from sage.misc.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
-from sage.rings.all import (ZZ, QQ, RR, RLF, CLF,
+from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
Ensure that it says what we think it says::
- sage: JordanSpinEJA(2, field=QQ)
- Euclidean Jordan algebra of dimension 2 over Rational Field
+ sage: JordanSpinEJA(2, field=AA)
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
sage: JordanSpinEJA(3, field=RDF)
Euclidean Jordan algebra of dimension 3 over Real Double Field
Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
- [1 0] [ 0 1/2*sqrt2] [0 0]
- [0 0], [1/2*sqrt2 0], [0 1]
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
)
::
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
- for (eigval, eigspace) in c.operator().matrix().left_eigenspaces():
+ for (eigval, eigspace) in c.operator().matrix().right_eigenspaces():
if eigval == ~(self.base_ring()(2)):
J5 = eigspace
else:
return (J0, J5, J1)
- def orthogonal_idempotents(self):
- r"""
- Generate a set of `r` orthogonal idempotents for this algebra,
- where `r` is its rank.
-
- This implementation is based on the so-called "central
- orthogonal idempotents" implemented for (semisimple) centers
- of SageMath ``FiniteDimensionalAlgebrasWithBasis``. Since all
- Euclidean Jordan algebas are commutative (and thus equal to
- their own centers) and semisimple, the method should work more
- or less as implemented, if it ever worked in the first place.
- (I don't know the justification for the original implementation.
- yet).
-
- How it works: we loop through the algebras generators, looking
- for their eigenspaces. If there's more than one eigenspace,
- and if they result in more than one subalgebra, then we split
- those subalgebras recursively until we get to subalgebras of
- dimension one (whose idempotent is the unit element). Why does
- some generator have to produce at least two subalgebras? I
- dunno. But it seems to work.
-
- Beware that Koecher defines the "center" of a Jordan algebra to
- be something else, because the usual definition is stupid in a
- (necessarily commutative) Jordan algebra.
- """
- if self.dimension() == 1:
- return [self.one()]
-
- for g in self.gens():
- eigenpairs = g.operator().matrix().right_eigenspaces()
- if len(eigenpairs) >= 2:
- subalgebras = []
- for eigval, eigspace in eigenpairs:
- # Make sub-EJAs from the matrix eigenspaces...
- sb = tuple( self.from_vector(b) for b in eigspace.basis() )
- try:
- # This will fail if e.g. the eigenspace basis
- # contains two elements and their product
- # isn't a linear combination of the two of
- # them (i.e. the generated EJA isn't actually
- # two dimensional).
- s = FiniteDimensionalEuclideanJordanSubalgebra(self, sb)
- subalgebras.append(s)
- except:
- pass
- if len(subalgebras) >= 2:
- # apply this method recursively.
- return tuple( c.superalgebra_element()
- for subalgebra in subalgebras
- for c in subalgebra.orthogonal_idempotents() )
-
- # If we got here, the algebra didn't decompose, at least not when we looked at
- # the eigenspaces corresponding only to basis elements of the algebra. The
- # implementation I stole says that this should work because of Schur's Lemma,
- # so I personally blame Schur's Lemma if it does not.
- raise Exception("Schur's Lemma didn't work!")
-
-
def random_elements(self, count):
"""
Return ``count`` random elements as a tuple.
return tuple( self.random_element() for idx in range(count) )
+ def _rank_computation(self):
+ r"""
+ Compute the rank of this algebra using highly suspicious voodoo.
+
+ ALGORITHM:
+
+ We first compute the basis representation of the operator L_x
+ using polynomial indeterminates are placeholders for the
+ coordinates of "x", which is arbitrary. We then use that
+ matrix to compute the (polynomial) entries of x^0, x^1, ...,
+ x^d,... for increasing values of "d", starting at zero. The
+ idea is that. If we also add "coefficient variables" a_0,
+ a_1,... to the ring, we can form the linear combination
+ a_0*x^0 + ... + a_d*x^d = 0, and ask what dimension the
+ solution space has as an affine variety. When "d" is smaller
+ than the rank, we expect that dimension to be the number of
+ coordinates of "x", since we can set *those* to whatever we
+ want, but linear independence forces the coefficients a_i to
+ be zero. Eventually, when "d" passes the rank, the dimension
+ of the solution space begins to grow, because we can *still*
+ set the coordinates of "x" arbitrarily, but now there are some
+ coefficients that make the sum zero as well. So, when the
+ dimension of the variety jumps, we return the corresponding
+ "d" as the rank of the algebra. This appears to work.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(5)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = JordanSpinEJA(5)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = RealSymmetricEJA(4)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = ComplexHermitianEJA(3)
+ sage: J._rank_computation() == J.rank()
+ True
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._rank_computation() == J.rank()
+ True
+
+ """
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ d = 0
+ ideal_dim = len(var_names)
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( vars[d+k]*self.monomial(k).operator().matrix()[i,j]
+ for k in range(n) )
+
+ while ideal_dim == len(var_names):
+ coeff_names = [ "a" + str(z) for z in range(d) ]
+ R = PolynomialRing(self.base_ring(), coeff_names + var_names)
+ vars = R.gens()
+ L_x = matrix(R, n, n, L_x_i_j)
+ x_powers = [ vars[k]*(L_x**k)*self.one().to_vector()
+ for k in range(d) ]
+ eqs = [ sum(x_powers[k][j] for k in range(d)) for j in range(n) ]
+ ideal_dim = R.ideal(eqs).dimension()
+ d += 1
+
+ # Subtract one because we increment one too many times, and
+ # subtract another one because "d" is one greater than the
+ # answer anyway; when d=3, we go up to x^2.
