from sage.matrix.constructor import matrix
-from sage.structure.category_object import normalize_names
from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+ """
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ The natural representation of an element in the subalgebra is
+ the same as its natural representation in the superalgebra::
+
+ sage: set_random_seed()
+ sage: A = random_eja().random_element().subalgebra_generated_by()
+ sage: y = A.random_element()
+ sage: actual = y.natural_representation()
+ sage: expected = y.superalgebra_element().natural_representation()
+ sage: actual == expected
+ True
+
+ """
+ def __init__(self, A, elt):
+ """
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x = sum( i*J.gens()[i] for i in range(6) )
+ sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: [ K.element_class(K,x^k) for k in range(J.rank()) ]
+ [f0, f1, f2]
+
+ ::
+
+ """
+ if elt in A.superalgebra():
+ # Try to convert a parent algebra element into a
+ # subalgebra element...
+ try:
+ coords = A.vector_space().coordinate_vector(elt.to_vector())
+ elt = A.from_vector(coords).monomial_coefficients()
+ except AttributeError:
+ # Catches a missing method in elt.to_vector()
+ pass
+
+ s = super(FiniteDimensionalEuclideanJordanElementSubalgebraElement,
+ self)
+
+ s.__init__(A, elt)
+
+
+ def superalgebra_element(self):
+ """
+ Return the object in our algebra's superalgebra that corresponds
+ to myself.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: random_eja)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x = sum(J.gens())
+ sage: x
+ e0 + e1 + e2 + e3 + e4 + e5
+ sage: A = x.subalgebra_generated_by()
+ sage: A.element_class(A,x)
+ f1
+ sage: A.element_class(A,x).superalgebra_element()
+ e0 + e1 + e2 + e3 + e4 + e5
+
+ TESTS:
+
+ We can convert back and forth faithfully::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: A.element_class(A,x).superalgebra_element() == x
+ True
+ sage: y = A.random_element()
+ sage: A.element_class(A,y.superalgebra_element()) == y
+ True
+
+ """
+ return self.parent().superalgebra().linear_combination(
+ zip(self.parent()._superalgebra_basis, self.to_vector()) )
+
+
+
+
class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
The subalgebra of an EJA generated by a single element.
"""
- @staticmethod
- def __classcall_private__(cls, elt):
+ 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().vector()]
+ 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.vector() )
+ basis_vectors.append( 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.
- F = superalgebra.base_ring()
+ field = superalgebra.base_ring()
mult_table = []
for b_right in superalgebra_basis:
b_right_rows = []
# 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).vector())
+ this_row = W.coordinates((b_left*b_right).to_vector())
b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
+ 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'
+
# 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()
- # EJAs are power-associative, and this algebra is nothin but
- # powers.
- assume_associative=True
-
- # TODO: Un-hard-code this. It should be possible to get the "next"
- # name based on the parent's generator names.
- names = 'f'
- names = normalize_names(W.dimension(), names)
-
- cat = superalgebra.category().Associative()
-
- # TODO: compute this and actually specify it.
- natural_basis = None
-
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls)
- return fdeja.__classcall__(cls,
- F,
- mult_table,
- rank,
- superalgebra_basis,
- W,
- assume_associative=assume_associative,
- names=names,
- category=cat,
- natural_basis=natural_basis)
-
- def __init__(self,
- field,
- mult_table,
- rank,
- superalgebra_basis,
- vector_space,
- assume_associative=True,
- names='f',
- category=None,
- natural_basis=None):
-
- self._superalgebra = superalgebra_basis[0].parent()
- self._vector_space = vector_space
+ 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
+
fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
- fdeja.__init__(field,
- mult_table,
- rank,
- assume_associative=assume_associative,
- names=names,
- category=category,
- natural_basis=natural_basis)
+ return fdeja.__init__(field,
+ mult_table,
+ rank,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis)
+
def superalgebra(self):
[ 1 0 0 1 0 1]
[ 0 1 2 3 4 5]
[ 5 11 14 26 34 45]
- sage: (x^0).vector()
+ sage: (x^0).to_vector()
(1, 0, 0, 1, 0, 1)
- sage: (x^1).vector()
+ sage: (x^1).to_vector()
(0, 1, 2, 3, 4, 5)
- sage: (x^2).vector()
+ sage: (x^2).to_vector()
(5, 11, 14, 26, 34, 45)
"""
return self._vector_space
- class Element(FiniteDimensionalEuclideanJordanAlgebraElement):
- def __init__(self, A, elt=None):
- """
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
-
- EXAMPLES::
-
- sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
- sage: [ K(x^k) for k in range(J.rank()) ]
- [f0, f1, f2]
-
- ::
-
- """
- if elt in A.superalgebra():
- # Try to convert a parent algebra element into a
- # subalgebra element...
- try:
- coords = A.vector_space().coordinates(elt.vector())
- elt = A(coords)
- except AttributeError:
- # Catches a missing method in elt.vector()
- pass
-
- FiniteDimensionalEuclideanJordanAlgebraElement.__init__(self,
- A,
- elt)
-
- def superalgebra_element(self):
- """
- Return the object in our algebra's superalgebra that corresponds
- to myself.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
- ....: random_eja)
-
- EXAMPLES::
-
- sage: J = RealSymmetricEJA(3)
- sage: x = sum(J.gens())
- sage: x
- e0 + e1 + e2 + e3 + e4 + e5
- sage: A = x.subalgebra_generated_by()
- sage: A(x)
- f1
- sage: A(x).superalgebra_element()
- e0 + e1 + e2 + e3 + e4 + e5
-
- TESTS:
-
- We can convert back and forth faithfully::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: A = x.subalgebra_generated_by()
- sage: A(x).superalgebra_element() == x
- True
- sage: y = A.random_element()
- sage: A(y.superalgebra_element()) == y
- True
-
- """
- return self.parent().superalgebra().linear_combination(
- zip(self.vector(), self.parent()._superalgebra_basis) )
+ Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement
projects / sage.d.git / blobdiff