X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=5ac0a77c7a9ab27929f4c3e1d73238d93e1ccd1b;hb=a7cf81951d0cb51ea40d9362a75385204596df42;hp=7c883d92fab3f3f2ee158737bdb0ec12bdf7effd;hpb=40fe88c9c758ef6468bf67acd6da9c4333b755f9;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 7c883d9..5ac0a77 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -8,6 +8,28 @@ from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): """ The subalgebra of an EJA generated by a single element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra + + 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') + """ @staticmethod def __classcall_private__(cls, elt): @@ -16,7 +38,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # First compute the vector subspace spanned by the powers of # the given element. V = superalgebra.vector_space() - eja_basis = [superalgebra.one()] + superalgebra_basis = [superalgebra.one()] basis_vectors = [superalgebra.one().vector()] W = V.span_of_basis(basis_vectors) for exponent in range(1, V.dimension()): @@ -24,21 +46,21 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide basis_vectors.append( new_power.vector() ) try: W = V.span_of_basis(basis_vectors) - eja_basis.append( new_power ) + 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. - eja_basis = tuple(eja_basis) + 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 eja_basis: + 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 @@ -47,7 +69,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # # IMPORTANT: this assumes that all vectors are COLUMN # vectors, unlike our superclass (which uses row vectors). - for b_left in eja_basis: + for b_left in superalgebra_basis: # Multiply in the original EJA, but then get the # coordinates from the subalgebra in terms of its # basis. @@ -72,22 +94,27 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # powers. assume_associative=True - # TODO: Un-hard-code this. It should be possible to get the "next" - # name based on the parent's generator names. - names = 'f' - names = normalize_names(W.dimension(), names) + # 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() - - # TODO: compute this and actually specify it. - natural_basis = None + natural_basis = tuple( b.natural_representation() + for b in superalgebra_basis ) fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls) return fdeja.__classcall__(cls, F, mult_table, rank, - eja_basis, + superalgebra_basis, W, assume_associative=assume_associative, names=names, @@ -98,16 +125,16 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide field, mult_table, rank, - eja_basis, + superalgebra_basis, vector_space, assume_associative=True, names='f', category=None, natural_basis=None): - self._superalgebra = eja_basis[0].parent() + self._superalgebra = superalgebra_basis[0].parent() self._vector_space = vector_space - self._eja_basis = eja_basis + self._superalgebra_basis = superalgebra_basis fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) fdeja.__init__(field, @@ -119,6 +146,13 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide natural_basis=natural_basis) + def superalgebra(self): + """ + Return the superalgebra that this algebra was generated from. + """ + return self._superalgebra + + def vector_space(self): """ SETUP:: @@ -149,6 +183,26 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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:: @@ -167,7 +221,7 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide :: """ - if elt in A._superalgebra: + if elt in A.superalgebra(): # Try to convert a parent algebra element into a # subalgebra element... try: @@ -180,3 +234,43 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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) )