Example of a finite dimensional algebra with basis¶
- sage.categories.examples.finite_dimensional_algebras_with_basis.Example[source]¶
- alias of - KroneckerQuiverPathAlgebra
- class sage.categories.examples.finite_dimensional_algebras_with_basis.KroneckerQuiverPathAlgebra(base_ring)[source]¶
- 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.PathAlgebrafor a full-featured implementation of path algebras.- algebra_generators()[source]¶
- Return algebra generators for this algebra. - See also - Algebras.ParentMethods.algebra_generators().- 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: A.algebra_generators() Finite family {'x': x, 'y': y, 'a': a, 'b': b} - >>> 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()[source]¶
- Return the unit of this algebra. - See also - AlgebrasWithBasis.ParentMethods.one_basis()- EXAMPLES: - sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example() sage: A.one() x + y - >>> from sage.all import * >>> A = FiniteDimensionalAlgebrasWithBasis(QQ).example() >>> A.one() x + y 
 - product_on_basis(w1, w2)[source]¶
- Return the product of the two basis elements indexed by - w1and- w2.- See also - AlgebrasWithBasis.ParentMethods.product_on_basis().- EXAMPLES: - sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example() - >>> from sage.all import * >>> A = FiniteDimensionalAlgebrasWithBasis(QQ).example() - Here is the multiplication table for the algebra: - 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] - >>> 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: 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 - >>> 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