X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=389efbf77219acff73db032e848e34d8da1670e9;hb=9632a965029feadbb70d18574f8704812ed71321;hp=8bee7297fa170c0faf47610dc0ac435a881531fb;hpb=7dd1a157df55b7aad13baf051d2930524d300c78;p=sage.d.git

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 8bee729..389efbf 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -773,6 +773,108 @@ 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.