gh-40819: various details in algebras, categories
    
after some suggestions by `ruff`

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #40819
Reported by: Frédéric Chapoton
Reviewer(s): Martin Rubey

Author: Release Manager

src/sage/algebras/lie_algebras/quotient.py

@@ -21,7 +21,6 @@
 from sage.categories.homset import Hom
 from sage.categories.lie_algebras import LieAlgebras
 from sage.categories.morphism import SetMorphism
-from sage.structure.element import Element
 from sage.structure.indexed_generators import standardize_names_index_set
 
 
@@ -218,7 +217,6 @@ def __classcall_private__(cls, ambient, I, names=None, index_set=None,
 
         # extract an index set from a complementary basis to the ideal
         I_supp = [X.leading_support() for X in I.leading_monomials()]
-        inv = ambient.basis().inverse_family()
         IA = I.ambient()
         B = ambient.basis()
         if index_set_mapping is None:
@@ -264,7 +262,6 @@ def __init__(self, L, I, names, index_set, index_set_mapping, category=None):
             sage: TestSuite(K).run()
         """
         B = L.basis()
-        IA = I.ambient()
         self._index_set_mapping = dict(index_set_mapping)
         sm = L.module().submodule_with_basis([I.reduce(B[k]).to_vector()
                                               for k in self._index_set_mapping.values()])

src/sage/categories/coxeter_groups.py

@@ -1924,10 +1924,10 @@ def reduced_word_graph(self):
 
             P = self.parent()
             edges = []
-            for i, x in enumerate(R):
-                x = tuple(x)
-                for y in R[i:]:
-                    y = tuple(y)
+            for i, _x in enumerate(R):
+                x = tuple(_x)
+                for _y in R[i:]:
+                    y = tuple(_y)
                     # Check that the reduced expressions differ by only
                     #   a single braid move
                     j = 0

src/sage/categories/crystals.py

@@ -1250,7 +1250,7 @@ def direct_sum(self, X):
         __add__ = direct_sum
 
         @abstract_method(optional=True)
-        def connected_components_generators(self):
+        def connected_components_generators(self) -> tuple:
             """
             Return a tuple of generators for each of the connected components
             of ``self``.

src/sage/categories/finite_dimensional_lie_algebras_with_basis.py

@@ -1075,7 +1075,7 @@ def ideal(self, *gens, **kwds):
                 [x, y]
             """
             from sage.algebras.lie_algebras.subalgebra import LieSubalgebra_finite_dimensional_with_basis
-            from sage.structure.element import parent, Element
+            from sage.structure.element import Element
             if len(gens) == 1 and not isinstance(gens[0], Element):
                 gens = gens[0]
             category = kwds.pop('category', None)
@@ -2720,17 +2720,17 @@ def reduce(self, X):
                 P = X.parent()
                 X = self.ambient()(X)  # make sure it is in the ambient space
                 for Y in self.basis():
-                    Y = self.lift(Y)
-                    k, c = Y.leading_item(key=self._order)
+                    lY = self.lift(Y)
+                    k, c = lY.leading_item(key=self._order)
                     if not X[k]:  # scalar will be 0
                         continue
 
                     if is_field:
-                        X -= (X[k] / c) * Y
+                        X -= (X[k] / c) * lY
                     else:
                         try:
                             q, _ = X[k].quo_rem(c)
-                            X -= q * Y
+                            X -= q * lY
                         except AttributeError:
                             break
 

src/sage/categories/finite_lattice_posets.py

@@ -10,7 +10,6 @@
 # *****************************************************************************
 
 from sage.categories.category_with_axiom import CategoryWithAxiom
-from sage.misc.cachefunc import cached_method
 
 
 class FiniteLatticePosets(CategoryWithAxiom):

src/sage/categories/highest_weight_crystals.py

@@ -114,7 +114,7 @@ def additional_structure(self):
     class ParentMethods:
 
         @cached_method
-        def highest_weight_vectors(self):
+        def highest_weight_vectors(self) -> tuple:
             r"""
             Return the highest weight vectors of ``self``.
 
@@ -674,7 +674,7 @@ class ParentMethods:
             Implement operations on tensor products of crystals.
             """
             @cached_method
-            def highest_weight_vectors(self):
+            def highest_weight_vectors(self) -> tuple:
                 r"""
                 Return the highest weight vectors of ``self``.
 

src/sage/categories/ore_modules.py

@@ -1,8 +1,5 @@
-from sage.misc.lazy_attribute import lazy_attribute
-
 from sage.categories.modules import Modules
 from sage.categories.category_types import Category_over_base_ring
-from sage.categories.homsets import Homsets
 from sage.rings.polynomial.ore_polynomial_ring import OrePolynomialRing
 
 

src/sage/categories/pushout.py

@@ -1955,7 +1955,7 @@ def _apply_functor(self, R):
             return FreeModule(R, self.n, sparse=self.is_sparse, inner_product_matrix=self.inner_product_matrix,
                               with_basis=self.with_basis, basis_keys=self.basis_keys)
         return FreeModule(R, self.n, sparse=self.is_sparse, inner_product_matrix=self.inner_product_matrix,
-                              with_basis=self.with_basis, basis_keys=self.basis_keys, name=name, latex_name=latex_name)
+                          with_basis=self.with_basis, basis_keys=self.basis_keys, name=name, latex_name=latex_name)
 
     def _apply_functor_to_morphism(self, f):
         """

src/sage/categories/rings.py

@@ -20,8 +20,6 @@
 from sage.misc.prandom import randint
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.rngs import Rngs
-from sage.structure.element import Element
-from sage.structure.parent import Parent
 
 
 class Rings(CategoryWithAxiom):

src/sage/categories/supercrystals.py

@@ -146,7 +146,7 @@ def genuine_highest_weight_vectors(self):
 
             connected_components_generators = genuine_highest_weight_vectors
 
-            def connected_components(self):
+            def connected_components(self) -> list:
                 r"""
                 Return the connected components of ``self`` as subcrystals.
 
@@ -174,17 +174,15 @@ def connected_components(self):
                 CCs = []
 
                 for mg in self.connected_components_generators():
-                    if not isinstance(mg, tuple):
-                        mg = (mg,)
-                    subcrystal = self.subcrystal(generators=mg,
+                    subcrystal = self.subcrystal(generators=(mg,),
                                                  index_set=index_set,
                                                  cartan_type=cartan_type,
                                                  category=category)
                     CCs.append(subcrystal)
 
                 return CCs
 
-            def genuine_lowest_weight_vectors(self):
+            def genuine_lowest_weight_vectors(self) -> tuple:
                 r"""
                 Return the tuple of genuine lowest weight elements of ``self``.
 
@@ -205,7 +203,7 @@ def genuine_lowest_weight_vectors(self):
                 return tuple([x[1] for x in self._genuine_highest_lowest_weight_vectors()])
 
             @cached_method
-            def _genuine_highest_lowest_weight_vectors(self):
+            def _genuine_highest_lowest_weight_vectors(self) -> tuple:
                 r"""
                 Return the genuine lowest and highest weight elements of ``self``.
 
@@ -267,7 +265,7 @@ def character(self):
                 return A.sum(A(x.weight()) for x in self)
 
             @cached_method
-            def highest_weight_vectors(self):
+            def highest_weight_vectors(self) -> tuple:
                 """
                 Return the highest weight vectors of ``self``.