X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=276eab040db20d1df1715eed782db7dfe2acd481;hb=269f872019b616287c4cf1878477879019f8bbd0;hp=166c9d217cf71d43fe4bfc0911e6fe6c3e1ec9d2;hpb=5d646c586de50b571d2983b546a05899bf0c20c2;p=sage.d.git

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 166c9d2..276eab0 100644
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -9,7 +9,7 @@ from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
 # TODO: make this unnecessary somehow.
 from sage.misc.lazy_import import lazy_import
 lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
-lazy_import('mjo.eja.eja_subalgebra',
+lazy_import('mjo.eja.eja_element_subalgebra',
             'FiniteDimensionalEuclideanJordanElementSubalgebra')
 from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
 from mjo.eja.eja_utils import _mat2vec
@@ -546,6 +546,115 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
         return not (p(zero) == zero)
 
 
+    def is_primitive_idempotent(self):
+        """
+        Return whether or not this element is a primitive (or minimal)
+        idempotent.
+
+        A primitive idempotent is a non-zero idempotent that is not
+        the sum of two other non-zero idempotents. Remark 2.7.15 in
+        Baes shows that this is what he refers to as a "minimal
+        idempotent."
+
+        An element of a Euclidean Jordan algebra is a minimal idempotent
+        if it :meth:`is_idempotent` and if its Peirce subalgebra
+        corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
+        Proposition 2.7.17).
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+            ....:                                  RealSymmetricEJA,
+            ....:                                  TrivialEJA,
+            ....:                                  random_eja)
+
+        WARNING::
+
+        This method is sloooooow.
+
+        EXAMPLES:
+
+        The spectral decomposition of a non-regular element should always
+        contain at least one non-minimal idempotent::
+
+            sage: J = RealSymmetricEJA(3, AA)
+            sage: x = sum(J.gens())
+            sage: x.is_regular()
+            False
+            sage: [ c.is_primitive_idempotent()
+            ....:   for (l,c) in x.spectral_decomposition() ]
+            [False, True]
+
+        On the other hand, the spectral decomposition of a regular
+        element should always be in terms of minimal idempotents::
+
+            sage: J = JordanSpinEJA(4, AA)
+            sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
+            sage: x.is_regular()
+            True
+            sage: [ c.is_primitive_idempotent()
+            ....:   for (l,c) in x.spectral_decomposition() ]
+            [True, True]
+
+        TESTS:
+
+        The identity element is minimal only in an EJA of rank one::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
+            True
+
+        A non-idempotent cannot be a minimal idempotent::
+
+            sage: set_random_seed()
+            sage: J = JordanSpinEJA(4)
+            sage: x = J.random_element()
+            sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
+            False
+
+        Proposition 2.7.19 in Baes says that an element is a minimal
+        idempotent if and only if it's idempotent with trace equal to
+        unity::
+
+            sage: set_random_seed()
+            sage: J = JordanSpinEJA(4)
+            sage: x = J.random_element()
+            sage: expected = (x.is_idempotent() and x.trace() == 1)
+            sage: actual = x.is_primitive_idempotent()
+            sage: actual == expected
+            True
+
+        Primitive idempotents must be non-zero::
+
+            sage: set_random_seed()
+            sage: J = random_eja()
+            sage: J.zero().is_idempotent()
+            True
+            sage: J.zero().is_primitive_idempotent()
+            False
+
+        As a consequence of the fact that primitive idempotents must
+        be non-zero, there are no primitive idempotents in a trivial
+        Euclidean Jordan algebra::
+
+            sage: J = TrivialEJA()
+            sage: J.one().is_idempotent()
+            True
+            sage: J.one().is_primitive_idempotent()
+            False
+
+        """
+        if not self.is_idempotent():
+            return False
+
+        if self.is_zero():
+            return False
+
+        (_,_,J1) = self.parent().peirce_decomposition(self)
+        return (J1.dimension() == 1)
+
+
     def is_nilpotent(self):
         """
         Return whether or not some power of this element is zero.
@@ -1180,10 +1289,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
 
             sage: from mjo.eja.eja_algebra import random_eja
 
-        TESTS::
+        TESTS:
+
+        Ensure that we can find an idempotent in a non-trivial algebra
+        where there are non-nilpotent elements::
 
             sage: set_random_seed()
-            sage: J = random_eja()
+            sage: J = random_eja(nontrivial=True)
             sage: x = J.random_element()
             sage: while x.is_nilpotent():
             ....:     x = J.random_element()
@@ -1192,6 +1304,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement):
             True
 
         """
+        if self.parent().is_trivial():
+            return self
+
         if self.is_nilpotent():
             raise ValueError("this only works with non-nilpotent elements!")