sagemathgh-37984: Ruff details in algebras and categories
    
various small fixes for ruff suggestions in algebras and categories

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#37984
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

Author: Release Manager

src/sage/algebras/cluster_algebra.py

@@ -1769,7 +1769,7 @@ def d_vector_to_g_vector(self, d) -> tuple:
             sage: A.d_vector_to_g_vector((1,0,-1))
             (-1, 1, 2)
         """
-        dm = vector((x if x < 0 else 0 for x in d))
+        dm = vector(x if x < 0 else 0 for x in d)
         dp = vector(d) - dm
         return tuple(- dm - self.euler_matrix() * dp)
 

src/sage/algebras/fusion_rings/f_matrix.py

@@ -41,6 +41,7 @@
 from sage.rings.polynomial.polydict import ETuple
 from sage.rings.qqbar import AA, QQbar, number_field_elements_from_algebraics
 
+
 class FMatrix(SageObject):
     r"""
     An F-matrix for a :class:`FusionRing`.
@@ -297,7 +298,7 @@ def __init__(self, fusion_ring, fusion_label="f", var_prefix='fx', inject_variab
         self.pool = None
 
     #######################
-    ### Class utilities ###
+    #   Class utilities   #
     #######################
 
     def _repr_(self):
@@ -434,35 +435,31 @@ def fmat(self, a, b, c, d, x, y, data=True):
              (-zeta60^14 + zeta60^6 + zeta60^4 - 1),
              (zeta60^14 - zeta60^6 - zeta60^4 + 1)]
         """
-        if (self._FR.Nk_ij(a, b, x) == 0 or self._FR.Nk_ij(x, c, d) == 0
-            or self._FR.Nk_ij(b, c, y) == 0 or self._FR.Nk_ij(a, y, d) == 0):
+        if (self._FR.Nk_ij(a, b, x) == 0
+                or self._FR.Nk_ij(x, c, d) == 0
+                or self._FR.Nk_ij(b, c, y) == 0
+                or self._FR.Nk_ij(a, y, d) == 0):
             return 0
 
         # Some known zero F-symbols
         if a == self._FR.one():
-            if x == b and y == d:
-                return 1
-            else:
-                return 0
+            return 1 if x == b and y == d else 0
+
         if b == self._FR.one():
-            if x == a and y == c:
-                return 1
-            else:
-                return 0
+            return 1 if x == a and y == c else 0
+
         if c == self._FR.one():
-            if x == d and y == b:
-                return 1
-            else:
-                return 0
+            return 1 if x == d and y == b else 0
+
         if data:
             # Better to use try/except for speed. Somewhat trivial, but worth
             # hours when method is called ~10^11 times
             try:
                 return self._fvars[a, b, c, d, x, y]
             except KeyError:
                 return 0
-        else:
-            return (a, b, c, d, x, y)
+
+        return (a, b, c, d, x, y)
 
     def fmatrix(self, a, b, c, d):
         r"""
@@ -583,7 +580,7 @@ def findcases(self, output=False):
             idx_map = {}
             ret = {}
         id_anyon = self._FR.one()
-        for (a, b, c, d) in product(self._FR.basis(), repeat=4):
+        for a, b, c, d in product(self._FR.basis(), repeat=4):
             if a == id_anyon or b == id_anyon or c == id_anyon:
                 continue
             for x in self.f_from(a, b, c, d):
@@ -593,10 +590,8 @@ def findcases(self, output=False):
                         ret[(a, b, c, d, x, y)] = v
                         idx_map[i] = (a, b, c, d, x, y)
                     i += 1
-        if output:
-            return idx_map, ret
-        else:
-            return i
+
+        return (idx_map, ret) if output else i
 
     def f_from(self, a, b, c, d):
         r"""
@@ -649,7 +644,7 @@ def f_to(self, a, b, c, d):
                 if self._FR.Nk_ij(b, c, y) != 0 and self._FR.Nk_ij(a, y, d) != 0]
 
