112112from sage .misc .cachefunc import cached_method
113113from sage .categories .pushout import ConstructionFunctor
114114from sage .combinat .free_module import CombinatorialFreeModule
115- from sage .categories .all import Rings , HopfAlgebrasWithBasis , Hom
115+ from sage .categories .all import Rings , GroupAlgebras , Hom
116116from sage .categories .morphism import SetMorphism
117117
118118
@@ -133,7 +133,7 @@ class GroupAlgebraFunctor (ConstructionFunctor) :
133133 sage: loads(dumps(F)) == F
134134 True
135135 sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
136- Category of hopf algebras with basis over Ring of integers modulo 12
136+ Category of group algebras over Ring of integers modulo 12
137137 """
138138 def __init__ (self , group ) :
139139 r"""
@@ -143,7 +143,7 @@ def __init__(self, group) :
143143
144144 sage: from sage.algebras.group_algebra_new import GroupAlgebraFunctor
145145 sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
146- Category of hopf algebras with basis over Ring of integers modulo 12
146+ Category of group algebras over Ring of integers modulo 12
147147 """
148148 self .__group = group
149149
@@ -206,7 +206,7 @@ def _apply_functor_to_morphism(self, f) :
206206 codomain = self (f .codomain ())
207207 return SetMorphism (Hom (self (f .domain ()), codomain , Rings ()), lambda x : sum (codomain (g ) * f (c ) for (g , c ) in dict (x ).iteritems ()))
208208
209- class GroupAlgebra (CombinatorialFreeModule , Algebra ):
209+ class GroupAlgebra (CombinatorialFreeModule ):
210210 r"""
211211 Create the given group algebra.
212212
@@ -231,10 +231,51 @@ class GroupAlgebra(CombinatorialFreeModule, Algebra):
231231 TypeError: "1" is not a group
232232
233233 sage: GroupAlgebra(SU(2, GF(4, 'a')), IntegerModRing(12)).category()
234- Category of hopf algebras with basis over Ring of integers modulo 12
234+ Category of group algebras over Ring of integers modulo 12
235235 sage: GroupAlgebra(KleinFourGroup()) is GroupAlgebra(KleinFourGroup())
236236 True
237237
238+ The one of the group indexes the one of this algebra::
239+
240+ sage: A = GroupAlgebra(DihedralGroup(6), QQ)
241+ sage: A.one_basis()
242+ ()
243+ sage: A.one()
244+ ()
245+
246+ The product of two basis elements is induced by the product of the
247+ corresponding elements of the group::
248+
249+ sage: A = GroupAlgebra(DihedralGroup(3), QQ)
250+ sage: (a, b) = A._group.gens()
251+ sage: a*b
252+ (1,2)
253+ sage: A.product_on_basis(a, b)
254+ (1,2)
255+
256+ The basis elements are group-like for the coproduct:
257+ `\Delta(g) = g \otimes g`::
258+
259+ sage: A = GroupAlgebra(DihedralGroup(3), QQ)
260+ sage: (a, b) = A._group.gens()
261+ sage: A.coproduct_on_basis(a)
262+ (1,2,3) # (1,2,3)
263+
264+ The counit on the basis elements is 1::
265+
266+ sage: A = GroupAlgebra(DihedralGroup(6), QQ)
267+ sage: (a, b) = A._group.gens()
268+ sage: A.counit_on_basis(a)
269+ 1
270+
271+ The antipode on basis elements is given by `\chi(g) = g^{-1}`::
272+
273+ sage: A = GroupAlgebra(DihedralGroup(3), QQ)
274+ sage: (a, b) = A._group.gens(); a
275+ (1,2,3)
276+ sage: A.antipode_on_basis(a)
277+ (1,3,2)
278+
238279 TESTS::
239280
240281 sage: A = GroupAlgebra(GL(3, GF(7)))
@@ -266,56 +307,17 @@ def __init__(self, group, base_ring=IntegerRing()):
266307 CombinatorialFreeModule .__init__ (self , base_ring , group ,
