X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=0f641416366ef7ee72d4703e070c69e2d60e30a9;hb=08aba469c5f8d8947a543f8882fa676ed165e7ee;hp=5ac0a77c7a9ab27929f4c3e1d73238d93e1ccd1b;hpb=57f22a182c4de750b0a03dac257143e1dca04341;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 5ac0a77..0f64141 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,8 +1,82 @@ from sage.matrix.constructor import matrix -from sage.structure.category_object import normalize_names 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): + """ + 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): @@ -11,76 +85,90 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide SETUP:: - sage: from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: JordanSpinEJA) TESTS: - Ensure that non-clashing names are chosen:: - - sage: m1 = matrix.identity(QQ,2) - sage: m2 = matrix(QQ, [[0,1], - ....: [1,0]]) - sage: J = FiniteDimensionalEuclideanJordanAlgebra(QQ, - ....: [m1,m2], - ....: 2, - ....: names='f') - sage: J.variable_names() - ('f0', 'f1') - sage: A = sum(J.gens()).subalgebra_generated_by() - sage: A.variable_names() - ('g0', 'g1') + 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 """ - @staticmethod - def __classcall_private__(cls, 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().vector()] - W = V.span_of_basis(basis_vectors) - for exponent in range(1, V.dimension()): - new_power = elt**exponent - basis_vectors.append( new_power.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. - F = superalgebra.base_ring() - mult_table = [] - for b_right in superalgebra_basis: - b_right_rows = [] - # The first row of the right-multiplication matrix by - # b1 is what we get if we apply that matrix to b1. The - # second row of the right multiplication matrix by b1 - # is what we get when we apply that matrix to b2... - # - # IMPORTANT: this assumes that all vectors are COLUMN - # vectors, unlike our superclass (which uses row vectors). - 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_row = W.coordinates((b_left*b_right).vector()) - b_right_rows.append(this_row) - b_right_matrix = matrix(F, b_right_rows) - mult_table.append(b_right_matrix) - - for m in mult_table: - m.set_immutable() - mult_table = tuple(mult_table) + 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 @@ -90,71 +178,53 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # its rank too. rank = W.dimension() - # EJAs are power-associative, and this algebra is nothin but - # powers. - assume_associative=True - - # Figure out a non-conflicting set of names to use. - valid_names = ['f','g','h','a','b','c','d'] - name_idx = 0 - names = normalize_names(W.dimension(), valid_names[0]) - # This loops so long as the list of collisions is nonempty. - # Just crash if we run out of names without finding a set that - # don't conflict with the parent algebra. - while [y for y in names if y in superalgebra.variable_names()]: - name_idx += 1 - names = normalize_names(W.dimension(), valid_names[name_idx]) - - cat = superalgebra.category().Associative() natural_basis = tuple( b.natural_representation() for b in superalgebra_basis ) - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls) - return fdeja.__classcall__(cls, - F, - mult_table, - rank, - superalgebra_basis, - W, - assume_associative=assume_associative, - names=names, - category=cat, - natural_basis=natural_basis) - - def __init__(self, - field, - mult_table, - rank, - superalgebra_basis, - vector_space, - assume_associative=True, - names='f', - category=None, - natural_basis=None): - - self._superalgebra = superalgebra_basis[0].parent() - self._vector_space = vector_space + + self._vector_space = W self._superalgebra_basis = superalgebra_basis + fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - fdeja.__init__(field, - mult_table, - rank, - assume_associative=assume_associative, - names=names, - category=category, - natural_basis=natural_basis) + return fdeja.__init__(field, + mult_table, + rank, + prefix=prefix, + category=category, + natural_basis=natural_basis) - def superalgebra(self): + def _a_regular_element(self): """ - Return the superalgebra that this algebra was generated from. + 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 + """ - return self._superalgebra + if self.dimension() == 0: + return self.zero() + else: + return self.monomial(1) - def vector_space(self): + 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 @@ -164,113 +234,150 @@ 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.vector_space() - Vector space of degree 6 and dimension 3 over Rational Field - User basis matrix: - [ 1 0 0 1 0 1] - [ 0 1 2 3 4 5] - [ 5 11 14 26 34 45] - sage: (x^0).vector() - (1, 0, 0, 1, 0, 1) - sage: (x^1).vector() - (0, 1, 2, 3, 4, 5) - sage: (x^2).vector() - (5, 11, 14, 26, 34, 45) + sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) + sage: [ K(x^k) for k in range(J.rank()) ] + [f0, f1, f2] + + :: """ - return self._vector_space + 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) - class Element(FiniteDimensionalEuclideanJordanAlgebraElement): + + 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 random_eja + sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + ....: random_eja) - TESTS:: + EXAMPLES:: - The natural representation of an element in the subalgebra is - the same as its natural representation in the superalgebra:: + 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: 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: 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. + """ + return self._superalgebra + + + def vector_space(self): """ - def __init__(self, A, elt=None): - """ - SETUP:: - - sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra - - EXAMPLES:: - - sage: J = RealSymmetricEJA(3) - sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) - sage: [ K(x^k) for k in range(J.rank()) ] - [f0, f1, f2] - - :: - - """ - if elt in A.superalgebra(): - # Try to convert a parent algebra element into a - # subalgebra element... - try: - coords = A.vector_space().coordinates(elt.vector()) - elt = A(coords) - except AttributeError: - # Catches a missing method in elt.vector() - pass - - FiniteDimensionalEuclideanJordanAlgebraElement.__init__(self, - A, - elt) - - 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.vector(), self.parent()._superalgebra_basis) ) + SETUP:: + + sage: from mjo.eja.eja_algebra import RealSymmetricEJA + sage: from mjo.eja.eja_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 + + + Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement