Algebras With Basis#
- class sage.categories.algebras_with_basis.AlgebrasWithBasis(base_category)#
Bases:
CategoryWithAxiom_over_base_ringThe category of algebras with a distinguished basis.
EXAMPLES:
sage: C = AlgebrasWithBasis(QQ); C Category of algebras with basis over Rational Field sage: sorted(C.super_categories(), key=str) [Category of algebras over Rational Field, Category of unital algebras with basis over Rational Field]
We construct a typical parent in this category, and do some computations with it:
sage: A = C.example(); A # needs sage.combinat sage.modules An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: A.category() # needs sage.combinat sage.modules Category of algebras with basis over Rational Field sage: A.one_basis() # needs sage.combinat sage.modules word: sage: A.one() # needs sage.combinat sage.modules B[word: ] sage: A.base_ring() # needs sage.combinat sage.modules Rational Field sage: A.basis().keys() # needs sage.combinat sage.modules Finite words over {'a', 'b', 'c'} sage: (a,b,c) = A.algebra_generators() # needs sage.combinat sage.modules sage: a^3, b^2 # needs sage.combinat sage.modules (B[word: aaa], B[word: bb]) sage: a * c * b # needs sage.combinat sage.modules B[word: acb] sage: A.product # needs sage.combinat sage.modules <bound method MagmaticAlgebras.WithBasis.ParentMethods._product_from_product_on_basis_multiply of An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field> sage: A.product(a * b, b) # needs sage.combinat sage.modules B[word: abb] sage: TestSuite(A).run(verbose=True) # needs sage.combinat sage.modules running ._test_additive_associativity() . . . pass running ._test_an_element() . . . pass running ._test_associativity() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_characteristic() . . . pass running ._test_construction() . . . pass running ._test_distributivity() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_nonzero_equal() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_one() . . . pass running ._test_pickling() . . . pass running ._test_prod() . . . pass running ._test_some_elements() . . . pass running ._test_zero() . . . pass sage: A.__class__ # needs sage.combinat sage.modules <class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category'> sage: A.element_class # needs sage.combinat sage.modules <class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category.element_class'>
Please see the source code of \(A\) (with
A??) for how to implement other algebras with basis.- class CartesianProducts(category, *args)#
Bases:
CartesianProductsCategoryThe category of algebras with basis, constructed as Cartesian products of algebras with basis.
Note: this construction give the direct products of algebras with basis. See comment in
Algebras.CartesianProducts- class ParentMethods#
Bases:
object- one()#
- one_from_cartesian_product_of_one_basis()#
Return the one of this Cartesian product of algebras, as per
Monoids.ParentMethods.oneIt is constructed as the Cartesian product of the ones of the summands, using their
one_basis()methods.This implementation does not require multiplication by scalars nor calling cartesian_product. This might help keeping things as lazy as possible upon initialization.
EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example(); A # needs sage.combinat sage.modules An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: A.one_basis() # needs sage.combinat sage.modules word: sage: B = cartesian_product((A, A, A)) # needs sage.combinat sage.modules sage: B.one_from_cartesian_product_of_one_basis() # needs sage.combinat sage.modules B[(0, word: )] + B[(1, word: )] + B[(2, word: )] sage: B.one() # needs sage.combinat sage.modules B[(0, word: )] + B[(1, word: )] + B[(2, word: )] sage: cartesian_product([SymmetricGroupAlgebra(QQ, 3), # needs sage.combinat sage.modules ....: SymmetricGroupAlgebra(QQ, 4)]).one() B[(0, [1, 2, 3])] + B[(1, [1, 2, 3, 4])]
- extra_super_categories()#
A Cartesian product of algebras with basis is endowed with a natural algebra with basis structure.
