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::
-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::
sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
- ....: JordanSpinEJA)
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+ sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
+
+ EXAMPLES:
+
+ The following Peirce subalgebras of the 2-by-2 real symmetric
+ matrices do not contain the superalgebra's identity element::
+
+ sage: J = RealSymmetricEJA(2)
+ sage: E11 = matrix(AA, [ [1,0],
+ ....: [0,0] ])
+ sage: E22 = matrix(AA, [ [0,0],
+ ....: [0,1] ])
+ sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),))
+ sage: K1.one().natural_representation()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2.one().natural_representation()
+ [0 0]
+ [0 1]
TESTS:
1
"""
- def __init__(self, elt, orthonormalize_basis):
- self._superalgebra = elt.parent()
- category = self._superalgebra.category().Associative()
+ def __init__(self, superalgebra, basis, 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)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- 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
+ fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
+ fdeja.__init__(field,
+ mult_table,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis)
- """
- 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,False)
- sage: [ K(x^k) for k in range(J.rank()) ]
- [f0, f1, f2]
+ sage: X = matrix(AA, [ [0,0,1],
+ ....: [0,1,0],
+ ....: [1,0,0] ])
+ sage: x = J(X)
+ sage: basis = ( x, x^2 ) # x^2 is the identity matrix
+ sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J, basis)
+ sage: K(J.one())
+ f1
+ sage: K(J.one() + x)
+ f0 + f1
::
"""
- 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::
+ if elt not in self.superalgebra():
+ raise ValueError("not an element of this subalgebra")
- 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
+ # The extra hackery is because foo.to_vector() might not
+ # live in foo.parent().vector_space()!
+ coords = sum( a*b for (a,b)
+ in zip(elt.to_vector(),
+ self.superalgebra().vector_space().basis()) )
+ return self.from_vector(self.vector_space().coordinate_vector(coords))
- """
- 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):
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,False)
+ sage: E11 = matrix(ZZ, [ [1,0,0],
+ ....: [0,0,0],
+ ....: [0,0,0] ])
+ sage: E22 = matrix(ZZ, [ [0,0,0],
+ ....: [0,1,0],
+ ....: [0,0,0] ])
+ sage: b1 = J(E11)
+ sage: b2 = J(E22)
+ sage: basis = (b1, b2)
+ sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis)
sage: K.vector_space()
- Vector space of degree 6 and dimension 3 over...
+ Vector space of degree 6 and dimension 2 over...
User basis matrix:
- [ 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, 1, 0, 0, 1)
- sage: (x^1).to_vector()
- (1, 0, 2, 0, 0, 5)
- sage: (x^2).to_vector()
- (1, 0, 4, 0, 0, 25)
+ [1 0 0 0 0 0]
+ [0 0 1 0 0 0]
+ sage: b1.to_vector()
+ (1, 0, 0, 0, 0, 0)
+ sage: b2.to_vector()
+ (0, 0, 1, 0, 0, 0)
"""
return self._vector_space
- Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement
+ Element = FiniteDimensionalEuclideanJordanSubalgebraElement
projects / sage.d.git / blobdiff