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]

Sage Live

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

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)

Sage Live

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

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

Sage Live

from sage.categories.kac_moody_algebras import KacMoodyAlgebras
KacMoodyAlgebras(QQ).example()                                        # needs sage.combinat sage.modules

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

Sage Live

KacMoodyAlgebras(QQ).example(4)                                       # needs sage.combinat sage.modules

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]

Sage Live

from sage.categories.kac_moody_algebras import KacMoodyAlgebras
KacMoodyAlgebras(QQ).super_categories()

Make live