     ####################
-    ### Data getters ###
+    #   Data getters   #
     ####################
 
     def get_fvars(self):
@@ -855,7 +850,7 @@ def get_radical_expression(self):
         return {sextuple: val.radical_expression() for sextuple, val in self.get_fvars_in_alg_field().items()}
 
     #######################
-    ### Private helpers ###
+    #   Private helpers   #
     #######################
 
     def _get_known_vals(self):
@@ -894,12 +889,12 @@ def _get_known_nonz(self):
              100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100)
         """
         nonz = {idx: 100 for idx in self._singles}
-        for idx, v in self._ks.items():
+        for idx, _ in self._ks.items():
             nonz[idx] = 100
         return ETuple(nonz, self._poly_ring.ngens())
 
     ##############################
-    ### Variables partitioning ###
+    #   Variables partitioning   #
     ##############################
 
     def largest_fmat_size(self):
@@ -967,7 +962,7 @@ def get_fvars_by_size(self, n, indices=False):
         return var_set
 
     ############################
-    ### Checkpoint utilities ###
+    #   Checkpoint utilities   #
     ############################
 
     def save_fvars(self, filename):
@@ -1009,12 +1004,10 @@ def save_fvars(self, filename):
             True
             sage: os.remove(filename)
         """
-        final_state = [
-            self._fvars,
-            self._non_cyc_roots,
-            self.get_coerce_map_from_fr_cyclotomic_field(),
-            self._qqbar_embedding,
-            ]
+        final_state = [self._fvars,
+                       self._non_cyc_roots,
+                       self.get_coerce_map_from_fr_cyclotomic_field(),
+                       self._qqbar_embedding]
         with open(filename, 'wb') as f:
             pickle.dump(final_state, f)
 
@@ -1212,7 +1205,7 @@ def _restore_state(self, filename):
         self._update_reduction_params()
 
     #################
-    ### MapReduce ###
+    #   MapReduce   #
     #################
 
     def start_worker_pool(self, processes=None):
@@ -1275,7 +1268,7 @@ class methods.
             self._reset_solver_state()
         # Set up shared memory resource handlers
         n_proc = cpu_count() if processes is None else processes
-        self._pid_list = shared_memory.ShareableList([0]*(n_proc+1))
+        self._pid_list = shared_memory.ShareableList([0] * (n_proc+1))
         pids_name = self._pid_list.shm.name
         self._solved = shared_memory.ShareableList(self._solved)
         s_name = self._solved.shm.name
@@ -1304,7 +1297,7 @@ def init(fmats_id, solved_name, vd_name, ks_names, fvar_names, n_proc, pids_name
         self.pool = Pool(processes=n_proc, initializer=init, initargs=args)
         self._pid_list[0] = getpid()
         for i, p in enumerate(self.pool._pool):
-            self._pid_list[i+1] = p.pid
+            self._pid_list[i + 1] = p.pid
         # return True
 
     def shutdown_worker_pool(self):
@@ -1394,7 +1387,7 @@ def _map_triv_reduce(self, mapper, input_iter, worker_pool=None, chunksize=None,
         return results
 
     ########################
-    ### Equations set up ###
+    #   Equations set up   #
     ########################
 
     def get_orthogonality_constraints(self, output=True):
@@ -1494,7 +1487,9 @@ def get_defining_equations(self, option, output=True):
             self._reset_solver_state()
         n_proc = self.pool._processes if self.pool is not None else 1
         params = [(child_id, n_proc, output) for child_id in range(n_proc)]
-        eqns = self._map_triv_reduce('get_reduced_'+option, params, worker_pool=self.pool, chunksize=1, mp_thresh=0)
+        eqns = self._map_triv_reduce('get_reduced_' + option, params,
+                                     worker_pool=self.pool, chunksize=1,
+                                     mp_thresh=0)
         if output:
             F = self._field
             for i, eq_tup in enumerate(eqns):
@@ -1503,7 +1498,7 @@ def get_defining_equations(self, option, output=True):
         self.ideal_basis.extend(eqns)
 
     ############################
-    ### Equations processing ###
+    #   Equations processing   #
     ############################
 
     def _tup_to_fpoly(self, eq_tup):
@@ -1611,7 +1606,7 @@ def _triangular_elim(self, eqns=None, verbose=True):
         self.ideal_basis = eqns
 
     #####################
-    ### Graph methods ###
+    #   Graph methods   #
     #####################
 
     def equations_graph(self, eqns=None):
@@ -1820,7 +1815,7 @@ def _get_component_variety(self, var, eqns):
         return [{inv_idx_map[i]: value for i, (key, value) in enumerate(sorted(soln.items()))} for soln in var_in_R]
 
     #######################
-    ### Solution method ###
+    #   Solution method   #
     #######################
 
     # TODO: this can probably be improved by constructing a set of defining polynomials
@@ -1919,7 +1914,7 @@ def _get_explicit_solution(self, eqns=None, verbose=True):
             for fx, rhs in self._ks.items():
                 if not self._solved[fx]:
                     lt = (ETuple({fx: 2}, n), one)
-                    eqns.append(((lt, (ETuple({}, n), -rhs))))
+                    eqns.append((lt, (ETuple({}, n), -rhs)))
         eqns_partition = self._partition_eqns(verbose=verbose)
 
         F = self._field
@@ -1956,20 +1951,20 @@ def _get_explicit_solution(self, eqns=None, verbose=True):
             if self.attempt_number_field_computation():
                 if verbose:
                     print("Computing appropriate NumberField...")
-                roots = [self._FR.field().gen()]+[r[1] for r in non_cyclotomic_roots]
+                roots = [self._FR.field().gen()] + [r[1] for r in non_cyclotomic_roots]
                 self._field, bf_elts, self._qqbar_embedding = number_field_elements_from_algebraics(roots, minimal=True)
             else:
                 self._field = QQbar
                 bf_elts = [self._qqbar_embedding(F.gen())]
                 bf_elts += [rhs for fx, rhs in non_cyclotomic_roots]
-                self._qqbar_embedding = lambda x : x
+                self._qqbar_embedding = lambda x: x
             self._non_cyc_roots = bf_elts[1:]
 
             # Embed cyclotomic field into newly constructed base field
             cyc_gen_as_bf_elt = bf_elts.pop(0)
             phi = self._FR.field().hom([cyc_gen_as_bf_elt], self._field)
             self._coerce_map_from_cyc_field = phi
-            numeric_fvars = {k : phi(v) for k, v in numeric_fvars.items()}
+            numeric_fvars = {k: phi(v) for k, v in numeric_fvars.items()}
             for i, elt in enumerate(bf_elts):
                 numeric_fvars[non_cyclotomic_roots[i][0]] = elt
             # Update polynomial ring
@@ -2114,7 +2109,7 @@ def find_orthogonal_solution(self, checkpoint=False, save_results="", warm_start
                 print("Set up {} hex and orthogonality constraints...".format(len(self.ideal_basis)))
 
         # Unzip _fvars and link to shared_memory structure if using multiprocessing
-        if use_mp:# and loads_shared_memory:
+        if use_mp:  # and loads_shared_memory:
             self._fvars = self._shared_fvars
         else:
             n = self._poly_ring.ngens()
@@ -2164,12 +2159,12 @@ def find_orthogonal_solution(self, checkpoint=False, save_results="", warm_start
         self._chkpt_status = 7
         self.clear_equations()
         if checkpoint:
-            remove("fmatrix_solver_checkpoint_"+self.get_fr_str()+".pickle")
+            remove("fmatrix_solver_checkpoint_" + self.get_fr_str() + ".pickle")
         if save_results:
             self.save_fvars(save_results)
 
     #########################
-    ### Cyclotomic method ###
+    #   Cyclotomic method   #
     #########################
 
     def _fix_gauge(self, algorithm=""):
@@ -2202,8 +2197,8 @@ def _fix_gauge(self, algorithm=""):
                     break
 
