- 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()
- mult_table = []
- for b_right in superalgebra_basis:
- b_right_cols = []
- # The first column of the left-multiplication matrix by
- # b1 is what we get if we apply that matrix to b1. The
- # second column of the left-multiplication matrix by b1
- # is what we get when we apply that matrix to b2...
- 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_col = W.coordinates((b_left*b_right).to_vector())
- b_right_cols.append(this_col)
- b_right_matrix = matrix.column(field, b_right_cols)
- 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'
+ def __init__(self, elt, orthonormalize_basis):
+ 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)
+
+
+ # 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):
+ 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)