class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
The subalgebra of an EJA generated by a single element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
+
+ TESTS:
+
+ Ensure that non-clashing names are chosen::
+
+ sage: m1 = matrix.identity(QQ,2)
+ sage: m2 = matrix(QQ, [[0,1],
+ ....: [1,0]])
+ sage: J = FiniteDimensionalEuclideanJordanAlgebra(QQ,
+ ....: [m1,m2],
+ ....: 2,
+ ....: names='f')
+ sage: J.variable_names()
+ ('f0', 'f1')
+ sage: A = sum(J.gens()).subalgebra_generated_by()
+ sage: A.variable_names()
+ ('g0', 'g1')
+
"""
@staticmethod
def __classcall_private__(cls, elt):
# First compute the vector subspace spanned by the powers of
# the given element.
V = superalgebra.vector_space()
- eja_basis = [superalgebra.one()]
+ superalgebra_basis = [superalgebra.one()]
basis_vectors = [superalgebra.one().vector()]
W = V.span_of_basis(basis_vectors)
for exponent in range(1, V.dimension()):
basis_vectors.append( new_power.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:
+ for b_right in superalgebra_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
#
# IMPORTANT: this assumes that all vectors are COLUMN
# vectors, unlike our superclass (which uses row vectors).
- for b_left in eja_basis:
+ for b_left in superalgebra_basis:
# Multiply in the original EJA, but then get the
# coordinates from the subalgebra in terms of its
# basis.
# 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)
+ # Figure out a non-conflicting set of names to use.
+ valid_names = ['f','g','h','a','b','c','d']
+ name_idx = 0
+ names = normalize_names(W.dimension(), valid_names[0])
+ # This loops so long as the list of collisions is nonempty.
+ # Just crash if we run out of names without finding a set that
+ # don't conflict with the parent algebra.
+ while [y for y in names if y in superalgebra.variable_names()]:
+ name_idx += 1
+ names = normalize_names(W.dimension(), valid_names[name_idx])
cat = superalgebra.category().Associative()
-
- # TODO: compute this and actually specify it.
- natural_basis = None
+ natural_basis = tuple( b.natural_representation()
+ for b in superalgebra_basis )
fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls)
return fdeja.__classcall__(cls,
F,
mult_table,
rank,
- eja_basis,
+ superalgebra_basis,
W,
assume_associative=assume_associative,
names=names,
field,
mult_table,
rank,
- eja_basis,
+ superalgebra_basis,
vector_space,
assume_associative=True,
names='f',
category=None,
natural_basis=None):
- self._superalgebra = eja_basis[0].parent()
+ self._superalgebra = superalgebra_basis[0].parent()
self._vector_space = vector_space
- self._eja_basis = eja_basis
+ self._superalgebra_basis = superalgebra_basis
fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
fdeja.__init__(field,
natural_basis=natural_basis)
+ def superalgebra(self):
+ """
+ Return the superalgebra that this algebra was generated from.
+ """
+ return self._superalgebra
+
+
def vector_space(self):
"""
SETUP::
class Element(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=None):
"""
SETUP::
::
"""
- if elt in A._superalgebra:
+ if elt in A.superalgebra():
# Try to convert a parent algebra element into a
# subalgebra element...
try:
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) )
projects / sage.d.git / blobdiff