Kac-Moody Algebras#
AUTHORS:
Travis Scrimshaw (07-15-2017): Initial implementation
- class sage.categories.kac_moody_algebras.KacMoodyAlgebras(base, name=None)[source]#
Bases:
Category_over_base_ring
Category of Kac-Moody algebras.
- class ParentMethods[source]#
Bases:
object
- cartan_type()[source]#
Return the Cartan type of
self
.EXAMPLES:
sage: L = LieAlgebra(QQ, cartan_type=['A', 2]) # needs sage.combinat sage.modules sage: L.cartan_type() # needs sage.combinat sage.modules ['A', 2]
>>> 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()[source]#
Return the Weyl group of
self
.EXAMPLES:
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)
>>> 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)[source]#
Return an example of a Kac-Moody algebra as per
Category.example
.EXAMPLES:
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
>>> 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: KacMoodyAlgebras(QQ).example(4) # needs sage.combinat sage.modules Lie algebra of ['A', 4] in the Chevalley basis
>>> 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()[source]#
EXAMPLES:
sage: from sage.categories.kac_moody_algebras import KacMoodyAlgebras sage: KacMoodyAlgebras(QQ).super_categories() [Category of Lie algebras over Rational Field]
>>> from sage.all import * >>> from sage.categories.kac_moody_algebras import KacMoodyAlgebras >>> KacMoodyAlgebras(QQ).super_categories() [Category of Lie algebras over Rational Field]