X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=ca8efa1fd410b8f5f3f6d177b62779b1a24ccaf5;hp=5ac0a77c7a9ab27929f4c3e1d73238d93e1ccd1b;hb=HEAD;hpb=57f22a182c4de750b0a03dac257143e1dca04341 diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 5ac0a77..97a7978 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,276 +1,307 @@ from sage.matrix.constructor import matrix -from sage.structure.category_object import normalize_names +from sage.misc.cachefunc import cached_method -from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra -from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement +from mjo.eja.eja_algebra import EJA +from mjo.eja.eja_element import (EJAElement, + CartesianProductParentEJAElement) - -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): +class EJASubalgebraElement(EJAElement): """ - 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: + 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') + The matrix representation of an element in the subalgebra is + the same as its matrix representation in the superalgebra:: - """ - @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) - - # 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 - # polynomial has the same degree as the space's dimension - # (remember how we constructed the space?), so that must be - # 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._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) + sage: x = random_eja(field=QQ,orthonormalize=False).random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: y = A.random_element() + sage: actual = y.to_matrix() + sage: expected = y.superalgebra_element().to_matrix() + 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:: - def superalgebra(self): + sage: x = random_eja(field=AA).random_element() # long time + sage: A = x.subalgebra_generated_by(orthonormalize=True) # long time + sage: y = A.random_element() # long time + sage: y.operator()(A.one()) == y # long time + True + + """ + + def superalgebra_element(self): """ - Return the superalgebra that this algebra was generated from. + 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 + b0 + b1 + b2 + b3 + b4 + b5 + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A(x) + c1 + sage: A(x).superalgebra_element() + b0 + b1 + b2 + b3 + b4 + b5 + sage: y = sum(A.gens()) + sage: y + c0 + c1 + sage: B = y.subalgebra_generated_by(orthonormalize=False) + sage: B(y) + d1 + sage: B(y).superalgebra_element() + c0 + c1 + + TESTS: + + We can convert back and forth faithfully:: + + sage: J = random_eja(field=QQ, orthonormalize=False) + sage: x = J.random_element() + sage: A = x.subalgebra_generated_by(orthonormalize=False) + sage: A(x).superalgebra_element() == x + True + sage: y = A.random_element() + sage: A(y.superalgebra_element()) == y + True + sage: B = y.subalgebra_generated_by(orthonormalize=False) + sage: B(y).superalgebra_element() == y + True + """ - return self._superalgebra + return self.parent().superalgebra_embedding()(self) + - def vector_space(self): + +class EJASubalgebra(EJA): + """ + A subalgebra of an EJA with a given basis. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: JordanSpinEJA, + ....: RealSymmetricEJA) + sage: from mjo.eja.eja_subalgebra import EJASubalgebra + + EXAMPLES: + + The following Peirce subalgebras of the 2-by-2 real symmetric + matrices do not contain the superalgebra's identity element:: + + sage: J = RealSymmetricEJA(2) + sage: E11 = matrix(AA, [ [1,0], + ....: [0,0] ]) + sage: E22 = matrix(AA, [ [0,0], + ....: [0,1] ]) + sage: K1 = EJASubalgebra(J, (J(E11),), associative=True) + sage: K1.one().to_matrix() + [1 0] + [0 0] + sage: K2 = EJASubalgebra(J, (J(E22),), associative=True) + sage: K2.one().to_matrix() + [0 0] + [0 1] + + TESTS: + + Ensure that our generator names don't conflict with the + superalgebra:: + + sage: J = JordanSpinEJA(3) + sage: J.one().subalgebra_generated_by().gens() + (c0,) + sage: J = JordanSpinEJA(3, prefix='f') + sage: J.one().subalgebra_generated_by().gens() + (g0,) + sage: J = JordanSpinEJA(3, prefix='a') + sage: J.one().subalgebra_generated_by().gens() + (b0,) + + 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, superalgebra, basis, **kwargs): + self._superalgebra = superalgebra + 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 = ["b","c","d","e","f","g","h","l","m"] + try: + prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] + except ValueError: + prefix = prefixen[0] + + # The superalgebra constructor expects these to be in original matrix + # form, not algebra-element form. + matrix_basis = tuple( b.to_matrix() for b in basis ) + def jordan_product(x,y): + return (self._superalgebra(x)*self._superalgebra(y)).to_matrix() + + def inner_product(x,y): + return self._superalgebra(x).inner_product(self._superalgebra(y)) + + super().