Example of a finite dimensional algebra with basis

sage.categories.examples.finite_dimensional_algebras_with_basis.Example

alias of KroneckerQuiverPathAlgebra

class sage.categories.examples.finite_dimensional_algebras_with_basis.KroneckerQuiverPathAlgebra(base_ring)

Bases: CombinatorialFreeModule

An example of a finite dimensional algebra with basis: the path algebra of the Kronecker quiver.

This class illustrates a minimal implementation of a finite dimensional algebra with basis. See sage.quivers.algebra.PathAlgebra for a full-featured implementation of path algebras.

algebra_generators()

Return algebra generators for this algebra.

See also

Algebras.ParentMethods.algebra_generators().

EXAMPLES:

Sage

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: A.algebra_generators()
Finite family {'x': x, 'y': y, 'a': a, 'b': b}

Python

>>> from sage.all import *
>>> 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
>>> A.algebra_generators()
Finite family {'x': x, 'y': y, 'a': a, 'b': b}

one()

Return the unit of this algebra.

See also

AlgebrasWithBasis.ParentMethods.one_basis()

EXAMPLES:

Sage

sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example()
sage: A.one()
x + y

Python

>>> from sage.all import *
>>> A = FiniteDimensionalAlgebrasWithBasis(QQ).example()
>>> A.one()
x + y

product_on_basis(w1, w2)

Return the product of the two basis elements indexed by w1 and w2.

See also

AlgebrasWithBasis.ParentMethods.product_on_basis().

EXAMPLES:

Sage

sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example()

Python

>>> from sage.all import *
>>> A = FiniteDimensionalAlgebrasWithBasis(QQ).example()

Here is the multiplication table for the algebra:

Sage

sage: matrix([[p*q for q in A.basis()] for p in A.basis()])
[x 0 a b]
[0 y 0 0]
[0 a 0 0]
[0 b 0 0]

Python

>>> from sage.all import *
>>> matrix([[p*q for q in A.basis()] for p in A.basis()])
[x 0 a b]
[0 y 0 0]
[0 a 0 0]
[0 b 0 0]

Here we take some products of linear combinations of basis elements:

Sage

sage: x, y, a, b = A.basis()
sage: a * (1-b)^2 * x
0
sage: x*a + b*y
a + b
sage: x*x
x
sage: x*y
0
sage: x*a*y
a

Python

>>> from sage.all import *
>>> x, y, a, b = A.basis()
>>> a * (Integer(1)-b)**Integer(2) * x
0
>>> x*a + b*y
a + b
>>> x*x
x
>>> x*y
0
>>> x*a*y
a