Kac-Moody Algebras - Category Framework

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# Kac-Moody Algebras

AUTHORS:

- Travis Scrimshaw (07-15-2017): Initial implementation

class sage.categories.kac_moody_algebras.KacMoodyAlgebras(base, name=None)

  Bases: `Category_over_base_ring`

  Category of Kac-Moody algebras.

  class ParentMethods

    Bases: `object`

    cartan_type()

      Return the Cartan type of `self`.

      EXAMPLES:

      Sage

          sage: L = LieAlgebra(QQ, cartan_type=['A', 2])                          # needs sage.combinat sage.modules
          sage: L.cartan_type()                                                   # needs sage.combinat sage.modules
          ['A', 2]

      Python

          >>> from sage.all import *
          >>> L = LieAlgebra(QQ, cartan_type=['A', Integer(2)])                          # needs sage.combinat sage.modules
          >>> L.cartan_type()                                                   # needs sage.combinat sage.modules
          ['A', 2]

    weyl_group()

      Return the Weyl group of `self`.

      EXAMPLES:

      Sage

          sage: L = LieAlgebra(QQ, cartan_type=['A', 2])                          # needs sage.combinat sage.modules
          sage: L.weyl_group()                                                    # needs sage.combinat sage.modules
          Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

      Python

          >>> from sage.all import *
          >>> L = LieAlgebra(QQ, cartan_type=['A', Integer(2)])                          # needs sage.combinat sage.modules
          >>> L.weyl_group()                                                    # needs sage.combinat sage.modules
          Weyl Group of type ['A', 2] (as a matrix group acting on the ambient space)

  example(n=2)

    Return an example of a Kac-Moody algebra as per `Category.example`.

    EXAMPLES:

    Sage

        sage: from sage.categories.kac_moody_algebras import KacMoodyAlgebras
        sage: KacMoodyAlgebras(QQ).example()                                        # needs sage.combinat sage.modules
        Lie algebra of ['A', 2] in the Chevalley basis

    Python

        >>> from sage.all import *
        >>> from sage.categories.kac_moody_algebras import KacMoodyAlgebras
        >>> KacMoodyAlgebras(QQ).example()                                        # needs sage.combinat sage.modules
        Lie algebra of ['A', 2] in the Chevalley basis

    We can specify the rank of the example:

    Sage

        sage: KacMoodyAlgebras(QQ).example(4)                                       # needs sage.combinat sage.modules
        Lie algebra of ['A', 4] in the Chevalley basis

    [Python]

        >>> from sage.all import *
        >>> KacMoodyAlgebras(QQ).example(Integer(4))                                       # needs sage.combinat sage.modules
        Lie algebra of ['A', 4] in the Chevalley basis

  super_categories()

    EXAMPLES:

    Sage

        sage: from sage.categories.kac_moody_algebras import KacMoodyAlgebras
        sage: KacMoodyAlgebras(QQ).super_categories()
        [Category of Lie algebras over Rational Field]

    Python

        >>> from sage.all import *
        >>> from sage.categories.kac_moody_algebras import KacMoodyAlgebras
        >>> KacMoodyAlgebras(QQ).super_categories()
        [Category of Lie algebras over Rational Field]