from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-
-class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
SETUP::
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: 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
+ True
+
"""
def superalgebra_element(self):
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()
+ 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()) )
+ # 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)
-class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
- 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 JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+
+ 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 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
+ sage: K1.one().natural_representation()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2.one().natural_representation()
+ [0 0]
+ [0 1]
TESTS:
sage: J.one().subalgebra_generated_by().gens()
(c0,)
+ 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, 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()
- 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]
- mult_table[i][j] = W.coordinate_vector(product.to_vector())
+ def __init__(self, superalgebra, basis, category=None, check_axioms=True):
+ 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
# are off-limits.
prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
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()
+ # 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 )
+
+ 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)
+
+ natural_basis = tuple( b.natural_representation() for b in basis )
- 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
+ fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
+ fdeja.__init__(field,
+ mult_table,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis,
+ check_field=False,
+ check_axioms=check_axioms)
- 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 FiniteDimensionalEuclideanJordanSubalgebra
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 = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
+ sage: K(J.one())
+ f1
+ sage: K(J.one() + x)
+ f0 + f1
::
"""
- 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 not in self.superalgebra():
+ raise ValueError("not an element of this subalgebra")
- if elt in self.superalgebra():
- coords = self.vector_space().coordinate_vector(elt.to_vector())
- return self.from_vector(coords)
+ # 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 one_basis(self):
- """
- Return the basis-element-index of this algebra's unit element.
- """
- return 0
- def one(self):
+ def natural_basis_space(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.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
- ....: random_eja)
-
- 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
+ Return the natural basis space of this algebra, which is identical
+ to that of its superalgebra.
- TESTS:
-
- The identity element acts like the identity::
-
- 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
+ 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.monomial(self.one_basis())
+ return self.superalgebra().natural_basis_space()
def superalgebra(self):
SETUP::
sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ 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 3 over Rational Field
+ Vector space of degree 6 and dimension 2 over...
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)
+ [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 = FiniteDimensionalEuclideanJordanElementSubalgebraElement
+ Element = FiniteDimensionalEuclideanJordanSubalgebraElement