FiniteRankFreeModule_abstract.isomorphism_with_fixed_basis: Return a SetIsomorphism

Author: Matthias Koeppe

src/sage/tensor/modules/finite_rank_free_module.py

@@ -539,7 +539,9 @@ class :class:`~sage.modules.free_module.FreeModule_generic`
 from typing import Generator, Optional
 
 from sage.categories.fields import Fields
+from sage.categories.homset import Hom
 from sage.categories.modules import Modules
+from sage.categories.morphism import SetIsomorphism
 from sage.categories.rings import Rings
 from sage.misc.cachefunc import cached_method
 from sage.rings.integer import Integer
@@ -755,15 +757,15 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
         - ``codomain`` -- (default: ``None``) the codomain of the
           isomorphism represented by a free module within the category
           :class:`~sage.categories.modules_with_basis.ModulesWithBasis` with
-          the same rank and base ring as ``self``; if ``None`` a free module
+          the same rank and base ring as ``self``; if ``None``, a free module
           represented by
           :class:`~sage.combinat.free_module.CombinatorialFreeModule` is
           constructed
 
         OUTPUT:
 
-        - a module morphism represented by
-          :class:`~sage.modules.with_basis.morphism.ModuleMorphismFromFunction`
+        - a module isomorphism represented by
+          :class:`~sage.categories.morphism.SetIsomorphism`
 
         EXAMPLES::
 
@@ -800,6 +802,12 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
             To:   Free module generated by {'a', 'b', 'c'} over Rational Field
             sage: phi_eW(e[1] + 2 * e[2])
             B['a'] + 2*B['b']
+            sage: ~phi_eW
+            Generic morphism:
+            From: Free module generated by {'a', 'b', 'c'} over Rational Field
+            To:   3-dimensional vector space over the Rational Field
+            sage: (~phi_eW)(W.basis()['b']).display()
+            e_2
 
         Providing a :class:`~sage.modules.free_module.Module_free_ambient` as the codomain::
 
@@ -814,44 +822,60 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
         Sending (1,1)-tensors to matrices::
 
             sage: T11 = V.tensor_module(1, 1); T11
-            Free module of type-(1,1) tensors on the 3-dimensional vector space over the Rational Field
+            Free module of type-(1,1) tensors on the
+             3-dimensional vector space over the Rational Field
             sage: e_T11 = T11.basis("e"); e_T11
             Standard basis on the
              Free module of type-(1,1) tensors on the 3-dimensional vector space over the Rational Field
              induced by Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field
             sage: W = MatrixSpace(QQ, 3)
             sage: phi_e_T11 = T11.isomorphism_with_fixed_basis(e_T11, codomain=W); phi_e_T11
             Generic morphism:
-            From: Free module of type-(1,1) tensors on the 3-dimensional vector space over the Rational Field
+            From: Free module of type-(1,1) tensors on the
+                  3-dimensional vector space over the Rational Field
             To:   Full MatrixSpace of 3 by 3 dense matrices over Rational Field
             sage: t = T11.an_element(); t.display()
             1/2 e_1⊗e^1
             sage: phi_e_T11(t)
             [1/2   0   0]
             [  0   0   0]
             [  0   0   0]
+            sage: ~phi_e_T11
+            Generic morphism:
+            From: Full MatrixSpace of 3 by 3 dense matrices over Rational Field
+            To:   Free module of type-(1,1) tensors on the
+                  3-dimensional vector space over the Rational Field
+            sage: (~phi_e_T11)(W([[0, 1/2, 1/3],
+            ....:                 [-1/2, 0, 0],
+            ....:                 [-1/3, 0, 0]])).display()
+            1/2 e_1⊗e^2 + 1/3 e_1⊗e^3 - 1/2 e_2⊗e^1 - 1/3 e_3⊗e^1
 
