X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=9e5b010145b99f9442b5a86df8db3cd7d25048ad;hb=0ac5b5a3c8eb960b0f3c20a28c4d4bc0e33e5294;hp=7c883d92fab3f3f2ee158737bdb0ec12bdf7effd;hpb=40fe88c9c758ef6468bf67acd6da9c4333b755f9;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 7c883d9..9e5b010 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,64 +1,157 @@ 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 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 + + """ + + 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): """ 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 + """ - @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() - eja_basis = [superalgebra.one()] - basis_vectors = [superalgebra.one().vector()] + superalgebra_basis = [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(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( V.from_vector(new_power.to_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: - 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 eja_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) + field = superalgebra.base_ring() + 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) + + # 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(superalgebra.prefix()) + 1] + except ValueError: + prefix = prefixen[0] # The rank is the highest possible degree of a minimal # polynomial, and is bounded above by the dimension. We know @@ -68,55 +161,118 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # its rank too. rank = W.dimension() - # EJAs are power-associative, and this algebra is nothin but - # 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) - - cat = superalgebra.category().Associative() - - # TODO: compute this and actually specify it. - natural_basis = None - - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, cls) - return fdeja.__classcall__(cls, - F, - mult_table, - rank, - eja_basis, - W, - assume_associative=assume_associative, - names=names, - category=cat, - natural_basis=natural_basis) - - def __init__(self, - field, - mult_table, - rank, - eja_basis, - vector_space, - assume_associative=True, - names='f', - category=None, - natural_basis=None): - - self._superalgebra = eja_basis[0].parent() - self._vector_space = vector_space - self._eja_basis = eja_basis + 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 + 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 _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 + 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 == 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 + """ + return self.monomial(self.one_basis()) + + + def superalgebra(self): + """ + Return the superalgebra that this algebra was generated from. + """ + return self._superalgebra def vector_space(self): @@ -134,49 +290,18 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide sage: K.vector_space() Vector space of degree 6 and dimension 3 over Rational Field User basis matrix: - [ 1 0 0 1 0 1] + [ 1 0 1 0 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() + [10 14 21 19 31 50] + sage: (x^0).to_vector() + (1, 0, 1, 0, 0, 1) + sage: (x^1).to_vector() (0, 1, 2, 3, 4, 5) - sage: (x^2).vector() - (5, 11, 14, 26, 34, 45) + sage: (x^2).to_vector() + (10, 14, 21, 19, 31, 50) """ return self._vector_space - class Element(FiniteDimensionalEuclideanJordanAlgebraElement): - 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) + Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement