# Kac-Moody Lie Algebras

- [Introduction](introduction.md)

- [Generalized Cartan Matrices](cartan.md)

  - [`IsGeneralizedCartanMatrix(C): AlgMatElt → BoolElt`](cartan.md#function-isgeneralizedcartanmatrix-algmatelt)

  - [`KacMoodyClass(C): AlgMatElt → MonStgElt, ModMatRngElt`](cartan.md#function-kacmoodyclass)

  - [`KacMoodyClasses(C): AlgMatElt → SeqEnum, SeqEnum, SeqEnum`](cartan.md#function-kacmoodyclasses-algmatelt)

  - [`Example: Generalized Cartan`](cartan.md#example-ex-89fcf7)

- [Affine Kac–Moody Lie Algebras](affine.md)

  - [Constructing Affine Kac–Moody Lie Algebras](affine.md#constructing-affine-kac-moody-lie-algebras)

    - [`AffineLieAlgebra(N, F): MonStgElt, Fld → AlgKac`](affine.md#function-affineliealgebra-monstgelt-fld)

    - [`AffineLieAlgebra(C, F): AlgMatElt, Fld → AlgKac`](affine.md#function-affineliealgebra-algmatelt-fld)

    - [`Example: construct`](affine.md#example-ex-055fd7)

  - [Properties of Affine Kac–Moody Lie Algebras](affine.md#properties-of-affine-kac-moody-lie-algebras)

    - [`CartanMatrix(L): AlgKac → AlgMatElt`](affine.md#function-cartanmatrix-algkac)

    - [`CartanName(L): AlgKac → MonStgElt`](affine.md#function-cartanname-algkac)

    - [`Dimension(L): AlgKac → Infty`](affine.md#function-dimension-algkac)

    - [`CoefficientRing(L): AlgKac → Rng`](affine.md#function-coefficientring-algkac)

    - [`BaseRing(L): AlgKac → Rng`](affine.md#function-basering-algkac)

    - [`FiniteLieAlgebra(L): AlgKac → AlgLie`](affine.md#function-finiteliealgebra-algkac)

    - [`LaurentSeriesRing(L): AlgKac → RngSerLaur`](affine.md#function-laurentseriesring-algkac)

    - [`StandardGenerators(L): AlgKac → SeqEnum[AlgKacElt], SeqEnum[AlgKacElt], SeqEnum[AlgKacElt]`](affine.md#function-standardgenerators-algkac)

    - [`Example: construct`](affine.md#example-ex-17cbd6)

  - [Constructing Elements of Affine Kac–Moody Lie Algebras](affine.md#constructing-elements-of-affine-kac-moody-lie-algebras)

    - [`L . i: AlgKac, RngIntElt → AlgKacElt`](affine.md#operation-operation-algkac-rngintelt-algkacelt)

    - [`HasAttribute(L, "c"): AlgKac, MonStgElt → BoolElt, AlgKacElt`](affine.md#function-hasattribute-algkac-monstgelt)

    - [`HasAttribute(L, "d"): AlgKac, MonStgElt → BoolElt, AlgKacElt`](affine.md#function-hasattribute-algkac-monstgelt-2)

    - [`elt<L | {<} [ {<} p₁, y₁ {>}, ... ], λ, μ {>} >: AlgKac, Tup → AlgKacElt`](affine.md#constructor-constructor-elt-algkac-tup-algkacelt)

  - [Properties of Elements of Affine Kac–Moody Lie Algebras](affine.md#properties-of-elements-of-affine-kac-moody-lie-algebras)

    - [`EltTup(x): AlgKacElt → Tup`](affine.md#function-kmaff-elttup)

    - [`IsZero(x): AlgKacElt → BoolElt`](affine.md#function-iszero-algkacelt)

    - [`x eq y: AlgKacElt, AlgKacElt → BoolElt`](affine.md#operation-op-eq-algkacelt-algkacelt)

    - [`x + y: AlgKacElt, AlgKacElt → AlgKacElt`](affine.md#operation-op-plus-algkacelt-algkacelt)

    - [`x - y: AlgKacElt, AlgKacElt → AlgKacElt`](affine.md#operation-op-minus-algkacelt-algkacelt)

    - [`x * y: AlgKacElt, AlgKacElt → AlgKacElt`](affine.md#operation-op-times-algkacelt-algkacelt)

    - [`c * y: RngElt, AlgKacElt → AlgKacElt`](affine.md#operation-op-times-rngelt-algkacelt)

    - [`-x: AlgKacElt → AlgKacElt`](affine.md#operation-operation-algkacelt-algkacelt)

    - [`Example: construct`](affine.md#example-ex-1599ed)