# Lie Algebras

- [Introduction](introduction.md)

  - [Guide for the Reader](introduction.md#guide-for-the-reader)

- [Constructors for Lie Algebras](construct.md)

  - [`LieAlgebra<R, n | Q : parameters >: Rng, RngIntElt, SeqEnum → AlgLie`](construct.md#constructor-liealgebrarnq)

  - [`LieAlgebra<M | Q : parameters >: ModTupRng, SeqEnum → AlgLie`](construct.md#constructor-constructor-liealgebra-modtuprng-seqenum-alglie)

  - [`LieAlgebra<R, n | T : parameters >: Rng, RngIntElt, SeqEnum → AlgLie`](construct.md#constructor-constructor-liealgebra-rng-rngintelt-seqenum-alglie)

  - [`LieAlgebra< t | T : parameters >: SeqEnum, SeqEnum → AlgLie`](construct.md#constructor-constructor-liealgebra-seqenum-seqenum-alglie)

  - [`LieAlgebra(A): AlgAss → AlgLie, Map`](construct.md#function-liealgebra-algass)

  - [`LieAlgebra(A): AlgMat → AlgLie`](construct.md#function-liealgebra-algmat)

  - [`AbelianLieAlgebra(R, n): Rng, RngIntElt → AlgLie`](construct.md#function-abelianliealgebra-rng-rngintelt)

  - [`Example: Lie Algebra Cons`](construct.md#example-ex-b11bef)

  - [`ChangeBasis(L, B): AlgLie, {[AlgLieElt]} → AlgLie`](construct.md#function-changebasis-alglie-alglieelt)

  - [`ChangeBasis(L, B): AlgLie, {[ModTupFldElt]} → AlgLie`](construct.md#function-changebasis-alglie-modtupfldelt)

  - [`ChangeBasis(L, B): AlgLie, Mtrx → AlgLie`](construct.md#function-changebasis-alglie-mtrx)

  - [`MatrixLieAlgebra(R, n): Rng, RngIntElt → AlgMatLie`](construct.md#function-matrixliealgebra-rng-rngintelt)

  - [`MatrixLieAlgebra(A): AlgMat → AlgMatLie`](construct.md#function-matrixliealgebra-algmat)

  - [`Algebra(M): AlgMatLie → AlgLie, Map`](construct.md#function-algebra-algmatlie)

  - [`LieAlgebra(M): AlgMatLie → AlgLie, Map`](construct.md#function-liealgebra-algmatlie)

  - [`Example: Matrix Lie Algebra`](construct.md#example-ex-297c34)

- [Finitely Presented Lie Algebras](fp-construct.md)

  - [Construction of the Free Lie Algebra](fp-construct.md#construction-of-the-free-lie-algebra)

    - [`FreeLieAlgebra(F, n): Rng, RngIntElt → AlgFPLie`](fp-construct.md#function-freeliealgebra-rng-rngintelt)

    - [`Example: Free Lie`](fp-construct.md#example-ex-49252a)

  - [Properties of the Free Lie Algebra](fp-construct.md#properties-of-the-free-lie-algebra)

    - [`Rank(L): AlgFPLie → RngIntElt`](fp-construct.md#function-algfplierank)

    - [`CoefficientRing(L): AlgFPLie → Rng`](fp-construct.md#function-coefficientring-algfplie)

    - [`BaseRing(L): AlgFPLie → Rng`](fp-construct.md#function-basering-algfplie)

  - [Operations on Elements of the Free Lie Algebra](fp-construct.md#operations-on-elements-of-the-free-lie-algebra)

    - [`x + y: AlgFPLieElt, AlgFPLieElt → AlgFPLieElt`](fp-construct.md#operation-op-plus-algfplieelt-algfplieelt)

    - [`x - y: AlgFPLieElt, AlgFPLieElt → AlgFPLieElt`](fp-construct.md#operation-op-minus-algfplieelt-algfplieelt)

    - [`x * y: AlgFPLieElt, AlgFPLieElt → AlgFPLieElt`](fp-construct.md#operation-op-times-algfplieelt-algfplieelt)

    - [`L ! 0: AlgFPLie, RngIntElt → AlgFPLieElt`](fp-construct.md#operation-op-algfplie-rngintelt)

    - [`Zero(L): AlgFPLie → AlgFPLieElt`](fp-construct.md#function-zero-algfplie)

    - [`Example: Free Lie Algebra`](fp-construct.md#example-ex-0ec7d4)

    - [`IsLeaf(m): AlgFPLieElt → BoolElt, AlgFPLieElt, AlgFPLieElt`](fp-construct.md#function-isleaf-algfplieelt)

    - [`Example: Is Leaf`](fp-construct.md#example-ex-8f1849)

  - [Construction of a Finitely-Presented Lie Algebra](fp-construct.md#construction-of-a-finitely-presented-lie-algebra)

    - [`LieAlgebra(R): [ AlgFPLieElt ] → AlgLie, SeqEnum, SeqEnum, Map`](fp-construct.md#function-fpl-liealgebra)

    - [`LieAlgebra(R): { AlgFPLieElt } → AlgLie, SeqEnum, SeqEnum, Map`](fp-construct.md#function-liealgebra-algfplieelt)

    - [`Example: Lie Algebra`](fp-construct.md#example-ex-92e129)

    - [`Example: Lie Ring`](fp-construct.md#example-ex-0d1f8a)

    - [`quo<L | R>: AlgFPLie, [ AlgFPLieElt ] → AlgLie, Map`](fp-construct.md#constructor-constructor-quo-algfplie-algfplieelt-alglie-map)

    - [`quo<L | R>: AlgFPLie, { AlgFPLieElt } → AlgLie, Map`](fp-construct.md#constructor-constructor-quo-algfplie-algfplieelt-alglie-map-2)

    - [`Example: Quo Alg FP Lie`](fp-construct.md#example-ex-b4f6d4)

    - [`NilpotentQuotient(R, d): [ AlgFPLieElt ], RngIntElt → AlgLie, SeqEnum, SeqEnum, UserProgram`](fp-construct.md#function-nilpotentquotient-algfplieelt-rngintelt)

    - [`NilpotentQuotient(R, d): { AlgFPLieElt, RngIntElt } → AlgLie, SeqEnum, SeqEnum, UserProgram`](fp-construct.md#function-nilpotentquotient-algfplieelt-rngintelt-2)

    - [`Example: Nilpotent Quotient`](fp-construct.md#example-ex-ed7e0d)

  - [Homomorphisms of the Free Lie Algebra](fp-construct.md#homomorphisms-of-the-free-lie-algebra)

    - [`hom<L -> M | Q>: AlgFPLie, AlgFPLie, [ AlgFPLieElt ] → Map`](fp-construct.md#constructor-constructor-hom-algfplie-algfplie-algfplieelt-map)

    - [`hom<L -> M | Q>: AlgFPLie, TupMod, [ TupModElt ] → Map`](fp-construct.md#constructor-constructor-hom-algfplie-tupmod-tupmodelt-map)

    - [`Example: Alg FPL Hom`](fp-construct.md#example-ex-8cb631)

- [Lie Algebras Generated by Extremal Elements](alglieextr.md)

  - [Constructing Lie Algebras Generated by Extremal Elements](alglieextr.md#constructing-lie-algebras-generated-by-extremal-elements)

    - [`ExtremalLieAlgebra(K, n): Rng, RngIntElt → AlgLieExtr`](alglieextr.md#function-extremalliealgebrakn)

    - [`ExtremalLieAlgebra(K, G): Rng, GrphUnd → AlgLieExtr`](alglieextr.md#function-extremalliealgebra-rng-grphund)

  - [Properties of Lie Algebras Generated by Extremal Elements](alglieextr.md#properties-of-lie-algebras-generated-by-extremal-elements)

    - [`NumberOfGenerators(L): AlgLieExtr → RngIntElt`](alglieextr.md#function-numberofgenerators-alglieextr)

    - [`Ngens(L): AlgLieExtr → RngIntElt`](alglieextr.md#function-ngens-alglieextr)

    - [`CoefficientRing(L): AlgLieExtr → Rng`](alglieextr.md#function-coefficientring-alglieextr)

    - [`BaseRing(L): AlgLieExtr → Rng`](alglieextr.md#function-basering-alglieextr)

    - [`CommutatorGraph(L): AlgLieExtr → GrphUnd`](alglieextr.md#function-commutatorgraph-alglieextr)

    - [`Example: Alg Lie Extr Constr`](alglieextr.md#example-ex-3fa178)

    - [`Basis(L): AlgLieExtr → SeqEnum[AlgFPLieElt], SeqEnum[UserProgram]`](alglieextr.md#function-basis-alglieextr)

    - [`ZBasis(L): AlgLieExtr → SeqEnum[AlgFPLieElt], SeqEnum[RngIntElt], SeqEnum[AlgFPLieElt]`](alglieextr.md#function-zbasis-alglieextr)

    - [`Dimension(L): AlgLieExtr → RngIntElt`](alglieextr.md#function-dimension-alglieextr)

    - [`Example: Alg Lie Extr Basis`](alglieextr.md#example-ex-0b6eca)

    - [`MultiplicationTable(~L): AlgLieExtr`](alglieextr.md#function-multiplicationtablelproc)

    - [`MultiplicationTable(L): AlgLieExtr → SeqEnum`](alglieextr.md#function-multiplicationtablel)

    - [`Example: Alg Lie Extr Mult Table`](alglieextr.md#example-ex-7dacec)

  - [Instances of Lie Algebras Generated by Extremal Elements](alglieextr.md#instances-of-lie-algebras-generated-by-extremal-elements)

    - [`Instance(L): AlgLieExtr → AlgLie`](alglieextr.md#function-instance-alglieextr)

    - [`Instance(L, Q): AlgLieExtr, SeqEnum → AlgLie, Map`](alglieextr.md#function-instance-alglieextr-seqenum)

    - [`Example: Alg Lie Extr Mult Instance`](alglieextr.md#example-ex-658a16)

  - [Studying the Parameter Space](alglieextr.md#studying-the-parameter-space)

    - [`FreefValues(L): AlgLieExtr → SeqEnum, SeqEnum`](alglieextr.md#function-freefvalues)

    - [`fValue(L, x, b): AlgLieExtr, RngIntElt, RngIntElt → RngElt`](alglieextr.md#function-fvalue-alglieextr-rngintelt-rngintelt)

    - [`fValueProof(L, x, b): AlgLieExtr, RngIntElt, RngIntElt`](alglieextr.md#function-fvalueproof-alglieextr-rngintelt-rngintelt)

    - [`Example: Alg Lie Extrf Val`](alglieextr.md#example-ex-dc2192)

    - [`DimensionsEstimate(L, g): AlgLieExtr, UserProgram → SeqEnum, SetMulti`](alglieextr.md#function-dimensionsestimate)

    - [`InstancesForDimensions(L, g, D): AlgLieExtr, UserProgram, SetEnum[RngIntElt] → Assoc`](alglieextr.md#function-instancesfordimensions-alglieextr-userprogram-setenum-rngintelt)

    - [`Example: Alg Lie Extr Variety Dims`](alglieextr.md#example-ex-44228f)

- [Families of Lie Algebras](family-construct.md)

  - [Almost Reductive Lie Algebras](family-construct.md#almost-reductive-lie-algebras)

    - [`LieAlgebra(T, k): MonStgElt, Rng → AlgLie`](family-construct.md#function-liealgebra)

    - [`LieAlgebra(R, k): RootDtm, Rng → AlgLie`](family-construct.md#function-liealgebra-rootdtm-rng)

    - [`LieAlgebra(R, k): RootSys, Rng → AlgLie`](family-construct.md#function-liealgebra-rootsys-rng)

    - [`LieAlgebra(D, k): GrphDir, Rng → AlgLie`](family-construct.md#function-liealgebra-grphdir-rng)

    - [`LieAlgebra(C, k): AlgMatElt, Rng → AlgLie`](family-construct.md#function-liealgebra-algmatelt-rng)

    - [`MatrixLieAlgebra(T, k): MonStgElt, Rng → AlgLie`](family-construct.md#function-matrixliealgebra)

    - [`MatrixLieAlgebra(R, k): RootDtm, Rng → AlgLie`](family-construct.md#function-matrixliealgebra-rootdtm-rng)

    - [`MatrixLieAlgebra(R, k): RootSys, Rng → AlgLie`](family-construct.md#function-matrixliealgebra-rootsys-rng)

    - [`MatrixLieAlgebra(D, k): GrphDir, Rng → AlgLie`](family-construct.md#function-matrixliealgebra-grphdir-rng)

    - [`MatrixLieAlgebra(C, k): AlgMatElt, Rng → AlgLie`](family-construct.md#function-matrixliealgebra-algmatelt-rng)

    - [`Example: Reductive Lie Algebra`](family-construct.md#example-ex-05c3a3)

    - [`Example: Lie Algebra Isogeny`](family-construct.md#example-ex-ef718c)

    - [`LieAlgebra(N, k, p): MonStgElt, Rng, GrpPermElt → AlgLie`](family-construct.md#function-liealgebra-monstgelt-rng-grppermelt)

    - [`LieAlgebra(R, k, p): RootDtm, Rng, GrpPermElt → AlgLie`](family-construct.md#function-liealgebra-rootdtm-rng-grppermelt)

    - [`TwistedLieAlgebra(R, k): RootDtm, Rng → AlgLie`](family-construct.md#function-twistedliealgebra-rootdtm-rng)

    - [`Example: Twisted Lie Algebra`](family-construct.md#example-ex-0173e2)

  - [Cartan-Type Lie Algebras](family-construct.md#cartan-type-lie-algebras)

    - [`WittLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, Map`](family-construct.md#function-wittliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`Example: Witt Alg Ex`](family-construct.md#example-ex-ea6512)

    - [`SpecialLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, Map, Map`](family-construct.md#function-specialliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`ConformalSpecialLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, AlgLie, Map, Map`](family-construct.md#function-conformalspecialliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`Example: Special Lie Alg Ex`](family-construct.md#example-ex-c2ee06)

    - [`HamiltonianLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie`](family-construct.md#function-hamiltonianliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`ConformalHamiltonianLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie, AlgLie`](family-construct.md#function-conformalhamiltonianliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`Example: Hamilton Alg Ex`](family-construct.md#example-ex-2e2daf)

    - [`ContactLieAlgebra(F, m, n): Fld, RngIntElt, SeqEnum[RngIntElt] → AlgLie, AlgLie`](family-construct.md#function-contactliealgebra-fld-rngintelt-seqenum-rngintelt)

    - [`Example: Contact Alg Ex`](family-construct.md#example-ex-983c1d)

