f1
sage: A(x).superalgebra_element()
e0 + e1 + e2 + e3 + e4 + e5
+ sage: y = sum(A.gens())
+ sage: y
+ f0 + f1
+ sage: B = y.subalgebra_generated_by()
+ sage: B(y)
+ g1
+ sage: B(y).superalgebra_element()
+ f0 + f1
TESTS:
sage: y = A.random_element()
sage: A(y.superalgebra_element()) == y
True
+ sage: B = y.subalgebra_generated_by()
+ sage: B(y).superalgebra_element() == y
+ True
"""
- return self.parent().superalgebra().linear_combination(
- zip(self.parent()._superalgebra_basis, self.to_vector()) )
+ W = self.parent().vector_space()
+ V = self.parent().superalgebra().vector_space()
+ A = W.basis_matrix().transpose()
+ W_coords = A*self.to_vector()
+ V_coords = V.coordinate_vector(W_coords)
+ return self.parent().superalgebra().from_vector(V_coords)
SETUP::
sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: JordanSpinEJA)
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+
+ EXAMPLES:
+
+ The following Peirce subalgebras of the 2-by-2 real symmetric
+ matrices do not contain the superalgebra's identity element::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: E11 = matrix(AA, [ [1,0],
+ ....: [0,0] ])
+ sage: E22 = matrix(AA, [ [0,0],
+ ....: [0,1] ])
+ sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
+ sage: K1.one().natural_representation()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2.one().natural_representation()
+ [0 0]
+ [0 1]
TESTS:
1
"""
- def __init__(self, superalgebra, basis, rank=None, category=None):
+ def __init__(self, superalgebra, basis, category=None, check_axioms=True):
self._superalgebra = superalgebra
V = self._superalgebra.vector_space()
field = self._superalgebra.base_ring()
superalgebra_basis = [ self._superalgebra.from_vector(b)
for b in basis_vectors ]
+ # If our superalgebra is a subalgebra of something else, then
+ # these vectors won't have the right coordinates for
+ # V.span_of_basis() unless we use V.from_vector() on them.
W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+
n = len(superalgebra_basis)
mult_table = [[W.zero() for i in range(n)] for j in range(n)]
for i in range(n):
self._vector_space = W
- self._superalgebra_basis = superalgebra_basis
-
fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
+ fdeja.__init__(field,
+ mult_table,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis,
+ check_field=False,
+ check_axioms=check_axioms)
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: basis = tuple( x^k for k in range(J.rank()) )
- sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
- sage: [ K(x^k) for k in range(J.rank()) ]
- [f0, f1, f2]
+ sage: X = matrix(AA, [ [0,0,1],
+ ....: [0,1,0],
+ ....: [1,0,0] ])
+ sage: x = J(X)
+ sage: basis = ( x, x^2 ) # x^2 is the identity matrix
+ sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
+ sage: K(J.one())
+ f1
+ sage: K(J.one() + x)
+ f0 + f1
::
if elt not in self.superalgebra():
raise ValueError("not an element of this subalgebra")
- coords = self.vector_space().coordinate_vector(elt.to_vector())
- return self.from_vector(coords)
-
-
- def one(self):
- """
- Return the multiplicative identity element of this algebra.
-
- The superclass method computes the identity element, which is
- beyond overkill in this case: the superalgebra identity
- restricted to this algebra is its identity. Note that we can't
- count on the first basis element being the identity -- it migth
- have been scaled if we orthonormalized the basis.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
- ....: random_eja)
-
- EXAMPLES::
-
- sage: J = RealCartesianProductEJA(5)
- sage: J.one()
- e0 + e1 + e2 + e3 + e4
- sage: x = sum(J.gens())
- sage: A = x.subalgebra_generated_by()
- sage: A.one()
- f0
- sage: A.one().superalgebra_element()
- e0 + e1 + e2 + e3 + e4
-
- TESTS:
+ # The extra hackery is because foo.to_vector() might not
+ # live in foo.parent().vector_space()!
+ coords = sum( a*b for (a,b)
+ in zip(elt.to_vector(),
+ self.superalgebra().vector_space().basis()) )
+ return self.from_vector(self.vector_space().coordinate_vector(coords))
- The identity element acts like the identity over the rationals::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: A = x.subalgebra_generated_by()
- sage: x = A.random_element()
- sage: A.one()*x == x and x*A.one() == x
- True
-
- The identity element acts like the identity over the algebraic
- reals with an orthonormal basis::
-
- sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
- sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
- sage: x = A.random_element()
- sage: A.one()*x == x and x*A.one() == x
- True
-
- The matrix of the unit element's operator is the identity over
- the rationals::
-
- sage: set_random_seed()
- sage: x = random_eja().random_element()
- sage: A = x.subalgebra_generated_by()
- sage: actual = A.one().operator().matrix()
- sage: expected = matrix.identity(A.base_ring(), A.dimension())
- sage: actual == expected
- True
-
- The matrix of the unit element's operator is the identity over
- the algebraic reals with an orthonormal basis::
-
- sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
- sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
- sage: actual = A.one().operator().matrix()
- sage: expected = matrix.identity(A.base_ring(), A.dimension())
- sage: actual == expected
- True
-
- """
- if self.dimension() == 0:
- return self.zero()
- else:
- sa_one = self.superalgebra().one().to_vector()
- sa_coords = self.vector_space().coordinate_vector(sa_one)
- return self.from_vector(sa_coords)
def natural_basis_space(self):
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
- sage: basis = (x^0, x^1, x^2)
+ sage: E11 = matrix(ZZ, [ [1,0,0],
+ ....: [0,0,0],
+ ....: [0,0,0] ])
+ sage: E22 = matrix(ZZ, [ [0,0,0],
+ ....: [0,1,0],
+ ....: [0,0,0] ])
+ sage: b1 = J(E11)
+ sage: b2 = J(E22)
+ sage: basis = (b1, b2)
sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over...
+ Vector space of degree 6 and dimension 2 over...
User basis matrix:
- [ 1 0 1 0 0 1]
- [ 1 0 2 0 0 5]
- [ 1 0 4 0 0 25]
- sage: (x^0).to_vector()
- (1, 0, 1, 0, 0, 1)
- sage: (x^1).to_vector()
- (1, 0, 2, 0, 0, 5)
- sage: (x^2).to_vector()
- (1, 0, 4, 0, 0, 25)
+ [1 0 0 0 0 0]
+ [0 0 1 0 0 0]
+ sage: b1.to_vector()
+ (1, 0, 0, 0, 0, 0)
+ sage: b2.to_vector()
+ (0, 0, 1, 0, 0, 0)
"""
return self._vector_space
projects / sage.d.git / blobdiff