gh-40325: minor details in doc and code of algebras/
    
mostly about spaces around = in the doctests

also pep8-cleaning one of the modified files

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #40325
Reported by: Frédéric Chapoton
Reviewer(s): David Coudert

Author: Release Manager

src/sage/algebras/affine_nil_temperley_lieb.py

@@ -221,7 +221,7 @@ def has_no_braid_relation(self, w, i) -> bool:
 
             sage: A = AffineNilTemperleyLiebTypeA(5)
             sage: W = A.weyl_group()
-            sage: s=W.simple_reflections()
+            sage: s = W.simple_reflections()
             sage: A.has_no_braid_relation(s[2]*s[1]*s[0]*s[4]*s[3],0)
             False
             sage: A.has_no_braid_relation(s[2]*s[1]*s[0]*s[4]*s[3],2)

src/sage/algebras/free_algebra_element.py

@@ -18,8 +18,7 @@
     sage: (x*y)^3
     x*y*x*y*x*y
 """
-
-#*****************************************************************************
+# ***************************************************************************
 #  Copyright (C) 2005 David Kohel <[email protected]>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
@@ -31,8 +30,8 @@
 #  See the GNU General Public License for more details; the full text
 #  is available at:
 #
-#                  http://www.gnu.org/licenses/
-#*****************************************************************************
+#                  https://www.gnu.org/licenses/
+# ***************************************************************************
 
 from sage.misc.repr import repr_lincomb
 from sage.monoids.free_monoid_element import FreeMonoidElement
@@ -56,7 +55,7 @@ class FreeAlgebraElement(IndexedFreeModuleElement, AlgebraElement):
         sage: y * x < x * y
         False
     """
-    def __init__(self, A, x):
+    def __init__(self, A, x) -> None:
         """
         Create the element ``x`` of the FreeAlgebra ``A``.
 
@@ -68,7 +67,7 @@ def __init__(self, A, x):
         """
         if isinstance(x, FreeAlgebraElement):
             # We should have an input for when we know we don't need to
-            #   convert the keys/values
+            # convert the keys/values
             x = x._monomial_coefficients
         R = A.base_ring()
         if isinstance(x, AlgebraElement):  # and x.parent() == A.base_ring():
@@ -131,7 +130,7 @@ def __call__(self, *x, **kwds):
             7
             sage: (2*x+y).subs({x:1,y:z})
             2 + z
-            sage: f=x+3*y+z
+            sage: f = x+3*y+z
             sage: f(1,2,1/2)
             15/2
             sage: f(1,2)
@@ -157,7 +156,8 @@ def extract_from(kwds, g):
                         pass
                 return None
 
-            x = [extract_from(kwds,(p.gen(i),p.variable_name(i))) for i in range(p.ngens())]
+            x = [extract_from(kwds, (p.gen(i), p.variable_name(i)))
+                 for i in range(p.ngens())]
         elif isinstance(x[0], tuple):
             x = x[0]
 
@@ -169,18 +169,19 @@ def extract_from(kwds, g):
         result = None
         for m, c in self._monomial_coefficients.items():
             if result is None:
-                result = c*m(x)
+                result = c * m(x)
             else:
-                result += c*m(x)
+                result += c * m(x)
 
         if result is None:
             return self.parent().zero()
         return result
 
     def _mul_(self, y):
         """
-        Return the product of ``self`` and ``y`` (another free algebra
-        element with the same parent).
+        Return the product of ``self`` and ``y``.
+
+        This is another free algebra element with the same parent.
 
         EXAMPLES::
 
@@ -192,16 +193,16 @@ def _mul_(self, y):
         z_elt = {}
         for mx, cx in self:
             for my, cy in y:
-                key = mx*my
+                key = mx * my
                 if key in z_elt:
-                    z_elt[key] += cx*cy
+                    z_elt[key] += cx * cy
                 else:
-                    z_elt[key] = cx*cy
+                    z_elt[key] = cx * cy
                 if not z_elt[key]:
                     del z_elt[key]
         return A._from_dict(z_elt)
 
-    def is_unit(self):
+    def is_unit(self) -> bool:
         r"""
         Return ``True`` if ``self`` is invertible.
 
