what can be supported in a general Jordan Algebra.
"""
-from itertools import izip, repeat
+from itertools import repeat
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
+from sage.misc.lazy_import import lazy_import
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
+lazy_import('mjo.eja.eja_subalgebra',
+ 'FiniteDimensionalEuclideanJordanSubalgebra')
from mjo.eja.eja_utils import _mat2vec
class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
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
# long run to have the multiplication table be in terms of
# algebra elements. We do this after calling the superclass
# constructor so that from_vector() knows what to do.
- self._multiplication_table = [ map(lambda x: self.from_vector(x), ls)
- for ls in mult_table ]
+ self._multiplication_table = [
+ list(map(lambda x: self.from_vector(x), ls))
+ for ls in mult_table
+ ]
def _element_constructor_(self, elt):
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: RealCartesianProductEJA,
+ ....: HadamardEJA,
....: RealSymmetricEJA)
EXAMPLES:
vector representations) back and forth faithfully::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA.random_instance()
+ sage: J = HadamardEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
"""
(A_of_x, x, xr, detA) = self._charpoly_matrix_system()
R = A_of_x.base_ring()
- if i >= self.rank():
+
+ if i == self.rank():
+ return R.one()
+ if i > self.rank():
# Guaranteed by theory
return R.zero()
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
EXAMPLES:
sage: p(*xvec)
t^2 - 2*t + 1
+ By definition, the characteristic polynomial is a monic
+ degree-zero polynomial in a rank-zero algebra. Note that
+ Cayley-Hamilton is indeed satisfied since the polynomial
+ ``1`` evaluates to the identity element of the algebra on
+ any argument::
+
+ sage: J = TrivialEJA()
+ sage: J.characteristic_polynomial()
+ 1
+
"""
r = self.rank()
n = self.dimension()
- # The list of coefficient polynomials a_1, a_2, ..., a_n.
- a = [ self._charpoly_coeff(i) for i in range(n) ]
+ # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
+ a = [ self._charpoly_coeff(i) for i in range(r+1) ]
# We go to a bit of trouble here to reorder the
# indeterminates, so that it's easier to evaluate the
S = PolynomialRing(S, R.variable_names())
t = S(t)
- # Note: all entries past the rth should be zero. The
- # coefficient of the highest power (x^r) is 1, but it doesn't
- # appear in the solution vector which contains coefficients
- # for the other powers (to make them sum to x^r).
- if (r < n):
- a[r] = 1 # corresponds to x^r
- else:
- # When the rank is equal to the dimension, trying to
- # assign a[r] goes out-of-bounds.
- a.append(1) # corresponds to x^r
-
- return sum( a[k]*(t**k) for k in xrange(len(a)) )
+ return sum( a[k]*(t**k) for k in range(len(a)) )
def inner_product(self, x, y):
SETUP::
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: TrivialEJA)
EXAMPLES::
sage: J = ComplexHermitianEJA(3)
sage: J.is_trivial()
False
- sage: A = J.zero().subalgebra_generated_by()
- sage: A.is_trivial()
+
+ ::
+
+ sage: J = TrivialEJA()
+ sage: J.is_trivial()
True
"""
"""
M = list(self._multiplication_table) # copy
- for i in xrange(len(M)):
+ for i in range(len(M)):
# M had better be "square"
M[i] = [self.monomial(i)] + M[i]
M = [["*"] + list(self.gens())] + M
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
EXAMPLES::
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
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 peirce_decomposition(self, c):
+ """
+ The Peirce decomposition of this algebra relative to the
+ idempotent ``c``.
+
+ In the future, this can be extended to a complete system of
+ orthogonal idempotents.
+
+ INPUT:
+
+ - ``c`` -- an idempotent of this algebra.
+
+ OUTPUT:
+
+ A triple (J0, J5, J1) containing two subalgebras and one subspace
+ of this algebra,
+
+ - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
+ corresponding to the eigenvalue zero.
+
+ - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
+ corresponding to the eigenvalue one-half.
+
+ - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
+ corresponding to the eigenvalue one.
