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 FiniteDimensionalEuclideanJordanElementSubalgebraElement(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
- """
- def __init__(self, A, elt):
- """
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
-
- EXAMPLES::
-
- sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
- sage: [ K.element_class(K,x^k) for k in range(J.rank()) ]
- [f0, f1, f2]
-
- ::
-
- """
- if elt in A.superalgebra():
- # Try to convert a parent algebra element into a
- # subalgebra element...
- try:
- coords = A.vector_space().coordinate_vector(elt.to_vector())
- elt = A.from_vector(coords).monomial_coefficients()
- except AttributeError:
- # Catches a missing method in elt.to_vector()
- pass
-
- s = super(FiniteDimensionalEuclideanJordanElementSubalgebraElement,
- self)
+ The left-multiplication-by operator for elements in the subalgebra
+ works like it does in the superalgebra, even if we orthonormalize
+ our basis::
- s.__init__(A, elt)
+ 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
+ """
def superalgebra_element(self):
"""
sage: J = RealSymmetricEJA(3)
sage: x = sum(J.gens())
sage: x
- e0 + e1 + e2 + e3 + e4 + e5
- sage: A = x.subalgebra_generated_by()
- sage: A.element_class(A,x)
- f1
- sage: A.element_class(A,x).superalgebra_element()
- e0 + e1 + e2 + e3 + e4 + e5
+ b0 + b1 + b2 + b3 + b4 + b5
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A(x)
+ c1
+ sage: A(x).superalgebra_element()
+ b0 + b1 + b2 + b3 + b4 + b5
+ sage: y = sum(A.gens())
+ sage: y
+ c0 + c1
+ sage: B = y.subalgebra_generated_by(orthonormalize=False)
+ sage: B(y)
+ d1
+ sage: B(y).superalgebra_element()
+ 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.element_class(A,x).superalgebra_element() == x
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A(x).superalgebra_element() == x
True
sage: y = A.random_element()
- sage: A.element_class(A,y.superalgebra_element()) == y
+ sage: A(y.superalgebra_element()) == y
+ True
+ sage: B = y.subalgebra_generated_by(orthonormalize=False)
+ 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_embedding()(self)
-class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
- """
- The subalgebra of an EJA generated by a single element.
+class EJASubalgebra(EJA):
"""
- def __init__(self, elt):
- superalgebra = elt.parent()
-
- # First compute the vector subspace spanned by the powers of
- # the given element.
- V = superalgebra.vector_space()
- superalgebra_basis = [superalgebra.one()]
- basis_vectors = [superalgebra.one().to_vector()]
- W = V.span_of_basis(basis_vectors)
- for exponent in range(1, V.dimension()):
- new_power = elt**exponent
- basis_vectors.append( new_power.to_vector() )
- try:
- W = V.span_of_basis(basis_vectors)
- superalgebra_basis.append( new_power )
- except ValueError:
- # Vectors weren't independent; bail and keep the
- # last subspace that worked.
- break
-
- # Make the basis hashable for UniqueRepresentation.
- superalgebra_basis = tuple(superalgebra_basis)
-
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
- field = superalgebra.base_ring()
- mult_table = []
- for b_right in superalgebra_basis:
- b_right_rows = []
- # The first row of the right-multiplication matrix by
- # b1 is what we get if we apply that matrix to b1. The
- # second row of the right multiplication matrix by b1
- # is what we get when we apply that matrix to b2...
- #
- # IMPORTANT: this assumes that all vectors are COLUMN
- # vectors, unlike our superclass (which uses row vectors).
- for b_left in superalgebra_basis:
- # Multiply in the original EJA, but then get the
- # coordinates from the subalgebra in terms of its
- # basis.
- this_row = W.coordinates((b_left*b_right).to_vector())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(field, b_right_rows)
- mult_table.append(b_right_matrix)
-
- for m in mult_table:
- m.set_immutable()
- mult_table = tuple(mult_table)
-
- # TODO: We'll have to redo this and make it unique again...
- prefix = 'f'
-
- # The rank is the highest possible degree of a minimal
- # polynomial, and is bounded above by the dimension. We know
- # in this case that there's an element whose minimal
- # polynomial has the same degree as the space's dimension
- # (remember how we constructed the space?), so that must be
- # its rank too.
- rank = W.dimension()
-
- category = superalgebra.category().Associative()
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
+ A subalgebra of an EJA with a given basis.
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+ sage: from mjo.eja.eja_subalgebra import EJASubalgebra
+
+ 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 = EJASubalgebra(J, (J(E11),), associative=True)
+ sage: K1.one().to_matrix()
+ [1 0]
+ [0 0]
+ 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::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one().subalgebra_generated_by().gens()
+ (c0,)
+ sage: J = JordanSpinEJA(3, prefix='f')
+ sage: J.one().subalgebra_generated_by().gens()
+ (g0,)
+ sage: J = JordanSpinEJA(3, prefix='a')
+ sage: J.one().subalgebra_generated_by().gens()
+ (b0,)
+
+ Ensure that we can find subalgebras of subalgebras::
+
+ sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
+ sage: B = A.one().subalgebra_generated_by()
+ sage: B.dimension()
+ 1
+ """
+ def __init__(self, superalgebra, basis, **kwargs):
self._superalgebra = superalgebra
- self._vector_space = W
- self._superalgebra_basis = superalgebra_basis
+ V = self._superalgebra.vector_space()
+ field = self._superalgebra.base_ring()
+
+ # A half-assed attempt to ensure that we don't collide with
+ # the superalgebra's prefix (ignoring the fact that there
+ # could be super-superelgrbas in scope). If possible, we
+ # try to "increment" the parent algebra's prefix, although
+ # this idea goes out the window fast because some prefixen
+ # are off-limits.
+ prefixen = ["b","c","d","e","f","g","h","l","m"]
+ try:
+ prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
+ except ValueError:
+ prefix = prefixen[0]
+
+ # 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))
+
+ super().__init__(matrix_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=superalgebra.matrix_space(),
+ prefix=prefix,
+ **kwargs)
+
+
+
+ def _element_constructor_(self, elt):
+ """
+ Construct an element of this subalgebra from the given one.
+ The only valid arguments are elements of the parent algebra
+ that happen to live in this subalgebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+ sage: from mjo.eja.eja_subalgebra import EJASubalgebra
+ EXAMPLES::
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
+ sage: J = RealSymmetricEJA(3)
+ 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 = EJASubalgebra(J,
+ ....: basis,
+ ....: associative=True,
+ ....: orthonormalize=False)
+ sage: K(J.one())
+ c1
+ sage: K(J.one() + x)
+ c0 + c1
+ ::
+
+ """
+ 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 FiniteDimensionalEuclideanJordanElementSubalgebra
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES::
- sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
- sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over Rational Field
- User basis matrix:
- [ 1 0 0 1 0 1]
- [ 0 1 2 3 4 5]
- [ 5 11 14 26 34 45]
- sage: (x^0).to_vector()
- (1, 0, 0, 1, 0, 1)
- sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
- sage: (x^2).to_vector()
- (5, 11, 14, 26, 34, 45)
+ 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 = FiniteDimensionalEuclideanJordanElementSubalgebraElement
+
+
+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