gh-36642: `sage.algebras`: Update `# needs`, modularization fixes, doctest cosmetics
    
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URL: #36642
Reported by: Matthias Köppe
Reviewer(s): Kwankyu Lee, Matthias Köppe

Author: Release Manager

src/sage/algebras/affine_nil_temperley_lieb.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.combinat sage.modules
 """
 Affine nilTemperley Lieb Algebra of type A
 """

src/sage/algebras/algebra.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.combinat sage.modules
 """
 Abstract base class for algebras
 """

src/sage/algebras/askey_wilson.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.combinat sage.modules
 """
 Askey-Wilson Algebras
 

src/sage/algebras/associated_graded.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.modules
 r"""
 Associated Graded Algebras To Filtered Algebras
 

src/sage/algebras/catalog.py

@@ -84,17 +84,18 @@
   <sage.algebras.yokonuma_hecke_algebra.YokonumaHeckeAlgebra>`
 """
 
-from sage.algebras.free_algebra import FreeAlgebra as Free
-from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra as Quaternion
-from sage.algebras.steenrod.steenrod_algebra import SteenrodAlgebra as Steenrod
-from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra as FiniteDimensional
-from sage.algebras.group_algebra import GroupAlgebra as Group
-from sage.algebras.clifford_algebra import CliffordAlgebra as Clifford
-from sage.algebras.clifford_algebra import ExteriorAlgebra as Exterior
-from sage.algebras.weyl_algebra import DifferentialWeylAlgebra as DifferentialWeyl
-from sage.algebras.lie_algebras.lie_algebra import LieAlgebra as Lie
-
 from sage.misc.lazy_import import lazy_import
+lazy_import('sage.algebras.free_algebra', 'FreeAlgebra', as_='Free')
+lazy_import('sage.algebras.quatalg.quaternion_algebra', 'QuaternionAlgebra', as_='Quaternion')
+lazy_import('sage.algebras.steenrod.steenrod_algebra', 'SteenrodAlgebra', as_='Steenrod')
+lazy_import('sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra',
+            'FiniteDimensionalAlgebra', as_='FiniteDimensional')
+lazy_import('sage.algebras.group_algebra', 'GroupAlgebra', as_='Group')
+lazy_import('sage.algebras.clifford_algebra', 'CliffordAlgebra', as_='Clifford')
+lazy_import('sage.algebras.clifford_algebra', 'ExteriorAlgebra', as_='Exterior')
+lazy_import('sage.algebras.weyl_algebra', 'DifferentialWeylAlgebra', as_='DifferentialWeyl')
+lazy_import('sage.algebras.lie_algebras.lie_algebra', 'LieAlgebra', as_='Lie')
+
 lazy_import('sage.algebras.iwahori_hecke_algebra', 'IwahoriHeckeAlgebra', 'IwahoriHecke')
 lazy_import('sage.algebras.nil_coxeter_algebra', 'NilCoxeterAlgebra', 'NilCoxeter')
 lazy_import('sage.algebras.free_zinbiel_algebra', 'FreeZinbielAlgebra', 'FreeZinbiel')

src/sage/algebras/cellular_basis.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.combinat sage.modules
 r"""
 Cellular Basis
 ==============

src/sage/algebras/clifford_algebra.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.modules
 r"""
 Clifford Algebras
 
@@ -1770,6 +1771,7 @@ def interior_product_on_basis(self, a, b):
 
         Check :trac:`34694`::
 
+            sage: # needs sage.symbolic
             sage: E = ExteriorAlgebra(SR,'e',3)
             sage: E.inject_variables()
             Defining e0, e1, e2

src/sage/algebras/clifford_algebra_element.pyx

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.modules
 """
 Clifford algebra elements
 
@@ -944,6 +945,7 @@ cdef class CohomologyRAAGElement(CliffordAlgebraElement):
 
         EXAMPLES::
 
+            sage: # needs sage.graphs sage.groups
             sage: C4 = graphs.CycleGraph(4)
             sage: A = groups.misc.RightAngledArtin(C4)
             sage: H = A.cohomology()

src/sage/algebras/cluster_algebra.py

@@ -1,3 +1,4 @@
+# sage.doctest: needs sage.graphs sage.modules
 r"""
 Cluster algebras
 
@@ -361,12 +362,11 @@
 from sage.categories.rings import Rings
 from sage.combinat.cluster_algebra_quiver.quiver import ClusterQuiver
 from sage.combinat.permutation import Permutation
-from sage.geometry.cone import Cone
-from sage.geometry.fan import Fan
 from sage.graphs.digraph import DiGraph
 from sage.matrix.constructor import identity_matrix, matrix
 from sage.matrix.special import block_matrix
 from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import lazy_import
 from sage.misc.misc_c import prod
 from sage.modules.free_module_element import vector
 from sage.rings.infinity import infinity
@@ -381,6 +381,8 @@
 from sage.structure.sage_object import SageObject
 from sage.structure.unique_representation import UniqueRepresentation
 
+lazy_import('sage.geometry.cone', 'Cone')
+lazy_import('sage.geometry.fan', 'Fan')
 
 ##############################################################################
 # Elements of a cluster algebra
@@ -2373,7 +2375,7 @@ def cluster_fan(self, depth=infinity):
         EXAMPLES::
 
             sage: A = ClusterAlgebra(['A', 2])
-            sage: A.cluster_fan()
+            sage: A.cluster_fan()                                                       # needs sage.geometry.polyhedron
             Rational polyhedral fan in 2-d lattice N
         """
         seeds = self.seeds(depth=depth, mutating_F=False)