TESTS::
- The natural representation of an element in the subalgebra is
- the same as its natural representation in the superalgebra::
+ The matrix representation of an element in the subalgebra is
+ the same as its matrix representation in the superalgebra::
sage: set_random_seed()
sage: A = random_eja().random_element().subalgebra_generated_by()
sage: y = A.random_element()
- sage: actual = y.natural_representation()
- sage: expected = y.superalgebra_element().natural_representation()
+ sage: actual = y.to_matrix()
+ sage: expected = y.superalgebra_element().to_matrix()
sage: actual == expected
True
our basis::
sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
+ sage: x = random_eja(field=AA).random_element()
sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
sage: y = A.random_element()
sage: y.operator()(A.one()) == y
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()) )
+ # As with the _element_constructor_() method on the
+ # algebra... even in a subspace of a subspace, the basis
+ # elements belong to the ambient space. As a result, only one
+ # level of coordinate_vector() is needed, regardless of how
+ # deeply we're nested.
+ W = self.parent().vector_space()
+ V = self.parent().superalgebra().vector_space()
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = self.to_vector()*W.basis_matrix()
+ V_coords = V.coordinate_vector(ambient_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().to_matrix()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2.one().to_matrix()
+ [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()
except ValueError:
prefix = prefixen[0]
- basis_vectors = [ b.to_vector() for b in basis ]
- 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(b.to_vector()) for b in basis),
+ check=check_axioms)
+
+ n = len(basis)
+ if check_axioms:
+ # The tables are square if we're verifying that they
+ # are commutative.
+ mult_table = [[W.zero() for j in range(n)] for i in range(n)]
+ ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
+ for j in range(n) ]
+ for i in range(n) ]
+ else:
+ mult_table = [[W.zero() for j in range(i+1)] for i in range(n)]
+ ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
+ for j in range(i+1) ]
+ for i in range(n) ]
- 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):
- for j in range(n):
- product = superalgebra_basis[i]*superalgebra_basis[j]
+ for j in range(i+1):
+ product = basis[i]*basis[j]
# product.to_vector() might live in a vector subspace
# if our parent algebra is already a subalgebra. We
# use V.from_vector() to make it "the right size" in
# that case.
product_vector = V.from_vector(product.to_vector())
mult_table[i][j] = W.coordinate_vector(product_vector)
+ if check_axioms:
+ mult_table[j][i] = mult_table[i][j]
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
+ matrix_basis = tuple( b.to_matrix() for b in basis )
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,
+ ip_table,
+ prefix=prefix,
+ category=category,
+ matrix_basis=matrix_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)
+ # The extra hackery is because foo.to_vector() might not live
+ # in foo.parent().vector_space()! Subspaces of subspaces still
+ # have user bases in the ambient space, though, so only one
+ # level of coordinate_vector() is needed. In other words, if V
+ # is itself a subspace, the basis elements for W will be of
+ # the same length as the basis elements for V -- namely
+ # whatever the dimension of the ambient (parent of V?) space is.
+ V = self.superalgebra().vector_space()
+ W = self.vector_space()
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = elt.to_vector()*V.basis_matrix()
+ W_coords = W.coordinate_vector(ambient_coords)
+ return self.from_vector(W_coords)
- def natural_basis_space(self):
+ def matrix_space(self):
"""
- Return the natural basis space of this algebra, which is identical
- to that of its superalgebra.
+ Return the matrix space of this algebra, which is identical to
+ that of its superalgebra.
- This is correct "by definition," and avoids a mismatch when the
- subalgebra is trivial (with no natural basis to infer anything
- from) and the parent is not.
+ This is correct "by definition," and avoids a mismatch when
+ the subalgebra is trivial (with no matrix basis elements to
+ infer anything from) and the parent is not.
"""
- return self.superalgebra().natural_basis_space()
+ return self.superalgebra().matrix_space()
def superalgebra(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