from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-
class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
SETUP::
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
+ The left-multiplication-by operator for elements in the subalgebra
+ works like it does in the superalgebra, even if we orthonormalize
+ our basis::
- s = super(FiniteDimensionalEuclideanJordanElementSubalgebraElement,
- self)
-
- s.__init__(A, elt)
+ sage: set_random_seed()
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: y = A.random_element()
+ sage: y.operator()(A.one()) == y
+ True
+ """
def superalgebra_element(self):
"""
sage: x
e0 + e1 + e2 + e3 + e4 + e5
sage: A = x.subalgebra_generated_by()
- sage: A.element_class(A,x)
+ sage: A(x)
f1
- sage: A.element_class(A,x).superalgebra_element()
+ sage: A(x).superalgebra_element()
e0 + e1 + e2 + e3 + e4 + e5
TESTS:
sage: J = random_eja()
sage: x = J.random_element()
sage: A = x.subalgebra_generated_by()
- sage: A.element_class(A,x).superalgebra_element() == x
+ sage: A(x).superalgebra_element() == x
True
sage: y = A.random_element()
- sage: A.element_class(A,y.superalgebra_element()) == y
+ sage: A(y.superalgebra_element()) == y
True
"""
class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
The subalgebra of an EJA generated by a single element.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA)
+
+ TESTS:
+
+ Ensure that our generator names don't conflict with the superalgebra::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one().subalgebra_generated_by().gens()
+ (f0,)
+ sage: J = JordanSpinEJA(3, prefix='f')
+ sage: J.one().subalgebra_generated_by().gens()
+ (g0,)
+ sage: J = JordanSpinEJA(3, prefix='b')
+ sage: J.one().subalgebra_generated_by().gens()
+ (c0,)
+
+ Ensure that we can find subalgebras of subalgebras::
+
+ sage: A = ComplexHermitianEJA(3).one().subalgebra_generated_by()
+ sage: B = A.one().subalgebra_generated_by()
+ sage: B.dimension()
+ 1
+
"""
- 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().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.to_vector() )
- try:
- W = V.span_of_basis(basis_vectors)
- 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.
- 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.
- field = superalgebra.base_ring()
- mult_table = []
- 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
- # 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 superalgebra_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).to_vector())
- b_right_rows.append(this_row)
- 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'
+ def __init__(self, elt, orthonormalize_basis):
+ self._superalgebra = elt.parent()
+ category = self._superalgebra.category().Associative()
+ V = self._superalgebra.vector_space()
+ field = self._superalgebra.base_ring()
+
+ # A half-assed attempt to ensure that we don't collide with
+ # the superalgebra's prefix (ignoring the fact that there
+ # could be super-superelgrbas in scope). If possible, we
+ # try to "increment" the parent algebra's prefix, although
+ # this idea goes out the window fast because some prefixen
+ # are off-limits.
+ prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
+ try:
+ prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
+ except ValueError:
+ prefix = prefixen[0]
+
+ # This list is guaranteed to contain all independent powers,
+ # because it's the maximal set of powers that could possibly
+ # be independent (by a dimension argument).
+ powers = [ elt**k for k in range(V.dimension()) ]
+ power_vectors = [ p.to_vector() for p in powers ]
+ P = matrix(field, power_vectors)
+
+ if orthonormalize_basis == False:
+ # In this case, we just need to figure out which elements
+ # of the "powers" list are redundant... First compute the
+ # vector subspace spanned by the powers of the given
+ # element.
+
+ # Figure out which powers form a linearly-independent set.
+ ind_rows = P.pivot_rows()
+
+ # Pick those out of the list of all powers.
+ superalgebra_basis = tuple(map(powers.__getitem__, ind_rows))
+
+ # 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.
+ basis_vectors = map(power_vectors.__getitem__, ind_rows)
+ else:
+ # If we're going to orthonormalize the basis anyway, we
+ # might as well just do Gram-Schmidt on the whole list of
+ # powers. The redundant ones will get zero'd out. If this
+ # looks like a roundabout way to orthonormalize, it is.
+ # But converting everything from algebra elements to vectors
+ # to matrices and then back again turns out to be about
+ # as fast as reimplementing our own Gram-Schmidt that
+ # works in an EJA.
