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:
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 github issue #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]
- class CartesianProducts(category, *args)#
Bases:
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]
- Commutative#
alias of
CommutativeAlgebras
- class DualObjects(category, *args)#
Bases:
DualObjectsCategory
- extra_super_categories()#
Return 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 github issue #15647.
- class ElementMethods#
Bases:
object
- Filtered#
alias of
FilteredAlgebras
- Graded#
alias of
GradedAlgebras
- class Quotients(category, *args)#
Bases:
QuotientsCategory
- class ParentMethods#
Bases:
object
- 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 {'x': B['x'], 'y': B['y'], 'a': 0, 'b': 0}
Todo
this could possibly remove the elements that retract to zero.
- Semisimple#
alias of
SemisimpleAlgebras
- class SubcategoryMethods#
Bases:
object
- 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
- Supercommutative()#
Return the full subcategory of the supercommutative objects of
self
.This is shorthand for creating the corresponding super category.
EXAMPLES:
sage: Algebras(ZZ).Supercommutative() Category of supercommutative algebras over Integer Ring sage: Algebras(ZZ).WithBasis().Supercommutative() Category of supercommutative super algebras with basis over Integer Ring sage: Cat = Algebras(ZZ).Supercommutative() sage: Cat is Algebras(ZZ).Super().Supercommutative() True
- Super#
alias of
SuperAlgebras
- class TensorProducts(category, *args)#
Bases:
TensorProductsCategory
- class ElementMethods#
Bases:
object
- class ParentMethods#
Bases:
object
- 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
- WithBasis#
alias of
AlgebrasWithBasis