from sage.matrix.constructor import matrix
-from sage.structure.category_object import normalize_names
from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
+ """
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ The matrix representation of an element in the subalgebra is
+ the same as its matrix 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.to_matrix()
+ sage: expected = y.superalgebra_element().to_matrix()
+ 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
-class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
- The subalgebra of an EJA generated by a single element.
+
+ 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
+ sage: y = sum(A.gens())
+ sage: y
+ f0 + f1
+ sage: B = y.subalgebra_generated_by()
+ sage: B(y)
+ g1
+ sage: B(y).superalgebra_element()
+ f0 + f1
+
+ 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
+ sage: B = y.subalgebra_generated_by()
+ sage: B(y).superalgebra_element() == y
+ True
+
+ """
+ # As with the _element_constructor_() method on the
+ # algebra... even in a subspace of a subspace, the basis
+ # elements belong to the ambient space. As a result, only one
+ # level of coordinate_vector() is needed, regardless of how
+ # deeply we're nested.
+ W = self.parent().vector_space()
+ V = self.parent().superalgebra().vector_space()
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = self.to_vector()*W.basis_matrix()
+ V_coords = V.coordinate_vector(ambient_coords)
+ return self.parent().superalgebra().from_vector(V_coords)
+
+
+
+
+class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
"""
- @staticmethod
- def __classcall_private__(cls, 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().vector()]
- W = V.span_of_basis(basis_vectors)
- for exponent in range(1, V.dimension()):
- new_power = elt**exponent
- basis_vectors.append( new_power.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.
- F = 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).vector())
- b_right_rows.append(this_row)
- b_right_matrix = matrix(F, b_right_rows)
- mult_table.append(b_right_matrix)
-
- for m in mult_table:
- m.set_immutable()
- mult_table = tuple(mult_table)
-
- # 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()
-
- # EJAs are power-associative, and this algebra is nothin but
- # 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)
-
- cat = superalgebra.category().Associative()
-
- # TODO: compute this and actually specify it.
- natural_basis = None
-
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls)
- return fdeja.__classcall__(cls,
- F,
- mult_table,
- rank,
- superalgebra_basis,
- W,
- assume_associative=assume_associative,
- names=names,
- category=cat,
- natural_basis=natural_basis)
-
- def __init__(self,
- field,
- mult_table,
- rank,
- superalgebra_basis,
- vector_space,
- assume_associative=True,
- names='f',
- category=None,
- natural_basis=None):
-
- self._superalgebra = superalgebra_basis[0].parent()
- self._vector_space = vector_space
- self._superalgebra_basis = superalgebra_basis
-
- fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
+ A subalgebra of an EJA with a given basis.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: 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().to_matrix()
+ [1 0]
+ [0 0]
+ sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),))
+ sage: K2.one().to_matrix()
+ [0 0]
+ [0 1]
+
+ 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, superalgebra, basis, category=None, check_axioms=True):
+ 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
+ # 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]
+
+ # 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.
+ W = V.span_of_basis( V.from_vector(b.to_vector()) for b in basis )
+
+ n = len(basis)
+ mult_table = [[W.zero() for i in range(n)] for j in range(n)]
+ ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j])
+ for i in range(n) ]
+ for j in range(n) ]
+
+ for i in range(n):
+ for j in range(n):
+ product = basis[i]*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)
+
+ self._inner_product_matrix = matrix(field, ip_table)
+ matrix_basis = tuple( b.to_matrix() for b in basis )
+
+
+ self._vector_space = W
+
+ fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
fdeja.__init__(field,
mult_table,
- rank,
- assume_associative=assume_associative,
- names=names,
+ prefix=prefix,
category=category,
- natural_basis=natural_basis)
+ matrix_basis=matrix_basis,
+ check_field=False,
+ check_axioms=check_axioms)
+
+
+
+ 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 FiniteDimensionalEuclideanJordanSubalgebra
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(3)
+ 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 not in self.superalgebra():
+ raise ValueError("not an element of this subalgebra")
+
+ # The extra hackery is because foo.to_vector() might not live
+ # in foo.parent().vector_space()! Subspaces of subspaces still
+ # have user bases in the ambient space, though, so only one
+ # level of coordinate_vector() is needed. In other words, if V
+ # is itself a subspace, the basis elements for W will be of
+ # the same length as the basis elements for V -- namely
+ # whatever the dimension of the ambient (parent of V?) space is.
+ V = self.superalgebra().vector_space()
+ W = self.vector_space()
+
+ # Multiply on the left because basis_matrix() is row-wise.
+ ambient_coords = elt.to_vector()*V.basis_matrix()
+ W_coords = W.coordinate_vector(ambient_coords)
+ return self.from_vector(W_coords)
+
+
+
+ def matrix_space(self):
+ """
+ Return the matrix 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 matrix basis elements to
+ infer anything from) and the parent is not.
+ """
+ return self.superalgebra().matrix_space()
def superalgebra(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 = sum( i*J.gens()[i] for i in range(6) )
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x)
+ 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 Rational Field
+ Vector space of degree 6 and dimension 2 over...
User basis matrix:
- [ 1 0 0 1 0 1]
- [ 0 1 2 3 4 5]
- [ 5 11 14 26 34 45]
- sage: (x^0).vector()
- (1, 0, 0, 1, 0, 1)
- sage: (x^1).vector()
- (0, 1, 2, 3, 4, 5)
- sage: (x^2).vector()
- (5, 11, 14, 26, 34, 45)
+ [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
- class Element(FiniteDimensionalEuclideanJordanAlgebraElement):
- def __init__(self, A, elt=None):
- """
- 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(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().coordinates(elt.vector())
- elt = A(coords)
- except AttributeError:
- # Catches a missing method in elt.vector()
- pass
-
- 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) )
+ Element = FiniteDimensionalEuclideanJordanSubalgebraElement