+ G,_ = P.gram_schmidt(orthonormal=True)
+ basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
+ superalgebra_basis = [ self._superalgebra.from_vector(b)
+ for b in basis_vectors ]
+
+ W = V.span_of_basis( V.from_vector(v) for v in basis_vectors )
+ 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]
+ # 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
+ # that case.
+ product_vector = V.from_vector(product.to_vector())
+ mult_table[i][j] = W.coordinate_vector(product_vector)
# 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()
- 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
natural_basis=natural_basis)
+ def _a_regular_element(self):
+ """
+ Override the superalgebra method to return the one
+ regular element that is sure to exist in this
+ subalgebra, namely the element that generated it.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: J._a_regular_element().is_regular()
+ True
+
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(1)
+
+
+ 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,False)
+ sage: [ K(x^k) for k in range(J.rank()) ]
+ [f0, f1, f2]
+
+ ::
+
+ """
+ if elt == 0:
+ # Just as in the superalgebra class, we need to hack
+ # this special case to ensure that random_element() can
+ # coerce a ring zero into the algebra.
+ return self.zero()
+
+ if elt in self.superalgebra():
+ coords = self.vector_space().coordinate_vector(elt.to_vector())
+ return self.from_vector(coords)
+
+
+
+ def one(self):
+ """
+ Return the multiplicative identity element of this algebra.
+
+ The superclass method computes the identity element, which is
+ beyond overkill in this case: the superalgebra identity
+ restricted to this algebra is its identity. Note that we can't
+ count on the first basis element being the identity -- it migth
+ have been scaled if we orthonormalized the basis.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ ....: random_eja)
+
+ EXAMPLES::
+
+ sage: J = RealCartesianProductEJA(5)
+ sage: J.one()
+ e0 + e1 + e2 + e3 + e4
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A.one()
+ f0
+ sage: A.one().superalgebra_element()
+ e0 + e1 + e2 + e3 + e4
+
+ TESTS:
+
+ The identity element acts like the identity over the rationals::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: x = A.random_element()
+ sage: A.one()*x == x and x*A.one() == x
+ True
+
+ The identity element acts like the identity over the algebraic
+ reals with an orthonormal basis::
+
+ sage: set_random_seed()
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: x = A.random_element()
+ sage: A.one()*x == x and x*A.one() == x
+ True
+
+ The matrix of the unit element's operator is the identity over
+ the rationals::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: A = x.subalgebra_generated_by()
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
+
+ The matrix of the unit element's operator is the identity over
+ the algebraic reals with an orthonormal basis::
+
+ sage: set_random_seed()
+ sage: x = random_eja(AA).random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize_basis=True)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
+
+ """
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ sa_one = self.superalgebra().one().to_vector()
+ sa_coords = self.vector_space().coordinate_vector(sa_one)
+ return self.from_vector(sa_coords)
+
+
+ def natural_basis_space(self):
+ """
+ Return the natural basis space of this algebra, which is identical
+ to that of its superalgebra.
+
+ This is correct "by definition," and avoids a mismatch when the
+ subalgebra is trivial (with no natural basis to infer anything
+ from) and the parent is not.
+ """
+ return self.superalgebra().natural_basis_space()
+
def superalgebra(self):
"""
EXAMPLES::
sage: J = RealSymmetricEJA(3)
- sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
+ sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over Rational Field
+ Vector space of degree 6 and dimension 3 over...
User basis matrix:
- [ 1 0 0 1 0 1]
- [ 0 1 2 3 4 5]
- [ 5 11 14 26 34 45]
+ [ 1 0 1 0 0 1]
+ [ 1 0 2 0 0 5]
+ [ 1 0 4 0 0 25]
sage: (x^0).to_vector()
- (1, 0, 0, 1, 0, 1)
+ (1, 0, 1, 0, 0, 1)
sage: (x^1).to_vector()
- (0, 1, 2, 3, 4, 5)
+ (1, 0, 2, 0, 0, 5)
sage: (x^2).to_vector()
- (5, 11, 14, 26, 34, 45)
+ (1, 0, 4, 0, 0, 25)
"""
return self._vector_space
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