gh-38382: refining the category of all-commuting g-algebras

fixing #38047

### 📝 Checklist

- [x] The title is concise and informative.
- [ ] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [ ] I have updated the documentation and checked the documentation
preview.

URL: #38382
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,3 +1,3 @@
 tarball=configure-VERSION.tar.gz
-sha1=282c4b485888da0c9e2ce72c1c2b08f2df47480b
-sha256=ddb46580851222af6d0f3bc8f997c47792cccb5a875dd22d130bb5b1acb8731e
+sha1=7b7c79fc15432a1cc3d08c2f2fc610391970a727
+sha256=732dc8f003f58952b811302a48ad743dab86208d236df8843fd6b904c46aa474

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-3c279ec5712e0fa35c5733e03e010970727d7189
+46d6ebe05583047bc9cc89904af075483e78fdae

src/sage/algebras/free_algebra.py

@@ -915,7 +915,7 @@ def g_algebra(self, relations, names=None, order='degrevlex', check=True):
         """
         The `G`-Algebra derived from this algebra by relations.
 
-        By default is assumed, that two variables commute.
+        By default it is assumed that any two variables commute.
 
         .. TODO::
 
@@ -958,6 +958,7 @@ def g_algebra(self, relations, names=None, order='degrevlex', check=True):
             (-t)*x*y + t*y + (t + 1)
         """
         from sage.matrix.constructor import Matrix
+        commutative = not relations
 
         base_ring = self.base_ring()
         polynomial_ring = PolynomialRing(base_ring, self.gens())
@@ -993,7 +994,7 @@ def g_algebra(self, relations, names=None, order='degrevlex', check=True):
         from sage.rings.polynomial.plural import g_Algebra
         return g_Algebra(base_ring, cmat, dmat,
                          names=names or self.variable_names(),
-                         order=order, check=check)
+                         order=order, check=check, commutative=commutative)
 
     def poincare_birkhoff_witt_basis(self):
         """

src/sage/rings/polynomial/plural.pyx

@@ -157,7 +157,7 @@ class G_AlgFactory(UniqueFactory):
         - ``key`` -- a 6-tuple, formed by a base ring, a tuple of names, two
           matrices over a polynomial ring over the base ring with the given
           variable names, a term order, and a category
-        - ``extra_args`` -- a dictionary, whose only relevant key is 'check'.
+        - ``extra_args`` -- a dictionary, whose only relevant key is 'check'
 
         TESTS::
 
@@ -174,7 +174,7 @@ class G_AlgFactory(UniqueFactory):
                                        category, check)
 
     def create_key_and_extra_args(self, base_ring, c, d, names=None, order=None,
-                                  category=None, check=None):
+                                  category=None, check=None, commutative=None):
         """
         Create a unique key for g-algebras.
 
@@ -186,13 +186,18 @@ class G_AlgFactory(UniqueFactory):
         - ``order`` -- (optional) term order
         - ``category`` -- (optional) category
         - ``check`` -- optional bool
+        - ``commutative`` -- optional bool
 
         TESTS::
 
             sage: A.<x,y,z> = FreeAlgebra(QQ, 3)
             sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y})
             sage: H is A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) # indirect doctest
             True
+
+            sage: P = A.g_algebra(relations={}, order='lex')
+            sage: P.category()
+            Category of commutative algebras over Rational Field
         """
         if names is None:
             raise ValueError("The generator names must be provided")
@@ -213,7 +218,11 @@ class G_AlgFactory(UniqueFactory):
         d.set_immutable()
 
         # Get the correct category
-        category = check_default_category(Algebras(base_ring), category)
+        if commutative:
+            usualcat = Algebras(base_ring).Commutative()
+        else:
+            usualcat = Algebras(base_ring)
+        category = check_default_category(usualcat, category)
 
         # Extra arg
         if check is None:

src/sage/structure/factory.pyx

@@ -171,9 +171,9 @@ cdef class UniqueFactory(SageObject):
         ....:         return args, {'impl':kwds.get('impl', None)}
         ....:     def create_object(self, version, key, **extra_args):
         ....:         impl = extra_args['impl']
-        ....:         if impl=='C':
+        ....:         if impl == 'C':
         ....:             return C(*key)
-        ....:         if impl=='D':
+        ....:         if impl == 'D':
         ....:             return D(*key)
         ....:         return E(*key)
         ....: