisimple and Reductive Lie Algebras
            SemisimpleType(L) : AlgLie -> MonStgElt
            Example AlgLie_SemisimpleType (H100E30)
            ReductiveType(L) : AlgLie -> RootDtm, MonStgElt, SeqEnum, SeqEnum
            Example AlgLie_ReductiveType (H100E31)
            RootSystem(L) : AlgLie -> [ ModTupRngElt ], [ AlgLieElt ], [ ModTupRngElt ], AlgMatElt
            Example AlgLie_RootSystem (H100E32)
            RootDatum(L) : AlgLie -> RootDtm
            Example AlgLie_RootDatum (H100E33)
            ChevalleyBasis(L) : AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
            Example AlgLie_ChevalleyBasis (H100E34)
            ChevalleyBasis(L, H, R) : AlgLie, AlgLie, RootDtm -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
            IsChevalleyBasis(L, R, x, y, h) : AlgLie, RootDtm, [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ] -> BoolElt, [ Tup ]
            Example AlgLie_ChevalleyBasisSmallChar (H100E35)
            TwistedBasis(L, H, R) : AlgLie, AlgLie, RootDtm -> AlgLie, AlgLie, Rec, AlgMatElt
            Example AlgLie_TwistedBasis (H100E36)
            WeylGroup(L) : AlgLie -> GrpPermCox
            WeylGroup(GrpFPCox, L) : Cat, AlgLie -> GrpPermCox
            WeylGroup(GrpMat, L) : Cat, AlgLie -> GrpPermCox

 
Operations on Subalgebras and Ideals
      DirectSum(L, M) : AlgLie, AlgLie -> AlgLie
      IndecomposableSummands(L) : AlgLie -> [ AlgLie ]
      Example AlgLie_DirectSumDecomposition (H100E37)

      Standard Ideals and Subalgebras
            Centre(L) : AlgLie -> AlgLie
            Centraliser(L, K) : AlgLie, AlgLie -> AlgLie, Map
            Centraliser(L, x) : AlgLie, AlgLieElt -> AlgLie, Map
            Normaliser(L, K) : AlgLie, AlgLie -> AlgLie
            SolubleRadical(L) : AlgLie -> AlgLie
            Nilradical(L) : AlgLie -> AlgLie
            Example AlgLie_Operations (H100E38)

      Cartan and Toral Subalgebras
            CartanSubalgebra(L) : AlgLie -> AlgLie
            IsCartanSubalgebra(L, H) : AlgLie, AlgLie -> BoolElt
            Example AlgLie_CartanSubalgebra (H100E39)
            SplittingCartanSubalgebra(L) : AlgLie -> AlgLie
            IsSplittingCartanSubalgebra(L, H) : AlgLie, AlgLie -> BoolElt
            SplitToralSubalgebra(L) : AlgLie -> AlgLie
            IsSplitToralSubalgebra(L, H) : AlgLie, AlgLie -> BoolElt
            Example AlgLie_SplitToral (H100E40)

      Standard Series
            CompositionSeries(L) : AlgLie -> [ Alg ], [ AlgLie ], AlgMatElt
            CompositionFactors(L) : AlgLie -> [ AlgLie ]
            MinimalIdeals(L : parameters) : AlgLie -> [ AlgLie ], BoolElt
            MaximalIdeals(L : parameters) : AlgLie -> [ AlgLie ], BoolElt
            DerivedSeries(L) : AlgLie -> [ AlgLie ]
            LowerCentralSeries(L) : AlgLie -> [ AlgLie ]
            UpperCentralSeries(L) : AlgLie -> [ AlgLie ]
            Example AlgLie_Series (H100E41)

      The Lie Algebra of Derivations
            LieAlgebraOfDerivations(L) : AlgLie -> AlgLie, Rec
            Example AlgLie_LieAlgebraOfDerivations (H100E42)

