from sage.matrix.constructor import matrix
+from sage.misc.cachefunc import cached_method
-from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+from mjo.eja.eja_algebra import EJA
+from mjo.eja.eja_element import (EJAElement,
+ CartesianProductParentEJAElement)
-class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+class EJASubalgebraElement(EJAElement):
"""
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
works like it does in the superalgebra, even if we orthonormalize
our basis::
- sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
- sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
- sage: y = A.random_element()
- sage: y.operator()(A.one()) == y
+ sage: x = random_eja(field=AA).random_element() # long time
+ sage: A = x.subalgebra_generated_by(orthonormalize=True) # long time
+ sage: y = A.random_element() # long time
+ sage: y.operator()(A.one()) == y # long time
True
"""
sage: J = RealSymmetricEJA(3)
sage: x = sum(J.gens())
sage: x
- e0 + e1 + e2 + e3 + e4 + e5
- sage: A = x.subalgebra_generated_by()
+ b0 + b1 + b2 + b3 + b4 + b5
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: A(x)
- f1
+ c1
sage: A(x).superalgebra_element()
- e0 + e1 + e2 + e3 + e4 + e5
+ b0 + b1 + b2 + b3 + b4 + b5
sage: y = sum(A.gens())
sage: y
- f0 + f1
- sage: B = y.subalgebra_generated_by()
+ c0 + c1
+ sage: B = y.subalgebra_generated_by(orthonormalize=False)
sage: B(y)
- g1
+ d1
sage: B(y).superalgebra_element()
- f0 + f1
+ c0 + c1
TESTS:
We can convert back and forth faithfully::
- sage: set_random_seed()
- sage: J = random_eja()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
sage: x = J.random_element()
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: A(x).superalgebra_element() == x
True
sage: y = A.random_element()
sage: A(y.superalgebra_element()) == y
True
- sage: B = y.subalgebra_generated_by()
+ sage: B = y.subalgebra_generated_by(orthonormalize=False)
sage: B(y).superalgebra_element() == y
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()
-
- # 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)
+ return self.parent().superalgebra_embedding()(self)
-class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class EJASubalgebra(EJA):
"""
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 EJASubalgebra
EXAMPLES:
....: [0,0] ])
sage: E22 = matrix(AA, [ [0,0],
....: [0,1] ])
- sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
- sage: K1.one().natural_representation()
+ sage: K1 = EJASubalgebra(J, (J(E11),), associative=True)
+ sage: K1.one().to_matrix()
[1 0]
[0 0]
- sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
- sage: K2.one().natural_representation()
+ sage: K2 = EJASubalgebra(J, (J(E22),), associative=True)
+ sage: K2.one().to_matrix()
[0 0]
[0 1]
TESTS:
- Ensure that our generator names don't conflict with the superalgebra::
+ Ensure that our generator names don't conflict with the
+ superalgebra::
sage: J = JordanSpinEJA(3)
sage: J.one().subalgebra_generated_by().gens()
- (f0,)
+ (c0,)
sage: J = JordanSpinEJA(3, prefix='f')
sage: J.one().subalgebra_generated_by().gens()
(g0,)
- sage: J = JordanSpinEJA(3, prefix='b')
+ sage: J = JordanSpinEJA(3, prefix='a')
sage: J.one().subalgebra_generated_by().gens()
- (c0,)
+ (b0,)
Ensure that we can find subalgebras of subalgebras::
sage: B = A.one().subalgebra_generated_by()
sage: B.dimension()
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
# try to "increment" the parent algebra's prefix, although
# this idea goes out the window fast because some prefixen
# are off-limits.
- prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
+ prefixen = ["b","c","d","e","f","g","h","l","m"]
try:
prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
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 )
+ # 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()
- n = len(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 = 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)
+ def inner_product(x,y):
+ return self._superalgebra(x).inner_product(self._superalgebra(y))
- natural_basis = tuple( b.natural_representation() for b in basis )
-
-
- self._vector_space = W
-
- fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
- fdeja.__init__(field,
- mult_table,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis,
- check_field=False,
- check_axioms=check_axioms)
+ super().__init__(matrix_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=superalgebra.matrix_space(),
+ 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 EJASubalgebra
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 = EJASubalgebra(J,
+ ....: basis,
+ ....: associative=True,
+ ....: orthonormalize=False)
sage: K(J.one())
- f1
+ c1
sage: K(J.one() + x)
- f0 + f1
+ c0 + c1
::
"""
- 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)
-
-
-
- def natural_basis_space(self):
- """
- Return the natural basis 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.
- """
- return self.superalgebra().natural_basis_space()
+ if elt in self.superalgebra():
+ # If the subalgebra is trivial, its _matrix_span will be empty
+ # but we still want to be able convert the superalgebra's zero()
+ # element into the subalgebra's zero() element. There's no great
+ # workaround for this because sage checks that your basis is
+ # linearly-independent everywhere, so we can't just give it a
+ # basis consisting of the zero element.
+ m = elt.to_matrix()
+ if self.is_trivial() and m.is_zero():
+ return self.zero()
+ else:
+ return super()._element_constructor_(m)
+ else:
+ return super()._element_constructor_(elt)
def superalgebra(self):
return self._superalgebra
- def vector_space(self):
- """
+ @cached_method
+ def superalgebra_embedding(self):
+ r"""
+ Return the embedding from this subalgebra into the superalgebra.
+
SETUP::
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+ sage: from mjo.eja.eja_algebra import HadamardEJA
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)
+ sage: J = HadamardEJA(4)
+ sage: A = J.one().subalgebra_generated_by()
+ sage: iota = A.superalgebra_embedding()
+ sage: iota
+ Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix:
+ [1/2]
+ [1/2]
+ [1/2]
+ [1/2]
+ Domain: Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
+ sage: iota(A.one()) == J.one()
+ True
"""
- return self._vector_space
+ from mjo.eja.eja_operator import EJAOperator
+ mm = self._module_morphism(lambda j: self.superalgebra()(self.monomial(j).to_matrix()),
+ codomain=self.superalgebra())
+ return EJAOperator(self,
+ self.superalgebra(),
+ mm.matrix())
+
+
+
+ Element = EJASubalgebraElement
- Element = FiniteDimensionalEuclideanJordanSubalgebraElement
+
+class CartesianProductEJASubalgebraElement(EJASubalgebraElement,
+ CartesianProductParentEJAElement):
+ r"""
+ The class for elements that both belong to a subalgebra and
+ have a Cartesian product algebra as their parent. By inheriting
+ :class:`CartesianProductParentEJAElement` in addition to
+ :class:`EJASubalgebraElement`, we allow the
+ ``to_matrix()`` method to be overridden with the version that
+ works on Cartesian products.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA)
+
+ TESTS:
+
+ This used to fail when ``subalgebra_idempotent()`` tried to
+ embed the subalgebra element back into the original EJA::
+
+ sage: J1 = HadamardEJA(0, field=QQ, orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.one().subalgebra_idempotent() == J.one()
+ True
+
+ """
+ pass
+
+class CartesianProductEJASubalgebra(EJASubalgebra):
+ r"""
+ Subalgebras whose parents are Cartesian products. Exists only
+ to specify a special element class that will (in addition)
+ inherit from ``CartesianProductParentEJAElement``.
+ """
+ Element = CartesianProductEJASubalgebraElement
projects / sage.d.git / blobdiff