         Sending symmetric bilinear forms to matrices (note that they are currently elements
         of `T^{(0,2)}(M)`, not the symmetric power of `M`)::
 
             sage: T02 = V.tensor_module(0, 2); T02
-            Free module of type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
+            Free module of type-(0,2) tensors on the
+             3-dimensional vector space over the Rational Field
             sage: e_T02 = T02.basis("e"); e_T02
             Standard basis on the
-             Free module of type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
-             induced by Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field
+             Free module of type-(0,2) tensors on the
+              3-dimensional vector space over the Rational Field
+             induced by Basis (e_1,e_2,e_3) on the
+              3-dimensional vector space over the Rational Field
             sage: W = MatrixSpace(QQ, 3)
             sage: phi_e_T02 = T02.isomorphism_with_fixed_basis(e_T02, codomain=W); phi_e_T02
             Generic morphism:
-            From: Free module of type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
+            From: Free module of type-(0,2) tensors on the
+                  3-dimensional vector space over the Rational Field
             To:   Full MatrixSpace of 3 by 3 dense matrices over Rational Field
 
             sage: a = V.sym_bilinear_form()
             sage: a[1,1], a[1,2], a[1,3] = 1, 2, 3
             sage: a[2,2], a[2,3] = 4, 5
             sage: a[3,3] = 6
             sage: a.display()
-            e^1⊗e^1 + 2 e^1⊗e^2 + 3 e^1⊗e^3 + 2 e^2⊗e^1 + 4 e^2⊗e^2 + 5 e^2⊗e^3 + 3 e^3⊗e^1 + 5 e^3⊗e^2 + 6 e^3⊗e^3
+            e^1⊗e^1 + 2 e^1⊗e^2 + 3 e^1⊗e^3 + 2 e^2⊗e^1 + 4 e^2⊗e^2 + 5 e^2⊗e^3
+             + 3 e^3⊗e^1 + 5 e^3⊗e^2 + 6 e^3⊗e^3
             sage: phi_e_T02(a)
             [1 2 3]
             [2 4 5]
@@ -860,15 +884,18 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
         Same but explicitly in the subspace of symmetric bilinear forms::
 
             sage: Sym2Vdual = V.dual_symmetric_power(2); Sym2Vdual
-            Free module of fully symmetric type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
+            Free module of fully symmetric type-(0,2) tensors on the
+             3-dimensional vector space over the Rational Field
             sage: Sym2Vdual.is_submodule(T02)
             True
             sage: Sym2Vdual.rank()
             6
             sage: e_Sym2Vdual = Sym2Vdual.basis("e"); e_Sym2Vdual
             Standard basis on the
-             Free module of fully symmetric type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
-             induced by Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field
+             Free module of fully symmetric type-(0,2) tensors on the
+              3-dimensional vector space over the Rational Field
+             induced by Basis (e_1,e_2,e_3) on the
+              3-dimensional vector space over the Rational Field
             sage: W_basis = [phi_e_T02(b) for b in e_Sym2Vdual]; W_basis
             [
             [1 0 0]  [0 1 0]  [0 0 1]  [0 0 0]  [0 0 0]  [0 0 0]
@@ -877,25 +904,34 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
             ]
             sage: W = MatrixSpace(QQ, 3).submodule(W_basis); W
             Free module generated by {0, 1, 2, 3, 4, 5} over Rational Field
-            sage: phi_e_Sym2Vdual = Sym2Vdual.isomorphism_with_fixed_basis(e_Sym2Vdual, codomain=W); phi_e_Sym2Vdual
+            sage: phi_e_Sym2Vdual = Sym2Vdual.isomorphism_with_fixed_basis(e_Sym2Vdual,
+            ....:                                                          codomain=W)
+            sage: phi_e_Sym2Vdual
             Generic morphism:
-              From: Free module of fully symmetric type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
+              From: Free module of fully symmetric type-(0,2) tensors on the
+                    3-dimensional vector space over the Rational Field
               To:   Free module generated by {0, 1, 2, 3, 4, 5} over Rational Field
 
