Lie Algebras
- Introduction
- Constructors for Lie Algebras
LieAlgebra<R, n | Q : parameters >: Rng, RngIntElt, SeqEnum → AlgLieLieAlgebra<M | Q : parameters >: ModTupRng, SeqEnum → AlgLieLieAlgebra<R, n | T : parameters >: Rng, RngIntElt, SeqEnum → AlgLieLieAlgebra< t | T : parameters >: SeqEnum, SeqEnum → AlgLieLieAlgebra(A): AlgAss → AlgLie, MapLieAlgebra(A): AlgMat → AlgLieAbelianLieAlgebra(R, n): Rng, RngIntElt → AlgLieExample: Lie Algebra ConsChangeBasis(L, B): AlgLie, {[AlgLieElt]} → AlgLieChangeBasis(L, B): AlgLie, {[ModTupFldElt]} → AlgLieChangeBasis(L, B): AlgLie, Mtrx → AlgLieMatrixLieAlgebra(R, n): Rng, RngIntElt → AlgMatLieMatrixLieAlgebra(A): AlgMat → AlgMatLieAlgebra(M): AlgMatLie → AlgLie, MapLieAlgebra(M): AlgMatLie → AlgLie, MapExample: Matrix Lie Algebra
- 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
LieAlgebra(R): [ AlgFPLieElt ] → AlgLie, SeqEnum, SeqEnum, MapLieAlgebra(R): { AlgFPLieElt } → AlgLie, SeqEnum, SeqEnum, MapExample: Lie AlgebraExample: Lie Ringquo<L | R>: AlgFPLie, [ AlgFPLieElt ] → AlgLie, Mapquo<L | R>: AlgFPLie, { AlgFPLieElt } → AlgLie, MapExample: Quo Alg FP LieNilpotentQuotient(R, d): [ AlgFPLieElt ], RngIntElt → AlgLie, SeqEnum, SeqEnum, UserProgramNilpotentQuotient(R, d): { AlgFPLieElt, RngIntElt } → AlgLie, SeqEnum, SeqEnum, UserProgramExample: Nilpotent Quotient
- 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
FreefValues(L): AlgLieExtr → SeqEnum, SeqEnumfValue(L, x, b): AlgLieExtr, RngIntElt, RngIntElt → RngEltfValueProof(L, x, b): AlgLieExtr, RngIntElt, RngIntEltExample: Alg Lie Extrf ValDimensionsEstimate(L, g): AlgLieExtr, UserProgram → SeqEnum, SetMultiInstancesForDimensions(L, g, D): AlgLieExtr, UserProgram, SetEnum[RngIntElt] → AssocExample: Alg Lie Extr Variety Dims
- Families of Lie Algebras
- Almost Reductive Lie Algebras
LieAlgebra(T, k): MonStgElt, Rng → AlgLieLieAlgebra(R, k): RootDtm, Rng → AlgLieLieAlgebra(R, k): RootSys, Rng → AlgLieLieAlgebra(D, k): GrphDir, Rng → AlgLieLieAlgebra(C, k): AlgMatElt, Rng → AlgLieMatrixLieAlgebra(T, k): MonStgElt, Rng → AlgLieMatrixLieAlgebra(R, k): RootDtm, Rng → AlgLieMatrixLieAlgebra(R, k): RootSys, Rng → AlgLieMatrixLieAlgebra(D, k): GrphDir, Rng → AlgLieMatrixLieAlgebra(C, k): AlgMatElt, Rng → AlgLieExample: Reductive Lie AlgebraExample: Lie Algebra IsogenyLieAlgebra(N, k, p): MonStgElt, Rng, GrpPermElt → AlgLieLieAlgebra(R, k, p): RootDtm, Rng, GrpPermElt → AlgLieTwistedLieAlgebra(R, k): RootDtm, Rng → AlgLieExample: Twisted Lie Algebra
- Cartan-Type Lie Algebras
WittLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, MapExample: Witt Alg ExSpecialLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, Map, MapConformalSpecialLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, AlgLie, Map, MapExample: Special Lie Alg ExHamiltonianLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLieConformalHamiltonianLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, AlgLieExample: Hamilton Alg ExContactLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLieExample: Contact Alg Ex
