Examples of algebras with basis¶
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sage.categories.examples.algebras_with_basis.
Example
¶ alias of
sage.categories.examples.algebras_with_basis.FreeAlgebra
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class
sage.categories.examples.algebras_with_basis.
FreeAlgebra
(R, alphabet='a', 'b', 'c')¶ Bases:
sage.combinat.free_module.CombinatorialFreeModule
An example of an algebra with basis: the free algebra
This class illustrates a minimal implementation of an algebra with basis.
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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])
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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: ]
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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]
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