Supercommutative Algebras

class sage.categories.supercommutative_algebras.SupercommutativeAlgebras(base_category)

Bases: CategoryWithAxiom_over_base_ring

The category of supercommutative algebras.

An \(R\)-supercommutative algebra is an \(R\)-super algebra \(A = A_0 \oplus A_1\) endowed with an \(R\)-super algebra structure satisfying:

\[x_0 x'_0 = x'_0 x_0, \qquad x_1 x'_1 = -x'_1 x_1, \qquad x_0 x_1 = x_1 x_0,\]

for all \(x_0, x'_0 \in A_0\) and \(x_1, x'_1 \in A_1\).

EXAMPLES:

Sage

sage: Algebras(ZZ).Supercommutative()
Category of supercommutative algebras over Integer Ring

Python

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

class SignedTensorProducts(category, *args)

Bases: SignedTensorProductsCategory

extra_super_categories()

Return the extra super categories of self.

A signed tensor product of supercommutative algebras is a supercommutative algebra.

EXAMPLES:

Sage

sage: C = Algebras(ZZ).Supercommutative().SignedTensorProducts()
sage: C.extra_super_categories()
[Category of supercommutative algebras over Integer Ring]

Python

>>> from sage.all import *
>>> C = Algebras(ZZ).Supercommutative().SignedTensorProducts()
>>> C.extra_super_categories()
[Category of supercommutative algebras over Integer Ring]

class WithBasis(base_category)

Bases: CategoryWithAxiom_over_base_ring

class ParentMethods

Bases: object