eja: begin refactoring to allow noncanonical inner products.

[sage.d.git] / mjo / eja / eja_subalgebra.py

diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py
index 292c770c8b2bb8d52ecc38c25c1170a36ded59bc..6a9d10f65b7164627394c5ddb32bd682c323c5b2 100644 (file)
--- a/mjo/eja/eja_subalgebra.py
+++ b/mjo/eja/eja_subalgebra.py
@@ -83,11 +83,17 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide
             True
 
         """
+        # As with the _element_constructor_() method on the
+        # algebra... even in a subspace of a subspace, the basis
+        # elements belong to the ambient space. As a result, only one
+        # level of coordinate_vector() is needed, regardless of how
+        # deeply we're nested.
         W = self.parent().vector_space()
         V = self.parent().superalgebra().vector_space()
-        A = W.basis_matrix().transpose()
-        W_coords = A*self.to_vector()
-        V_coords = V.coordinate_vector(W_coords)
+
+        # Multiply on the left because basis_matrix() is row-wise.
+        ambient_coords = self.to_vector()*W.basis_matrix()
+        V_coords = V.coordinate_vector(ambient_coords)
         return self.parent().superalgebra().from_vector(V_coords)
 
 
@@ -164,20 +170,20 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda
         except ValueError:
             prefix = prefixen[0]
 
-        basis_vectors = [ b.to_vector() for b in basis ]
-        superalgebra_basis = [ self._superalgebra.from_vector(b)
-                               for b in basis_vectors ]
-
         # If our superalgebra is a subalgebra of something else, then
         # these vectors won't have the right coordinates for
         # V.span_of_basis() unless we use V.from_vector() on them.
-        W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+        W = V.span_of_basis( V.from_vector(b.to_vector()) for b in basis )
 
-        n = len(superalgebra_basis)
+        n = len(basis)
         mult_table = [[W.zero() for i in range(n)] for j in range(n)]
+        ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
+                       for i in range(n) ]
+                     for j in range(n) ]
+
         for i in range(n):
             for j in range(n):
-                product = superalgebra_basis[i]*superalgebra_basis[j]
+                product = basis[i]*basis[j]
                 # product.to_vector() might live in a vector subspace
                 # if our parent algebra is already a subalgebra. We
                 # use V.from_vector() to make it "the right size" in
@@ -185,8 +191,8 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda
                 product_vector = V.from_vector(product.to_vector())
                 mult_table[i][j] = W.coordinate_vector(product_vector)
 
-        natural_basis = tuple( b.natural_representation()
-                               for b in superalgebra_basis )
+        self._inner_product_matrix = matrix(field, ip_table)
+        natural_basis = tuple( b.natural_representation() for b in basis )
 
 
         self._vector_space = W
@@ -233,12 +239,20 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda
         if elt not in self.superalgebra():
             raise ValueError("not an element of this subalgebra")
 
-        # The extra hackery is because foo.to_vector() might not
-        # live in foo.parent().vector_space()!
-        coords = sum( a*b for (a,b)
-                          in zip(elt.to_vector(),
-                                 self.superalgebra().vector_space().basis()) )
-        return self.from_vector(self.vector_space().coordinate_vector(coords))
+        # The extra hackery is because foo.to_vector() might not live
+        # in foo.parent().vector_space()! Subspaces of subspaces still
+        # have user bases in the ambient space, though, so only one
+        # level of coordinate_vector() is needed. In other words, if V
+        # is itself a subspace, the basis elements for W will be of
+        # the same length as the basis elements for V -- namely
+        # whatever the dimension of the ambient (parent of V?) space is.
+        V = self.superalgebra().vector_space()
+        W = self.vector_space()
+
+        # Multiply on the left because basis_matrix() is row-wise.
+        ambient_coords = elt.to_vector()*V.basis_matrix()
+        W_coords = W.coordinate_vector(ambient_coords)
+        return self.from_vector(W_coords)