- Melikian Lie Algebras
- Construction of Elements
Zero(L): AlgLie → AlgLieEltZero(L): AlgMatLie → AlgMatLieEltL ! 0: AlgLie, RngIntElt → AlgLieEltL ! 0: AlgMatLie, RngIntElt → AlgMatLieEltRandom(L): AlgLie → AlgLieEltRandom(L): AlgMatLie → AlgMatLieElt- Construction of Elements of Structure Constant Algebras
elt<L | r₁, r₂, ..., rₙ>: AlgLie, RngElt, RngElt, ..., RngElt → AlgLieEltL ! Q: AlgLie, SeqEnum[RngElt] → AlgLieEltBasisProduct(L, i, j): AlgLie, RngIntElt, RngIntElt → AlgLieEltBasisProducts(L): AlgLie → SeqEnum
- Construction of Matrix Elements
- Construction of Subalgebras, Ideals and Quotients
sub<L | A>: AlgLie, List → AlgLie, Mapsub<L | A>: AlgMatLie, List → AlgMatLie, Mapideal<L | A>: AlgLie, List → AlgLie, Mapideal<L | A>: AlgMatLie, List → AlgMatLie, Mapquo<L | A>: AlgLie, List → AlgLie, Mapquo<L | A>: AlgMatLie, List → AlgMatLie, MapL / S: AlgLie, AlgLie → AlgLieL / S: AlgMatLie, AlgMatLie → AlgLieExample: Lie Algebra QuotientQuotientWithPullback(L, I): AlgLie, AlgLie → AlgLie, Map, UserProgram, UserProgramExample: Lie Algebra Quotient Pullback
- Operations on Lie Algebras
L eq K: AlgLie, AlgLie → BoolEltL eq K: AlgMatLie, AlgMatLie → BoolEltL ne K: AlgLie, AlgLie → BoolEltL ne K: AlgMatLie, AlgMatLie → BoolEltL subset K: AlgLie, AlgLie → BoolEltL subset K: AlgMatLie, AlgMatLie → BoolEltL notsubset K: AlgLie, AlgLie → BoolEltL meet M: AlgLie, AlgLie → AlgLieL meet M: AlgMatLie, AlgMatLie → AlgMatLieL * M: AlgLie, AlgLie → AlgLieL * M: AlgMatLie, AlgMatLie → AlgMatLieL ^ n: AlgLie, RngIntElt → AlgLieMorphism(L, M): AlgLie, AlgLie → MapIsIsomorphic(L, M): AlgLie, AlgLie → BoolElt, .IsKnownIsomorphic(L, M): AlgLie, AlgLie → BoolElt, BoolElt, .IsIsomorphism(m): Map[AlgLie, AlgLie] → BoolEltExample: IsomorphicExample: Isomorphic G2- Basic Invariants
- Changing Base Rings
ChangeRing(L, S): AlgLie, Rng → AlgLie, MapChangeRing(L, S): AlgMatLie, Rng → AlgMatLie, MapChangeRing(L, S, f): AlgLie, Rng, Map → AlgLie, MapChangeRing(L, S, f): AlgMatLie, Rng, Map → AlgMatLie, Map
- Bases
BasisElement(A, i): AlgLie, RngIntElt → AlgLieEltBasisElement(A, i): AlgMatLie, RngIntElt → AlgMatLieEltA . i: AlgLie, RngIntElt → AlgLieEltA . i: AlgMatLie, RngIntElt → AlgMatLieEltBasis(A): AlgLie → [ AlgLieElt ]Basis(A): AlgMatLie → [ AlgMatLieElt ]IsIndependent(Q): [ AlgLieElt ] → BoolEltIsIndependent(Q): [ AlgMatLieElt ] → BoolEltIsIndependent(Q): { AlgLieElt } → BoolEltIsIndependent(Q): { AlgMatLieElt } → BoolEltExtendBasis(S, L): AlgLie, AlgLie → [ AlgElt ]ExtendBasis(S, L): AlgMatLie, AlgMatLie → [ AlgElt ]ExtendBasis(Q, L): [ AlgLieElt ], AlgLie → [ AlgElt ]ExtendBasis(Q, L): [ AlgMatLie ], AlgMatLie → [ AlgElt ]
- Operations for Semisimple and Reductive Lie Algebras