         Sending tensors to elements of the tensor square of :class:`CombinatorialFreeModule`::
 
             sage: T20 = V.tensor_module(2, 0); T20
-            Free module of type-(2,0) tensors on the 3-dimensional vector space over the Rational Field
+            Free module of type-(2,0) tensors on the
+             3-dimensional vector space over the Rational Field
             sage: e_T20 = T02.basis("e"); e_T20
             Standard basis on the
-             Free module of type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
-             induced by Basis (e_1,e_2,e_3) on the 3-dimensional vector space over the Rational Field
+             Free module of type-(0,2) tensors on the
+              3-dimensional vector space over the Rational Field
+             induced by Basis (e_1,e_2,e_3) on the
+              3-dimensional vector space over the Rational Field
             sage: W = CombinatorialFreeModule(QQ, [1, 2, 3]).tensor_square(); W
-            Free module generated by {1, 2, 3} over Rational Field # Free module generated by {1, 2, 3} over Rational Field
+            Free module generated by {1, 2, 3} over Rational Field
+             # Free module generated by {1, 2, 3} over Rational Field
             sage: phi_e_T20 = T20.isomorphism_with_fixed_basis(e_T20, codomain=W); phi_e_T20
             Generic morphism:
-            From: Free module of type-(2,0) tensors on the 3-dimensional vector space over the Rational Field
-            To:   Free module generated by {1, 2, 3} over Rational Field # Free module generated by {1, 2, 3} over Rational Field
+            From: Free module of type-(2,0) tensors on the
+                  3-dimensional vector space over the Rational Field
+            To:   Free module generated by {1, 2, 3} over Rational Field
+                  # Free module generated by {1, 2, 3} over Rational Field
             sage: t = T20.an_element(); t.display()
             1/2 e_1⊗e_1
             sage: phi_e_T20(t)
@@ -940,13 +976,17 @@ def isomorphism_with_fixed_basis(self, basis=None, codomain=None):
 
         codomain_basis = Family(codomain.basis())
         if isinstance(codomain_basis, TrivialFamily):
-            # assume that codomain basis keys are to be ignored
-            key_pairs = enumerate(basis.keys())
+            # assume that codomain basis keys are to be ignored;
+            # need them several times, can't keep as generators
+            key_pairs = tuple(enumerate(basis.keys()))
+            basis_by_codomain_key = basis
         else:
             # assume that the keys of the codomain should be used
-            key_pairs = zip(codomain_basis.keys(), basis.keys())
-        # Need them several times, can't keep as generators
-        key_pairs = tuple(key_pairs)
+            # need them several times, can't keep as generators
+            key_pairs = tuple(zip(codomain_basis.keys(), basis.keys()))
+            basis_by_codomain_key = {}
+            for codomain_key, domain_key in key_pairs:
+                basis_by_codomain_key[codomain_key] = basis[domain_key]
 
         def _isomorphism(x):
             r"""
@@ -955,7 +995,21 @@ def _isomorphism(x):
             return codomain.sum(x[basis, domain_key] * codomain_basis[codomain_key]
                                 for codomain_key, domain_key in key_pairs)
 
-        return self.module_morphism(function=_isomorphism, codomain=codomain)
+        def _inverse(y):
+            r"""
+            Concrete isomorphism from ``codomain`` to ``self``.
+            """
+            return self.linear_combination(
+                (basis_by_codomain_key[codomain_key], coefficient)
+                for codomain_key, coefficient in y.monomial_coefficients().items())
+
+        category = Modules(self.base_ring())
+        homset = Hom(self, codomain, category)
+        isomorphism = SetIsomorphism(homset, _isomorphism)
+        inverse = SetIsomorphism(homset.reversed(), _inverse)
+        isomorphism._set_inverse(inverse)
+        inverse._set_inverse(isomorphism)
+        return isomorphism
 
     def _test_isomorphism_with_fixed_basis(self, **options):
         r"""
@@ -3160,7 +3214,6 @@ def hom(self, codomain, matrix_rep, bases=None, name=None,
         for more documentation.
 
         """
-        from sage.categories.homset import Hom
         homset = Hom(self, codomain)
         return homset(matrix_rep, bases=bases, name=name,
                       latex_name=latex_name)