X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=0f641416366ef7ee72d4703e070c69e2d60e30a9;hb=08aba469c5f8d8947a543f8882fa676ed165e7ee;hp=95534db842408f08480d012d6464fadf0c3e7fd4;hpb=b40f0964ea523f9063d62ec1772a5d698bf9c26a;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 95534db..0f64141 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -2,7 +2,7 @@ from sage.matrix.constructor import matrix from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement - +from mjo.eja.eja_utils import gram_schmidt class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): """ @@ -23,6 +23,17 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional 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): @@ -71,56 +82,93 @@ 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() - - # 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()] - 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() ) - try: - W = V.span_of_basis(basis_vectors) - superalgebra_basis.append( new_power ) - except ValueError: - # Vectors weren't independent; bail and keep the - # last subspace that worked. - break - - # Make the basis hashable for UniqueRepresentation. - superalgebra_basis = tuple(superalgebra_basis) - - # 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' + 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()) ] + + 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. + power_vectors = [ p.to_vector() for p in powers ] + + # Figure out which powers form a linearly-independent set. + ind_rows = matrix(field, power_vectors).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. + superalgebra_basis = gram_schmidt(powers) + basis_vectors = [ b.to_vector() for b in superalgebra_basis ] + + 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 @@ -130,11 +178,10 @@ 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 +195,30 @@ 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. @@ -163,18 +234,116 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide sage: J = RealSymmetricEJA(3) sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) + 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) + + 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. @@ -192,20 +361,20 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) + 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 Rational Field + Vector space of degree 6 and dimension 3 over... User basis matrix: [ 1 0 1 0 0 1] - [ 0 1 2 3 4 5] - [10 14 21 19 31 50] + [ 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() - (0, 1, 2, 3, 4, 5) + (1, 0, 2, 0, 0, 5) sage: (x^2).to_vector() - (10, 14, 21, 19, 31, 50) + (1, 0, 4, 0, 0, 25) """ return self._vector_space