eja: fix element powers.

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

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 603ef29021d41f53404a84513c2d854f8e4a8c64..ef5249bc6abe4f9d4dd62bf0f5de07caa85fed26 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -101,6 +101,21 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             Jordan algebras are always power-associative; see for
             example Faraut and Koranyi, Proposition II.1.2 (ii).
+
+            .. WARNING:
+
+                We have to override this because our superclass uses row vectors
+                instead of column vectors! We, on the other hand, assume column
+                vectors everywhere.
+
+            EXAMPLES:
+
+                sage: set_random_seed()
+                sage: J = eja_ln(5)
+                sage: x = J.random_element()
+                sage: x.matrix()*x.vector() == (x**2).vector()
+                True
+
             """
             A = self.parent()
             if n == 0:
@@ -108,7 +123,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             elif n == 1:
                 return self
             else:
-                return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+                return A.element_class(A, (self.matrix()**(n-1))*self.vector())
 
 
         def span_of_powers(self):
@@ -171,6 +186,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 sage: x.subalgebra_generated_by().is_associative()
                 True
 
+            Squaring in the subalgebra should be the same thing as
+            squaring in the superalgebra::
+
+                sage: J = eja_ln(5)
+                sage: x = J.random_element()
+                sage: u = x.subalgebra_generated_by().random_element()
+                sage: u.matrix()*u.vector() == (u**2).vector()
+                True
+
             """
             # First get the subspace spanned by the powers of myself...
             V = self.span_of_powers()
@@ -187,6 +211,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 # b1 is what we get if we apply that matrix to b1. The
                 # second row of the right multiplication matrix by b1
                 # is what we get when we apply that matrix to b2...
+                #
+                # IMPORTANT: this assumes that all vectors are COLUMN
+                # vectors, unlike our superclass (which uses row vectors).
                 for b_left in V.basis():
                     eja_b_left = self.parent()(b_left)
                     # Multiply in the original EJA, but then get the
@@ -199,7 +226,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
 
             # It's an algebra of polynomials in one element, and EJAs
             # are power-associative.
-            return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True)
+            #
+            # TODO: choose generator names intelligently.
+            return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
 
 
         def minimal_polynomial(self):
@@ -318,6 +347,56 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             return elt.is_nilpotent()
 
 
+        def subalgebra_idempotent(self):
+            """
+            Find an idempotent in the associative subalgebra I generate
+            using Proposition 2.3.5 in Baes.
+            """
+            if self.is_nilpotent():
+                raise ValueError("this only works with non-nilpotent elements!")
+
+            V = self.span_of_powers()
+            J = self.subalgebra_generated_by()
+            # Mis-design warning: the basis used for span_of_powers()
+            # and subalgebra_generated_by() must be the same, and in
+            # the same order!
+            u = J(V.coordinates(self.vector()))
+
+            # The image of the matrix of left-u^m-multiplication
+            # will be minimal for some natural number s...
+            s = 0
+            minimal_dim = V.dimension()
+            for i in xrange(1, V.dimension()):
+                this_dim = (u**i).matrix().image().dimension()
+                if this_dim < minimal_dim:
+                    minimal_dim = this_dim
+                    s = i
+
+            # Now minimal_matrix should correspond to the smallest
+            # non-zero subspace in Baes's (or really, Koecher's)
+            # proposition.
+            #
+            # However, we need to restrict the matrix to work on the
+            # subspace... or do we? Can't we just solve, knowing that
+            # A(c) = u^(s+1) should have a solution in the big space,
+            # too?
+            #
+            # Beware, solve_right() means that we're using COLUMN vectors.
+            # Our FiniteDimensionalAlgebraElement superclass uses rows.
+            u_next = u**(s+1)
+            A = u_next.matrix()
+            c_coordinates = A.solve_right(u_next.vector())
+
+            # Now c_coordinates is the idempotent we want, but it's in
+            # the coordinate system of the subalgebra.
+            #
+            # We need the basis for J, but as elements of the parent algebra.
+            #
+            basis = [self.parent(v) for v in V.basis()]
+            return self.parent().linear_combination(zip(c_coordinates, basis))
+
+
+
         def characteristic_polynomial(self):
             return self.matrix().characteristic_polynomial()