Algebras

AUTHORS:

  • David Kohel & William Stein (2005): initial revision
  • Nicolas M. Thiery (2008): rewrote for new category framework
class sage.categories.algebras.Algebras(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of algebras over a given base ring.

An algebra over a ring \(R\) is a module over \(R\) which is itself a ring.

TODO: should \(R\) be a commutative ring?

EXAMPLES:

sage: Algebras(ZZ)
Category of algebras over Integer Ring
sage: Algebras(ZZ).super_categories()
[Category of rings, Category of modules over Integer Ring]

TESTS:

sage: TestSuite(Algebras(ZZ)).run()
class CartesianProducts(category, *args)

Bases: sage.categories.cartesian_product.CartesianProductsCategory

The category of algebras constructed as cartesian products of algebras

This construction gives the direct product of algebras. See discussion on:

extra_super_categories()

A cartesian product of algebras is endowed with a natural algebra structure.

EXAMPLES:

sage: Algebras(QQ).CartesianProducts().extra_super_categories()
[Category of algebras over Rational Field]
sage: Algebras(QQ).CartesianProducts().super_categories()
[Category of algebras over Rational Field, Category of Cartesian products of monoids]
class Algebras.DualObjects(category, *args)

Bases: sage.categories.dual.DualObjectsCategory

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
extra_super_categories()

Returns the dual category

EXAMPLES:

The category of algebras over the Rational Field is dual to the category of coalgebras over the same field:

sage: C = Algebras(QQ)
sage: C.dual()
Category of duals of algebras over Rational Field
sage: C.dual().extra_super_categories()
[Category of coalgebras over Rational Field]
class Algebras.ElementMethods
class Algebras.ParentMethods
from_base_ring(r)

Canonical embedding from base ring

INPUT:

  • r – an element of self.base_ring()

Returns the canonical embedding of \(r\) into self.

EXAMPLES:

sage: A = AlgebrasWithBasis(QQ).example(); A
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: A.from_base_ring(1)
B[word: ]
class Algebras.TensorProducts(category, *args)

Bases: sage.categories.tensor.TensorProductsCategory

TESTS:

sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
sage: class FooBars(CovariantConstructionCategory):
...       _functor_category = "FooBars"
sage: Category.FooBars = lambda self: FooBars.category_of(self)
sage: C = FooBars(ModulesWithBasis(ZZ))
sage: C
Category of foo bars of modules with basis over Integer Ring
sage: C.base_category()
Category of modules with basis over Integer Ring
sage: latex(C)
\mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}})
sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module
sage: TestSuite(C).run()
ElementMethods

alias of TensorProducts.ElementMethods

ParentMethods

alias of TensorProducts.ParentMethods

extra_super_categories()

EXAMPLES:

sage: Algebras(QQ).TensorProducts().extra_super_categories()
[Category of algebras over Rational Field]
sage: Algebras(QQ).TensorProducts().super_categories()
[Category of algebras over Rational Field]

Meaning: a tensor product of algebras is an algebra

Algebras.super_categories()

EXAMPLES:

sage: Algebras(ZZ).super_categories()
[Category of rings, Category of modules over Integer Ring]

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