Regular Supercrystals¶
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class
sage.categories.regular_supercrystals.
RegularSuperCrystals
(s=None)¶ Bases:
sage.categories.category_singleton.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 methodcartan_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']}
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class
ElementMethods
¶ Bases:
object
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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
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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
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class
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
The category of regular crystals constructed by tensor product of regular crystals.
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extra_super_categories
()¶ EXAMPLES:
sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: RegularSuperCrystals().TensorProducts().extra_super_categories() [Category of regular super crystals]
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super_categories
()¶ EXAMPLES:
sage: from sage.categories.regular_supercrystals import RegularSuperCrystals sage: C = RegularSuperCrystals() sage: C.super_categories() [Category of finite super crystals]