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sage.categories: Update # needs, use block tags #38071

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4 changes: 2 additions & 2 deletions src/sage/categories/action.pyx
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Expand Up @@ -435,7 +435,7 @@ cdef class PrecomposedAction(Action):
We demonstrate that an example discussed on :issue:`14711` did not become a
problem::

sage: # needs sage.modular
sage: # needs sage.libs.flint sage.modular
sage: E = ModularSymbols(11).2
sage: s = E.modular_symbol_rep()
sage: del E,s
Expand Down Expand Up @@ -487,7 +487,7 @@ cdef class PrecomposedAction(Action):

Check that this action can be pickled (:issue:`29031`)::

sage: # needs sage.modular
sage: # needs sage.libs.flint sage.modular
sage: E = ModularSymbols(11).2
sage: v = E.manin_symbol_rep()
sage: c,x = v[0]
Expand Down
26 changes: 14 additions & 12 deletions src/sage/categories/algebra_functor.py
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Original file line number Diff line number Diff line change
@@ -1,4 +1,4 @@
# sage.doctest: needs sage.groups
# sage.doctest: needs sage.groups sage.modules
r"""
Group algebras and beyond: the Algebra functorial construction

Expand Down Expand Up @@ -205,7 +205,7 @@
divide the characteristic of the base field, the algebra is
semisimple::

sage: SymmetricGroup(5).algebra(QQ) in Algebras(QQ).Semisimple()
sage: SymmetricGroup(5).algebra(QQ) in Algebras(QQ).Semisimple() # needs sage.combinat
True
sage: CyclicPermutationGroup(10).algebra(FiniteField(7)) in Algebras.Semisimple
True
Expand All @@ -225,6 +225,7 @@
since there is a natural map from the integers to the rationals, there
is a natural map from `\ZZ[D_2]` to `\QQ[S_4]`::

sage: # needs sage.combinat
sage: A = DihedralGroup(2).algebra(ZZ)
sage: B = SymmetricGroup(4).algebra(QQ)
sage: a = A.an_element(); a
Expand Down Expand Up @@ -308,12 +309,12 @@

Properties of group algebras::

sage: SU(2, GF(4, 'a')).algebra(IntegerModRing(12)).category()
sage: SU(2, GF(4, 'a')).algebra(IntegerModRing(12)).category() # needs sage.rings.finite_rings
Category of finite group algebras over Ring of integers modulo 12

sage: SymmetricGroup(2).algebra(QQ).is_commutative()
sage: SymmetricGroup(2).algebra(QQ).is_commutative() # needs sage.combinat
True
sage: SymmetricGroup(3).algebra(QQ).is_commutative()
sage: SymmetricGroup(3).algebra(QQ).is_commutative() # needs sage.combinat
False

sage: G = DihedralGroup(4)
Expand Down Expand Up @@ -364,7 +365,7 @@

sage: G = SymmetricGroup(5)
sage: x,y = G.gens()
sage: A = G.algebra(QQ)
sage: A = G.algebra(QQ) # needs sage.combinat
sage: A( A(x) )
(1,2,3,4,5)

Expand All @@ -379,6 +380,7 @@
Coercion from the base ring takes precedences over coercion from the
group::

sage: # needs sage.rings.number_field
sage: G = GL(2,7)
sage: OG = GroupAlgebra(G, ZZ[AA(5).sqrt()])
sage: OG(2)
Expand All @@ -387,7 +389,6 @@
sage: OG(G(2))
[2 0]
[0 2]

sage: OG(FormalSum([ (1, G(2)), (2, RR(0.77)) ]) )
Traceback (most recent call last):
...
Expand All @@ -411,7 +412,7 @@
morphism::

sage: G = SymmetricGroup(3)
sage: A = G.algebra(ZZ)
sage: A = G.algebra(ZZ) # needs sage.combinat
sage: h = GF(5).coerce_map_from(ZZ)

sage: functor = A.construction()[0]; functor
Expand All @@ -420,7 +421,7 @@
sage: hh
Generic morphism:
From: Symmetric group algebra of order 3 over Integer Ring
To: Symmetric group algebra of order 3 over Finite Field of size 5
To: Symmetric group algebra of order 3 over Finite Field of size 5
sage: a = 2 * A.an_element(); a
2*() + 2*(2,3) + 6*(1,2,3) + 4*(1,3,2)

Expand Down Expand Up @@ -561,7 +562,7 @@ def __init__(self, group):
EXAMPLES::

sage: from sage.categories.algebra_functor import GroupAlgebraFunctor
sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category() # needs sage.rings.finite_rings
Category of finite group algebras over Ring of integers modulo 12
"""
self.__group = group
Expand Down Expand Up @@ -617,14 +618,14 @@ def _apply_functor_to_morphism(self, f):

EXAMPLES::

sage: # needs sage.combinat
sage: G = SymmetricGroup(3)
sage: A = GroupAlgebra(G, ZZ)
sage: h = GF(5).coerce_map_from(ZZ)
sage: hh = A.construction()[0](h); hh
Generic morphism:
From: Symmetric group algebra of order 3 over Integer Ring
To: Symmetric group algebra of order 3 over Finite Field of size 5

sage: a = 2 * A.an_element(); a
2*() + 2*(2,3) + 6*(1,2,3) + 4*(1,3,2)
sage: hh(a)
Expand Down Expand Up @@ -673,7 +674,7 @@ def _repr_object_names(self):
"""
EXAMPLES::

sage: Semigroups().Algebras(QQ) # indirect doctest
sage: Semigroups().Algebras(QQ) # indirect doctest
Category of semigroup algebras over Rational Field
"""
return "{} algebras over {}".format(self.base_category()._repr_object_names()[:-1],
Expand Down Expand Up @@ -726,6 +727,7 @@ def coproduct_on_basis(self, g):

