from sage.matrix.constructor import matrix
+from sage.misc.cachefunc import cached_method
+from sage.rings.all import QQ
-from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
+from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra
-class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement):
- """
- SETUP::
- sage: from mjo.eja.eja_algebra import random_eja
+class FiniteDimensionalEJAElementSubalgebra(FiniteDimensionalEJASubalgebra):
+ def __init__(self, elt, orthonormalize=True, **kwargs):
+ superalgebra = elt.parent()
- TESTS::
+ # TODO: going up to the superalgebra dimension here is
+ # overkill. We should append p vectors as rows to a matrix
+ # and continually rref() it until the rank stops going
+ # up. When n=10 but the dimension of the algebra is 1, that
+ # can save a shitload of time (especially over AA).
+ powers = tuple( elt**k for k in range(elt.degree()) )
- 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
-
- """
- 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).
- powers = [ elt**k for k in range(V.dimension()) ]
- power_vectors = [ p.to_vector() for p in powers ]
- P = matrix(field, power_vectors)
-
- 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.
-
- # Figure out which powers form a linearly-independent set.
- ind_rows = P.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. If this
- # looks like a roundabout way to orthonormalize, it is.
- # But converting everything from algebra elements to vectors
- # to matrices and then back again turns out to be about
- # as fast as reimplementing our own Gram-Schmidt that
- # works in an EJA.
- G,_ = P.gram_schmidt(orthonormal=True)
- basis_vectors = [ g for g in G.rows() if not g.is_zero() ]
- 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)
+ super().__init__(superalgebra,
+ powers,
+ associative=True,
+ **kwargs)
# The rank is the highest possible degree of a minimal
# polynomial, and is bounded above by the dimension. We know
# 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._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)
-
-
- 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):
- """
- 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_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
-
- 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]
-
- ::
-
- """
- 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)
-
+ self.rank.set_cache(self.dimension())
+ @cached_method
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
+ count on the first basis element being the identity -- it might
have been scaled if we orthonormalized the basis.
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
EXAMPLES::
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
sage: x = sum(J.gens())
The identity element acts like the identity over the rationals::
sage: set_random_seed()
- sage: x = random_eja().random_element()
+ sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
sage: A = x.subalgebra_generated_by()
sage: x = A.random_element()
sage: A.one()*x == x and x*A.one() == x
reals with an orthonormal basis::
sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
+ sage: x = random_eja().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
the rationals::
sage: set_random_seed()
- sage: x = random_eja().random_element()
+ sage: x = random_eja(field=QQ,orthonormalize=False).random_element()
sage: A = x.subalgebra_generated_by()
sage: actual = A.one().operator().matrix()
sage: expected = matrix.identity(A.base_ring(), A.dimension())
the algebraic reals with an orthonormal basis::
sage: set_random_seed()
- sage: x = random_eja(AA).random_element()
+ sage: x = random_eja().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())
"""
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_element_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra
-
- EXAMPLES::
-
- sage: J = RealSymmetricEJA(3)
- sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5)
- sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False)
- 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
+ return self(self.superalgebra().one())
- Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement