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
projects / sage.d.git / blobdiff