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::
True
"""
- # 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()
+ return self._superalgebra(self.to_matrix())
- # 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)
-
-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 = FiniteDimensionalEJASubalgebra(J, (J(E11),))
sage: K1.one().to_matrix()
[1 0]
[0 0]
- sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2 = FiniteDimensionalEJASubalgebra(J, (J(E22),))
sage: K2.one().to_matrix()
[0 0]
[0 1]
1
"""
- def __init__(self, superalgebra, basis, category=None, check_axioms=True):
+ 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]
- # 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) ]
-
- for i in range(n):
- 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]
-
+ # 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
-
- fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
- fdeja.__init__(field,
- mult_table,
- ip_table,
- prefix=prefix,
- category=category,
- matrix_basis=matrix_basis,
- check_field=False,
- check_axioms=check_axioms)
+ 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")
-
- # 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)
+ if elt in self.superalgebra():
+ return super()._element_constructor_(elt.to_matrix())
+ else:
+ return super()._element_constructor_(elt)
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