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()) )
+ W = self.parent().vector_space()
+ V = self.parent().superalgebra().vector_space()
+ A = W.basis_matrix().transpose()
+ W_coords = A*self.to_vector()
+ V_coords = V.coordinate_vector(W_coords)
+ return self.parent().superalgebra().from_vector(V_coords)
1
"""
- def __init__(self, superalgebra, basis, rank=None, category=None):
+ def __init__(self, superalgebra, basis, category=None, check_axioms=True):
self._superalgebra = superalgebra
V = self._superalgebra.vector_space()
field = self._superalgebra.base_ring()
prefix = prefixen[0]
basis_vectors = [ b.to_vector() for b in basis ]
- superalgebra_basis = [ self._superalgebra.from_vector(b)
- for b in basis_vectors ]
+ # 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(v) for v in basis_vectors )
- n = len(superalgebra_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 = superalgebra_basis[i]*superalgebra_basis[j]
+ 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
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 superalgebra_basis )
+ natural_basis = tuple( b.natural_representation() for b in basis )
self._vector_space = W
- self._superalgebra_basis = superalgebra_basis
-
fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
+ fdeja.__init__(field,
+ mult_table,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis,
+ check_field=False,
+ check_axioms=check_axioms)