Regular Supercrystals#

class sage.categories.regular_supercrystals.RegularSuperCrystals(s=None)#

Bases: Category_singleton

The category of crystals for super Lie algebras.

EXAMPLES:

sage: from sage.categories.regular_supercrystals import RegularSuperCrystals
sage: C = RegularSuperCrystals()
sage: C
Category of regular super crystals
sage: C.super_categories()
[Category of finite super crystals]

Parents in this category should implement the following methods:

  • either an attribute _cartan_type or a method cartan_type

  • module_generators: a list (or container) of distinct elements that generate the crystal using \(f_i\) and \(e_i\)

Furthermore, their elements x should implement the following methods:

  • x.e(i) (returning \(e_i(x)\))

  • x.f(i) (returning \(f_i(x)\))

  • x.weight() (returning \(\operatorname{wt}(x)\))

EXAMPLES:

sage: from sage.misc.abstract_method import abstract_methods_of_class
sage: from sage.categories.regular_supercrystals import RegularSuperCrystals
sage: abstract_methods_of_class(RegularSuperCrystals().element_class)
{'optional': [], 'required': ['e', 'f', 'weight']}
class ElementMethods#

Bases: object

epsilon(i)#

Return \(\varepsilon_i\) of self.

EXAMPLES:

sage: C = crystals.Tableaux(['A',[1,2]], shape = [2,1])
sage: c = C.an_element(); c
[[-2, -2], [-1]]
sage: c.epsilon(2)
0
sage: c.epsilon(0)
0
sage: c.epsilon(-1)
0
phi(i)#

Return \(\varphi_i\) of self.

EXAMPLES:

sage: C = crystals.Tableaux(['A',[1,2]], shape = [2,1])
sage: c = C.an_element(); c
[[-2, -2], [-1]]
sage: c.phi(1)
0
sage: c.phi(2)
0
sage: c.phi(0)
1
class TensorProducts(category, *args)#

Bases: TensorProductsCategory

The category of regular crystals constructed by tensor product of regular crystals.

extra_super_categories()#

EXAMPLES:

sage: from sage.categories.regular_supercrystals import RegularSuperCrystals
sage: RegularSuperCrystals().TensorProducts().extra_super_categories()
[Category of regular super crystals]
super_categories()#

EXAMPLES:

sage: from sage.categories.regular_supercrystals import RegularSuperCrystals
sage: C = RegularSuperCrystals()
sage: C.super_categories()
[Category of finite super crystals]