  - [Melikian Lie Algebras](family-construct.md#melikian-lie-algebras)

    - [`MelikianLieAlgebra(F, n1, n2): Fld, RngIntElt, RngIntElt → AlgLie, Map`](family-construct.md#function-melikianliealgebra-fld-rngintelt-rngintelt)

    - [`Example: Melikian Alg Ex`](family-construct.md#example-ex-c70322)

- [Construction of Elements](element-construct.md)

  - [`Zero(L): AlgLie → AlgLieElt`](element-construct.md#function-zero-alglie)

  - [`Zero(L): AlgMatLie → AlgMatLieElt`](element-construct.md#function-zero-algmatlie)

  - [`L ! 0: AlgLie, RngIntElt → AlgLieElt`](element-construct.md#operation-op-alglie-rngintelt)

  - [`L ! 0: AlgMatLie, RngIntElt → AlgMatLieElt`](element-construct.md#operation-op-algmatlie-rngintelt)

  - [`Random(L): AlgLie → AlgLieElt`](element-construct.md#function-random-alglie)

  - [`Random(L): AlgMatLie → AlgMatLieElt`](element-construct.md#function-random-algmatlie)

  - [Construction of Elements of Structure Constant Algebras](element-construct.md#construction-of-elements-of-structure-constant-algebras)

    - [`elt<L | r₁, r₂, ..., rₙ>: AlgLie, RngElt, RngElt, ..., RngElt → AlgLieElt`](element-construct.md#constructor-constructor-elt-alglie-rngelt-rngelt-rngelt-alglieelt)

    - [`L ! Q: AlgLie, SeqEnum[RngElt] → AlgLieElt`](element-construct.md#operation-op-alglie-seqenum-rngelt)

    - [`BasisProduct(L, i, j): AlgLie, RngIntElt, RngIntElt → AlgLieElt`](element-construct.md#function-basisproduct-alglie-rngintelt-rngintelt)

    - [`BasisProducts(L): AlgLie → SeqEnum`](element-construct.md#function-basisproducts-alglie)

  - [Construction of Matrix Elements](element-construct.md#construction-of-matrix-elements)

    - [`elt<R | L>: AlgMatLie, [ RngElt ] → AlgMatLieElt`](element-construct.md#constructor-constructor-elt-algmatlie-rngelt-algmatlieelt)

    - [`elt<R | L>: AlgMatLie, RngElt, ..., RngElt → AlgMatLieElt`](element-construct.md#constructor-constructor-elt-algmatlie-rngelt-rngelt-algmatlieelt)

    - [`R ! L: AlgMatLie, [RngElt] → AlgMatLieElt`](element-construct.md#operation-op-algmatlie-rngelt)

    - [`DiagonalMatrix(L, Q): AlgMatLie, [RngElt] → AlgMatLieElt`](element-construct.md#function-diagonalmatrix-algmatlie-rngelt)

    - [`ScalarMatrix(L, r): AlgMatLie, RngElt → AlgMatLieElt`](element-construct.md#function-scalarmatrix-algmatlie-rngelt)

- [Construction of Subalgebras, Ideals and Quotients](sub-construct.md)

  - [`sub<L | A>: AlgLie, List → AlgLie, Map`](sub-construct.md#constructor-constructor-sub-alglie-list-alglie-map)

  - [`sub<L | A>: AlgMatLie, List → AlgMatLie, Map`](sub-construct.md#constructor-constructor-sub-algmatlie-list-algmatlie-map)

  - [`ideal<L | A>: AlgLie, List → AlgLie, Map`](sub-construct.md#constructor-constructor-ideal-alglie-list-alglie-map)

  - [`ideal<L | A>: AlgMatLie, List → AlgMatLie, Map`](sub-construct.md#constructor-constructor-ideal-algmatlie-list-algmatlie-map)

  - [`quo<L | A>: AlgLie, List → AlgLie, Map`](sub-construct.md#constructor-constructor-quo-alglie-list-alglie-map)

  - [`quo<L | A>: AlgMatLie, List → AlgMatLie, Map`](sub-construct.md#constructor-constructor-quo-algmatlie-list-algmatlie-map)

  - [`L / S: AlgLie, AlgLie → AlgLie`](sub-construct.md#operation-op-div-alglie-alglie)

  - [`L / S: AlgMatLie, AlgMatLie → AlgLie`](sub-construct.md#operation-op-div-algmatlie-algmatlie)

  - [`Example: Lie Algebra Quotient`](sub-construct.md#example-ex-bb05e7)

  - [`QuotientWithPullback(L, I): AlgLie, AlgLie → AlgLie, Map, UserProgram, UserProgram`](sub-construct.md#function-quotientwithpullback-alglie-alglie)

  - [`Example: Lie Algebra Quotient Pullback`](sub-construct.md#example-ex-83d77d)

- [Operations on Lie Algebras](operation.md)

  - [`L eq K: AlgLie, AlgLie → BoolElt`](operation.md#operation-op-eq-alglie-alglie)

  - [`L eq K: AlgMatLie, AlgMatLie → BoolElt`](operation.md#operation-op-eq-algmatlie-algmatlie)

  - [`L ne K: AlgLie, AlgLie → BoolElt`](operation.md#operation-op-ne-alglie-alglie)

  - [`L ne K: AlgMatLie, AlgMatLie → BoolElt`](operation.md#operation-op-ne-algmatlie-algmatlie)

  - [`L subset K: AlgLie, AlgLie → BoolElt`](operation.md#operation-op-subset-alglie-alglie)

  - [`L subset K: AlgMatLie, AlgMatLie → BoolElt`](operation.md#operation-op-subset-algmatlie-algmatlie)

  - [`L notsubset K: AlgLie, AlgLie → BoolElt`](operation.md#operation-operation-notsubset-alglie-alglie-boolelt)

  - [`L meet M: AlgLie, AlgLie → AlgLie`](operation.md#operation-op-meet-alglie-alglie)

  - [`L meet M: AlgMatLie, AlgMatLie → AlgMatLie`](operation.md#operation-op-meet-algmatlie-algmatlie)

  - [`L * M: AlgLie, AlgLie → AlgLie`](operation.md#operation-op-times-alglie-alglie)

  - [`L * M: AlgMatLie, AlgMatLie → AlgMatLie`](operation.md#operation-op-times-algmatlie-algmatlie)

  - [`L ^ n: AlgLie, RngIntElt → AlgLie`](operation.md#operation-op-pow-alglie-rngintelt)

  - [`Morphism(L, M): AlgLie, AlgLie → Map`](operation.md#function-morphism-alglie-alglie)

  - [`IsIsomorphic(L, M): AlgLie, AlgLie → BoolElt, .`](operation.md#function-isisomorphiclie)

  - [`IsKnownIsomorphic(L, M): AlgLie, AlgLie → BoolElt, BoolElt, .`](operation.md#function-isknownisomorphic-alglie-alglie)

  - [`IsIsomorphism(m): Map[AlgLie, AlgLie] → BoolElt`](operation.md#function-isisomorphism-map-alglie-alglie)

  - [`Example: Isomorphic`](operation.md#example-ex-4fbecb)

  - [`Example: Isomorphic G2`](operation.md#example-ex-3b3b3e)

  - [Basic Invariants](operation.md#basic-invariants)

    - [`CoefficientRing(L): AlgLie → Rng`](operation.md#function-coefficientring-alglie)

    - [`CoefficientRing(L): AlgMatLie → Rng`](operation.md#function-coefficientring-algmatlie)

