class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
The subalgebra of an EJA generated by a single element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA)
+
+ TESTS:
+
+ Ensure that our generator names don't conflict with the superalgebra::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one().subalgebra_generated_by().gens()
+ (f0,)
+ sage: J = JordanSpinEJA(3, prefix='f')
+ sage: J.one().subalgebra_generated_by().gens()
+ (g0,)
+ sage: J = JordanSpinEJA(3, prefix='b')
+ 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()
+ self._superalgebra = elt.parent()
+ category = self._superalgebra.category().Associative()
+ 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 = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
+ try:
+ prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
+ except ValueError:
+ prefix = prefixen[0]
+
+ if elt.is_zero():
+ # Short circuit because 0^0 == 1 is going to make us
+ # think we have a one-dimensional algebra otherwise.
+ natural_basis = tuple()
+ mult_table = tuple()
+ rank = 0
+ self._vector_space = V.zero_subspace()
+ self._superalgebra_basis = []
+ fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra,
+ self)
+ return fdeja.__init__(field,
+ mult_table,
+ rank,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis)
+
# 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()]
+ superalgebra_basis = [self._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(self._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() )
+ basis_vectors.append( V.from_vector(new_power.to_vector()) )
try:
W = V.span_of_basis(basis_vectors)
superalgebra_basis.append( new_power )
# 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'
+ 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)
# The rank is the highest possible degree of a minimal
# polynomial, and is bounded above by the dimension. We know
# its rank too.
rank = W.dimension()
- 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
natural_basis=natural_basis)
+ def _a_regular_element(self):
+ """
+ Override the superalgebra method to return the one
+ regular element that is sure to exist in this
+ subalgebra, namely the element that generated it.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: J._a_regular_element().is_regular()
+ True
+
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(1)
+
+
def _element_constructor_(self, elt):
"""
Construct an element of this subalgebra from the given one.
::
"""
+ 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)
+ def one_basis(self):
+ """
+ Return the basis-element-index of this algebra's unit element.
+ """
+ return 0
+
+
+ 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.
+
+ 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
+
+ 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
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(self.one_basis())
+
+
+ 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()
+
+
def superalgebra(self):
"""
Return the superalgebra that this algebra was generated from.
projects / sage.d.git / blobdiff