EXAMPLES::

sage: # needs sage.combinat
sage: PF = NonDecreasingParkingFunctions(4)
sage: A = PF.algebra(ZZ); A
Algebra of Non-decreasing parking functions of size 4 over Integer Ring
Expand Down
75 changes: 35 additions & 40 deletions src/sage/categories/algebras_with_basis.py
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Original file line number Diff line number Diff line change
Expand Up @@ -33,37 +33,32 @@ class AlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
We construct a typical parent in this category, and do some
computations with it::

sage: A = C.example(); A # needs sage.combinat sage.modules
sage: # needs sage.combinat sage.modules
sage: A = C.example(); A
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
sage: A.category()
Category of algebras with basis over Rational Field

sage: A.one_basis() # needs sage.combinat sage.modules
sage: A.one_basis()
word:
sage: A.one() # needs sage.combinat sage.modules
sage: A.one()
B[word: ]

sage: A.base_ring() # needs sage.combinat sage.modules
sage: A.base_ring()
Rational Field
sage: A.basis().keys() # needs sage.combinat sage.modules
sage: A.basis().keys()
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
sage: (a,b,c) = A.algebra_generators()
sage: a^3, b^2
(B[word: aaa], B[word: bb])
sage: a * c * b # needs sage.combinat sage.modules
sage: a * c * b
B[word: acb]

sage: A.product # needs sage.combinat sage.modules
sage: A.product
<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
sage: A.product(a * b, b)
B[word: abb]

sage: TestSuite(A).run(verbose=True) # needs sage.combinat sage.modules
sage: TestSuite(A).run(verbose=True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
Expand Down Expand Up @@ -93,9 +88,9 @@ class AlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass
sage: A.__class__ # needs sage.combinat sage.modules
sage: A.__class__
<class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category'>
sage: A.element_class # needs sage.combinat sage.modules
sage: A.element_class
<class 'sage.categories.examples.algebras_with_basis.FreeAlgebra_with_category.element_class'>

Please see the source code of `A` (with ``A??``) for how to
Expand Down Expand Up @@ -150,9 +145,10 @@ def hochschild_complex(self, M):
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
sage: # needs sage.combinat sage.groups sage.modules
sage: SGA = SymmetricGroupAlgebra(QQ, 3)
sage: T = SGA.trivial_representation()
sage: H = SGA.hochschild_complex(T)
"""
from sage.homology.hochschild_complex import HochschildComplex
return HochschildComplex(self, M)
Expand Down Expand Up @@ -243,19 +239,19 @@ def one_from_cartesian_product_of_one_basis(self):

EXAMPLES::

sage: A = AlgebrasWithBasis(QQ).example(); A # needs sage.combinat sage.modules
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() # needs sage.combinat sage.modules
sage: A.one_basis()
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
sage: B = cartesian_product((A, A, A))
sage: B.one_from_cartesian_product_of_one_basis()
B[(0, word: )] + B[(1, word: )] + B[(2, word: )]
sage: B.one() # needs sage.combinat sage.modules
sage: B.one()
B[(0, word: )] + B[(1, word: )] + B[(2, word: )]

sage: cartesian_product([SymmetricGroupAlgebra(QQ, 3), # needs sage.combinat sage.modules
sage: cartesian_product([SymmetricGroupAlgebra(QQ, 3), # needs sage.combinat sage.groups sage.modules
....: SymmetricGroupAlgebra(QQ, 4)]).one()
B[(0, [1, 2, 3])] + B[(1, [1, 2, 3, 4])]
"""
Expand Down Expand Up @@ -347,23 +343,22 @@ def product_on_basis(self, t1, t2):

EXAMPLES::

sage: A = AlgebrasWithBasis(QQ).example(); A # needs sage.combinat sage.modules
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,b,c) = A.algebra_generators() # needs sage.combinat sage.modules

sage: x = tensor((a, b, c)); x # needs sage.combinat sage.modules
sage: (a,b,c) = A.algebra_generators()
sage: x = tensor((a, b, c)); x
B[word: a] # B[word: b] # B[word: c]
sage: y = tensor((c, b, a)); y # needs sage.combinat sage.modules
sage: y = tensor((c, b, a)); y
B[word: c] # B[word: b] # B[word: a]
sage: x * y # needs sage.combinat sage.modules
sage: x * y
B[word: ac] # B[word: bb] # B[word: ca]

sage: x = tensor(((a + 2*b), c)); x # needs sage.combinat sage.modules
sage: x = tensor(((a + 2*b), c)); x
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
sage: y = tensor((c, a)) + 1; y
B[word: ] # B[word: ] + B[word: c] # B[word: a]
sage: x * y # needs sage.combinat sage.modules
sage: x * y
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]

Expand Down
6 changes: 3 additions & 3 deletions src/sage/categories/bialgebras.py
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Expand Up @@ -71,7 +71,7 @@ def is_primitive(self):

sage: # needs sage.modules
sage: s = SymmetricFunctions(QQ).schur()
sage: s([5]).is_primitive()
sage: s([5]).is_primitive() # needs lrcalc_python
False
sage: p = SymmetricFunctions(QQ).powersum()
sage: p([5]).is_primitive()
Expand All @@ -87,9 +87,9 @@ def is_grouplike(self):
EXAMPLES::

sage: s = SymmetricFunctions(QQ).schur() # needs sage.modules
sage: s([5]).is_grouplike() # needs sage.modules
sage: s([5]).is_grouplike() # needs lrcalc_python sage.modules
False
sage: s([]).is_grouplike() # needs sage.modules
sage: s([]).is_grouplike() # needs lrcalc_python sage.modules
True
"""
return self.coproduct() == self.tensor(self)
Expand Down
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