    - [`BaseRing(L): AlgLie → Rng`](operation.md#function-basering-alglie)

    - [`BaseRing(L): AlgMatLie → Rng`](operation.md#function-basering-algmatlie)

    - [`Dimension(L): AlgLie → RngIntElt`](operation.md#function-dimension-alglie)

    - [`Dimension(L): AlgMatLie → RngIntElt`](operation.md#function-dimension-algmatlie)

    - [`# L: AlgLie → RngIntElt`](operation.md#operation-operation-alglie-rngintelt)

    - [`# L: AlgMatLie → RngIntElt`](operation.md#operation-operation-algmatlie-rngintelt)

    - [`Moduli(L): AlgLie → SeqEnum`](operation.md#function-moduli-alglie)

    - [`Example: Moduli Of Lie Algebra`](operation.md#example-ex-29430b)

  - [Changing Base Rings](operation.md#changing-base-rings)

    - [`ChangeRing(L, S): AlgLie, Rng → AlgLie, Map`](operation.md#function-changering-alglie-rng)

    - [`ChangeRing(L, S): AlgMatLie, Rng → AlgMatLie, Map`](operation.md#function-changering-algmatlie-rng)

    - [`ChangeRing(L, S, f): AlgLie, Rng, Map → AlgLie, Map`](operation.md#function-changering-alglie-rng-map)

    - [`ChangeRing(L, S, f): AlgMatLie, Rng, Map → AlgMatLie, Map`](operation.md#function-changering-algmatlie-rng-map)

  - [Bases](operation.md#bases)

    - [`BasisElement(A, i): AlgLie, RngIntElt → AlgLieElt`](operation.md#function-basiselement-alglie-rngintelt)

    - [`BasisElement(A, i): AlgMatLie, RngIntElt → AlgMatLieElt`](operation.md#function-basiselement-algmatlie-rngintelt)

    - [`A . i: AlgLie, RngIntElt → AlgLieElt`](operation.md#operation-operation-alglie-rngintelt-alglieelt)

    - [`A . i: AlgMatLie, RngIntElt → AlgMatLieElt`](operation.md#operation-operation-algmatlie-rngintelt-algmatlieelt)

    - [`Basis(A): AlgLie → [ AlgLieElt ]`](operation.md#function-basis-alglie)

    - [`Basis(A): AlgMatLie → [ AlgMatLieElt ]`](operation.md#function-basis-algmatlie)

    - [`IsIndependent(Q): [ AlgLieElt ] → BoolElt`](operation.md#function-isindependent-alglieelt)

    - [`IsIndependent(Q): [ AlgMatLieElt ] → BoolElt`](operation.md#function-isindependent-algmatlieelt)

    - [`IsIndependent(Q): { AlgLieElt } → BoolElt`](operation.md#function-isindependent-alglieelt-2)

    - [`IsIndependent(Q): { AlgMatLieElt } → BoolElt`](operation.md#function-isindependent-algmatlieelt-2)

    - [`ExtendBasis(S, L): AlgLie, AlgLie → [ AlgElt ]`](operation.md#function-extendbasis-alglie-alglie)

    - [`ExtendBasis(S, L): AlgMatLie, AlgMatLie → [ AlgElt ]`](operation.md#function-extendbasis-algmatlie-algmatlie)

    - [`ExtendBasis(Q, L): [ AlgLieElt ], AlgLie → [ AlgElt ]`](operation.md#function-extendbasis-alglieelt-alglie)

    - [`ExtendBasis(Q, L): [ AlgMatLie ], AlgMatLie → [ AlgElt ]`](operation.md#function-extendbasis-algmatlie-algmatlie-2)

  - [Operations for Semisimple and Reductive Lie Algebras](operation.md#operations-for-semisimple-and-reductive-lie-algebras)

    - [`SemisimpleType(L): AlgLie → MonStgElt`](operation.md#function-semisimpletype-alglie)

    - [`SemisimpleType(L): AlgMatLie → MonStgElt`](operation.md#function-semisimpletype-algmatlie)

    - [`CartanName(L): AlgLie → MonStgElt`](operation.md#function-cartanname-alglie)

    - [`CartanName(L): AlgMatLie → MonStgElt`](operation.md#function-cartanname-algmatlie)

    - [`Example: Semisimple Type`](operation.md#example-ex-6372db)

    - [`ReductiveType(L): AlgLie → RootDtm, MonStgElt, SeqEnum, SeqEnum`](operation.md#function-reductivetype-alglie)

    - [`ReductiveType(L, H): AlgLie, AlgLie → RootDtm, MonStgElt, SeqEnum, SeqEnum`](operation.md#function-reductivetype-alglie-alglie)

    - [`Example: Reductive Type`](operation.md#example-ex-c3afbb)

    - [`RootSystem(L): AlgLie → [ ModTupRngElt ], [ AlgLieElt ], [ ModTupRngElt ], AlgMatElt`](operation.md#function-rootsystem-alglie)

    - [`RootSystem(L): AlgMatLie → [ ModTupRngElt ], [ AlgMatLieElt ], [ ModTupRngElt ], AlgMatElt`](operation.md#function-rootsystem-algmatlie)

    - [`Example: Root System`](operation.md#example-ex-cfabbc)

    - [`RootDatum(L): AlgLie → RootDtm`](operation.md#function-rootdatum-alglie)

    - [`RootDatum(L): AlgMatLie → RootDtm`](operation.md#function-rootdatum-algmatlie)

    - [`Example: Root Datum`](operation.md#example-ex-afa536)

    - [`ChevalleyBasis(L): AlgLie → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]`](operation.md#function-chevalleybasis)

    - [`ChevalleyBasis(L, H): AlgLie, AlgLie → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]`](operation.md#function-chevalleybasis-alglie-alglie)

    - [`ChevalleyBasis(L): AlgMatLie → [ AlgMatLieElt ], [ AlgMatLieElt ], [ AlgMatLieElt ]`](operation.md#function-chevalleybasis-algmatlie)

    - [`Example: Chevalley Basis`](operation.md#example-ex-fc1498)

    - [`ChevalleyBasis(L, H, R): AlgLie, AlgLie, RootDtm → [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ]`](operation.md#function-chevalleybasis-alglie-alglie-rootdtm)

    - [`IsChevalleyBasis(L, R, x, y, h): AlgLie, RootDtm, [ AlgLieElt ], [ AlgLieElt ], [ AlgLieElt ] → BoolElt, [ Tup ]`](operation.md#function-ischevalleybasis-alglie-rootdtm-alglieelt-alglieelt-alglieelt)

    - [`Example: Chevalley Basis Small Char`](operation.md#example-ex-f1d947)

    - [`TwistedBasis(L, H, R): AlgLie, AlgLie, RootDtm → AlgLie, AlgLie, Rec, AlgMatElt`](operation.md#function-twistedbasis)

    - [`Example: Twisted Basis`](operation.md#example-ex-cf55bc)

    - [`WeylGroup(L): AlgLie → GrpPermCox`](operation.md#function-weylgroup-alglie)

    - [`WeylGroup(GrpPermCox, L): Cat, AlgLie → GrpPermCox`](operation.md#function-weylgroup-cat-alglie)

    - [`WeylGroup(GrpFPCox, L): Cat, AlgLie → GrpPermCox`](operation.md#function-weylgroup-cat-alglie-2)

