X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=2ec45cf537661f359ef6cf4e34c578c238f80b31;hb=b479f3bb0d3aae8c598a6ec6459688c4be3202af;hp=bb460194970e3da92709082a81f78c86a6c88d5c;hpb=556690d808d614f2c271dc431dd909e353530594;p=sage.d.git

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index bb46019..2ec45cf 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -199,7 +199,9 @@ 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):
@@ -242,18 +244,131 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 True
 
             """
+            # The element we're going to call "minimal_polynomial()" on.
+            # Either myself, interpreted as an element of a finite-
+            # dimensional algebra, or an element of an associative
+            # subalgebra.
+            elt = None
+
             if self.parent().is_associative():
-                return self.matrix().minimal_polynomial()
+                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+            else:
+                V = self.span_of_powers()
+                assoc_subalg = 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!
+                elt = assoc_subalg(V.coordinates(self.vector()))
+
+            # Recursive call, but should work since elt lives in an
+            # associative algebra.
+            return elt.minimal_polynomial()
+
+
+        def is_nilpotent(self):
+            """
+            Return whether or not some power of this element is zero.
+
+            The superclass method won't work unless we're in an
+            associative algebra, and we aren't. However, we generate
+            an assocoative subalgebra and we're nilpotent there if and
+            only if we're nilpotent here (probably).
+
+            TESTS:
+
+            The identity element is never nilpotent::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(2,10).abs()
+                sage: J = eja_rn(n)
+                sage: J.one().is_nilpotent()
+                False
+                sage: J = eja_ln(n)
+                sage: J.one().is_nilpotent()
+                False
+
+            The additive identity is always nilpotent::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(2,10).abs()
+                sage: J = eja_rn(n)
+                sage: J.zero().is_nilpotent()
+                True
+                sage: J = eja_ln(n)
+                sage: J.zero().is_nilpotent()
+                True
+
+            """
+            # The element we're going to call "is_nilpotent()" on.
+            # Either myself, interpreted as an element of a finite-
+            # dimensional algebra, or an element of an associative
+            # subalgebra.
+            elt = None
+
+            if self.parent().is_associative():
+                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+            else:
+                V = self.span_of_powers()
+                assoc_subalg = 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!
+                elt = assoc_subalg(V.coordinates(self.vector()))
+
+            # Recursive call, but should work since elt lives in an
+            # associative algebra.
+            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()
-            assoc_subalg = self.subalgebra_generated_by()
+            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!
-            subalg_self = assoc_subalg(V.coordinates(self.vector()))
-            # Recursive call, but should work since the subalgebra is
-            # associative.
-            return subalg_self.minimal_polynomial()
+            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?
+            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.
+            #
+            #
+            # TODO: this is buggy, but it's probably because the
+            # multiplication table for the subalgebra is wrong! The
+            # matrices should be symmetric I bet.
+            basis = [self.parent(v) for v in V.basis()]
+            return self.parent().linear_combination(zip(c_coordinates, basis))
+
 
 
         def characteristic_polynomial(self):