# TODO: make this unnecessary somehow.
from sage.misc.lazy_import import lazy_import
lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
+lazy_import('mjo.eja.eja_subalgebra',
+ 'FiniteDimensionalEuclideanJordanElementSubalgebra')
from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
from mjo.eja.eja_utils import _mat2vec
True
"""
- return self.span_of_powers().dimension()
+ return self.subalgebra_generated_by().dimension()
def left_matrix(self):
0
"""
- V = self.span_of_powers()
- assoc_subalg = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- elt = assoc_subalg(V.coordinates(self.vector()))
- return elt.operator().minimal_polynomial()
+ A = self.subalgebra_generated_by()
+ return A(self).operator().minimal_polynomial()
return ( L*M + M*L - (self*other).operator() )
- def span_of_powers(self):
- """
- Return the vector space spanned by successive powers of
- this element.
- """
- # The dimension of the subalgebra can't be greater than
- # the big algebra, so just put everything into a list
- # and let span() get rid of the excess.
- #
- # We do the extra ambient_vector_space() in case we're messing
- # with polynomials and the direct parent is a module.
- V = self.parent().vector_space()
- return V.span( (self**d).vector() for d in xrange(V.dimension()) )
def subalgebra_generated_by(self):
sage: set_random_seed()
sage: x = random_eja().random_element()
- sage: u = x.subalgebra_generated_by().random_element()
- sage: u.operator()(u) == u^2
+ sage: A = x.subalgebra_generated_by()
+ sage: A(x^2) == A(x)*A(x)
True
"""
- # First get the subspace spanned by the powers of myself...
- V = self.span_of_powers()
- F = self.base_ring()
-
- # Now figure out the entries of the right-multiplication
- # matrix for the successive basis elements b0, b1,... of
- # that subspace.
- mats = []
- for b_right in V.basis():
- eja_b_right = self.parent()(b_right)
- 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 V.basis():
- eja_b_left = self.parent()(b_left)
- # Multiply in the original EJA, but then get the
- # coordinates from the subalgebra in terms of its
- # basis.
- this_row = V.coordinates((eja_b_left*eja_b_right).vector())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
- mats.append(b_right_matrix)
-
- # It's an algebra of polynomials in one element, and EJAs
- # are power-associative.
- #
- # TODO: choose generator names intelligently.
- #
- # 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 polynomial has the same degree
- # as the space's dimension, so that must be its rank too.
- return FiniteDimensionalEuclideanJordanAlgebra(
- F,
- mats,
- V.dimension(),
- assume_associative=True,
- names='f')
+ return FiniteDimensionalEuclideanJordanElementSubalgebra(self)
def subalgebra_idempotent(self):
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")
- V = self.span_of_powers()
J = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- u = J(V.coordinates(self.vector()))
+ u = J(self)
# The image of the matrix of left-u^m-multiplication
# will be minimal for some natural number s...
s = 0
- minimal_dim = V.dimension()
- for i in xrange(1, V.dimension()):
+ minimal_dim = J.dimension()
+ for i in xrange(1, minimal_dim):
this_dim = (u**i).operator().matrix().image().dimension()
if this_dim < minimal_dim:
minimal_dim = this_dim
# Our FiniteDimensionalAlgebraElement superclass uses rows.
u_next = u**(s+1)
A = u_next.operator().matrix()
- c_coordinates = A.solve_right(u_next.vector())
+ c = J(A.solve_right(u_next.vector()))
- # Now c_coordinates is the idempotent we want, but it's in
- # the coordinate system of the subalgebra.
- #
- # We need the basis for J, but as elements of the parent algebra.
- #
- basis = [self.parent(v) for v in V.basis()]
- return self.parent().linear_combination(zip(c_coordinates, basis))
+ # Now c is the idempotent we want, but it still lives in the subalgebra.
+ return c.superalgebra_element()
def trace(self):
# 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.
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::
::
"""
- 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 / commitdiff