    - [`WeylGroup(GrpMat, L): Cat, AlgLie → GrpPermCox`](operation.md#function-weylgroup-cat-alglie-3)

- [Operations on Subalgebras and Ideals](sub.md)

  - [`DirectSum(L, M): AlgLie, AlgLie → AlgLie`](sub.md#function-directsum-alglie-alglie)

  - [`DirectSum(L, M): AlgMatLie, AlgMatLie → AlgMatLie`](sub.md#function-directsum-algmatlie-algmatlie)

  - [`IndecomposableSummands(L): AlgLie → [ AlgLie ]`](sub.md#function-indecomposablesummands-alglie)

  - [`IndecomposableSummands(L): AlgMatLie → [ AlgMatLie ]`](sub.md#function-indecomposablesummands-algmatlie)

  - [`DirectSumDecomposition(L): AlgLie → [ AlgLie ]`](sub.md#function-directsumdecomposition-alglie)

  - [`DirectSumDecomposition(L): AlgMatLie → [ AlgMatLie ]`](sub.md#function-directsumdecomposition-algmatlie)

  - [`Example: Direct Sum Decomposition`](sub.md#example-ex-45b01f)

  - [Standard Ideals and Subalgebras](sub.md#standard-ideals-and-subalgebras)

    - [`Centre(L): AlgLie → AlgLie`](sub.md#function-centre-alglie)

    - [`Centre(L): AlgMatLie → AlgMatLie`](sub.md#function-centre-algmatlie)

    - [`Center(L): AlgLie → AlgLie`](sub.md#function-center-alglie)

    - [`Center(L): AlgMatLie → AlgMatLie`](sub.md#function-center-algmatlie)

    - [`Centraliser(L, K): AlgLie, AlgLie → AlgLie, Map`](sub.md#function-centraliser-alglie-alglie)

    - [`Centraliser(L, K): AlgMatLie, AlgMatLie → AlgMatLie, Map`](sub.md#function-centraliser-algmatlie-algmatlie)

    - [`Centralizer(L, K): AlgLie, AlgLie → AlgLie, Map`](sub.md#function-centralizer-alglie-alglie)

    - [`Centralizer(L, K): AlgMatLie, AlgMatLie → AlgMatLie, Map`](sub.md#function-centralizer-algmatlie-algmatlie)

    - [`Centraliser(L, x): AlgLie, AlgLieElt → AlgLie, Map`](sub.md#function-centraliser-alglie-alglieelt)

    - [`Centraliser(L, x): AlgMatLie, AlgMatLieElt → AlgMatLie, Map`](sub.md#function-centraliser-algmatlie-algmatlieelt)

    - [`Centralizer(L, x): AlgLie, AlgLieElt → AlgLie, Map`](sub.md#function-centralizer-alglie-alglieelt)

    - [`Centralizer(L, x): AlgMatLie, AlgMatLieElt → AlgMatLie, Map`](sub.md#function-centralizer-algmatlie-algmatlieelt)

    - [`Normaliser(L, K): AlgLie, AlgLie → AlgLie`](sub.md#function-normaliser-alglie-alglie)

    - [`Normaliser(L, K): AlgMatLie, AlgMatLie → AlgMatLie`](sub.md#function-normaliser-algmatlie-algmatlie)

    - [`Normalizer(L, K): AlgLie, AlgLie → AlgLie`](sub.md#function-normalizer-alglie-alglie)

    - [`Normalizer(L, K): AlgMatLie, AlgMatLie → AlgMatLie`](sub.md#function-normalizer-algmatlie-algmatlie)

    - [`SolubleRadical(L): AlgLie → AlgLie`](sub.md#function-solubleradical-alglie)

    - [`SolubleRadical(L): AlgMatLie → AlgMatLie`](sub.md#function-solubleradical-algmatlie)

    - [`SolvableRadical(L): AlgLie → AlgLie`](sub.md#function-solvableradical-alglie)

    - [`SolvableRadical(L): AlgMatLie → AlgMatLie`](sub.md#function-solvableradical-algmatlie)

    - [`Nilradical(L): AlgLie → AlgLie`](sub.md#function-nilradical-alglie)

    - [`Nilradical(L): AlgMatLie → AlgMatLie`](sub.md#function-nilradical-algmatlie)

    - [`Example: Operations`](sub.md#example-ex-d2d791)

  - [Cartan and Toral Subalgebras](sub.md#cartan-and-toral-subalgebras)

    - [`CartanSubalgebra(L): AlgLie → AlgLie`](sub.md#function-cartansubalgebra-alglie)

    - [`CartanSubalgebra(L): AlgMatLie → AlgMatLie`](sub.md#function-cartansubalgebra-algmatlie)

    - [`IsCartanSubalgebra(L, H): AlgLie, AlgLie → BoolElt`](sub.md#function-iscartansubalgebra-alglie-alglie)

    - [`IsCartanSubalgebra(L, H): AlgMatLie, AlgMatLie → BoolElt`](sub.md#function-iscartansubalgebra-algmatlie-algmatlie)

    - [`Example: Cartan Subalgebra`](sub.md#example-ex-675c63)

    - [`SplittingCartanSubalgebra(L): AlgLie → AlgLie`](sub.md#function-splittingcartansubalgebra-alglie)

    - [`SplitMaximalToralSubalgebra(L): AlgLie → AlgLie`](sub.md#function-splitmaximaltoralsubalgebra)

    - [`IsSplittingCartanSubalgebra(L, H): AlgLie, AlgLie → BoolElt`](sub.md#function-issplittingcartansubalgebra-alglie-alglie)

    - [`IsSplittingCartanSubalgebra(L, H): AlgMatLie, AlgMatLie → BoolElt`](sub.md#function-issplittingcartansubalgebra-algmatlie-algmatlie)

    - [`SplitToralSubalgebra(L): AlgLie → AlgLie`](sub.md#function-splittoralsubalgebra)

    - [`IsSplitToralSubalgebra(L, H): AlgLie, AlgLie → BoolElt`](sub.md#function-issplittoralsubalgebra-alglie-alglie)

    - [`Example: Split Toral`](sub.md#example-ex-3f1c22)

  - [Standard Series](sub.md#standard-series)

    - [`CompositionSeries(L): AlgLie → [ Alg ], [ AlgLie ], AlgMatElt`](sub.md#function-compositionseries-alglie)

    - [`CompositionFactors(L): AlgLie → [ AlgLie ]`](sub.md#function-compositionfactors-alglie)

    - [`MinimalIdeals(L : parameters): AlgLie → [ AlgLie ], BoolElt`](sub.md#function-minimalideals-alglie)

    - [`MaximalIdeals(L : parameters): AlgLie → [ AlgLie ], BoolElt`](sub.md#function-maximalideals-alglie)

    - [`DerivedSeries(L): AlgLie → [ AlgLie ]`](sub.md#function-derivedseries-alglie)

    - [`DerivedSeries(L): AlgMatLie → [ AlgMatLie ]`](sub.md#function-derivedseries-algmatlie)

    - [`LowerCentralSeries(L): AlgLie → [ AlgLie ]`](sub.md#function-lowercentralseries-alglie)

    - [`LowerCentralSeries(L): AlgMatLie → [ AlgMatLie ]`](sub.md#function-lowercentralseries-algmatlie)

    - [`UpperCentralSeries(L): AlgLie → [ AlgLie ]`](sub.md#function-uppercentralseries-alglie)

    - [`UpperCentralSeries(L): AlgMatLie → [ AlgMatLie ]`](sub.md#function-uppercentralseries-algmatlie)