             # Fix var = 1, substitute, and solve equations
-            self.ideal_basis.add(var-1)
-            print("adding equation...", var-1)
+            self.ideal_basis.add(var - 1)
+            print("adding equation...", var - 1)
             self.ideal_basis = set(Ideal(list(self.ideal_basis)).groebner_basis(algorithm=algorithm))
             self._substitute_degree_one()
             self._update_equations()
@@ -2336,7 +2331,7 @@ def find_cyclotomic_solution(self, equations=None, algorithm="", verbose=True, o
         if equations is None:
             if verbose:
                 print("Setting up hexagons and pentagons...")
-            equations = self.get_defining_equations("hexagons")+self.get_defining_equations("pentagons")
+            equations = self.get_defining_equations("hexagons") + self.get_defining_equations("pentagons")
         if verbose:
             print("Finding a Groebner basis...")
         self.ideal_basis = set(Ideal(equations).groebner_basis(algorithm=algorithm))
@@ -2352,7 +2347,7 @@ def find_cyclotomic_solution(self, equations=None, algorithm="", verbose=True, o
             return self._fvars
 
     #####################
-    ### Verifications ###
+    #   Verifications   #
     #####################
 
     def fmats_are_orthogonal(self):

src/sage/algebras/lie_algebras/classical_lie_algebra.py

@@ -447,8 +447,8 @@ def __init__(self, R, n):
         gens = []
         for i in range(n):
             for j in range(n):
-                names.append('E_{0}_{1}'.format(i,j))
-                mat = MS({(i,j):one})
+                names.append('E_{}_{}'.format(i, j))
+                mat = MS({(i, j): one})
                 mat.set_immutable()
                 gens.append(mat)
         self._n = n

src/sage/algebras/lie_algebras/free_lie_algebra.py

@@ -80,7 +80,7 @@ def _repr_(self):
             sage: L.Lyndon()
             Free Lie algebra generated by (x, y) over Rational Field in the Lyndon basis
         """
-        return "{0} in the {1} basis".format(self.realization_of(), self._basis_name)
+        return "{} in the {} basis".format(self.realization_of(), self._basis_name)
 
     def _repr_term(self, x):
         """

src/sage/algebras/lie_algebras/heisenberg.py

@@ -194,7 +194,8 @@ def step(self):
     class Element(LieAlgebraElement):
         pass
 
-class HeisenbergAlgebra_fd():
+
+class HeisenbergAlgebra_fd:
     """
     Common methods for finite-dimensional Heisenberg algebras.
     """
@@ -417,7 +418,8 @@ def _repr_(self):
             sage: lie_algebras.Heisenberg(QQ, 3)
             Heisenberg algebra of rank 3 over Rational Field
         """
-        return "Heisenberg algebra of rank {0} over {1}".format(self._n, self.base_ring())
+        return "Heisenberg algebra of rank {} over {}".format(self._n, self.base_ring())
+
 
 class InfiniteHeisenbergAlgebra(HeisenbergAlgebra_abstract, LieAlgebraWithGenerators):
     r"""

src/sage/algebras/lie_algebras/lie_algebra.py

@@ -894,7 +894,7 @@ def __init__(self, R, names=None, index_set=None, category=None):
         LieAlgebraWithGenerators.__init__(self, R, names, index_set, category)
         self.__ngens = len(self._indices)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         Return a string representation of ``self``.
 
@@ -905,9 +905,9 @@ def _repr_(self):
             Lie algebra on 2 generators (x, y) over Rational Field
         """
         if self.__ngens == 1:
-            return "Lie algebra on the generator {0} over {1}".format(
+            return "Lie algebra on the generator {} over {}".format(
                 self.gen(0), self.base_ring())
-        return "Lie algebra on {0} generators {1} over {2}".format(
+        return "Lie algebra on {} generators {} over {}".format(
             self.__ngens, self.gens(), self.base_ring())
 
     @lazy_attribute