Kac-Moody Lie Algebras

• Introduction
• Generalized Cartan Matrices
  • IsGeneralizedCartanMatrix(C): AlgMatElt → BoolElt
  • KacMoodyClass(C): AlgMatElt → MonStgElt, ModMatRngElt
  • KacMoodyClasses(C): AlgMatElt → SeqEnum, SeqEnum, SeqEnum
  • Example: Generalized Cartan
• Affine Kac–Moody Lie Algebras
  • Constructing Affine Kac–Moody Lie Algebras
    • AffineLieAlgebra(N, F): MonStgElt, Fld → AlgKac
    • AffineLieAlgebra(C, F): AlgMatElt, Fld → AlgKac
    • Example: construct
  • Properties of Affine Kac–Moody Lie Algebras
    • CartanMatrix(L): AlgKac → AlgMatElt
    • CartanName(L): AlgKac → MonStgElt
    • Dimension(L): AlgKac → Infty
    • CoefficientRing(L): AlgKac → Rng
    • BaseRing(L): AlgKac → Rng
    • FiniteLieAlgebra(L): AlgKac → AlgLie
    • LaurentSeriesRing(L): AlgKac → RngSerLaur
    • StandardGenerators(L): AlgKac → SeqEnum[AlgKacElt], SeqEnum[AlgKacElt], SeqEnum[AlgKacElt]
    • Example: construct
  • Constructing Elements of Affine Kac–Moody Lie Algebras
    • L . i: AlgKac, RngIntElt → AlgKacElt
    • HasAttribute(L, "c"): AlgKac, MonStgElt → BoolElt, AlgKacElt
    • HasAttribute(L, "d"): AlgKac, MonStgElt → BoolElt, AlgKacElt
    • elt<L | {<} [ {<} p₁, y₁ {>}, ... ], λ, μ {>} >: AlgKac, Tup → AlgKacElt
  • Properties of Elements of Affine Kac–Moody Lie Algebras
    • EltTup(x): AlgKacElt → Tup
    • IsZero(x): AlgKacElt → BoolElt
    • x eq y: AlgKacElt, AlgKacElt → BoolElt
    • x + y: AlgKacElt, AlgKacElt → AlgKacElt
    • x - y: AlgKacElt, AlgKacElt → AlgKacElt
    • x * y: AlgKacElt, AlgKacElt → AlgKacElt
    • c * y: RngElt, AlgKacElt → AlgKacElt
    • -x: AlgKacElt → AlgKacElt
    • Example: construct