 
Properties of Lie Algebras and Ideals
      KillingMatrix(L) : AlgLie -> ModMatFldElt
      Example AlgLie_Other (H100E43)
      IsAbelian(L) : AlgLie -> BoolElt
      IsSoluble(L) : AlgLie -> BoolElt
      IsNilpotent(L) : AlgLie -> BoolElt
      IsCentral(L, M) : AlgLie,AlgLie -> BoolElt
      IsSimple(L) : AlgLie -> BoolElt
      IsSemisimple(L) : AlgLie -> BoolElt
      IsReductive(L) : AlgLie -> BoolElt
      HasLeviSubalgebra(L) : AlgLie -> BoolElt
      IsClassicalType(L) : AlgLie -> BoolElt
      Example AlgLie_Predicates (H100E44)

 
Operations on Elements
      IsCentral(L, M) : AlgLie,AlgLieElt -> BoolElt
      NonNilpotentElement(L) : AlgLie -> AlgLieElt
      Example AlgLie_NonNilpotentElement (H100E45)
      AdjointMatrix(L, x) : AlgLie, AlgLieElt -> AlgMatLieElt
      Example AlgLie_Other (H100E46)

      Indexing
            a[i] : AlgLieElt, RngIntElt -> RngElt
            a[i] := r : AlgLieElt, RngIntElt, RngElt -> AlgLieElt
            a[i, j] : AlgMatLieElt, RngIntElt, RngIntElt -> RngElt

 
The Natural Module
      Module(L) : AlgLie -> ModTupRng
      RModule(L) : AlgMatLie -> ModTupRng
      BaseModule(L) : AlgMatLie -> ModTupRng
      Degree(L) : AlgLie -> RngIntElt
      Degree(a) : AlgLieElt -> RngIntElt
      ElementToSequence(a) : AlgLieElt -> SeqEnum
      Coordinates(M, a) : AlgLie, AlgLieElt -> SeqEnum
      InnerProduct(a, b) : AlgLieElt, AlgLieElt -> RngElt
      Support(a) : AlgLieElt -> SetEnum

 
Operations for Matrix Lie Algebras
      BaseModule(M) : AlgMatLie -> ModTupRng
      Generic(M) : AlgMatLie -> AlgMatLie
      Kernel(X) : AlgMatLieElt -> ModTupRng
      NullspaceOfTranspose(X) : AlgMatLieElt -> ModTupRng

 
Homomorphisms
      hom<L -> M | Q> : AlgLie, AlgLie, [ AlgLieElt ] -> Map

 
Automorphisms of Classical- type Reductive Algebras
      IdentityAutomorphism(L) : AlgLie -> Map
      InnerAutomorphism(L, x) : AlgLie, GrpLieElt -> Map
      InnerAutomorphismGroup(L) : AlgLie -> GrpLie, Map
      DiagonalAutomorphism(L, v) : AlgLie, ModTupRngElt -> Map
      GraphAutomorphism(L, p) : AlgLie, GrpPermElt -> Map
      Example AlgLie_GraphAutomorphism (H100E47)

 
Restrictable Lie Algebras
      IsRestrictable(L) : AlgLie -> BoolElt, Map
      RestrictionMap(L) : AlgLie -> Map
      Example AlgLie_IsRestricted (H100E48)
      RestrictedSubalgebra(Q) : SetEnum[AlgLieElt] -> AlgLie
      pClosure(L, M) : AlgLie, AlgLie -> AlgLie
      IsRestrictedSubalgebra(L, M) : AlgLie, AlgLie -> AlgLie
      pQuotient(L, M) : AlgLie, AlgLie -> AlgLie
      JenningsLieAlgebra(G) : Grp -> AlgLie, SeqEnum
      Example AlgLie_JenningsLie (H100E49)

 
Universal Enveloping Algebras

      Background

            Universal Enveloping Algebras

            The Integral Form of a Universal Enveloping Algebra

      Construction of Universal Enveloping Algebras
            UniversalEnvelopingAlgebra(L) : AlgLie -> AlgUE
            IntegralUEA(L) : AlgLie -> AlgIUE
            Example AlgLie_UEACon (H100E50)
            AssignNames(~U, Q) : AlgPBW, [ MonStgElt ] ->
            ChangeRing(U, S) : AlgUE, Rng -> AlgUE

