X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=ca8efa1fd410b8f5f3f6d177b62779b1a24ccaf5;hp=95534db842408f08480d012d6464fadf0c3e7fd4;hb=HEAD;hpb=b40f0964ea523f9063d62ec1772a5d698bf9c26a diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 95534db..97a7978 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,10 +1,11 @@ from sage.matrix.constructor import matrix +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 FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): +class EJASubalgebraElement(EJAElement): """ SETUP:: @@ -12,17 +13,27 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional TESTS:: - The natural representation of an element in the subalgebra is - the same as its natural representation in the superalgebra:: + The matrix representation of an element in the subalgebra is + the same as its matrix representation in the superalgebra:: - sage: set_random_seed() - sage: A = random_eja().random_element().subalgebra_generated_by() + 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.natural_representation() - sage: expected = y.superalgebra_element().natural_representation() + 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:: + + 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): @@ -40,112 +51,129 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional sage: J = RealSymmetricEJA(3) sage: x = sum(J.gens()) sage: x - e0 + e1 + e2 + e3 + e4 + e5 - sage: A = x.subalgebra_generated_by() + b0 + b1 + b2 + b3 + b4 + b5 + sage: A = x.subalgebra_generated_by(orthonormalize=False) sage: A(x) - f1 + c1 sage: A(x).superalgebra_element() - e0 + e1 + e2 + e3 + e4 + e5 + 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: set_random_seed() - sage: J = random_eja() + sage: J = random_eja(field=QQ, orthonormalize=False) sage: x = J.random_element() - sage: A = x.subalgebra_generated_by() + 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.parent().superalgebra().linear_combination( - zip(self.parent()._superalgebra_basis, self.to_vector()) ) + return self.parent().superalgebra_embedding()(self) -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): - """ - The subalgebra of an EJA generated by a single element. +class EJASubalgebra(EJA): """ - 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().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.to_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. - field = superalgebra.base_ring() - mult_table = [] - for b_right in superalgebra_basis: - b_right_cols = [] - # The first column of the left-multiplication matrix by - # b1 is what we get if we apply that matrix to b1. The - # second column of the left-multiplication matrix by b1 - # is what we get when we apply that matrix to b2... - 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_col = W.coordinates((b_left*b_right).to_vector()) - b_right_cols.append(this_col) - b_right_matrix = matrix.column(field, b_right_cols) - 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 - # 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() - - category = superalgebra.category().Associative() - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) + A subalgebra of an EJA with a given basis. - self._superalgebra = superalgebra - self._vector_space = W - self._superalgebra_basis = superalgebra_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) - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) def _element_constructor_(self, elt): @@ -157,22 +185,42 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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(x^k) for k in range(J.rank()) ] - [f0, f1, f2] + 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 :: """ if elt in self.superalgebra(): - coords = self.vector_space().coordinate_vector(elt.to_vector()) - return self.from_vector(coords) + # 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) def superalgebra(self): @@ -182,33 +230,78 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide return self._superalgebra - def vector_space(self): - """ + @cached_method + def superalgebra_embedding(self): + r""" + Return the embedding from this subalgebra into the superalgebra. + SETUP:: - sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanElementSubalgebra + sage: from mjo.eja.eja_algebra import HadamardEJA 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 1 0 0 1] - [ 0 1 2 3 4 5] - [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).to_vector() - (10, 14, 21, 19, 31, 50) + 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 """ - return self._vector_space + 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 = FiniteDimensionalEuclideanJordanElementSubalgebraElement + + 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