+
+ These are the only possible eigenspaces for that operator, and this
+ algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
+ orthogonal, and are subalgebras of this algebra with the appropriate
+ restrictions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
+
+ EXAMPLES:
+
+ The canonical example comes from the symmetric matrices, which
+ decompose into diagonal and off-diagonal parts::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: C = matrix(QQ, [ [1,0,0],
+ ....: [0,1,0],
+ ....: [0,0,0] ])
+ sage: c = J(C)
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: J0
+ Euclidean Jordan algebra of dimension 1...
+ sage: J5
+ Vector space of degree 6 and dimension 2...
+ sage: J1
+ Euclidean Jordan algebra of dimension 3...
+
+ TESTS:
+
+ Every algebra decomposes trivially with respect to its identity
+ element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J0,J5,J1 = J.peirce_decomposition(J.one())
+ sage: J0.dimension() == 0 and J5.dimension() == 0
+ True
+ sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
+ True
+
+ The identity elements in the two subalgebras are the
+ projections onto their respective subspaces of the
+ superalgebra's identity element::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: if not J.is_trivial():
+ ....: while x.is_nilpotent():
+ ....: x = J.random_element()
+ sage: c = x.subalgebra_idempotent()
+ sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: J1(c) == J1.one()
+ True
+ sage: J0(J.one() - c) == J0.one()
+ True
+
+ """
+ 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 random_elements(self, count):
"""
True
"""
- return tuple( self.random_element() for idx in xrange(count) )
+ return tuple( self.random_element() for idx in range(count) )
def rank(self):
TESTS:
Ensure that every EJA that we know how to construct has a
- positive integer rank::
+ positive integer rank, unless the algebra is trivial in
+ which case its rank will be zero::
sage: set_random_seed()
- sage: r = random_eja().rank()
- sage: r in ZZ and r > 0
+ sage: J = random_eja()
+ sage: r = J.rank()
+ sage: r in ZZ
+ True
+ sage: r > 0 or (r == 0 and J.is_trivial())
True
"""
Beware, this will crash for "most instances" because the
constructor below looks wrong.
"""
+ if cls is TrivialEJA:
+ # The TrivialEJA class doesn't take an "n" argument because
+ # there's only one.
+ return cls(field)
+
n = ZZ.random_element(cls._max_test_case_size()) + 1
return cls(n, field, **kwargs)
-class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra,
- KnownRankEJA):
+class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
"""
Return the Euclidean Jordan Algebra corresponding to the set
`R^n` under the Hadamard product.
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES:
This multiplication table can be verified by hand::
- sage: J = RealCartesianProductEJA(3)
+ sage: J = HadamardEJA(3)
sage: e0,e1,e2 = J.gens()
sage: e0*e0
e0
We can change the generator prefix::
- sage: RealCartesianProductEJA(3, prefix='r').gens()
+ sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
"""
def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)
- mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ]
- for i in xrange(n) ]
+ mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
+ for i in range(n) ]
- fdeja = super(RealCartesianProductEJA, self)
+ fdeja = super(HadamardEJA, self)
return fdeja.__init__(field, mult_table, rank=n, **kwargs)
def inner_product(self, x, y):
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
TESTS:
over `R^n`::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA.random_instance()
+ sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: X = x.natural_representation()
sage: Y = y.natural_representation()
return x.to_vector().inner_product(y.to_vector())
-def random_eja():
+def random_eja(field=QQ, nontrivial=False):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
- ALGORITHM:
-
- For now, we choose a random natural number ``n`` (greater than zero)
- and then give you back one of the following:
-
- * The cartesian product of the rational numbers ``n`` times; this is
- ``QQ^n`` with the Hadamard product.
-
- * The Jordan spin algebra on ``QQ^n``.
-
- * The ``n``-by-``n`` rational symmetric matrices with the symmetric
- product.
-
- * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
- in the space of ``2n``-by-``2n`` real symmetric matrices.
-
- * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
- in the space of ``4n``-by-``4n`` real symmetric matrices.
-
- Later this might be extended to return Cartesian products of the
- EJAs above.
-
SETUP::
sage: from mjo.eja.eja_algebra import random_eja
Euclidean Jordan algebra of dimension...
