Example of a finite dimensional algebra with basis¶
-
sage.categories.examples.finite_dimensional_algebras_with_basis.
Example
¶ alias of
sage.categories.examples.finite_dimensional_algebras_with_basis.KroneckerQuiverPathAlgebra
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class
sage.categories.examples.finite_dimensional_algebras_with_basis.
KroneckerQuiverPathAlgebra
(base_ring)¶ Bases:
sage.combinat.free_module.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: 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}
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one
()¶ Return the unit of this algebra.
See also
AlgebrasWithBasis.ParentMethods.one_basis()
EXAMPLES:
sage: A = FiniteDimensionalAlgebrasWithBasis(QQ).example() sage: A.one() x + y
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product_on_basis
(w1, w2)¶ Return the product of the two basis elements indexed by
w1
andw2
.See also
AlgebrasWithBasis.ParentMethods.product_on_basis()
.EXAMPLES:
sage: 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]
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
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