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)
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)
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
Mon Dec 17 14:40:36 EST 2012