"""
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()]
+ 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(superalgebra.one().to_vector())]
+ 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
# 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):
product_vector = V.from_vector(product.to_vector())
mult_table[i][j] = W.coordinate_vector(product_vector)
- # 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(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
# 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.
sage: actual == expected
True
"""
- return self.monomial(self.one_basis())
+ 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):
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
+ sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over Rational Field
+ Vector space of degree 6 and dimension 3 over...
User basis matrix:
[ 1 0 1 0 0 1]
- [ 0 1 2 3 4 5]
- [10 14 21 19 31 50]
+ [ 1 0 2 0 0 5]
+ [ 1 0 4 0 0 25]
sage: (x^0).to_vector()
(1, 0, 1, 0, 0, 1)
sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
+ (1, 0, 2, 0, 0, 5)
sage: (x^2).to_vector()
- (10, 14, 21, 19, 31, 50)
+ (1, 0, 4, 0, 0, 25)
"""
return self._vector_space
projects / sage.d.git / blobdiff