@@ -274,10 +275,6 @@ def _acted_upon_(self, scalar, self_on_left=False):
             return scalar * Factorization([(self, 1)])
         return super()._acted_upon_(scalar, self_on_left)
 
-    # For backward compatibility
-    # _lmul_ = _acted_upon_
-    # _rmul_ = _acted_upon_
-
     def _im_gens_(self, codomain, im_gens, base_map):
         """
         Apply a morphism defined by its values on the generators.
@@ -304,7 +301,7 @@ def _im_gens_(self, codomain, im_gens, base_map):
         return codomain.sum(base_map(c) * m(*im_gens)
                             for m, c in self._monomial_coefficients.items())
 
-    def variables(self):
+    def variables(self) -> list:
         """
         Return the variables used in ``self``.
 

src/sage/algebras/fusion_rings/fusion_double.py

@@ -286,7 +286,7 @@ def s_ijconj(self, i, j, unitary=False, base_coercion=True):
 
         EXAMPLES::
 
-            sage: P=FusionDouble(CyclicPermutationGroup(3),prefix='p',inject_variables=True)
+            sage: P = FusionDouble(CyclicPermutationGroup(3),prefix='p',inject_variables=True)
             sage: P.s_ij(p1,p3)
             zeta3
             sage: P.s_ijconj(p1,p3)
@@ -695,7 +695,7 @@ def product_on_basis(self, a, b):
 
         EXAMPLES::
 
-            sage: Q=FusionDouble(SymmetricGroup(3),prefix='q',inject_variables=True)
+            sage: Q = FusionDouble(SymmetricGroup(3),prefix='q',inject_variables=True)
             sage: q3*q4
             q1 + q2 + q5 + q6 + q7
             sage: Q._names
@@ -820,7 +820,7 @@ def twist(self, reduced=True):
 
             EXAMPLES::
 
-                sage: Q=FusionDouble(CyclicPermutationGroup(3))
+                sage: Q = FusionDouble(CyclicPermutationGroup(3))
                 sage: [x.twist() for x in Q.basis()]
                 [0, 0, 0, 0, 2/3, 4/3, 0, 4/3, 2/3]
                 sage: [x.ribbon() for x in Q.basis()]

src/sage/algebras/iwahori_hecke_algebra.py

@@ -233,7 +233,7 @@ class IwahoriHeckeAlgebra(Parent, UniqueRepresentation):
 
         sage: R.<q> = LaurentPolynomialRing(ZZ)
         sage: H = IwahoriHeckeAlgebra('A3', q^2)
-        sage: T=H.T(); Cp=H.Cp(); C=H.C()
+        sage: T = H.T(); Cp = H.Cp(); C = H.C()
         sage: C(T[1])
         q*C[1] + q^2
         sage: elt = Cp(T[1,2,1]); elt
@@ -245,7 +245,7 @@ class IwahoriHeckeAlgebra(Parent, UniqueRepresentation):
 
         sage: R.<q> = LaurentPolynomialRing(ZZ)
         sage: H = IwahoriHeckeAlgebra('A3', q, -q^-1)
-        sage: T=H.T(); Cp=H.Cp(); C=H.C()
+        sage: T = H.T(); Cp = H.Cp(); C = H.C()
         sage: C(T[1])
         C[1] + q
         sage: elt = Cp(T[1,2,1]); elt
@@ -256,7 +256,7 @@ class IwahoriHeckeAlgebra(Parent, UniqueRepresentation):
     In the group algebra, so that `(T_r-1)(T_r+1) = 0`::
 
         sage: H = IwahoriHeckeAlgebra('A3', 1)
-        sage: T=H.T(); Cp=H.Cp(); C=H.C()
+        sage: T = H.T(); Cp = H.Cp(); C = H.C()
         sage: C(T[1])
         C[1] + 1
         sage: Cp(T[1,2,1])
@@ -270,7 +270,7 @@ class IwahoriHeckeAlgebra(Parent, UniqueRepresentation):
 
