X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=a774b985ff54a2ceeb01468b5887849a22ef2d65;hb=9c35b8d70e384cd98b8ec7eb7a84cf84db1d1137;hp=5ac0a77c7a9ab27929f4c3e1d73238d93e1ccd1b;hpb=a7cf81951d0cb51ea40d9362a75385204596df42;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 5ac0a77..a774b98 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,49 +1,122 @@ 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 -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): +class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): """ - The subalgebra of an EJA generated by a single element. - SETUP:: - sage: from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra + 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 + + """ + def __init__(self, A, elt): + """ + 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.element_class(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().coordinate_vector(elt.to_vector()) + elt = A.from_vector(coords).monomial_coefficients() + except AttributeError: + # Catches a missing method in elt.to_vector() + pass + + s = super(FiniteDimensionalEuclideanJordanElementSubalgebraElement, + self) + + s.__init__(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) - TESTS: + 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.element_class(A,x) + f1 + sage: A.element_class(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.element_class(A,x).superalgebra_element() == x + True + sage: y = A.random_element() + sage: A.element_class(A,y.superalgebra_element()) == y + True + + """ + return self.parent().superalgebra().linear_combination( + zip(self.parent()._superalgebra_basis, self.to_vector()) ) - 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') + +class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): + """ + The subalgebra of an EJA generated by a single element. """ - @staticmethod - def __classcall_private__(cls, elt): + 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().vector()] + 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.vector() ) + basis_vectors.append( new_power.to_vector() ) try: W = V.span_of_basis(basis_vectors) superalgebra_basis.append( new_power ) @@ -58,7 +131,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # Now figure out the entries of the right-multiplication # matrix for the successive basis elements b0, b1,... of # that subspace. - F = superalgebra.base_ring() + field = superalgebra.base_ring() mult_table = [] for b_right in superalgebra_basis: b_right_rows = [] @@ -73,15 +146,18 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # 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()) + this_row = W.coordinates((b_left*b_right).to_vector()) b_right_rows.append(this_row) - b_right_matrix = matrix(F, b_right_rows) + b_right_matrix = matrix(field, b_right_rows) 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' + # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know # in this case that there's an element whose minimal @@ -90,60 +166,23 @@ 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() + category = 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._superalgebra = superalgebra + 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): @@ -171,106 +210,15 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide [ 1 0 0 1 0 1] [ 0 1 2 3 4 5] [ 5 11 14 26 34 45] - sage: (x^0).vector() + sage: (x^0).to_vector() (1, 0, 0, 1, 0, 1) - sage: (x^1).vector() + sage: (x^1).to_vector() (0, 1, 2, 3, 4, 5) - sage: (x^2).vector() + sage: (x^2).to_vector() (5, 11, 14, 26, 34, 45) """ return self._vector_space - class Element(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 - - """ - 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) ) + Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement