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 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.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()
- eja_basis = [superalgebra.one()]
- basis_vectors = [superalgebra.one().vector()]
+ 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.vector() )
+ basis_vectors.append( new_power.to_vector() )
try:
W = V.span_of_basis(basis_vectors)
- eja_basis.append( new_power )
+ 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.
- eja_basis = tuple(eja_basis)
+ 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.
- F = superalgebra.base_ring()
- mult_table = []
- for b_right in eja_basis:
- b_right_rows = []
- # The first row of the right-multiplication matrix by
- # b1 is what we get if we apply that matrix to b1. The
- # second row of the right multiplication matrix by b1
- # is what we get when we apply that matrix to b2...
- #
- # IMPORTANT: this assumes that all vectors are COLUMN
- # vectors, unlike our superclass (which uses row vectors).
- for b_left in eja_basis:
- # 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())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
- mult_table.append(b_right_matrix)
-
- for m in mult_table:
- m.set_immutable()
- mult_table = tuple(mult_table)
+ 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):
+ for j in range(n):
+ product = superalgebra_basis[i]*superalgebra_basis[j]
+ mult_table[i][j] = W.coordinate_vector(product.to_vector())
+
+ # 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
# 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,
- eja_basis,
- W,
- assume_associative=assume_associative,
- names=names,
- category=cat,
- natural_basis=natural_basis)
-
- def __init__(self,
- field,
- mult_table,
- rank,
- eja_basis,
- vector_space,
- assume_associative=True,
- names='f',
- category=None,
- natural_basis=None):
-
- self._superalgebra = eja_basis[0].parent()
- self._vector_space = vector_space
- self._eja_basis = eja_basis
+ 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 _element_constructor_(self, elt):
+ """
+ Construct an element of this subalgebra from the given one.
+ The only valid arguments are elements of the parent algebra
+ that happen to live in this subalgebra.
+
+ 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 self.superalgebra():
+ coords = self.vector_space().coordinate_vector(elt.to_vector())
+ return self.from_vector(coords)
+
+
+ def superalgebra(self):
+ """
+ Return the superalgebra that this algebra was generated from.
+ """
+ return self._superalgebra
def vector_space(self):
sage: K.vector_space()
Vector space of degree 6 and dimension 3 over Rational Field
User basis matrix:
- [ 1 0 0 1 0 1]
+ [ 1 0 1 0 0 1]
[ 0 1 2 3 4 5]
- [ 5 11 14 26 34 45]
- sage: (x^0).vector()
- (1, 0, 0, 1, 0, 1)
- sage: (x^1).vector()
+ [10 14 21 19 31 50]
+ sage: (x^0).to_vector()
+ (1, 0, 1, 0, 0, 1)
+ sage: (x^1).to_vector()
(0, 1, 2, 3, 4, 5)
- sage: (x^2).vector()
- (5, 11, 14, 26, 34, 45)
+ sage: (x^2).to_vector()
+ (10, 14, 21, 19, 31, 50)
"""
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)
+ Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement