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