         sage: R.<q>=LaurentPolynomialRing(ZZ)
         sage: H = IwahoriHeckeAlgebra('A3', q)
-        sage: C=H.C()
+        sage: C = H.C()
         Traceback (most recent call last):
         ...
         ValueError: the Kazhdan-Lusztig bases are defined only when -q_1*q_2 is a square
@@ -279,7 +279,7 @@ class IwahoriHeckeAlgebra(Parent, UniqueRepresentation):
 
         sage: R.<v> = LaurentPolynomialRing(ZZ)
         sage: H = IwahoriHeckeAlgebra(['A',2,1], v^2)
-        sage: T=H.T(); Cp=H.Cp(); C=H.C()
+        sage: T = H.T(); Cp = H.Cp(); C = H.C()
         sage: C(T[1,0,2])
         v^3*C[1,0,2] + v^4*C[1,0] + v^4*C[0,2] + v^4*C[1,2]
          + v^5*C[0] + v^5*C[2] + v^5*C[1] + v^6
@@ -1621,8 +1621,8 @@ def hash_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: T=H.T()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: T = H.T()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: T.hash_involution_on_basis(s)
                 -(v^-2)*T[1]
                 sage: T[s].hash_involution()
@@ -1661,8 +1661,8 @@ def goldman_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: T=H.T()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: T = H.T()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: T.goldman_involution_on_basis(s)
                 -T[1] - (1-v^2)
                 sage: T[s].goldman_involution()
@@ -1836,8 +1836,8 @@ def to_T_basis(self, w):
 
             EXAMPLES::
 
-                sage: H=IwahoriHeckeAlgebra("A3",1); Cp=H.Cp(); C=H.C()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: H = IwahoriHeckeAlgebra("A3",1); Cp = H.Cp(); C = H.C()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: C.to_T_basis(s)
                 T[1] - 1
                 sage: Cp.to_T_basis(s)
@@ -2032,8 +2032,8 @@ def hash_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: Cp=H.Cp()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: Cp = H.Cp()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: Cp.hash_involution_on_basis(s)
                 -Cp[1] + (v^-1+v)
                 sage: Cp[s].hash_involution()
@@ -2130,7 +2130,7 @@ def product_on_basis(self, w1, w2):
                 sage: # optional - coxeter3
                 sage: R.<v> = LaurentPolynomialRing(ZZ, 'v')
                 sage: W = CoxeterGroup('A3', implementation='coxeter3')
-                sage: H = IwahoriHeckeAlgebra(W, v**2); Cp=H.Cp()
+                sage: H = IwahoriHeckeAlgebra(W, v**2); Cp = H.Cp()
                 sage: Cp.product_on_basis(W([1,2,1]), W([3,1]))
                 (v^-1+v)*Cp[1,2,1,3]
                 sage: Cp.product_on_basis(W([1,2,1]), W([3,1,2]))
@@ -2267,7 +2267,7 @@ def _decompose_into_generators(self, u):
 
                 sage: R.<v> = LaurentPolynomialRing(ZZ, 'v')             # optional - coxeter3
                 sage: W = CoxeterGroup('A3', implementation='coxeter3')  # optional - coxeter3
-                sage: H = IwahoriHeckeAlgebra(W, v**2); Cp=H.Cp()        # optional - coxeter3
+                sage: H = IwahoriHeckeAlgebra(W, v**2); Cp = H.Cp()      # optional - coxeter3
 
             When `u` is itself a generator `s`, the decomposition is trivial::
 
