Super Algebras

class sage.categories.super_algebras.SuperAlgebras(base_category)

Bases: SuperModulesCategory

The category of super algebras.

An \(R\)-super algebra is an \(R\)-super module \(A\) endowed with an \(R\)-algebra structure satisfying

\[A_0 A_0 \subseteq A_0, \qquad A_0 A_1 \subseteq A_1, \qquad A_1 A_0 \subseteq A_1, \qquad A_1 A_1 \subseteq A_0\]

and \(1 \in A_0\).

EXAMPLES:

Sage

sage: Algebras(ZZ).Super()
Category of super algebras over Integer Ring

Python

>>> from sage.all import *
>>> Algebras(ZZ).Super()
Category of super algebras over Integer Ring

class ParentMethods

Bases: object

graded_algebra()

Return the associated graded algebra to self.

Warning

Because a super module \(M\) is naturally \(\ZZ / 2 \ZZ\)-graded, and graded modules have a natural filtration induced by the grading, if \(M\) has a different filtration, then the associated graded module \(\operatorname{gr} M \neq M\). This is most apparent with super algebras, such as the differential Weyl algebra, and the multiplication may not coincide.

tensor(*parents, **kwargs)

Return the tensor product of the parents.

EXAMPLES:

Sage

sage: # needs sage.combinat sage.modules
sage: A.<x,y,z> = ExteriorAlgebra(ZZ); A.rename('A')
sage: T = A.tensor(A,A); T
A # A # A
sage: T in Algebras(ZZ).Graded().SignedTensorProducts()
True
sage: T in Algebras(ZZ).Graded().TensorProducts()
False
sage: A.rename(None)

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> A = ExteriorAlgebra(ZZ, names=('x', 'y', 'z',)); (x, y, z,) = A._first_ngens(3); A.rename('A')
>>> T = A.tensor(A,A); T
A # A # A
>>> T in Algebras(ZZ).Graded().SignedTensorProducts()
True
>>> T in Algebras(ZZ).Graded().TensorProducts()
False
>>> A.rename(None)

This also works when the other elements do not have a signed tensor product (Issue #31266):

Sage

sage: # needs sage.combinat sage.modules
sage: a = SteenrodAlgebra(3).an_element()
sage: M = CombinatorialFreeModule(GF(3), ['s', 't', 'u'])
sage: s = M.basis()['s']
sage: tensor([a, s])                                                    # needs sage.rings.finite_rings
2*Q_1 Q_3 P(2,1) # B['s']

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.modules
>>> a = SteenrodAlgebra(Integer(3)).an_element()
>>> M = CombinatorialFreeModule(GF(Integer(3)), ['s', 't', 'u'])
>>> s = M.basis()['s']
>>> tensor([a, s])                                                    # needs sage.rings.finite_rings
2*Q_1 Q_3 P(2,1) # B['s']

class SignedTensorProducts(category, *args)

Bases: SignedTensorProductsCategory

extra_super_categories()

EXAMPLES:

Sage

sage: Coalgebras(QQ).Graded().SignedTensorProducts().extra_super_categories()
[Category of graded coalgebras over Rational Field]
sage: Coalgebras(QQ).Graded().SignedTensorProducts().super_categories()
[Category of graded coalgebras over Rational Field]

Python

>>> from sage.all import *
>>> Coalgebras(QQ).Graded().SignedTensorProducts().extra_super_categories()
[Category of graded coalgebras over Rational Field]
>>> Coalgebras(QQ).Graded().SignedTensorProducts().super_categories()
[Category of graded coalgebras over Rational Field]

Meaning: a signed tensor product of coalgebras is a coalgebra

class SubcategoryMethods

Bases: object

Supercommutative()

Return the full subcategory of the supercommutative objects of self.

A super algebra \(M\) is supercommutative if, for all homogeneous \(x,y\in M\),

\[x \cdot y = (-1)^{|x||y|} y \cdot x.\]

REFERENCES:

Wikipedia article Supercommutative_algebra

EXAMPLES:

Sage

sage: Algebras(ZZ).Super().Supercommutative()
Category of supercommutative algebras over Integer Ring
sage: Algebras(ZZ).Super().WithBasis().Supercommutative()
Category of supercommutative algebras with basis over Integer Ring

Python

>>> from sage.all import *
>>> Algebras(ZZ).Super().Supercommutative()
Category of supercommutative algebras over Integer Ring
>>> Algebras(ZZ).Super().WithBasis().Supercommutative()
Category of supercommutative algebras with basis over Integer Ring

Supercommutative

alias of SupercommutativeAlgebras

extra_super_categories()

EXAMPLES:

Sage

sage: Algebras(ZZ).Super().super_categories() # indirect doctest
[Category of graded algebras over Integer Ring,
 Category of super modules over Integer Ring]

Python

>>> from sage.all import *
>>> Algebras(ZZ).Super().super_categories() # indirect doctest
[Category of graded algebras over Integer Ring,
 Category of super modules over Integer Ring]