+ return d-2
+
def rank(self):
"""
Return the rank of this EJA.
return 5
@classmethod
- def random_instance(cls, field=QQ, **kwargs):
+ def random_instance(cls, field=AA, **kwargs):
"""
Return a random instance of this type of algebra.
(r0, r1, r2)
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
V = VectorSpace(field, n)
mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
for i in range(n) ]
return x.to_vector().inner_product(y.to_vector())
-def random_eja(field=QQ, nontrivial=False):
+def random_eja(field=AA, nontrivial=False):
"""
Return a "random" finite-dimensional Euclidean Jordan Algebra.
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, RDF)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
sage: RealSymmetricEJA(2, RR)
Euclidean Jordan algebra of dimension 3 over Real Field with
53 bits of precision
return 4 # Dimension 10
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n, field)
super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs)
EXAMPLES::
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
sage: set_random_seed()
sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
sage: n = ZZ.random_element(n_max)
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: X = random_matrix(F, n)
sage: Y = random_matrix(F, n)
sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
....: [ 9, 10, 11, 12],
....: [-10, 9, -12, 11] ])
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
- [ 2*i + 1 4*i + 3]
- [ 10*i + 9 12*i + 11]
+ [ 2*I + 1 4*I + 3]
+ [ 10*I + 9 12*I + 11]
TESTS:
Unembedding is the inverse of embedding::
sage: set_random_seed()
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: M = random_matrix(F, 3)
sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
field = M.base_ring()
R = PolynomialRing(field, 'z')
z = R.gen()
- F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ if field is AA:
+ # Sage doesn't know how to embed AA into QQbar, i.e. how
+ # to adjoin sqrt(-1) to AA.
+ F = QQbar
+ else:
+ 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: Ye = y.natural_representation()
sage: X = ComplexHermitianEJA.real_unembed(Xe)
sage: Y = ComplexHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().vector()[0]
+ sage: expected = (X*Y).trace().real()
sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
sage: actual == expected
True
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, RDF)
+ Euclidean Jordan algebra of dimension 4 over Real Double Field
sage: ComplexHermitianEJA(2, RR)
Euclidean Jordan algebra of dimension 4 over Real Field with
53 bits of precision
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs)
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- F = QuadraticField(-1, 'i')
+ F = QuadraticField(-1, 'I')
i = F.gen()
blocks = []
raise ValueError('bad on-diagonal submatrix')
if submat[0,1] != -submat[1,0].conjugate():
raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].vector()[0] # real part
- z += submat[0,0].vector()[1]*i # imag part
- z += submat[0,1].vector()[0]*j # real part
- z += submat[0,1].vector()[1]*k # imag part
+ z = submat[0,0].real()
+ z += submat[0,0].imag()*i
+ z += submat[0,1].real()*j
+ z += submat[0,1].imag()*k
elements.append(z)
return matrix(Q, n/4, elements)
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, RDF)
+ Euclidean Jordan algebra of dimension 6 over Real Double Field
sage: QuaternionHermitianEJA(2, RR)
Euclidean Jordan algebra of dimension 6 over Real Field with
53 bits of precision
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs)
sage: actual == expected
True
"""
- def __init__(self, n, field=QQ, B=None, **kwargs):
+ def __init__(self, n, field=AA, B=None, **kwargs):
if B is None:
self._B = matrix.identity(field, max(0,n-1))
else:
True
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
# This is a special case of the BilinearFormEJA with the identity
# matrix as its bilinear form.
return super(JordanSpinEJA, self).__init__(n, field, **kwargs)
sage: J.one().norm()
0
sage: J.one().subalgebra_generated_by()
- Euclidean Jordan algebra of dimension 0 over Rational Field
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
sage: J.rank()
0
"""
- def __init__(self, field=QQ, **kwargs):
+ def __init__(self, field=AA, **kwargs):
mult_table = []
fdeja = super(TrivialEJA, self)
# The rank is zero using my definition, namely the dimension of the
projects / sage.d.git / blobdiff