SemisimpleType(L): AlgLie → MonStgEltSemisimpleType(L): AlgMatLie → MonStgEltCartanName(L): AlgLie → MonStgEltCartanName(L): AlgMatLie → MonStgEltExample: Semisimple TypeReductiveType(L): AlgLie → RootDtm, MonStgElt, SeqEnum, SeqEnumReductiveType(L, H): AlgLie, AlgLie → RootDtm, MonStgElt, SeqEnum, SeqEnumExample: Reductive TypeRootSystem(L): AlgLie → [ ModTupRngElt ], [ AlgLieElt ], [ ModTupRngElt ], AlgMatEltRootSystem(L): AlgMatLie → [ ModTupRngElt ], [ AlgMatLieElt ], [ ModTupRngElt ], AlgMatEltExample: Root SystemRootDatum(L): AlgLie → RootDtmRootDatum(L): AlgMatLie → RootDtmExample: Root DatumChevalleyBasis(L): AlgLie → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]ChevalleyBasis(L, H): AlgLie, AlgLie → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]ChevalleyBasis(L): AlgMatLie → [ AlgMatLieElt ], [ AlgMatLieElt ], [ AlgMatLieElt ]Example: Chevalley BasisChevalleyBasis(L, H, R): AlgLie, AlgLie, RootDtm → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]IsChevalleyBasis(L, R, x, y, h): AlgLie, RootDtm, [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ] → BoolElt, [ Tup ]Example: Chevalley Basis Small CharTwistedBasis(L, H, R): AlgLie, AlgLie, RootDtm → AlgLie, AlgLie, Rec, AlgMatEltExample: Twisted BasisWeylGroup(L): AlgLie → GrpPermCoxWeylGroup(GrpPermCox, L): Cat, AlgLie → GrpPermCoxWeylGroup(GrpFPCox, L): Cat, AlgLie → GrpPermCoxWeylGroup(GrpMat, L): Cat, AlgLie → GrpPermCox
- Operations on Subalgebras and Ideals
DirectSum(L, M): AlgLie, AlgLie → AlgLieDirectSum(L, M): AlgMatLie, AlgMatLie → AlgMatLieIndecomposableSummands(L): AlgLie → [ AlgLie ]IndecomposableSummands(L): AlgMatLie → [ AlgMatLie ]DirectSumDecomposition(L): AlgLie → [ AlgLie ]DirectSumDecomposition(L): AlgMatLie → [ AlgMatLie ]Example: Direct Sum Decomposition- Standard Ideals and Subalgebras
Centre(L): AlgLie → AlgLieCentre(L): AlgMatLie → AlgMatLieCenter(L): AlgLie → AlgLieCenter(L): AlgMatLie → AlgMatLieCentraliser(L, K): AlgLie, AlgLie → AlgLie, MapCentraliser(L, K): AlgMatLie, AlgMatLie → AlgMatLie, MapCentralizer(L, K): AlgLie, AlgLie → AlgLie, MapCentralizer(L, K): AlgMatLie, AlgMatLie → AlgMatLie, MapCentraliser(L, x): AlgLie, AlgLieElt → AlgLie, MapCentraliser(L, x): AlgMatLie, AlgMatLieElt → AlgMatLie, MapCentralizer(L, x): AlgLie, AlgLieElt → AlgLie, MapCentralizer(L, x): AlgMatLie, AlgMatLieElt → AlgMatLie, MapNormaliser(L, K): AlgLie, AlgLie → AlgLieNormaliser(L, K): AlgMatLie, AlgMatLie → AlgMatLieNormalizer(L, K): AlgLie, AlgLie → AlgLieNormalizer(L, K): AlgMatLie, AlgMatLie → AlgMatLieSolubleRadical(L): AlgLie → AlgLieSolubleRadical(L): AlgMatLie → AlgMatLieSolvableRadical(L): AlgLie → AlgLieSolvableRadical(L): AlgMatLie → AlgMatLieNilradical(L): AlgLie → AlgLieNilradical(L): AlgMatLie → AlgMatLieExample: Operations
- Cartan and Toral Subalgebras
CartanSubalgebra(L): AlgLie → AlgLieCartanSubalgebra(L): AlgMatLie → AlgMatLieIsCartanSubalgebra(L, H): AlgLie, AlgLie → BoolEltIsCartanSubalgebra(L, H): AlgMatLie, AlgMatLie → BoolEltExample: Cartan SubalgebraSplittingCartanSubalgebra(L): AlgLie → AlgLieSplitMaximalToralSubalgebra(L): AlgLie → AlgLieIsSplittingCartanSubalgebra(L, H): AlgLie, AlgLie → BoolEltIsSplittingCartanSubalgebra(L, H): AlgMatLie, AlgMatLie → BoolEltSplitToralSubalgebra(L): AlgLie → AlgLieIsSplitToralSubalgebra(L, H): AlgLie, AlgLie → BoolEltExample: Split Toral
