# 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
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 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)
"""
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()
super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs)
+class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
+ r"""
+ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
+ with the half-trace inner product and jordan product ``x*y =
+ (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
+ symmetric positive-definite "bilinear form" matrix. It has
+ dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
+ when ``B`` is the identity matrix of order ``n-1``.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
+
+ EXAMPLES:
+
+ When no bilinear form is specified, the identity matrix is used,
+ and the resulting algebra is the Jordan spin algebra::
+
+ sage: J0 = BilinearFormEJA(3)
+ sage: J1 = JordanSpinEJA(3)
+ sage: J0.multiplication_table() == J0.multiplication_table()
+ True
+
+ TESTS:
+
+ We can create a zero-dimensional algebra::
+
+ sage: J = BilinearFormEJA(0)
+ sage: J.basis()
+ Finite family {}
+
+ We can check the multiplication condition given in the Jordan, von
+ Neumann, and Wigner paper (and also discussed on my "On the
+ symmetry..." paper). Note that this relies heavily on the standard
+ choice of basis, as does anything utilizing the bilinear form matrix::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(5)
+ sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
+ sage: B = M.transpose()*M
+ sage: J = BilinearFormEJA(n, B=B)
+ sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
+ sage: V = J.vector_space()
+ sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
+ ....: for ei in eis ]
+ sage: actual = [ sis[i]*sis[j]
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: expected = [ J.one() if i == j else J.zero()
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: actual == expected
+ True
+ """
+ def __init__(self, n, field=QQ, B=None, **kwargs):
+ if B is None:
+ self._B = matrix.identity(field, max(0,n-1))
+ else:
+ self._B = B
+
+ V = VectorSpace(field, 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]
+ xbar = x[1:]
+ y0 = y[0]
+ ybar = y[1:]
+ z0 = x0*y0 + (self._B*xbar).inner_product(ybar)
+ zbar = y0*xbar + x0*ybar
+ z = V([z0] + zbar.list())
+ mult_table[i][j] = z
+
+ # The rank of this algebra is two, unless we're in a
+ # one-dimensional ambient space (because the rank is bounded
+ # by the ambient dimension).
+ fdeja = super(BilinearFormEJA, self)
+ return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs)
+
+ def inner_product(self, x, y):
+ r"""
+ Half of the trace inner product.
+
+ This is defined so that the special case of the Jordan spin
+ algebra gets the usual inner product.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
+
+ TESTS:
+
+ Ensure that this is one-half of the trace inner-product::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(5)
+ sage: M = matrix.random(QQ, n-1, algorithm='unimodular')
+ sage: B = M.transpose()*M
+ sage: J = BilinearFormEJA(n, B=B)
+ sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
+ sage: V = J.vector_space()
+ sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
+ ....: for ei in eis ]
+ sage: actual = [ sis[i]*sis[j]
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: expected = [ J.one() if i == j else J.zero()
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+
+ """
+ xvec = x.to_vector()
+ xbar = xvec[1:]
+ yvec = y.to_vector()
+ ybar = yvec[1:]
+ return x[0]*y[0] + (self._B*xbar).inner_product(ybar)
+
class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the usual inner product and jordan product ``x*y =
- (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
+ (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
the reals.
SETUP::
projects / sage.d.git / blobdiff