generally rules out using the rationals as your ``field``, but
is required for spectral decompositions.
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS:
+
+ We should compute that an element subalgebra is associative even
+ if we circumvent the element method::
+
+ sage: set_random_seed()
+ sage: J = random_eja(field=QQ,orthonormalize=False)
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: basis = tuple(b.superalgebra_element() for b in A.basis())
+ sage: J.subalgebra(basis, orthonormalize=False).is_associative()
+ True
+
"""
Element = FiniteDimensionalEJAElement
inner_product,
field=AA,
orthonormalize=True,
- associative=False,
+ associative=None,
cartesian_product=False,
check_field=True,
check_axioms=True,
category = MagmaticAlgebras(field).FiniteDimensional()
category = category.WithBasis().Unital().Commutative()
+ if associative is None:
+ # We should figure it out. As with check_axioms, we have to do
+ # this without the help of the _jordan_product_is_associative()
+ # method because we need to know the category before we
+ # initialize the algebra.
+ associative = all( jordan_product(jordan_product(bi,bj),bk)
+ ==
+ jordan_product(bi,jordan_product(bj,bk))
+ for bi in basis
+ for bj in basis
+ for bk in basis)
+
if associative:
# Element subalgebras can take advantage of this.
category = category.Associative()
for i in range(self.dimension())
for j in range(self.dimension()) )
+ def _jordan_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is
+ associative; that is, whether or not `x*(y*z) = (x*y)*z`
+ for all `x,y,x`.
+
+ This method should agree with :meth:`is_associative` unless
+ you lied about the value of the ``associative`` parameter
+ when you constructed the algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(4, orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
+ True
+
+ TESTS:
+
+ The values we've presupplied to the constructors agree with
+ the computation::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J.is_associative() == J._jordan_product_is_associative()
+ True
+
+ """
+ R = self.base_ring()
+
+ # Used to check whether or not something is zero.
+ epsilon = R.zero()
+ if not R.is_exact():
+ # I don't know of any examples that make this magnitude
+ # necessary because I don't know how to make an
+ # associative algebra when the element subalgebra
+ # construction is unreliable (as it is over RDF; we can't
+ # find the degree of an element because we can't compute
+ # the rank of a matrix). But even multiplication of floats
+ # is non-associative, so *some* epsilon is needed... let's
+ # just take the one from _inner_product_is_associative?
+ epsilon = 1e-15
+
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.gens()[i]
+ y = self.gens()[j]
+ z = self.gens()[k]
+ diff = (x*y)*z - x*(y*z)
+
+ if diff.norm() > epsilon:
+ return False
+
+ return True
+
def _inner_product_is_associative(self):
r"""
Return whether or not this algebra's inner product `B` is
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
....: RealSymmetricEJA)
TESTS:
- Ensure that we can convert any element of the two non-matrix
- simple algebras (whose matrix representations are columns)
- back and forth faithfully::
+ Ensure that we can convert any element back and forth
+ faithfully between its matrix and algebra representations::
sage: set_random_seed()
- sage: J = HadamardEJA.random_instance()
- sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
- True
- sage: J = JordanSpinEJA.random_instance()
+ sage: J = random_eja()
sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
+ sage: J(x.to_matrix()) == x
True
We cannot coerce elements between algebras just because their
Traceback (most recent call last):
...
ValueError: not an element of this algebra
-
"""
msg = "not an element of this algebra"
if elt in self.base_ring():
if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
raise TypeError("basis not rational")
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ check_field=check_field,
+ **kwargs)
+
self._rational_algebra = None
if field is not QQ:
# There's no point in constructing the extra algebra if this
jordan_product,
inner_product,
field=QQ,
+ associative=self.is_associative(),
orthonormalize=False,
check_field=False,
check_axioms=False)
- super().__init__(basis,
- jordan_product,
- inner_product,
- field=field,
- check_field=check_field,
- **kwargs)
-
@cached_method
def _charpoly_coefficients(self):
r"""
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super(RealSymmetricEJA, self).__init__(self._denormalized_basis(n),
- self.jordan_product,
- self.trace_inner_product,
- **kwargs)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n),
+ self.jordan_product,
+ self.trace_inner_product,
+ associative=associative,
+ **kwargs)
# TODO: this could be factored out somehow, but is left here
# because the MatrixEJA is not presently a subclass of the
True
"""
- super(ComplexMatrixEJA,cls).real_embed(M)
+ super().real_embed(M)
n = M.nrows()
# We don't need any adjoined elements...
True
"""
- super(ComplexMatrixEJA,cls).real_unembed(M)
+ super().real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
F = cls.complex_extension(M.base_ring())
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super(ComplexHermitianEJA, self).__init__(self._denormalized_basis(n),
- self.jordan_product,
- self.trace_inner_product,
- **kwargs)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n),
+ self.jordan_product,
+ self.trace_inner_product,
+ associative=associative,
+ **kwargs)
# TODO: this could be factored out somehow, but is left here
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
True
"""
- super(QuaternionMatrixEJA,cls).real_embed(M)
+ super().real_embed(M)
quaternions = M.base_ring()
n = M.nrows()
True
"""
- super(QuaternionMatrixEJA,cls).real_unembed(M)
+ super().real_unembed(M)
n = ZZ(M.nrows())
d = cls.dimension_over_reals()
# if the user passes check_axioms=True.
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super(QuaternionHermitianEJA, self).__init__(self._denormalized_basis(n),
- self.jordan_product,
- self.trace_inner_product,
- **kwargs)
+ associative = False
+ if n <= 1:
+ associative = True
+
+ super().__init__(self._denormalized_basis(n),
+ self.jordan_product,
+ self.trace_inner_product,
+ associative=associative,
+ **kwargs)
+
# TODO: this could be factored out somehow, but is left here
# because the MatrixEJA is not presently a subclass of the
# FDEJA class that defines rank() and one().
n = B.nrows()
column_basis = tuple( b.column() for b in FreeModule(ZZ, n).basis() )
- super(BilinearFormEJA, self).__init__(column_basis,
- jordan_product,
- inner_product,
- **kwargs)
+
+ # TODO: I haven't actually checked this, but it seems legit.
+ associative = False
+ if n <= 2:
+ associative = True
+
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ associative=associative,
+ **kwargs)
# The rank of this algebra is two, unless we're in a
# one-dimensional ambient space (because the rank is bounded
# But also don't pass check_field=False here, because the user
# can pass in a field!
- super(JordanSpinEJA, self).__init__(B, **kwargs)
+ super().__init__(B, **kwargs)
@staticmethod
def _max_random_instance_size():
if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
- super(TrivialEJA, self).__init__(basis,
- jordan_product,
- inner_product,
- **kwargs)
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ **kwargs)
+
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
self.rank.set_cache(0)
projects / sage.d.git / blobdiff