"""
- classname = choice(KnownRankEJA.__subclasses__())
- return classname.random_instance()
+ eja_classes = KnownRankEJA.__subclasses__()
+ if nontrivial:
+ eja_classes.remove(TrivialEJA)
+ classname = choice(eja_classes)
+ 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 )
- basis = tuple(s*c for (s,c) in izip(basis,self._basis_normalizers))
+ basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers))
Qs = self.multiplication_table_from_matrix_basis(basis)
# with had entries in a nice field.
return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i)
else:
- basis = ( (b/n) for (b,n) in izip(self.natural_basis(),
- self._basis_normalizers) )
+ basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
+ self._basis_normalizers) )
# Do this over the rationals and convert back at the end.
J = MatrixEuclideanJordanAlgebra(QQ,
(_,x,_,_) = J._charpoly_matrix_system()
p = J._charpoly_coeff(i)
# p might be missing some vars, have to substitute "optionally"
- pairs = izip(x.base_ring().gens(), self._basis_normalizers)
+ pairs = zip(x.base_ring().gens(), self._basis_normalizers)
substitutions = { v: v*c for (v,c) in pairs }
result = p.subs(substitutions)
V = VectorSpace(field, dimension**2)
W = V.span_of_basis( _mat2vec(s) for s in basis )
n = len(basis)
- mult_table = [[W.zero() for j in xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(n):
+ mult_table = [[W.zero() for j in range(n)] for i in range(n)]
+ for i in range(n):
+ for j in range(n):
mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))
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`::
# The basis of symmetric matrices, as matrices, in their R^(n-by-n)
# coordinates.
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(field, n, lambda k,l: k==i and l==j)
if i == j:
Sij = Eij
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
# 2-by-2 block we see to a single complex element.
elements = []
- for k in xrange(n/2):
- for j in xrange(n/2):
+ for k in range(n/2):
+ for j in range(n/2):
submat = M[2*k:2*k+2,2*j:2*j+2]
if submat[0,0] != submat[1,1]:
raise ValueError('bad on-diagonal submatrix')
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:
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(F, n, lambda k,l: k==i and l==j)
if i == j:
Sij = cls.real_embed(Eij)
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
if not n.mod(4).is_zero():
- raise ValueError("the matrix 'M' must be a complex embedding")
+ raise ValueError("the matrix 'M' must be a quaternion embedding")
# Use the base ring of the matrix to ensure that its entries can be
# multiplied by elements of the quaternion algebra.
# 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
# quaternion block.
elements = []
- for l in xrange(n/4):
- for m in xrange(n/4):
+ for l in range(n/4):
+ for m in range(n/4):
submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed(
M[4*l:4*l+4,4*m:4*m+4] )
if submat[0,0] != submat[1,1].conjugate():
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`::
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ for i in range(n):
+ for j in range(i+1):
Eij = matrix(Q, n, lambda k,l: k==i and l==j)
if i == j:
Sij = cls.real_embed(Eij)
"""
def __init__(self, n, field=QQ, **kwargs):
V = VectorSpace(field, n)
- mult_table = [[V.zero() for j in xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(n):
+ mult_table = [[V.zero() for j in range(n)] for i in range(n)]
+ for i in range(n):
+ for j in range(n):
x = V.gen(i)
y = V.gen(j)
x0 = x[0]
"""
return x.to_vector().inner_product(y.to_vector())
+
+
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
+ """
+ The trivial Euclidean Jordan algebra consisting of only a zero element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import TrivialEJA
+
+ EXAMPLES::
+
+ sage: J = TrivialEJA()
+ sage: J.dimension()
+ 0
+ sage: J.zero()
+ 0
+ sage: J.one()
+ 0
+ sage: 7*J.one()*12*J.one()
+ 0
+ sage: J.one().inner_product(J.one())
+ 0
+ sage: J.one().norm()
+ 0
+ sage: J.one().subalgebra_generated_by()
+ Euclidean Jordan algebra of dimension 0 over Rational Field
+ sage: J.rank()
+ 0
+
+ """
+ def __init__(self, field=QQ, **kwargs):
+ mult_table = []
+ fdeja = super(TrivialEJA, self)
+ # The rank is zero using my definition, namely the dimension of the
+ # largest subalgebra generated by any element.
+ return fdeja.__init__(field, mult_table, rank=0, **kwargs)
projects / sage.d.git / blobdiff