+ self.rank.set_cache(0)
+ self.one.set_cache( self.zero() )
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ # We don't take a "size" argument so the superclass method is
+ # inappropriate for us.
+ return cls(field, **kwargs)
+
+class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ r"""
+ The external (orthogonal) direct sum of two other Euclidean Jordan
+ algebras. Essentially the Cartesian product of its two factors.
+ Every Euclidean Jordan algebra decomposes into an orthogonal
+ direct sum of simple Euclidean Jordan algebras, so no generality
+ is lost by providing only this construction.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.dimension()
+ 8
+ sage: J.rank()
+ 5
+
+ """
+ def __init__(self, J1, J2, field=AA, **kwargs):
+ self._factors = (J1, J2)
+ n1 = J1.dimension()
+ n2 = J2.dimension()
+ n = n1+n2
+ V = VectorSpace(field, n)
+ mult_table = [ [ V.zero() for j in range(n) ]
+ for i in range(n) ]
+ for i in range(n1):
+ for j in range(n1):
+ p = (J1.monomial(i)*J1.monomial(j)).to_vector()
+ mult_table[i][j] = V(p.list() + [field.zero()]*n2)
+
+ for i in range(n2):
+ for j in range(n2):
+ p = (J2.monomial(i)*J2.monomial(j)).to_vector()
+ mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
+
+ super(DirectSumEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(J1.rank() + J2.rank())
+
+
+ def factors(self):
+ r"""
+ Return the pair of this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2,QQ)
+ sage: J2 = JordanSpinEJA(3,QQ)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.factors()
+ (Euclidean Jordan algebra of dimension 2 over Rational Field,
+ Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+ """
+ return self._factors
+
+ def projections(self):
+ r"""
+ Return a pair of projections onto this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: ComplexHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = ComplexHermitianEJA(2)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: (pi_left, pi_right) = J.projections()
+ sage: J.one().to_vector()
+ (1, 0, 1, 0, 0, 1)
+ sage: pi_left(J.one()).to_vector()
+ (1, 0)
+ sage: pi_right(J.one()).to_vector()
+ (1, 0, 0, 1)
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ pi_left = lambda x: J1.from_vector(x.to_vector()[:n])
+ pi_right = lambda x: J2.from_vector(x.to_vector()[n:])
+ return (pi_left, pi_right)
+
+ def inclusions(self):
+ r"""
+ Return the pair of inclusion maps from our factors into us.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = JordanSpinEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: (iota_left, iota_right) = J.inclusions()
+ sage: iota_left(J1.zero()) == J.zero()
+ True
+ sage: iota_right(J2.zero()) == J.zero()
+ True
+ sage: J1.one().to_vector()
+ (1, 0, 0)
+ sage: iota_left(J1.one()).to_vector()
+ (1, 0, 0, 0, 0, 0)
+ sage: J2.one().to_vector()
+ (1, 0, 1)
+ sage: iota_right(J2.one()).to_vector()
+ (0, 0, 0, 1, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 0, 1, 0, 1)
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ V_basis = self.vector_space().basis()
+ I1 = matrix.column(self.base_ring(), V_basis[:n])
+ I2 = matrix.column(self.base_ring(), V_basis[n:])
+ iota_left = lambda x: self.from_vector(I1*x.to_vector())
+ iota_right = lambda x: self.from_vector(I2*+x.to_vector())
+ return (iota_left, iota_right)
+
+ def inner_product(self, x, y):
+ r"""
+ The standard Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ SETUP::
+
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: QuaternionHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLE::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: x1 = J1.one()
+ sage: x2 = x1
+ sage: y1 = J2.one()
+ sage: y2 = y1
+ sage: x1.inner_product(x2)
+ 3
+ sage: y1.inner_product(y2)
+ 2
+ sage: J.one().inner_product(J.one())
+ 5
+
+ """
+ (pi_left, pi_right) = self.projections()
+ x1 = pi_left(x)
+ x2 = pi_right(x)
+ y1 = pi_left(y)
+ y2 = pi_right(y)
+
+ return (x1.inner_product(y1) + x2.inner_product(y2))
+
+
+
+random_eja = ConcreteEuclideanJordanAlgebra.random_instance