X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=e39792a91732724b5fc7bc8b352e8fd977c80940;hb=6d30ad670e205bcfd299835ca67d93d7e1bfc2ec;hp=95534db842408f08480d012d6464fadf0c3e7fd4;hpb=b40f0964ea523f9063d62ec1772a5d698bf9c26a;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 95534db..e39792a 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -71,19 +71,81 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional 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): - 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 ) @@ -98,29 +160,17 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # Now figure out the entries of the right-multiplication # matrix for the successive basis elements b0, b1,... of # that subspace. - field = superalgebra.base_ring() - mult_table = [] - for b_right in superalgebra_basis: - b_right_cols = [] - # The first column of the left-multiplication matrix by - # b1 is what we get if we apply that matrix to b1. The - # second column of the left-multiplication matrix by b1 - # is what we get when we apply that matrix to b2... - 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_col = W.coordinates((b_left*b_right).to_vector()) - b_right_cols.append(this_col) - b_right_matrix = matrix.column(field, b_right_cols) - mult_table.append(b_right_matrix) - - for m in mult_table: - m.set_immutable() - mult_table = tuple(mult_table) - - # TODO: We'll have to redo this and make it unique again... - prefix = 'f' + 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 @@ -130,11 +180,10 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # 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 @@ -148,6 +197,30 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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. @@ -170,11 +243,76 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide :: """ + 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) + def one_basis(self): + """ + Return the basis-element-index of this algebra's unit element. + """ + return 0 + + + 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 algebra identity should be our + first basis element. We implement this via :meth:`one_basis` + because that method can optionally be used by other parts of the + category framework. + + 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:: + + sage: set_random_seed() + sage: J = random_eja().random_element().subalgebra_generated_by() + sage: x = J.random_element() + sage: J.one()*x == x and x*J.one() == x + True + + The matrix of the unit element's operator is the identity:: + + sage: set_random_seed() + sage: J = random_eja().random_element().subalgebra_generated_by() + sage: actual = J.one().operator().matrix() + sage: expected = matrix.identity(J.base_ring(), J.dimension()) + sage: actual == expected + True + """ + if self.dimension() == 0: + return self.zero() + else: + return self.monomial(self.one_basis()) + + def superalgebra(self): """ Return the superalgebra that this algebra was generated from.