__init__(matrix_basis, + jordan_product, + inner_product, + field=field, + matrix_space=superalgebra.matrix_space(), + prefix=prefix, + **kwargs) + + + + 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 + sage: from mjo.eja.eja_subalgebra import EJASubalgebra EXAMPLES:: 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: X = matrix(AA, [ [0,0,1], + ....: [0,1,0], + ....: [1,0,0] ]) + sage: x = J(X) + sage: basis = ( x, x^2 ) # x^2 is the identity matrix + sage: K = EJASubalgebra(J, + ....: basis, + ....: associative=True, + ....: orthonormalize=False) + sage: K(J.one()) + c1 + sage: K(J.one() + x) + c0 + c1 + + :: """ - return self._vector_space + if elt in self.superalgebra(): + # If the subalgebra is trivial, its _matrix_span will be empty + # but we still want to be able convert the superalgebra's zero() + # element into the subalgebra's zero() element. There's no great + # workaround for this because sage checks that your basis is + # linearly-independent everywhere, so we can't just give it a + # basis consisting of the zero element. + m = elt.to_matrix() + if self.is_trivial() and m.is_zero(): + return self.zero() + else: + return super()._element_constructor_(m) + else: + return super()._element_constructor_(elt) - class Element(FiniteDimensionalEuclideanJordanAlgebraElement): + def superalgebra(self): """ + Return the superalgebra that this algebra was generated from. + """ + return self._superalgebra - SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + @cached_method + def superalgebra_embedding(self): + r""" + Return the embedding from this subalgebra into the superalgebra. - TESTS:: + SETUP:: - The natural representation of an element in the subalgebra is - the same as its natural representation in the superalgebra:: + sage: from mjo.eja.eja_algebra import HadamardEJA - 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 + EXAMPLES:: + + sage: J = HadamardEJA(4) + sage: A = J.one().subalgebra_generated_by() + sage: iota = A.superalgebra_embedding() + sage: iota + Linear operator between finite-dimensional Euclidean Jordan algebras represented by the matrix: + [1/2] + [1/2] + [1/2] + [1/2] + Domain: Euclidean Jordan algebra of dimension 1 over Algebraic Real Field + Codomain: Euclidean Jordan algebra of dimension 4 over Algebraic Real Field + sage: iota(A.one()) == J.one() 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) ) + from mjo.eja.eja_operator import EJAOperator + mm = self._module_morphism(lambda j: self.superalgebra()(self.monomial(j).to_matrix()), + codomain=self.superalgebra()) + return EJAOperator(self, + self.superalgebra(), + mm.matrix()) + + + + Element = EJASubalgebraElement + + + +class CartesianProductEJASubalgebraElement(EJASubalgebraElement, + CartesianProductParentEJAElement): + r""" + The class for elements that both belong to a subalgebra and + have a Cartesian product algebra as their parent. By inheriting + :class:`CartesianProductParentEJAElement` in addition to + :class:`EJASubalgebraElement`, we allow the + ``to_matrix()`` method to be overridden with the version that + works on Cartesian products. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + TESTS: + + This used to fail when ``subalgebra_idempotent()`` tried to + embed the subalgebra element back into the original EJA:: + + sage: J1 = HadamardEJA(0, field=QQ, orthonormalize=False) + sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False) + sage: J = cartesian_product([J1,J2]) + sage: J.one().subalgebra_idempotent() == J.one() + True + + """ + pass + +class CartesianProductEJASubalgebra(EJASubalgebra): + r""" + Subalgebras whose parents are Cartesian products. Exists only + to specify a special element class that will (in addition) + inherit from ``CartesianProductParentEJAElement``. + """ + Element = CartesianProductEJASubalgebraElement