Allow GroupAlgebra to be used as base ring of a PolynomialRing

[Etn40ff]

The __init__ method of GroupAlgebra did not check for commutativity of the group so that this would happen:

sage: G = AbelianGroup(2)
sage: A = GroupAlgebra(G)
sage: A.is_commutative()
True
sage: A.is_ring()
True
sage: PolynomialRing(A,'x,y')

TypeError: The base ring Group algebra of group "Multiplicative Abelian group isomorphic to Z x Z " over base ring Integer Ring is not a commutative ring

Moreover the init function of PolynomialRing needs the method is_prime_field in order to work.

Component: group theory

Keywords: GroupAlgebra PolynomialRing

Author: Salvatore Stella

Branch/Commit: u/etn40ff/commutative_group_algebra @ 0936b72

Issue created by migration from https://trac.sagemath.org/ticket/19518

[Etn40ff]

Branch: u/etn40ff/commutative_group_algebra

[Etn40ff]

Commit: 491a113

[Etn40ff]

Changed keywords from none to GroupAlgebra PolynomialRing

[Etn40ff]

Author: Salvatore Stella

[Etn40ff]

Description changed:

--- 
+++ 
@@ -1 +1,17 @@
+The `__init__` method of GroupAlgebra did not check for commutativity of the group so that this would happen:

+```
+sage: G = AbelianGroup(2)
+sage: A = GroupAlgebra(G)
+sage: A.is_commutative()
+True
+sage: A.is_ring()
+True
+sage: PolynomialRing(A,'x,y')
+
+TypeError: The base ring Group algebra of group "Multiplicative Abelian group isomorphic to Z x Z " over base ring Integer Ring is not a commutative ring
+```
+
+Moreover the init function of PolynomialRing needs the method is_prime_field in
+order to work.
+

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 491a113 to c3ebef0

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

c3ebef0

Fixed typo

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Branch pushed to git repo; I updated commit sha1. New commits:

0936b72

Added missing :

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from c3ebef0 to 0936b72

[tscrim]

comment:5

This has uncovered a couple of different bugs:

1. We have an issue with translating to/from GAP:

sage: SU(2, 3).random_element()
[      1 2*a + 2]
[      0       1]
sage: SU(2, 4).random_element()
<repr(<sage.groups.matrix_gps.group_element.UnitaryMatrixGroup_gap_with_category.element_class at 0x7f826b84eaa0>) failed: ValueError: the given finite field has incompatible size>

2. Abelian groups .algebra aren't returning group algebras.
3. Commutative() does not commute with the Algebras functor construction.

sage: Groups().Commutative().Algebras(QQ)
Join of Category of group algebras over Rational Field and Category of commutative algebras over Rational Field and Category of commutative magma algebras over Rational Field
sage: Groups().Algebras(QQ).Commutative()
Category of commutative group algebras over Rational Field

The main problem for this ticket I think is the last one.

[cheuberg]

comment:6

doctests fail (cf. patchbot)