from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-
+from mjo.eja.eja_utils import gram_schmidt
class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
1
"""
- def __init__(self, elt):
+ def __init__(self, elt, orthonormalize_basis):
self._superalgebra = elt.parent()
category = self._superalgebra.category().Associative()
V = self._superalgebra.vector_space()
natural_basis=natural_basis)
- # First compute the vector subspace spanned by the powers of
- # the given element.
- 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( 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.
+ # This list is guaranteed to contain all independent powers,
+ # because it's the maximal set of powers that could possibly
+ # be independent (by a dimension argument).
+ powers = [ elt**k for k in range(V.dimension()) ]
+
+ if orthonormalize_basis == False:
+ # In this case, we just need to figure out which elements
+ # of the "powers" list are redundant... First compute the
+ # vector subspace spanned by the powers of the given
+ # element.
+ power_vectors = [ p.to_vector() for p in powers ]
+
+ # Figure out which powers form a linearly-independent set.
+ ind_rows = matrix(field, power_vectors).pivot_rows()
+
+ # Pick those out of the list of all powers.
+ superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
+
+ # 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.
+ basis_vectors = map(power_vectors.__getitem__, ind_rows)
+ else:
+ # If we're going to orthonormalize the basis anyway, we
+ # might as well just do Gram-Schmidt on the whole list of
+ # powers. The redundant ones will get zero'd out.
+ superalgebra_basis = gram_schmidt(powers)
+ basis_vectors = [ b.to_vector() for b in superalgebra_basis ]
+
+ W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
n = len(superalgebra_basis)
mult_table = [[W.zero() for i in range(n)] for j in range(n)]
for i in range(n):
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.
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
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