@@ -2440,8 +2440,8 @@ def hash_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: C=H.C()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: C = H.C()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: C.hash_involution_on_basis(s)
                 -C[1] - (v^-1+v)
                 sage: C[s].hash_involution()
@@ -2473,7 +2473,7 @@ class A(_Basis):
 
             sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
             sage: H = IwahoriHeckeAlgebra('A3', v**2)
-            sage: A=H.A(); T=H.T()
+            sage: A = H.A(); T = H.T()
             sage: T(A[1])
             T[1] + (1/2-1/2*v^2)
             sage: T(A[1,2])
@@ -2526,8 +2526,8 @@ def to_T_basis(self, w):
             EXAMPLES::
 
                 sage: R.<v> = LaurentPolynomialRing(QQ)
-                sage: H = IwahoriHeckeAlgebra('A3', v**2); A=H.A(); T=H.T()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: H = IwahoriHeckeAlgebra('A3', v**2); A = H.A(); T = H.T()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: A.to_T_basis(s)
                 T[1] + (1/2-1/2*v^2)
                 sage: T(A[1,2])
@@ -2550,8 +2550,8 @@ def goldman_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: A=H.A()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: A = H.A()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: A.goldman_involution_on_basis(s)
                 -A[1]
                 sage: A[1,2].goldman_involution()
@@ -2593,7 +2593,7 @@ class B(_Basis):
 
             sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
             sage: H = IwahoriHeckeAlgebra('A3', v**2)
-            sage: A=H.A(); T=H.T(); Cp=H.Cp()
+            sage: A = H.A(); T = H.T(); Cp = H.Cp()
             sage: T(A[1])
             T[1] + (1/2-1/2*v^2)
             sage: T(A[1,2])
@@ -2659,8 +2659,8 @@ def to_T_basis(self, w):
             EXAMPLES::
 
                 sage: R.<v> = LaurentPolynomialRing(QQ)
-                sage: H = IwahoriHeckeAlgebra('A3', v**2); B=H.B(); T=H.T()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: H = IwahoriHeckeAlgebra('A3', v**2); B = H.B(); T = H.T()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: B.to_T_basis(s)
                 T[1] + (1/2-1/2*v^2)
                 sage: T(B[1,2])
@@ -2687,8 +2687,8 @@ def goldman_involution_on_basis(self, w):
 
                 sage: R.<v> = LaurentPolynomialRing(QQ, 'v')
                 sage: H = IwahoriHeckeAlgebra('A3', v**2)
-                sage: B=H.B()
-                sage: s=H.coxeter_group().simple_reflection(1)
+                sage: B = H.B()
+                sage: s = H.coxeter_group().simple_reflection(1)
                 sage: B.goldman_involution_on_basis(s)
                 -B[1]
                 sage: B[1,2].goldman_involution()
@@ -2822,8 +2822,8 @@ def _bar_on_coefficients(self, c):
         EXAMPLES::
 
             sage: R.<q>=LaurentPolynomialRing(ZZ)
-            sage: H=IwahoriHeckeAlgebra("A3",q^2)
-            sage: GH=H._generic_iwahori_hecke_algebra
+            sage: H = IwahoriHeckeAlgebra("A3",q^2)
+            sage: GH = H._generic_iwahori_hecke_algebra
             sage: GH._bar_on_coefficients(GH.u_inv)
             u
             sage: GH._bar_on_coefficients(GH.v_inv)
@@ -2871,8 +2871,8 @@ def specialize_to(self, new_hecke):
                 EXAMPLES::
 
                     sage: R.<a,b>=LaurentPolynomialRing(ZZ,2)
-                    sage: H=IwahoriHeckeAlgebra("A3",a^2,-b^2)
-                    sage: GH=H._generic_iwahori_hecke_algebra
+                    sage: H = IwahoriHeckeAlgebra("A3",a^2,-b^2)
+                    sage: GH = H._generic_iwahori_hecke_algebra
                     sage: GH.T()(GH.C()[1])
                     (v^-1)*T[1] + (-u*v^-1)
                     sage: ( GH.T()(GH.C()[1]) ).specialize_to(H)

src/sage/algebras/steenrod/steenrod_algebra.py

@@ -2966,7 +2966,7 @@ def top_class(self):
 
         TESTS::
 
-            sage: A=SteenrodAlgebra(2, profile=(3,2,1), basis='pst')
+            sage: A = SteenrodAlgebra(2, profile=(3,2,1), basis='pst')
             sage: A.top_class().parent() is A
             True
         """