Construction of Lie Algebras
LieAlgebra< R, n | Q : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie
LieAlgebra< R, n | T : parameters > : Rng, RngIntElt, SeqEnum -> AlgLie
LieAlgebra(A) : AlgAss -> AlgLie
Construction of Semisimple and Reductive Lie Algebras
SemisimpleLieAlgebra( N, k ) : MonStrElt, Rng -> AlgLie
ReductiveLieAlgebra( R, k ) : RootDtm, Rng -> AlgLie
ReductiveLieAlgebra( R, k ) : RootSys, Rng -> AlgLie
Example AlgLie_ReductiveLieAlgebra (H81E1)
SemisimpleLieAlgebra( C, k ) : AlgMatElt, Rng -> AlgLie
SemisimpleLieAlgebra( D, k ) : GrphDir, Rng -> AlgLie
SimpleLieAlgebra(X, n, k) : MonStgElt, RngIntElt, Fld -> AlgLie
Example AlgLie_SimpleLieAlgebra (H81E2)
Construction of Elements
elt< L | r_1, r_2, ..., r_n > : AlgLie, RngElt, RngElt, ..., RngElt -> AlgLieElt
L ! Q : AlgLie, SeqEnum[RngElt] -> AlgLieElt
Zero(L) : AlgLie -> AlgLieElt
Random(L) : AlgLie -> AlgLieElt
BasisProduct(L, i, j) : AlgLie, RngIntElt, RngIntElt -> AlgLieElt
BasisProducts(L) : AlgLie -> [[ AlgLieElt ]]
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
Operations on Lie Algebras
L eq K : AlgLie, AlgLie -> BoolElt
L ne K : AlgLie, AlgLie -> BoolElt
L subset K : AlgLie, AlgLie -> BoolElt
L subset K : AlgLie, AlgLie -> BoolElt
L meet M : AlgGen, AlgGen -> AlgGen
L * M : AlgGen, AlgGen -> AlgGen
L ^ n : AlgGen, RngIntElt -> AlgGen
Morphism(L, M) : AlgGen, AlgGen -> Map
Basic Invariants
CoefficientRing(L) : AlgLie -> Rng
Dimension(L) : AlgLie -> RngIntElt
# L : AlgLie -> RngIntElt
Changing Base Rings
ChangeRing(L, S) : AlgLie, Rng -> AlgGen, Map
ChangeRing(L, S, f) : AlgGen, Rng, Map -> AlgGen, Map
Bases
BasisElement(A, i) : AlgGen, RngIntElt -> AlgGenElt
Basis(A) : AlgGen -> [ AlgGenElt ]
IsIndependent(Q) : [ AlgGen ] -> BoolElt
ExtendBasis(S, A) : AlgGen, AlgGen -> [ AlgElt ]
Operations for Semisimple and Reductive Lie Algebras
SemiSimpleType(L) : AlgLie -> AlgLie
Example AlgLie_SemiSimpleType (H81E3)
RootSystem(L) : AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ], AlgMatElt
Example AlgLie_RootSystem (H81E4)
RootDatum(L) : AlgLie -> RootDtm
Example AlgLie_RootDatum (H81E5)
ChevalleyBasis(L) : AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
Example AlgLie_ChevalleyBasis (H81E6)
WeylGroup(L) : AlgLie -> GrpPermCox
WeylGroup(GrpFPCox, L) : Cat, AlgLie -> GrpPermCox
WeylGroup(GrpMat, L) : Cat, AlgLie -> GrpPermCox
Operations on Subalgebras and Ideals
DirectSum(A, B) : AlgGen, AlgGen -> AlgGen
DirectSumDecomposition(L) : AlgLie -> [ AlgLie ]
Example AlgLie_DirectSumDecomposition (H81E7)
Standard Ideals and Series
Centre(L) : AlgLie -> AlgLie
Centraliser(L, K) : AlgLie, AlgLie -> AlgLie
Normaliser(L, K) : AlgLie, AlgLie -> AlgLie
SolubleRadical(L) : AlgLie -> AlgLie
NilRadical(L) : AlgLie -> AlgLie
Example AlgLie_Operations (H81E8)
CartanSubalgebra(L) : AlgLie -> AlgLie
Example AlgLie_CartanSubalgebra (H81E9)
KillingMatrix(L) : AlgLie -> AlgMatElt
AdjointMatrix(L, x) : AlgLie, AlgLieElt -> AlgMatElt
Example AlgLie_Other (H81E10)
CompositionSeries(L) : AlgLie -> [ Alg ], [ AlgGen ], AlgMatElt
MinimalIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt
MaximalIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt
DerivedSeries(L) : AlgLie -> [ AlgLie ]
LowerCentralSeries(L) : AlgLie -> [ AlgLie ]
UpperCentralSeries(L) : AlgLie -> [ AlgLie ]
Example AlgLie_Series (H81E11)
Properties of Lie Algebras and Ideals
IsAbelian(L) : AlgLie -> BoolElt
IsSoluble(L) : AlgLie -> BoolElt
IsNilpotent(L) : AlgLie -> BoolElt
IsRestrictedLieAlgebra(L) : AlgLie -> BoolElt
IsSimple(L) : AlgLie -> BoolElt
IsSemisimple(L) : AlgLie -> BoolElt
IsReductive(L) : AlgLie -> BoolElt
HasLeviSubalgebra(L) : AlgLie -> BoolElt
Example AlgLie_Predicates (H81E12)
Properties of Subalgebras and Ideals
Operations and Properties for Elements
Arithmetic
NonNilpotentElement(L) : AlgLie -> AlgLieElt
Example AlgLie_NonNilpotentElement (H81E13)
Centre(L) : AlgLie -> AlgLie
Centraliser(L, K) : AlgLie, AlgLie -> AlgLie
Normaliser(L, K) : AlgLie, AlgLie -> AlgLie
SolubleRadical(L) : AlgLie -> AlgLie
Nilradical(L) : AlgLie -> AlgLie
Example AlgLie_Operations (H81E14)
CartanSubalgebra(L) : AlgLie -> AlgLie
Example AlgLie_CartanSubalgebra (H81E15)
KillingMatrix(L) : AlgLie -> AlgMatElt
AdjointMatrix(L, x) : AlgLie, AlgLieElt -> AlgMatElt
Example AlgLie_Other (H81E16)
Indexing
a[i] : AlgLieElt, RngIntElt -> RngElt
a[i] := r : AlgLieElt, RngIntElt, RngElt -> AlgGenElt
The Natural Module
Module(L) : AlgLie -> 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) : AlgLienElt -> SetEnum
Homomorphisms
hom< L -> M | Q > : AlgLie, AlgGen, [ AlgGenElt ] -> Map
Representations of Lie Algebras
AdjointRepresentation( L ) : AlgLie -> Map
StandardRepresentation( L ) : AlgLie -> Map
Example AlgLie_StandardRepresentation (H81E17)
DominantCharacter(D, w) : RootDtm, [ ] -> [ ModTupRngElt ], [ RngIntElt ]
DimensionOfHighestWeightModule(D, w) : RootDtm, [ ] -> RngIntElt
Example AlgLie_DominantCharacter (H81E18)
DecomposeTensorProduct(D, w, x) : RootDtm, [ ], [ ] -> [ ModTupRngElt ], [ RngIntElt ]
Example AlgLie_DecomposeTensor (H81E19)
HighestWeightRepresentation(L, w) : AlgLie, [ ] -> UserProgram