SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA)
TESTS:
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):
- superalgebra = elt.parent()
+ 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]
+
+ 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)
+
# 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().to_vector()]
+ superalgebra_basis = [self._superalgebra.one()]
+ # If our superalgebra is a subalgebra of something else, then
+ # superalgebra.one().to_vector() won't have the right
+ # coordinates unless we use V.from_vector() below.
+ basis_vectors = [V.from_vector(self._superalgebra.one().to_vector())]
W = V.span_of_basis(basis_vectors)
for exponent in range(1, V.dimension()):
new_power = elt**exponent
- basis_vectors.append( new_power.to_vector() )
+ basis_vectors.append( V.from_vector(new_power.to_vector()) )
try:
W = V.span_of_basis(basis_vectors)
superalgebra_basis.append( new_power )
# Now figure out the entries of the right-multiplication
# matrix for the successive basis elements b0, b1,... of
# that subspace.
- field = superalgebra.base_ring()
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]
- mult_table[i][j] = W.coordinate_vector(product.to_vector())
-
- # 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(superalgebra.prefix()) + 1]
- except ValueError:
- prefix = prefixen[0]
+ # 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()
- category = superalgebra.category().Associative()
natural_basis = tuple( b.natural_representation()
for b in superalgebra_basis )
- self._superalgebra = superalgebra
+
self._vector_space = W
self._superalgebra_basis = superalgebra_basis
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.
sage: actual == expected
True
"""
- return self.monomial(self.one_basis())
+ if self.dimension() == 0:
+ return self.zero()
+ else:
+ return self.monomial(self.one_basis())
def superalgebra(self):