from sage.matrix.constructor import matrix
-from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
-class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
- """
- SETUP::
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS::
-
- The natural representation of an element in the subalgebra is
- the same as its natural 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.natural_representation()
- sage: expected = y.superalgebra_element().natural_representation()
- 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):
- """
- 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.parent()._superalgebra_basis, self.to_vector()) )
-
-
-
-
-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
-
- """
+class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanSubalgebra):
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).
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()
- natural_basis = tuple( b.natural_representation()
- for b in superalgebra_basis )
-
-
- self._vector_space = W
- self._superalgebra_basis = superalgebra_basis
-
-
fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self)
- return fdeja.__init__(field,
- mult_table,
- rank,
- prefix=prefix,
- category=category,
- natural_basis=natural_basis)
+ return fdeja.__init__(self._superalgebra,
+ superalgebra_basis,
+ rank=rank,
+ category=category)
def _a_regular_element(self):
"""
return self._vector_space
-
- Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement
--- /dev/null
+from sage.matrix.constructor import matrix
+
+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 natural representation of an element in the subalgebra is
+ the same as its natural 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.natural_representation()
+ sage: expected = y.superalgebra_element().natural_representation()
+ 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):
+ """
+ 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.parent()._superalgebra_basis, self.to_vector()) )
+
+
+
+
+class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra):
+ """
+ A subalgebra of an EJA with a given basis.
+
+ 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, 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
+ # 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]
+
+ 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)
+ 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)
+
+ natural_basis = tuple( b.natural_representation()
+ for b in superalgebra_basis )
+
+
+ self._vector_space = W
+ self._superalgebra_basis = superalgebra_basis
+
+
+ fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self)
+ return fdeja.__init__(field,
+ mult_table,
+ rank,
+ prefix=prefix,
+ category=category,
+ natural_basis=natural_basis)
+
+
+
+ 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 = sum( i*J.gens()[i] for i in range(6) )
+ 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 not in self.superalgebra():
+ raise ValueError("not an element of this subalgebra")
+
+ 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):
+ """
+ Return the superalgebra that this algebra was generated from.
+ """
+ return self._superalgebra
+
+
+ def vector_space(self):
+ """
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import RealSymmetricEJA
+ 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: 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:
+ [ 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)
+
+ """
+ return self._vector_space
+
+
+ Element = FiniteDimensionalEuclideanJordanSubalgebraElement
projects / sage.d.git / commitdiff