X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=ca8efa1fd410b8f5f3f6d177b62779b1a24ccaf5;hp=8f6e56b55f309d10967ee5aae2bb8b2fe7261566;hb=HEAD;hpb=32723f51147eb6260c8b41549208c851e54a4c56 diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 8f6e56b..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): @@ -40,56 +51,87 @@ 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): +class EJASubalgebra(EJA): """ - The subalgebra of an EJA generated by a single element. + A subalgebra of an EJA with a given basis. SETUP:: sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: JordanSpinEJA) + ....: 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:: + Ensure that our generator names don't conflict with the + superalgebra:: sage: J = JordanSpinEJA(3) sage: J.one().subalgebra_generated_by().gens() - (f0,) + (c0,) sage: J = JordanSpinEJA(3, prefix='f') sage: J.one().subalgebra_generated_by().gens() (g0,) - sage: J = JordanSpinEJA(3, prefix='b') + sage: J = JordanSpinEJA(3, prefix='a') sage: J.one().subalgebra_generated_by().gens() - (c0,) + (b0,) Ensure that we can find subalgebras of subalgebras:: @@ -97,11 +139,9 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide sage: B = A.one().subalgebra_generated_by() sage: B.dimension() 1 - """ - def __init__(self, elt): - self._superalgebra = elt.parent() - category = self._superalgebra.category().Associative() + def __init__(self, superalgebra, basis, **kwargs): + self._superalgebra = superalgebra V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() @@ -111,109 +151,29 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide # 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' ] + prefixen = ["b","c","d","e","f","g","h","l","m"] try: prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] except ValueError: prefix = prefixen[0] - if elt.is_zero(): - # Short circuit because 0^0 == 1 is going to make us - # think we have a one-dimensional algebra otherwise. - natural_basis = tuple() - mult_table = tuple() - rank = 0 - self._vector_space = V.zero_subspace() - self._superalgebra_basis = [] - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, - self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) - - - # First compute the vector subspace spanned by the powers of - # the given element. - powers = [ elt**k for k in range(V.dimension()) ] - power_vectors = [ p.to_vector() for p in powers ] - - # Figure out which powers form a linearly-independent set. - ind_rows = matrix(field, power_vectors).pivot_rows() - - # Pick those out of the list of all powers. - superalgebra_basis = tuple(map(powers.__getitem__, ind_rows)) - - # If our superalgebra is a subalgebra of something else, then - # these vectors won't have the right coordinates for - # V.span_of_basis() unless we use V.from_vector() on them. - basis_vectors = map(power_vectors.__getitem__, ind_rows) - W = V.span_of_basis( V.from_vector(v) for v in basis_vectors ) - - # Now figure out the entries of the right-multiplication - # matrix for the successive basis elements b0, b1,... of - # that subspace. - 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) - - # 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() - - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) - - - self._vector_space = W - self._superalgebra_basis = superalgebra_basis - - - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) - - - def _a_regular_element(self): - """ - Override the superalgebra method to return the one - regular element that is sure to exist in this - subalgebra, namely the element that generated it. + # 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() - SETUP:: + def inner_product(x,y): + return self._superalgebra(x).inner_product(self._superalgebra(y)) - sage: from mjo.eja.eja_algebra import random_eja + super().__init__(matrix_basis, + jordan_product, + inner_product, + field=field, + matrix_space=superalgebra.matrix_space(), + prefix=prefix, + **kwargs) - TESTS:: - - sage: set_random_seed() - sage: J = random_eja().random_element().subalgebra_generated_by() - sage: J._a_regular_element().is_regular() - True - - """ - if self.dimension() == 0: - return self.zero() - else: - return self.monomial(1) def _element_constructor_(self, elt): @@ -225,135 +185,123 @@ 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 == 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) + # 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 one_basis(self): + def superalgebra(self): """ - Return the basis-element-index of this algebra's unit element. + Return the superalgebra that this algebra was generated from. """ - return 0 - + return self._superalgebra - 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. + @cached_method + def superalgebra_embedding(self): + r""" + Return the embedding from this subalgebra into the superalgebra. SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, - ....: random_eja) + sage: from mjo.eja.eja_algebra import HadamardEJA 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 + 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 - 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 """ - if self.dimension() == 0: - return self.zero() - else: - return self.monomial(self.one_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()) - def natural_basis_space(self): - """ - Return the natural basis space of this algebra, which is identical - to that of its superalgebra. - This is correct "by definition," and avoids a mismatch when the - subalgebra is trivial (with no natural basis to infer anything - from) and the parent is not. - """ - return self.superalgebra().natural_basis_space() + Element = EJASubalgebraElement - def superalgebra(self): - """ - Return the superalgebra that this algebra was generated from. - """ - return self._superalgebra +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. - def vector_space(self): - """ - SETUP:: + 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, + ....: RealSymmetricEJA) - EXAMPLES:: + TESTS: - sage: J = RealSymmetricEJA(3) - sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) - sage: K.vector_space() - Vector space of degree 6 and dimension 3 over... - User basis matrix: - [ 1 0 1 0 0 1] - [ 1 0 2 0 0 5] - [ 1 0 4 0 0 25] - sage: (x^0).to_vector() - (1, 0, 1, 0, 0, 1) - sage: (x^1).to_vector() - (1, 0, 2, 0, 0, 5) - sage: (x^2).to_vector() - (1, 0, 4, 0, 0, 25) + This used to fail when ``subalgebra_idempotent()`` tried to + embed the subalgebra element back into the original EJA:: - """ - return self._vector_space + 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 - Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement +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