    - [`Example: Series`](sub.md#example-ex-9cb576)

  - [The Lie Algebra of Derivations](sub.md#the-lie-algebra-of-derivations)

    - [`LieAlgebraOfDerivations(L): AlgLie → AlgLie, Rec`](sub.md#function-liealgebraofderivations-alglie)

    - [`Example: Lie Algebra Of Derivations`](sub.md#example-ex-d37e8c)

- [Properties of Lie Algebras and Ideals](property.md)

  - [`KillingForm(L): AlgLie → ModMatFldElt`](property.md#function-killingform-alglie)

  - [`KillingForm(L): AlgMatLie → ModMatFldElt`](property.md#function-killingform-algmatlie)

  - [`KillingMatrix(L): AlgLie → ModMatFldElt`](property.md#function-killingmatrix-alglie)

  - [`KillingMatrix(L): AlgMatLie → ModMatFldElt`](property.md#function-killingmatrix-algmatlie)

  - [`Example: Other`](property.md#example-ex-53bd1a)

  - [`IsAbelian(L): AlgLie → BoolElt`](property.md#function-isabelian-alglie)

  - [`IsAbelian(L): AlgMatLie → BoolElt`](property.md#function-isabelian-algmatlie)

  - [`IsSoluble(L): AlgLie → BoolElt`](property.md#function-issoluble-alglie)

  - [`IsSoluble(L): AlgMatLie → BoolElt`](property.md#function-issoluble-algmatlie)

  - [`IsSolvable(L): AlgLie → BoolElt`](property.md#function-issolvable-alglie)

  - [`IsSolvable(L): AlgMatLie → BoolElt`](property.md#function-issolvable-algmatlie)

  - [`IsNilpotent(L): AlgLie → BoolElt`](property.md#function-isnilpotent-alglie)

  - [`IsNilpotent(L): AlgMatLie → BoolElt`](property.md#function-isnilpotent-algmatlie)

  - [`IsCentral(L, M): AlgLie, AlgLie → BoolElt`](property.md#function-iscentral-alglie-alglie)

  - [`IsCentral(L, M): AlgMatLie, AlgMatLie → BoolElt`](property.md#function-iscentral-algmatlie-algmatlie)

  - [`IsSimple(L): AlgLie → BoolElt`](property.md#function-issimple-alglie)

  - [`IsSimple(L): AlgMatLie → BoolElt`](property.md#function-issimple-algmatlie)

  - [`IsSemisimple(L): AlgLie → BoolElt`](property.md#function-issemisimple-alglie)

  - [`IsSemisimple(L): AlgMatLie → BoolElt`](property.md#function-issemisimple-algmatlie)

  - [`IsReductive(L): AlgLie → BoolElt`](property.md#function-isreductive-alglie)

  - [`IsReductive(L): AlgMatLie → BoolElt`](property.md#function-isreductive-algmatlie)

  - [`HasLeviSubalgebra(L): AlgLie → BoolElt`](property.md#function-haslevisubalgebra-alglie)

  - [`HasLeviSubalgebra(L): AlgMatLie → BoolElt`](property.md#function-haslevisubalgebra-algmatlie)

  - [`IsClassicalType(L): AlgLie → BoolElt`](property.md#function-isclassicaltype-alglie)

  - [`Example: Predicates`](property.md#example-ex-6ba33a)

- [Operations on Elements](elements.md)

  - [`x + y: AlgLieElt, AlgLieElt → AlgLieElt`](elements.md#operation-op-plus-alglieelt-alglieelt)

  - [`x + y: AlgMatLieElt, AlgMatLieElt → AlgMatLieElt`](elements.md#operation-op-plus-algmatlieelt-algmatlieelt)

  - [`x - y: AlgLieElt, AlgLieElt → AlgLieElt`](elements.md#operation-op-minus-alglieelt-alglieelt)

  - [`x - y: AlgMatLieElt, AlgMatLieElt → AlgMatLieElt`](elements.md#operation-op-minus-algmatlieelt-algmatlieelt)

  - [`x * y: AlgLieElt, AlgLieElt → AlgLieElt`](elements.md#operation-op-times-alglieelt-alglieelt)

  - [`x * y: AlgMatLieElt, AlgMatLieElt → AlgMatLieElt`](elements.md#operation-op-times-algmatlieelt-algmatlieelt)

  - [`IsCentral(L, M): AlgLie, AlgLieElt → BoolElt`](elements.md#function-iscentral-alglie-alglieelt)

  - [`IsCentral(L, M): AlgMatLie, AlgMatLieElt → BoolElt`](elements.md#function-iscentral-algmatlie-algmatlieelt)

  - [`NonNilpotentElement(L): AlgLie → AlgLieElt`](elements.md#function-nonnilpotentelement-alglie)

  - [`Example: Non Nilpotent Element`](elements.md#example-ex-a38e8a)

  - [`AdjointMatrix(L, x): AlgLie, AlgLieElt → AlgMatLieElt`](elements.md#function-adjointmatrix-alglie-alglieelt)

  - [`RightAdjointMatrix(L, x): AlgLie, AlgLieElt → AlgMatLieElt`](elements.md#function-rightadjointmatrix-alglie-alglieelt)

  - [`Example: Other`](elements.md#example-ex-c4c36f)

  - [Indexing](elements.md#indexing)

    - [`a[i]: AlgLieElt, RngIntElt → RngElt`](elements.md#indexed-indexed-a-alglieelt-rngintelt-rngelt)

    - [`a[i]: AlgMatLieElt, RngIntElt → ModTupRngElt`](elements.md#literal-literal-a-i-algmatlieelt-rngintelt-modtuprngelt)

    - [`a[i] := r: AlgLieElt, RngIntElt, RngElt → AlgLieElt`](elements.md#literal-literal-a-i-r-alglieelt-rngintelt-rngelt-alglieelt)

    - [`a[i] := r: AlgMatLieElt, RngIntElt, ModTupRngElt → AlgMatLieElt`](elements.md#literal-literal-a-i-r-algmatlieelt-rngintelt-modtuprngelt-algmatlieelt)

    - [`a[i, j]: AlgMatLieElt, RngIntElt, RngIntElt → RngElt`](elements.md#indexed-indexed-a-algmatlieelt-rngintelt-rngintelt-rngelt)

    - [`a[i, j] := r: AlgMatLieElt, RngIntElt, RngIntElt, RngElt → AlgMatLieElt`](elements.md#literal-literal-a-i-j-r-algmatlieelt-rngintelt-rngintelt-rngelt-algmatlieelt)

- [The Natural Module](natural-module.md)

  - [`Module(L): AlgLie → ModTupRng`](natural-module.md#function-module-alglie)

  - [`RModule(L): AlgMatLie → ModTupRng`](natural-module.md#function-rmodule-algmatlie)

  - [`BaseModule(L): AlgMatLie → ModTupRng`](natural-module.md#function-basemodule-algmatlie)

  - [`Degree(L): AlgLie → RngIntElt`](natural-module.md#function-degree-alglie)

  - [`Degree(L): AlgMatLie → RngIntElt`](natural-module.md#function-degree-algmatlie)

  - [`Degree(a): AlgLieElt → RngIntElt`](natural-module.md#function-degree-alglieelt)

  - [`ElementToSequence(a): AlgLieElt → SeqEnum`](natural-module.md#function-elementtosequence-alglieelt)

