from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
-from sage.rings.integer_ring import ZZ
-from sage.rings.number_field.number_field import NumberField, QuadraticField
-from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
-from sage.rings.rational_field import QQ
-from sage.rings.real_lazy import CLF, RLF
-
+from sage.rings.all import (ZZ, QQ, RR, RLF, CLF,
+ PolynomialRing,
+ QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
from mjo.eja.eja_utils import _mat2vec
rank,
prefix='e',
category=None,
- natural_basis=None):
+ natural_basis=None,
+ check=True):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import random_eja
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
EXAMPLES:
sage: x*y == y*x
True
+ TESTS:
+
+ The ``field`` we're given must be real::
+
+ sage: JordanSpinEJA(2,QQbar)
+ Traceback (most recent call last):
+ ...
+ ValueError: field is not real
+
"""
+ if check:
+ if not field.is_subring(RR):
+ # Note: this does return true for the real algebraic
+ # field, and any quadratic field where we've specified
+ # a real embedding.
+ raise ValueError('field is not real')
+
self._rank = rank
self._natural_basis = natural_basis
sage: J = ComplexHermitianEJA(3)
sage: J.is_trivial()
False
- sage: A = J.zero().subalgebra_generated_by()
- sage: A.is_trivial()
- True
"""
return self.dimension() == 0
return self.linear_combination(zip(self.gens(), coeffs))
- def random_element(self):
- # Temporary workaround for https://trac.sagemath.org/ticket/28327
- if self.is_trivial():
- return self.zero()
- else:
- s = super(FiniteDimensionalEuclideanJordanAlgebra, self)
- return s.random_element()
-
def random_elements(self, count):
"""
Return ``count`` random elements as a tuple.
return x.to_vector().inner_product(y.to_vector())
-def random_eja():
+def random_eja(field=QQ):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
"""
classname = choice(KnownRankEJA.__subclasses__())
- return classname.random_instance()
+ return classname.random_instance(field=field)
z = R.gen()
p = z**2 - 2
if p.is_irreducible():
- field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
+ field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
basis = tuple( s.change_ring(field) for s in basis )
self._basis_normalizers = tuple(
~(self.natural_inner_product(s,s).sqrt()) for s in basis )
Xu = cls.real_unembed(X)
Yu = cls.real_unembed(Y)
tr = (Xu*Yu).trace()
+
if tr in RLF:
# It's real already.
return tr
sage: e2*e2
e2
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: RealSymmetricEJA(2, AA)
+ Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
+ sage: RealSymmetricEJA(2, RR)
+ Euclidean Jordan algebra of dimension 3 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `(n^2 + n) / 2`::
n = M.nrows()
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- field = M.base_ring()
+
+ # We don't need any adjoined elements...
+ field = M.base_ring().base_ring()
+
blocks = []
for z in M.list():
- a = z.vector()[0] # real part, I guess
- b = z.vector()[1] # imag part, I guess
+ a = z.list()[0] # real part, I guess
+ b = z.list()[1] # imag part, I guess
blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
- # We can drop the imaginaries here.
- return matrix.block(field.base_ring(), n, blocks)
+ return matrix.block(field, n, blocks)
@staticmethod
if not n.mod(2).is_zero():
raise ValueError("the matrix 'M' must be a complex embedding")
- field = QQ
+ # If "M" was normalized, its base ring might have roots
+ # adjoined and they can stick around after unembedding.
+ field = M.base_ring()
R = PolynomialRing(field, 'z')
z = R.gen()
- F = NumberField(z**2 + 1,'i', embedding=CLF(-1).sqrt())
+ F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
i = F.gen()
# Go top-left to bottom-right (reading order), converting every
sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ EXAMPLES:
+
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: ComplexHermitianEJA(2, AA)
+ Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
+ sage: ComplexHermitianEJA(2, RR)
+ Euclidean Jordan algebra of dimension 4 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `n^2`::
True
"""
- R = PolynomialRing(QQ, 'z')
+ R = PolynomialRing(field, 'z')
z = R.gen()
- F = NumberField(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
+ F = field.extension(z**2 + 1, 'I')
I = F.gen()
# This is like the symmetric case, but we need to be careful:
sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
+ EXAMPLES:
+
+ In theory, our "field" can be any subfield of the reals::
+
+ sage: QuaternionHermitianEJA(2, AA)
+ Euclidean Jordan algebra of dimension 6 over Algebraic Real Field
+ sage: QuaternionHermitianEJA(2, RR)
+ Euclidean Jordan algebra of dimension 6 over Real Field with
+ 53 bits of precision
+
TESTS:
The dimension of this algebra is `2*n^2 - n`::
projects / sage.d.git / blobdiff