      Related Structures
            CoefficientRing(U) : AlgPBW -> Rng
            Algebra(U) : AlgUE -> AlgLie

      Elements of Universal Enveloping Algebras

            Creation of Elements
                  U ! 0 : AlgPBW, RngIntElt -> AlgPBWElt
                  U ! 1 : AlgPBW, RngIntElt -> AlgPBWElt
                  U . i : AlgPBW, RngIntElt -> AlgPBWElt
                  U ! r : AlgPBW, Any -> AlgPBWElt
                  HBinomial(U, i, n) : AlgIUE, RngIntElt, RngIntElt -> AlgIUEElt
                  Example AlgLie_HBinomial (H100E51)

            Operations on Elements
                  Monomials(u) : AlgPBWElt -> SeqEnum
                  Coefficients(u) : AlgPBWElt -> SeqEnum
                  Degree(u, i) : AlgPBWElt -> RngIntElt
                  Example AlgLie_EltOps (H100E52)

 
Solvable and Nilpotent Lie Algebras Classification

      The List of Solvable Lie Algebras

      Comments on the Classification over Finite Fields

      The List of Nilpotent Lie Algebras

      Intrinsics for Working with the Classifications
            SolvableLieAlgebra( F, n, k : parameters) : Fld, RngIntElt, RngIntElt -> AlgLie
            Example AlgLie_SLACLnk (H100E53)
            NilpotentLieAlgebra( F, r, k : parameters) : Fld, RngIntElt, RngIntElt -> AlgLie
            Example AlgLie_NLACNrk (H100E54)
            AllSolvableLieAlgebras(F, d) : Fld, RngIntElt -> SeqEnum
            AllNilpotentLieAlgebras(F, d) : Fld, RngIntElt -> SeqEnum
            IdDataSLAC(L) : AlgLie -> MonStgElt, SeqEnum, Map
            IdDataNLAC(L) : AlgLie -> MonStgElt, SeqEnum, Map
            MatrixOfIsomorphism(f) : Map -> AlgMatElt
            Example AlgLie_SLACIdData (H100E55)
            Example AlgLie_NLACIdData (H100E56)

 
Semisimple Subalgebras of Simple Lie Algebras
      SubalgebrasInclusionGraph( t ) : MonStgElt -> GrphDir
      Example AlgLie_SSSDB (H100E57)
      RestrictionMatrix( G, k ) : GrphDir, RngIntElt ) -> AlgMatElt
      Example AlgLie_SSSDBRestr (H100E58)

 
Nilpotent Orbits in Simple Lie Algebras
      IsGenuineWeightedDynkinDiagram( L, wd ) : AlgLie, SeqEnum -> BoolElt, SeqEnum
      Example AlgLie_NilpOrbGenuine (H100E59)
      NilpotentOrbit( L, wd ) : AlgLie, SeqEnum -> NilpOrbAlgLie
      NilpotentOrbit( L, e ) : AlgLie, AlgLieElt -> NilpOrbAlgLie
      Example AlgLie_NoByRep (H100E60)
      NilpotentOrbits( L ) : AlgLie -> SeqEnum
      Example AlgLie_NOD4 (H100E61)
      Partition( o ) : NilpOrbAlgLie -> SeqEnum
      Example AlgLie_PartEx (H100E62)
      SL2Triple( o ) : NilpOrbAlgLie -> SeqEnum
      SL2Triple( L, e ) : AlgLie, AlgLieElt -> SeqEnum
      Representative( o ) : NilpOrbAlgLie -> AlgLieElt
      WeightedDynkinDiagram( o ) : NilpOrbAlgLie -> SeqEnum
      Example AlgLie_WDD (H100E63)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013