# 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
return not (p(zero) == zero)
+ def is_minimal_idempotent(self):
+ """
+ Return whether or not this element is 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,
+ ....: 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_minimal_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_minimal_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_minimal_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_minimal_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_minimal_idempotent()
+ sage: actual == expected
+ True
+
+ """
+ if not self.is_idempotent():
+ 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.
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()
True
"""
+ if self.parent().is_trivial():
+ return self
+
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")