267308 prefix = '' ,
268309 bracket = False ,
269- category = HopfAlgebrasWithBasis (base_ring ))
310+ category = GroupAlgebras (base_ring ))
270311
271312 if not base_ring .has_coerce_map_from (group ) :
272313 ## some matrix groups assume that coercion is only valid to
273314 ## other matrix groups. This is a workaround
274315 ## call _element_constructor_ to coerce group elements
275316 #try :
276- self ._populate_coercion_lists_ (coerce_list = [base_ring , group ])
277- #except TypeError :
278- # self._populate_coercion_lists_( coerce_list = [base_ring] )
279- else :
280- self ._populate_coercion_lists_ (coerce_list = [base_ring ])
317+ self ._populate_coercion_lists_ (coerce_list = [group ])
281318
282319 # Methods taken from sage.categories.examples.hopf_algebras_with_basis:
283320
284- @cached_method
285- def one_basis (self ):
286- """
287- Returns the one of the group, which indexes the one of this
288- algebra, as per :meth:`AlgebrasWithBasis.ParentMethods.one_basis`.
289-
290- EXAMPLES::
291-
292- sage: A = GroupAlgebra(DihedralGroup(6), QQ)
293- sage: A.one_basis()
294- ()
295- sage: A.one()
296- ()
297- """
298- return self ._group .one ()
299-
300- def product_on_basis (self , g1 , g2 ):
301- r"""
302- Product, on basis elements, as per
303- :meth:`AlgebrasWithBasis.ParentMethods.product_on_basis`.
304-
305- The product of two basis elements is induced by the product of
306- the corresponding elements of the group.
307-
308- EXAMPLES::
309-
310- sage: A = GroupAlgebra(DihedralGroup(3), QQ)
311- sage: (a, b) = A._group.gens()
312- sage: a*b
313- (1,2)
314- sage: A.product_on_basis(a, b)
315- (1,2)
316- """
317- return self .basis ()[g1 * g2 ]
318-
319321 @cached_method
320322 def algebra_generators (self ):
321323 r"""
@@ -336,53 +338,6 @@ def algebra_generators(self):
336338
337339 gens = algebra_generators
338340
339- def coproduct_on_basis (self , g ):
340- r"""
341- Coproduct, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.coproduct_on_basis`.
342-
343- The basis elements are group-like: `\Delta(g) = g \otimes g`.
344-
345- EXAMPLES::
346-
347- sage: A = GroupAlgebra(DihedralGroup(3), QQ)
348- sage: (a, b) = A._group.gens()
349- sage: A.coproduct_on_basis(a)
350- (1,2,3) # (1,2,3)
351- """
352- from sage .categories .all import tensor
353- g = self .monomial (g )
354- return tensor ([g , g ])
355-
356- def counit_on_basis (self , g ):
357- r"""
358- Counit, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.counit_on_basis`.
359-
360- The counit on the basis elements is 1.
361-
362- EXAMPLES::
363-
364- sage: A = GroupAlgebra(DihedralGroup(6), QQ)
365- sage: (a, b) = A._group.gens()
366- sage: A.counit_on_basis(a)
367- 1
368- """
369- return self .base_ring ().one ()
370-
371- def antipode_on_basis (self , g ):
372- r"""
373- Antipode, on basis elements, as per :meth:`HopfAlgebrasWithBasis.ParentMethods.antipode_on_basis`.
374-
375- It is given, on basis elements, by `\chi(g) = g^{-1}`
376-
377- EXAMPLES::
378-
379- sage: A = GroupAlgebra(DihedralGroup(3), QQ)
380- sage: (a, b) = A._group.gens()
381- sage: A.antipode_on_basis(a)
382- (1,3,2)
383- """
384- return self .monomial (~ g )
385-
386341 # other methods:
387342
388343 def ngens (self ) :
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