Examples of algebras with basis#
- sage.categories.examples.algebras_with_basis.Example#
alias of
FreeAlgebra
- class sage.categories.examples.algebras_with_basis.FreeAlgebra(R, alphabet=('a', 'b', 'c'))#
Bases:
CombinatorialFreeModule
An example of an algebra with basis: the free algebra
This class illustrates a minimal implementation of an algebra with basis.
- algebra_generators()#
Return the generators of this algebra, as per
algebra_generators()
.EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example(); A An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: A.algebra_generators() Family (B[word: a], B[word: b], B[word: c])
- one_basis()#
Returns the empty word, which index the one of this algebra, as per
AlgebrasWithBasis.ParentMethods.one_basis()
.EXAMPLES::r
sage: A = AlgebrasWithBasis(QQ).example() sage: A.one_basis() word: sage: A.one() B[word: ]
- product_on_basis(w1, w2)#
Product of basis elements, as per
AlgebrasWithBasis.ParentMethods.product_on_basis()
.EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example() sage: Words = A.basis().keys() sage: A.product_on_basis(Words("acb"), Words("cba")) B[word: acbcba] sage: (a,b,c) = A.algebra_generators() sage: a * (1-b)^2 * c B[word: abbc] - 2*B[word: abc] + B[word: ac]