EXAMPLES:
sage: AlgebrasWithBasis(QQ).CartesianProducts().extra_super_categories() [Category of algebras with basis over Rational Field] sage: AlgebrasWithBasis(QQ).CartesianProducts().super_categories() [Category of algebras with basis over Rational Field, Category of Cartesian products of algebras over Rational Field, Category of Cartesian products of vector spaces with basis over Rational Field]
- class ElementMethods#
Bases:
object
- Filtered#
alias of
FilteredAlgebrasWithBasis
- FiniteDimensional#
alias of
FiniteDimensionalAlgebrasWithBasis
- Graded#
alias of
GradedAlgebrasWithBasis
- class ParentMethods#
Bases:
object- hochschild_complex(M)#
Return the Hochschild complex of
selfwith coefficients inM.See also
EXAMPLES:
sage: R.<x> = QQ[] sage: A = algebras.DifferentialWeyl(R) # needs sage.modules sage: H = A.hochschild_complex(A) # needs sage.modules sage: SGA = SymmetricGroupAlgebra(QQ, 3) # needs sage.combinat sage.modules sage: T = SGA.trivial_representation() # needs sage.combinat sage.modules sage: H = SGA.hochschild_complex(T) # needs sage.combinat sage.modules
- one()#
Return the multiplicative unit element.
EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example() # needs sage.combinat sage.modules sage: A.one_basis() # needs sage.combinat sage.modules word: sage: A.one() # needs sage.combinat sage.modules B[word: ]
- Super#
alias of
SuperAlgebrasWithBasis
- class TensorProducts(category, *args)#
Bases:
TensorProductsCategoryThe category of algebras with basis constructed by tensor product of algebras with basis
- class ElementMethods#
Bases:
objectImplements operations on elements of tensor products of algebras with basis
- class ParentMethods#
Bases:
objectimplements operations on tensor products of algebras with basis
- one_basis()#
Returns the index of the one of this tensor product of algebras, as per
AlgebrasWithBasis.ParentMethods.one_basisIt is the tuple whose operands are the indices of the ones of the operands, as returned by their
one_basis()methods.EXAMPLES:
sage: # needs sage.combinat sage.modules 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.one_basis() word: sage: B = tensor((A, A, A)) sage: B.one_basis() (word: , word: , word: ) sage: B.one() B[word: ] # B[word: ] # B[word: ]
- product_on_basis(t1, t2)#
The product of the algebra on the basis, as per
AlgebrasWithBasis.ParentMethods.product_on_basis.EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example(); A # needs sage.combinat sage.modules An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: (a,b,c) = A.algebra_generators() # needs sage.combinat sage.modules sage: x = tensor((a, b, c)); x # needs sage.combinat sage.modules B[word: a] # B[word: b] # B[word: c] sage: y = tensor((c, b, a)); y # needs sage.combinat sage.modules B[word: c] # B[word: b] # B[word: a] sage: x * y # needs sage.combinat sage.modules B[word: ac] # B[word: bb] # B[word: ca] sage: x = tensor(((a + 2*b), c)); x # needs sage.combinat sage.modules B[word: a] # B[word: c] + 2*B[word: b] # B[word: c] sage: y = tensor((c, a)) + 1; y # needs sage.combinat sage.modules B[word: ] # B[word: ] + B[word: c] # B[word: a] sage: x * y # needs sage.combinat sage.modules B[word: a] # B[word: c] + B[word: ac] # B[word: ca] + 2*B[word: b] # B[word: c] + 2*B[word: bc] # B[word: ca]
TODO: optimize this implementation!
- extra_super_categories()#
EXAMPLES:
sage: AlgebrasWithBasis(QQ).TensorProducts().extra_super_categories() [Category of algebras with basis over Rational Field] sage: AlgebrasWithBasis(QQ).TensorProducts().super_categories() [Category of algebras with basis over Rational Field, Category of tensor products of algebras over Rational Field, Category of tensor products of vector spaces with basis over Rational Field]
- example(alphabet=('a', 'b', 'c'))#
Return an example of algebra with basis.
EXAMPLES:
sage: AlgebrasWithBasis(QQ).example() # needs sage.combinat sage.modules An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
An other set of generators can be specified as optional argument:
sage: AlgebrasWithBasis(QQ).example((1,2,3)) # needs sage.combinat sage.modules An example of an algebra with basis: the free algebra on the generators (1, 2, 3) over Rational Field