- Standard Series
- The Lie Algebra of Derivations
- Properties of Lie Algebras and Ideals
- Operations on Elements
x + y: AlgLieElt, AlgLieElt → AlgLieEltx + y: AlgMatLieElt, AlgMatLieElt → AlgMatLieEltx - y: AlgLieElt, AlgLieElt → AlgLieEltx - y: AlgMatLieElt, AlgMatLieElt → AlgMatLieEltx * y: AlgLieElt, AlgLieElt → AlgLieEltx * y: AlgMatLieElt, AlgMatLieElt → AlgMatLieEltIsCentral(L, M): AlgLie, AlgLieElt → BoolEltIsCentral(L, M): AlgMatLie, AlgMatLieElt → BoolEltNonNilpotentElement(L): AlgLie → AlgLieEltExample: Non Nilpotent ElementAdjointMatrix(L, x): AlgLie, AlgLieElt → AlgMatLieEltRightAdjointMatrix(L, x): AlgLie, AlgLieElt → AlgMatLieEltExample: Other- Indexing
a[i]: AlgLieElt, RngIntElt → RngElta[i]: AlgMatLieElt, RngIntElt → ModTupRngElta[i] := r: AlgLieElt, RngIntElt, RngElt → AlgLieElta[i] := r: AlgMatLieElt, RngIntElt, ModTupRngElt → AlgMatLieElta[i, j]: AlgMatLieElt, RngIntElt, RngIntElt → RngElta[i, j] := r: AlgMatLieElt, RngIntElt, RngIntElt, RngElt → AlgMatLieElt
- The Natural Module
- Operations for Matrix Lie Algebras
- Homomorphisms
- Automorphisms of Classical-type Reductive Algebras
- Restrictable Lie Algebras
IsRestrictable(L): AlgLie → BoolElt, MapIsRestricted(L): AlgLie → BoolElt, MapIspLieAlgebra(L): AlgLie → BoolElt, MapRestrictionMap(L): AlgLie → MappMap(L): AlgLie → MapExample: Is RestrictedRestrictedSubalgebra(Q): SetEnum[AlgLieElt] → AlgLiepSubalgebra(Q): SetEnum[AlgLieElt] → AlgLiepClosure(L, M): AlgLie, AlgLie → AlgLieIsRestrictedSubalgebra(L, M): AlgLie, AlgLie → AlgLieIspSubalgebra(L, M): AlgLie, AlgLie → AlgLiepQuotient(L, M): AlgLie, AlgLie → AlgLieJenningsLieAlgebra(G): Grp → AlgLie, SeqEnumExample: Jennings Lie
- Universal Enveloping Algebras
- Background
- Construction of Universal Enveloping Algebras
- Related Structures
- Elements of Universal Enveloping Algebras
- Creation of Elements
U ! 0: AlgPBW, RngIntElt → AlgPBWEltZero(U): AlgPBW → AlgPBWEltU ! 1: AlgPBW, RngIntElt → AlgPBWEltOne(U): AlgPBW → AlgPBWEltU . i: AlgPBW, RngIntElt → AlgPBWEltU ! r: AlgPBW, Any → AlgPBWEltHBinomial(U, i, n): AlgIUE, RngIntElt, RngIntElt → AlgIUEEltHBinomial(h, n): AlgIUEElt, RngIntElt → AlgIUEEltExample: H Binomial
- 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
SolvableLieAlgebra( F, n, k : parameters): Fld, RngIntElt, RngIntElt → AlgLieExample: SLAC LnkNilpotentLieAlgebra( F, r, k : parameters): Fld, RngIntElt, RngIntElt → AlgLieExample: NLAC NrkAllSolvableLieAlgebras(F, d): Fld, RngIntElt → SeqEnumAllNilpotentLieAlgebras(F, d): Fld, RngIntElt → SeqEnumIdDataSLAC(L): AlgLie → MonStgElt, SeqEnum, MapIdDataNLAC(L): AlgLie → MonStgElt, SeqEnum, MapMatrixOfIsomorphism(f): Map → AlgMatEltExample: SLAC Id DataExample: NLAC Id Data
- Semisimple Subalgebras of Simple Lie Algebras
- Nilpotent Orbits in Simple Lie Algebras
