Algebras¶
AUTHORS:
- David Kohel & William Stein (2005): initial revision
- Nicolas M. Thiery (2008-2011): rewrote for the category framework
-
class
sage.categories.algebras.
Algebras
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
The category of associative and unital algebras over a given base ring.
An associative and unital algebra over a ring \(R\) is a module over \(R\) which is itself a ring.
Warning
Algebras
will be eventually be replaced bymagmatic_algebras.MagmaticAlgebras
for consistency with e.g. Wikipedia article Algebras which assumes neither associativity nor the existence of a unit (see trac ticket #15043).Todo
Should \(R\) be a commutative ring?
EXAMPLES:
sage: Algebras(ZZ) Category of algebras over Integer Ring sage: sorted(Algebras(ZZ).super_categories(), key=str) [Category of associative algebras over Integer Ring, Category of rings, Category of unital algebras 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: C = Algebras(QQ).CartesianProducts() sage: C.extra_super_categories() [Category of algebras over Rational Field] sage: sorted(C.super_categories(), key=str) [Category of Cartesian products of distributive magmas and additive magmas, Category of Cartesian products of monoids, Category of Cartesian products of vector spaces over Rational Field, Category of algebras over Rational Field]
-
-
Algebras.
Commutative
¶ alias of
CommutativeAlgebras
-
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" ....: _base_category_class = (Category,) 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]
Warning
This is only correct in certain cases (finite dimension, ...). See trac ticket #15647.
-
-
class
Algebras.
ElementMethods
¶
-
Algebras.
Filtered
¶ alias of
FilteredAlgebras
-
Algebras.
Graded
¶ alias of
GradedAlgebras
-
class
Algebras.
Quotients
(category, *args)¶ Bases:
sage.categories.quotients.QuotientsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) 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()
-
class
ParentMethods
¶ -
algebra_generators
()¶ Return algebra generators for
self
.This implementation retracts the algebra generators from the ambient algebra.
EXAMPLES:
sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example(); A An example of a finite dimensional algebra with basis: the path algebra of the Kronecker quiver (containing the arrows a:x->y and b:x->y) over Rational Field sage: S = A.semisimple_quotient() sage: S.algebra_generators() Finite family {'y': B['y'], 'x': B['x'], 'b': 0, 'a': 0}
Todo
this could possibly remove the elements that retract to zero.
-
-
class
-
Algebras.
Semisimple
¶ alias of
SemisimpleAlgebras
-
class
Algebras.
SubcategoryMethods
¶ -
Semisimple
()¶ Return the subcategory of semisimple objects of
self
.Note
This mimics the syntax of axioms for a smooth transition if
Semisimple
becomes one.EXAMPLES:
sage: Algebras(QQ).Semisimple() Category of semisimple algebras over Rational Field sage: Algebras(QQ).WithBasis().FiniteDimensional().Semisimple() Category of finite dimensional semisimple algebras with basis over Rational Field
-
-
Algebras.
Super
¶ alias of
SuperAlgebras
-
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" ....: _base_category_class = (Category,) 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()
-
class
ElementMethods
¶
-
class
Algebras.TensorProducts.
ParentMethods
¶
-
Algebras.TensorProducts.
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, Category of tensor products of vector spaces over Rational Field]
Meaning: a tensor product of algebras is an algebra
-
class
-
Algebras.
WithBasis
¶ alias of
AlgebrasWithBasis
-
class