+ if not c.is_idempotent():
+ raise ValueError("element is not idempotent: %s" % c)
+
+ # Default these to what they should be if they turn out to be
+ # trivial, because eigenspaces_left() won't return eigenvalues
+ # corresponding to trivial spaces (e.g. it returns only the
+ # eigenspace corresponding to lambda=1 if you take the
+ # decomposition relative to the identity element).
+ trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
+ J0 = trivial # eigenvalue zero
+ J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
+ J1 = trivial # eigenvalue one
+
+ for (eigval, eigspace) in c.operator().matrix().left_eigenspaces():
+ if eigval == ~(self.base_ring()(2)):
+ J5 = eigspace
+ else:
+ gens = tuple( self.from_vector(b) for b in eigspace.basis() )
+ subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens)
+ if eigval == 0:
+ J0 = subalg
+ elif eigval == 1:
+ J1 = subalg
+ else:
+ raise ValueError("unexpected eigenvalue: %s" % eigval)
+
+ 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