from sage.matrix.constructor import matrix
-from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+from mjo.eja.eja_algebra import FiniteDimensionalEJA
+from mjo.eja.eja_element import FiniteDimensionalEJAElement
-class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement):
"""
SETUP::
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: x = random_eja(field=QQ,orthonormalize=False).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
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: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: x = random_eja(field=AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=True)
sage: y = A.random_element()
sage: y.operator()(A.one()) == y
True
sage: x = sum(J.gens())
sage: x
e0 + e1 + e2 + e3 + e4 + e5
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: A(x)
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(orthonormalize=False)
+ 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()) )
+ return self.parent().superalgebra()(self.to_matrix())
-class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
"""
A subalgebra of an EJA with a given basis.
sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
....: JordanSpinEJA,
....: RealSymmetricEJA)
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
EXAMPLES:
....: [0,0] ])
sage: E22 = matrix(AA, [ [0,0],
....: [0,1] ])
- sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
- sage: K1.one().natural_representation()
+ sage: K1 = FiniteDimensionalEJASubalgebra(J, (J(E11),))
+ sage: K1.one().to_matrix()
[1 0]
[0 0]
- sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
- sage: K2.one().natural_representation()
+ sage: K2 = FiniteDimensionalEJASubalgebra(J, (J(E22),))
+ sage: K2.one().to_matrix()
[0 0]
[0 1]
1
"""
- def __init__(self, superalgebra, basis, rank=None, category=None):
+ def __init__(self, superalgebra, basis, **kwargs):
self._superalgebra = superalgebra
V = self._superalgebra.vector_space()
field = self._superalgebra.base_ring()
- if category is None:
- category = self._superalgebra.category()
# A half-assed attempt to ensure that we don't collide with
# the superalgebra's prefix (ignoring the fact that there
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 ]
-
- 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]
- # 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)
-
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
+ # The superalgebra constructor expects these to be in original matrix
+ # form, not algebra-element form.
+ matrix_basis = tuple( b.to_matrix() for b in basis )
+ def jordan_product(x,y):
+ return (self._superalgebra(x)*self._superalgebra(y)).to_matrix()
+ def inner_product(x,y):
+ return self._superalgebra(x).inner_product(self._superalgebra(y))
- 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)
+ super().__init__(matrix_basis,
+ jordan_product,
+ inner_product,
+ prefix=prefix,
+ **kwargs)
SETUP::
sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
EXAMPLES::
....: [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 = FiniteDimensionalEJASubalgebra(J, basis, orthonormalize=False)
sage: K(J.one())
f1
sage: K(J.one() + x)
::
"""
- 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)
+ if elt in self.superalgebra():
+ return super()._element_constructor_(elt.to_matrix())
+ else:
+ return super()._element_constructor_(elt)
- 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):
return self._superalgebra
- def vector_space(self):
- """
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
-
- EXAMPLES::
-
- sage: J = RealSymmetricEJA(3)
- 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 2 over...
- User basis matrix:
- [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
-
-
- Element = FiniteDimensionalEuclideanJordanSubalgebraElement
+ Element = FiniteDimensionalEJASubalgebraElement
projects / sage.d.git / blobdiff