from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-from mjo.eja.eja_utils import gram_schmidt
-class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
"""
SETUP::
sage: actual == expected
True
+ The left-multiplication-by operator for elements in the subalgebra
+ works like it does in the superalgebra, even if we orthonormalize
+ our basis::
+
+ 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):
-class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
+class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
- The subalgebra of an EJA generated by a single element.
+ A subalgebra of an EJA with a given basis.
SETUP::
1
"""
- def __init__(self, elt, orthonormalize_basis):
- self._superalgebra = elt.parent()
- category = self._superalgebra.category().Associative()
+ def __init__(self, superalgebra, basis, rank=None, category=None):
+ self._superalgebra = superalgebra
V = self._superalgebra.vector_space()
field = self._superalgebra.base_ring()
+ if category is None:
+ category = self._superalgebra.category()
# A half-assed attempt to ensure that we don't collide with
# the superalgebra's prefix (ignoring the fact that there
except ValueError:
prefix = prefixen[0]
- if elt.is_zero():
- # Short circuit because 0^0 == 1 is going to make us
- # think we have a one-dimensional algebra otherwise.
- natural_basis = tuple()
- mult_table = tuple()
- rank = 0
- self._vector_space = V.zero_subspace()
- self._superalgebra_basis = []
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra,
- self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
-
-
- # 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()) ]
-
- 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.
- power_vectors = [ p.to_vector() for p in powers ]
-
- # Figure out which powers form a linearly-independent set.
- ind_rows = matrix(field, power_vectors).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.
- superalgebra_basis = gram_schmidt(powers)
- basis_vectors = [ b.to_vector() for b in superalgebra_basis ]
+ basis_vectors = [ b.to_vector() for b in basis ]
+ 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)
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
- # in this case that there's an element whose minimal
- # polynomial has the same degree as the space's dimension
- # (remember how we constructed the space?), so that must be
- # its rank too.
- rank = W.dimension()
-
natural_basis = tuple( b.natural_representation()
for b in superalgebra_basis )
self._superalgebra_basis = superalgebra_basis
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
+ fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
return fdeja.__init__(field,
mult_table,
rank,
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):
"""
SETUP::
sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
EXAMPLES::
sage: J = RealSymmetricEJA(3)
sage: x = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: basis = tuple( x^k for k in range(J.rank()) )
+ sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
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)
+ if elt not in self.superalgebra():
+ raise ValueError("not an element of this subalgebra")
-
- def one_basis(self):
- """
- Return the basis-element-index of this algebra's unit element.
- """
- return 0
+ 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 algebra identity should be our
- first basis element. We implement this via :meth:`one_basis`
- because that method can optionally be used by other parts of the
- category framework.
+ 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::
TESTS:
- The identity element acts like the identity::
+ The identity element acts like the identity over the rationals::
sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: x = J.random_element()
- sage: J.one()*x == x and x*J.one() == x
+ 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 matrix of the unit element's operator is the identity::
+ The identity element acts like the identity over the algebraic
+ reals with an orthonormal basis::
sage: set_random_seed()
- sage: J = random_eja().random_element().subalgebra_generated_by()
- sage: actual = J.one().operator().matrix()
- sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ 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:
- return self.monomial(self.one_basis())
+ 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):
SETUP::
sage: from mjo.eja.eja_algebra import RealSymmetricEJA
- sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
EXAMPLES::
sage: J = RealSymmetricEJA(3)
sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ sage: basis = (x^0, x^1, x^2)
+ sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
sage: K.vector_space()
Vector space of degree 6 and dimension 3 over...
User basis matrix:
return self._vector_space
- Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement
+ Element = FiniteDimensionalEuclideanJordanSubalgebraElement
projects / sage.d.git / blobdiff