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
+ The left-multiplication-by operator for elements in the subalgebra
+ works like it does in the superalgebra, even if we orthonormalize
+ our basis::
+
+ 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()
+ 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
+ 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 = 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(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):
+class EJASubalgebra(EJA):
"""
- The subalgebra of an EJA generated by a single element.
+ A subalgebra of an EJA with a given basis.
SETUP::
sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: JordanSpinEJA)
+ ....: 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::
+ 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, 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()]
- # If our superalgebra is a subalgebra of something else, then
- # superalgebra.one().to_vector() won't have the right
- # coordinates unless we use V.from_vector() below.
- basis_vectors = [V.from_vector(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( V.from_vector(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()
- 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)
+ def __init__(self, superalgebra, basis, **kwargs):
+ self._superalgebra = superalgebra
+ 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
# 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(superalgebra.prefix()) + 1]
+ prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
except ValueError:
prefix = prefixen[0]
- # 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()
+ # 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()
- category = superalgebra.category().Associative()
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
-
- self._superalgebra = superalgebra
- self._vector_space = W
- self._superalgebra_basis = superalgebra_basis
+ 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)
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
def _element_constructor_(self, elt):
SETUP::
sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
+ sage: from mjo.eja.eja_subalgebra import EJASubalgebra
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
- 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 = EJASubalgebra(J,
+ ....: basis,
+ ....: associative=True,
+ ....: orthonormalize=False)
+ sage: K(J.one())
+ c1
+ sage: K(J.one() + x)
+ c0 + c1
::
"""
- if elt == 0:
- # Just as in the superalgebra class, we need to hack
- # this special case to ensure that random_element() can
- # coerce a ring zero into the algebra.
- return self.zero()
-
if elt in self.superalgebra():
- coords = self.vector_space().coordinate_vector(elt.to_vector())
- return self.from_vector(coords)
+ # 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 one_basis(self):
+ def superalgebra(self):
"""
- Return the basis-element-index of this algebra's unit element.
+ Return the superalgebra that this algebra was generated from.
"""
- return 0
-
+ return self._superalgebra
- def one(self):
- """
- Return the multiplicative identity element of this algebra.
- The superclass method computes the identity element, which is
- beyond overkill in this case: the algebra identity should be our
- first basis element. We implement this via :meth:`one_basis`
- because that method can optionally be used by other parts of the
- category framework.
+ @cached_method
+ def superalgebra_embedding(self):
+ r"""
+ Return the embedding from this subalgebra into the superalgebra.
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
- ....: random_eja)
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES::
- sage: J = RealCartesianProductEJA(5)
- sage: J.one()
- e0 + e1 + e2 + e3 + e4
- sage: x = sum(J.gens())
- sage: A = x.subalgebra_generated_by()
- sage: A.one()
- f0
- sage: A.one().superalgebra_element()
- e0 + e1 + e2 + e3 + e4
-
- TESTS:
+ 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
- The identity element acts like the identity::
+ """
+ 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())
- sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: x = J.random_element()
- sage: J.one()*x == x and x*J.one() == x
- True
- The matrix of the unit element's operator is the identity::
- sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: actual = J.one().operator().matrix()
- sage: expected = matrix.identity(J.base_ring(), J.dimension())
- sage: actual == expected
- True
- """
- return self.monomial(self.one_basis())
+ Element = EJASubalgebraElement
- def superalgebra(self):
- """
- Return the superalgebra that this algebra was generated from.
- """
- return self._superalgebra
+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.
- def vector_space(self):
- """
- SETUP::
+ 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,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ TESTS:
- 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 1 0 0 1]
- [ 0 1 2 3 4 5]
- [10 14 21 19 31 50]
- sage: (x^0).to_vector()
- (1, 0, 1, 0, 0, 1)
- sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
- sage: (x^2).to_vector()
- (10, 14, 21, 19, 31, 50)
+ This used to fail when ``subalgebra_idempotent()`` tried to
+ embed the subalgebra element back into the original EJA::
- """
- return self._vector_space
+ 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
- Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement
+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