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]