eja: drop the a_jordan_frame() method on EJAs.

I understand this better now, and don't expect this method to work
in general. It works for associative algebras, but we already know
how to find a Jordan frame in associative algebras: the spectral
decomposition, which works on the associative algebra alg({x}).

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 0a260653a7c5480acea6ffc00a9158169e4e7a18..0ef8bef1ab06bdb7dc08dd5785642f3c46b23369 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -773,108 +773,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
         return (J0, J5, J1)
 
 
-    def a_jordan_frame(self):
-        r"""
-        Generate a Jordan frame for this algebra.
-
-        This implementation is based on the so-called "central
-        orthogonal idempotents" implemented for (semisimple) centers
-        of SageMath ``FiniteDimensionalAlgebrasWithBasis``. Since all
-        Euclidean Jordan algebas are commutative (and thus equal to
-        their own centers) and semisimple, the method should work more
-        or less as implemented, if it ever worked in the first place.
-        (I don't know the justification for the original implementation.
-        yet).
-
-        How it works: we loop through the algebras generators, looking
-        for their eigenspaces. If there's more than one eigenspace,
-        and if they result in more than one subalgebra, then we split
-        those subalgebras recursively until we get to subalgebras of
-        dimension one (whose idempotent is the unit element). Why does
-        some generator have to produce at least two subalgebras? I
-        dunno. But it seems to work.
-
-        Beware that Koecher defines the "center" of a Jordan algebra to
-        be something else, because the usual definition is stupid in a
-        (necessarily commutative) Jordan algebra.
-
-        SETUP::
-
-            sage: from mjo.eja.eja_algebra import (random_eja,
-            ....:                                  JordanSpinEJA,
-            ....:                                  TrivialEJA)
-
-        EXAMPLES:
-
-        A Jordan frame for the trivial algebra has to be empty
-        (zero-length) since its rank is zero. More to the point, there
-        are no non-zero idempotents in the trivial EJA. This does not
-        cause any problems so long as we adopt the convention that the
-        empty sum is zero, since then the sole element of the trivial
-        EJA has an (empty) spectral decomposition::
-
-            sage: J = TrivialEJA()
-            sage: J.a_jordan_frame()
-            ()
-
-        A one-dimensional algebra has rank one (equal to its dimension),
-        and only one primitive idempotent, namely the algebra's unit
-        element::
-
-            sage: J = JordanSpinEJA(1)
-            sage: J.a_jordan_frame()
-            (e0,)
-
-        TESTS::
-
-            sage: J = random_eja()
-            sage: c = J.a_jordan_frame()
-            sage: all( x^2 == x for x in c )
-            True
-            sage: r = len(c)
-            sage: all( c[i]*c[j] == c[i]*(i==j) for i in range(r)
-            ....:                               for j in range(r) )
-            True
-
-        """
-        if self.dimension() == 0:
-            return ()
-        if self.dimension() == 1:
-            return (self.one(),)
-
-        for g in self.gens():
-            eigenpairs = g.operator().matrix().right_eigenspaces()
-            if len(eigenpairs) >= 2:
-                subalgebras = []
-                for eigval, eigspace in eigenpairs:
-                    # Make sub-EJAs from the matrix eigenspaces...
-                    sb = tuple( self.from_vector(b) for b in eigspace.basis() )
-                    try:
-                        # This will fail if e.g. the eigenspace basis
-                        # contains two elements and their product
-                        # isn't a linear combination of the two of
-                        # them (i.e. the generated EJA isn't actually
-                        # two dimensional).
-                        s = FiniteDimensionalEuclideanJordanSubalgebra(self, sb)
-                        subalgebras.append(s)
-                    except ArithmeticError as e:
-                        if str(e) == "vector is not in free module":
-                            # Ignore only the "not a sub-EJA" error
-                            pass
-
-                if len(subalgebras) >= 2:
-                    # apply this method recursively.
-                    return tuple( c.superalgebra_element()
-                                  for subalgebra in subalgebras
-                                  for c in subalgebra.a_jordan_frame() )
-
-        # If we got here, the algebra didn't decompose, at least not when we looked at
-        # the eigenspaces corresponding only to basis elements of the algebra. The
-        # implementation I stole says that this should work because of Schur's Lemma,
-        # so I personally blame Schur's Lemma if it does not.
-        raise Exception("Schur's Lemma didn't work!")
-
-
     def random_elements(self, count):
         """
         Return ``count`` random elements as a tuple.