  - [`Eltseq(a): AlgLieElt → SeqEnum`](natural-module.md#function-eltseq-alglieelt)

  - [`Coordinates(M, a): AlgLie, AlgLieElt → SeqEnum`](natural-module.md#function-coordinates-alglie-alglieelt)

  - [`Coordinates(M, a): AlgMatLie, AlgMatLieElt → SeqEnum`](natural-module.md#function-coordinates-algmatlie-algmatlieelt)

  - [`InnerProduct(a, b): AlgLieElt, AlgLieElt → RngElt`](natural-module.md#function-innerproduct-alglieelt-alglieelt)

  - [`Support(a): AlgLieElt → SetEnum`](natural-module.md#function-support-alglieelt)

- [Operations for Matrix Lie Algebras](matrix.md)

  - [`BaseModule(M): AlgMatLie → ModTupRng`](matrix.md#function-basemodule-algmatlie-2)

  - [`Generic(M): AlgMatLie → AlgMatLie`](matrix.md#function-generic-algmatlie)

  - [`Kernel(X): AlgMatLieElt → ModTupRng`](matrix.md#function-kernel-algmatlieelt)

  - [`Nullspace(X): AlgMatLieElt → ModTupRng`](matrix.md#function-nullspace-algmatlieelt)

  - [`NullspaceOfTranspose(X): AlgMatLieElt → ModTupRng`](matrix.md#function-nullspaceoftranspose-algmatlieelt)

  - [`RowNullSpace(X): AlgMatLieElt → ModTupRng`](matrix.md#function-rownullspace-algmatlieelt)

- [Homomorphisms](homomorphisms.md)

  - [`hom<L -> M | Q>: AlgLie, AlgLie, [ AlgLieElt ] → Map`](homomorphisms.md#constructor-constructor-hom-alglie-alglie-alglieelt-map)

  - [`hom<L -> M | Q>: AlgLie, TupMod, [ TupModElt ] → Map`](homomorphisms.md#constructor-constructor-hom-alglie-tupmod-tupmodelt-map)

- [Automorphisms of Classical-type Reductive Algebras](automorphisms-classical.md)

  - [`IdentityAutomorphism(L): AlgLie → Map`](automorphisms-classical.md#function-identityautomorphism-alglie)

  - [`InnerAutomorphism(L, x): AlgLie, GrpLieElt → Map`](automorphisms-classical.md#function-innerautomorphism-alglie-grplieelt)

  - [`InnerAutomorphismGroup(L): AlgLie → GrpLie, Map`](automorphisms-classical.md#function-innerautomorphismgroup-alglie)

  - [`DiagonalAutomorphism(L, v): AlgLie, ModTupRngElt → Map`](automorphisms-classical.md#function-diagonalautomorphism-alglie-modtuprngelt)

  - [`GraphAutomorphism(L, p): AlgLie, GrpPermElt → Map`](automorphisms-classical.md#function-graphautomorphism-alglie-grppermelt)

  - [`DiagramAutomorphism(L, p): AlgLie, GrpPermElt → Map`](automorphisms-classical.md#function-diagramautomorphism-alglie-grppermelt)

  - [`Example: Graph Automorphism`](automorphisms-classical.md#example-ex-d6f1f7)

- [Restrictable Lie Algebras](restrictable.md)

  - [`IsRestrictable(L): AlgLie → BoolElt, Map`](restrictable.md#function-isrestrictable-alglie)

  - [`IsRestricted(L): AlgLie → BoolElt, Map`](restrictable.md#function-isrestricted-alglie)

  - [`IspLieAlgebra(L): AlgLie → BoolElt, Map`](restrictable.md#function-ispliealgebra-alglie)

  - [`RestrictionMap(L): AlgLie → Map`](restrictable.md#function-restrictionmap-alglie)

  - [`pMap(L): AlgLie → Map`](restrictable.md#function-pmap-alglie)

  - [`Example: Is Restricted`](restrictable.md#example-ex-7334d1)

  - [`RestrictedSubalgebra(Q): SetEnum[AlgLieElt] → AlgLie`](restrictable.md#function-restrictedsubalgebra-setenum-alglieelt)

  - [`pSubalgebra(Q): SetEnum[AlgLieElt] → AlgLie`](restrictable.md#function-psubalgebra-setenum-alglieelt)

  - [`pClosure(L, M): AlgLie, AlgLie → AlgLie`](restrictable.md#function-pclosure-alglie-alglie)

  - [`IsRestrictedSubalgebra(L, M): AlgLie, AlgLie → AlgLie`](restrictable.md#function-isrestrictedsubalgebra-alglie-alglie)

  - [`IspSubalgebra(L, M): AlgLie, AlgLie → AlgLie`](restrictable.md#function-ispsubalgebra-alglie-alglie)

  - [`pQuotient(L, M): AlgLie, AlgLie → AlgLie`](restrictable.md#function-pquotient-alglie-alglie)

  - [`JenningsLieAlgebra(G): Grp → AlgLie, SeqEnum`](restrictable.md#function-jenningsliealgebra-grp)

  - [`Example: Jennings Lie`](restrictable.md#example-ex-2771f0)

- [Universal Enveloping Algebras](universal_enveloping_algebras.md)

  - [Background](universal_enveloping_algebras.md#background)

    - [Universal Enveloping Algebras](universal_enveloping_algebras.md#uea-background-1)

    - [The Integral Form of a Universal Enveloping Algebra](universal_enveloping_algebras.md#the-integral-form-of-a-universal-enveloping-algebra)

  - [Construction of Universal Enveloping Algebras](universal_enveloping_algebras.md#construction-of-universal-enveloping-algebras)

    - [`UniversalEnvelopingAlgebra(L): AlgLie → AlgUE`](universal_enveloping_algebras.md#function-universalenvelopingalgebra-alglie)

    - [`IntegralUEA(L): AlgLie → AlgIUE`](universal_enveloping_algebras.md#function-integraluea-alglie)

    - [`IntegralUEAlgebra(L): AlgLie → AlgIUE`](universal_enveloping_algebras.md#function-integraluealgebra-alglie)

    - [`IntegralUniversalEnvelopingAlgebra(L): AlgLie → AlgIUE`](universal_enveloping_algebras.md#function-integraluniversalenvelopingalgebra-alglie)

    - [`Example: UEA Con`](universal_enveloping_algebras.md#example-ex-c78d02)

    - [`AssignNames(~U, Q): AlgPBW, [ MonStgElt ]`](universal_enveloping_algebras.md#function-assignnames-algpbw-monstgelt-ref)

    - [`ChangeRing(U, S): AlgUE, Rng → AlgUE`](universal_enveloping_algebras.md#function-changering-algue-rng)

  - [Related Structures](universal_enveloping_algebras.md#related-structures)

    - [`CoefficientRing(U): AlgPBW → Rng`](universal_enveloping_algebras.md#function-coefficientring-algpbw)

    - [`BaseRing(U): AlgPBW → Rng`](universal_enveloping_algebras.md#function-basering-algpbw)

    - [`Algebra(U): AlgUE → AlgLie`](universal_enveloping_algebras.md#function-algebra-algue)

  - [Elements of Universal Enveloping Algebras](universal_enveloping_algebras.md#elements-of-universal-enveloping-algebras)

    - [Creation of Elements](universal_enveloping_algebras.md#creation-of-elements)

      - [`U ! 0: AlgPBW, RngIntElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-algpbw-rngintelt)

      - [`Zero(U): AlgPBW → AlgPBWElt`](universal_enveloping_algebras.md#function-zero-algpbw)

      - [`U ! 1: AlgPBW, RngIntElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-algpbw-rngintelt-2)

      - [`One(U): AlgPBW → AlgPBWElt`](universal_enveloping_algebras.md#function-one-algpbw)

      - [`U . i: AlgPBW, RngIntElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-operation-algpbw-rngintelt-algpbwelt)

      - [`U ! r: AlgPBW, Any → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-algpbw-any)

      - [`HBinomial(U, i, n): AlgIUE, RngIntElt, RngIntElt → AlgIUEElt`](universal_enveloping_algebras.md#function-hbinomial-algiue-rngintelt-rngintelt)

      - [`HBinomial(h, n): AlgIUEElt, RngIntElt → AlgIUEElt`](universal_enveloping_algebras.md#function-hbinomial-algiueelt-rngintelt)

      - [`Example: H Binomial`](universal_enveloping_algebras.md#example-ex-4c79c2)

    - [Operations on Elements](universal_enveloping_algebras.md#operations-on-elements)

      - [`x + y: AlgPBWElt, AlgPBWElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-plus-algpbwelt-algpbwelt)

      - [`x - y: AlgPBWElt, AlgPBWElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-minus-algpbwelt-algpbwelt)

      - [`x * y: AlgPBWElt, AlgPBWElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-times-algpbwelt-algpbwelt)

      - [`c * x: RngElt, AlgPBWElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-times-rngelt-algpbwelt)

      - [`x * c: AlgPBWElt, RngElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-times-algpbwelt-rngelt)

      - [`x ^ n: AlgPBWElt, RngIntElt → AlgPBWElt`](universal_enveloping_algebras.md#operation-op-pow-algpbwelt-rngintelt)

      - [`Monomials(u): AlgPBWElt → SeqEnum`](universal_enveloping_algebras.md#function-monomials-algpbwelt)

      - [`Coefficients(u): AlgPBWElt → SeqEnum`](universal_enveloping_algebras.md#function-coefficients-algpbwelt)

      - [`Degree(u, i): AlgPBWElt, RngIntElt → RngIntElt`](universal_enveloping_algebras.md#function-degree-algpbwelt-rngintelt)

      - [`Example: Elt Ops`](universal_enveloping_algebras.md#example-ex-81e4e4)

- [Solvable and Nilpotent Lie Algebras Classification](solvable-nilpotent.md)

  - [The List of Solvable Lie Algebras](solvable-nilpotent.md#the-list-of-solvable-lie-algebras)

  - [Comments on the Classification over Finite Fields](solvable-nilpotent.md#comments-on-the-classification-over-finite-fields)

  - [The List of Nilpotent Lie Algebras](solvable-nilpotent.md#the-list-of-nilpotent-lie-algebras)

  - [Intrinsics for Working with the Classifications](solvable-nilpotent.md#intrinsics-for-working-with-the-classifications)

    - [`SolvableLieAlgebra( F, n, k : parameters): Fld, RngIntElt, RngIntElt → AlgLie`](solvable-nilpotent.md#function-solvableliealgebra-fld-rngintelt-rngintelt)

    - [`Example: SLAC Lnk`](solvable-nilpotent.md#example-ex-72dcd0)

    - [`NilpotentLieAlgebra( F, r, k : parameters): Fld, RngIntElt, RngIntElt → AlgLie`](solvable-nilpotent.md#function-nilpotentliealgebra-fld-rngintelt-rngintelt)

    - [`Example: NLAC Nrk`](solvable-nilpotent.md#example-ex-3827f2)

    - [`AllSolvableLieAlgebras(F, d): Fld, RngIntElt → SeqEnum`](solvable-nilpotent.md#function-allsolvableliealgebras-fld-rngintelt)

    - [`AllNilpotentLieAlgebras(F, d): Fld, RngIntElt → SeqEnum`](solvable-nilpotent.md#function-allnilpotentliealgebras-fld-rngintelt)

    - [`IdDataSLAC(L): AlgLie → MonStgElt, SeqEnum, Map`](solvable-nilpotent.md#function-iddataslac-alglie)

    - [`IdDataNLAC(L): AlgLie → MonStgElt, SeqEnum, Map`](solvable-nilpotent.md#function-iddatanlac-alglie)

    - [`MatrixOfIsomorphism(f): Map → AlgMatElt`](solvable-nilpotent.md#function-matrixofisomorphism-map)

    - [`Example: SLAC Id Data`](solvable-nilpotent.md#example-ex-b0dc46)

    - [`Example: NLAC Id Data`](solvable-nilpotent.md#example-ex-7b45ed)

- [Semisimple Subalgebras of Simple Lie Algebras](sub-sssdb.md)

  - [`SubalgebrasInclusionGraph( t ): MonStgElt → GrphDir`](sub-sssdb.md#function-subalgebrasinclusiongraph-monstgelt)

  - [`Example: SSSDB`](sub-sssdb.md#example-ex-7eb7e1)

  - [`RestrictionMatrix( G, k ): GrphDir, RngIntElt ) → AlgMatElt`](sub-sssdb.md#function-restrictionmatrix-grphdir-rngintelt)

  - [`Example: SSSDB Restr`](sub-sssdb.md#example-ex-03a6eb)

- [Nilpotent Orbits in Simple Lie Algebras](nilpotent-orbits.md)

  - [`IsGenuineWeightedDynkinDiagram( L, wd ): AlgLie, SeqEnum → BoolElt, SeqEnum`](nilpotent-orbits.md#function-isgenuineweighteddynkindiagram-alglie-seqenum)

  - [`Example: Nilp Orb Genuine`](nilpotent-orbits.md#example-ex-762477)

  - [`NilpotentOrbit( L, wd ): AlgLie, SeqEnum → NilpOrbAlgLie`](nilpotent-orbits.md#function-nilpotentorbit-alglie-seqenum)

  - [`NilpotentOrbit( L, e ): AlgLie, AlgLieElt → NilpOrbAlgLie`](nilpotent-orbits.md#function-nilpotentorbit-alglie-alglieelt)

  - [`Example: No By Rep`](nilpotent-orbits.md#example-ex-ab5b8b)

  - [`NilpotentOrbits( L ): AlgLie → SeqEnum`](nilpotent-orbits.md#function-nilpotentorbits-alglie)

  - [`Example: NOD4`](nilpotent-orbits.md#example-ex-1ce736)

  - [`Partition( o ): NilpOrbAlgLie → SeqEnum`](nilpotent-orbits.md#function-partition-nilporbalglie)

  - [`Example: Part Ex`](nilpotent-orbits.md#example-ex-52861d)

  - [`SL2Triple( o ): NilpOrbAlgLie → SeqEnum`](nilpotent-orbits.md#function-sl2triple-nilporbalglie)

  - [`SL2Triple( L, e ): AlgLie, AlgLieElt → SeqEnum`](nilpotent-orbits.md#function-sl2triple-alglie-alglieelt)

  - [`Representative( o ): NilpOrbAlgLie → AlgLieElt`](nilpotent-orbits.md#function-representative-nilporbalglie)

  - [`WeightedDynkinDiagram( o ): NilpOrbAlgLie → SeqEnum`](nilpotent-orbits.md#function-weighteddynkindiagram-nilporbalglie)

  - [`Example: WDD`](nilpotent-orbits.md#example-ex-eb9e7c)