- Introduction
- 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 (H107E1)
- ChangeBasis(L, B) : AlgLie, [AlgLieElt] -> AlgLie
- MatrixLieAlgebra(R, n) : Rng, RngIntElt -> AlgMatLie
- MatrixLieAlgebra(A) : AlgMat -> AlgMatLie
- Algebra(M) : AlgMatLie -> AlgLie, Map
- Example AlgLie_MatrixLieAlgebra (H107E2)
- 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, Map
- Example AlgLie_LieAlgebra (H107E6)
- Example AlgLie_LieRing (H107E7)
- quo<L | R> : AlgFPLie, [ AlgFPLieElt ] -> AlgLie, Map
- Example AlgLie_QuoAlgFPLie (H107E8)
- NilpotentQuotient(R, d) : [ AlgFPLieElt ], RngIntElt -> AlgLie, SeqEnum, SeqEnum, UserProgram
- Example AlgLie_NilpotentQuotient (H107E9)
- 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, SeqEnum
- fValue(L, x, b) : AlgLieExtr, RngIntElt, RngIntElt -> RngElt
- fValueProof(L, x, b) : AlgLieExtr, RngIntElt, RngIntElt ->
- Example AlgLie_AlgLieExtrfVal (H107E15)
- DimensionsEstimate(L, g) : AlgLieExtr, UserProgram -> SeqEnum, SetMulti
- InstancesForDimensions(L, g, D) : AlgLieExtr, UserProgram, SetEnum[RngIntElt] -> Assoc
- Example AlgLie_AlgLieExtrVarietyDims (H107E16)
- Families of Lie Algebras
- Almost Reductive Lie Algebras
- LieAlgebra(T, k) : MonStgElt, Rng -> AlgLie
- MatrixLieAlgebra(T, k) : MonStgElt, Rng -> AlgLie
- Example AlgLie_ReductiveLieAlgebra (H107E17)
- Example AlgLie_LieAlgebraIsogeny (H107E18)
- LieAlgebra(N, k, p) : MonStgElt, Rng, GrpPermElt -> AlgLie
- TwistedLieAlgebra(R, k) : RootDtm, Rng -> AlgLie
- Example AlgLie_TwistedLieAlgebra (H107E19)
- Cartan-Type Lie Algebras
- WittLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, Map
- Example AlgLie_witt-alg-ex (H107E20)
- SpecialLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie, Map, Map
- Example AlgLie_special-lie-alg-ex (H107E21)
- HamiltonianLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie
- Example AlgLie_hamilton-alg-ex (H107E22)
- ContactLieAlgebra(F, m, n) : Fld, RngIntElt, SeqEnum[RngIntElt] -> AlgLie, AlgLie
- Example AlgLie_contact-alg-ex (H107E23)
- Melikian Lie Algebras
- 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
- 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 (H107E25)
- QuotientWithPullback(L, I) : AlgLie, AlgLie -> AlgLie, Map, UserProgram, UserProgram
- Example AlgLie_LieAlgebraQuotientPullback (H107E26)
- 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 (H107E27)
- Example AlgLie_IsomorphicG2 (H107E28)
- Basic Invariants
- Changing Base Rings
- Bases
- Operations for Semisimple and Reductive Lie Algebras
- SemisimpleType(L) : AlgLie -> MonStgElt
- Example AlgLie_SemisimpleType (H107E30)
- ReductiveType(L) : AlgLie -> RootDtm, MonStgElt, SeqEnum, SeqEnum
- Example AlgLie_ReductiveType (H107E31)
- RootSystem(L) : AlgLie -> [ ModTupRngElt ], [ AlgLieElt ], [ ModTupRngElt ], AlgMatElt
- Example AlgLie_RootSystem (H107E32)
- RootDatum(L) : AlgLie -> RootDtm
- Example AlgLie_RootDatum (H107E33)
- ChevalleyBasis(L) : AlgLie -> [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]
- Example AlgLie_ChevalleyBasis (H107E34)
- 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 (H107E35)
- TwistedBasis(L, H, R) : AlgLie, AlgLie, RootDtm -> AlgLie, AlgLie, Rec, AlgMatElt
- Example AlgLie_TwistedBasis (H107E36)
- WeylGroup(L) : AlgLie -> GrpPermCox
- WeylGroup(GrpFPCox, L) : Cat, AlgLie -> GrpPermCox
- WeylGroup(GrpMat, L) : Cat, AlgLie -> GrpPermCox
- Operations on Subalgebras and Ideals
- Properties of Lie Algebras and Ideals
- Operations on Elements
- The Natural Module
- Operations for Matrix Lie Algebras
- Homomorphisms
- Automorphisms of Classical- type Reductive Algebras
- Restrictable Lie Algebras
- Universal Enveloping Algebras
- 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 (H107E53)
- NilpotentLieAlgebra( F, r, k : parameters) : Fld, RngIntElt, RngIntElt -> AlgLie
- Example AlgLie_NLACNrk (H107E54)
- 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 (H107E55)
- Example AlgLie_NLACIdData (H107E56)
- Semisimple Subalgebras of Simple Lie Algebras
- Nilpotent Orbits in Simple Lie Algebras
- Bibliography
V2.28, 13 July 2023