# 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)

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 by magmatic_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]
class CartesianProducts(category, *args)

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
class DualObjects(category, *args)
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 trac ticket #15647.

class ElementMethods

Bases: object

Filtered
class Quotients(category, *args)
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
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
class TensorProducts(category, *args)
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