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]