X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=cb9631df18b5a0749dda75855682c40a689cd5b3;hb=c089560955d306b4c2408b222012747c8fe3bddc;hp=0f641416366ef7ee72d4703e070c69e2d60e30a9;hpb=08aba469c5f8d8947a543f8882fa676ed165e7ee;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 0f64141..cb9631d 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -2,9 +2,8 @@ from sage.matrix.constructor import matrix from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement -from mjo.eja.eja_utils import gram_schmidt -class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): +class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): """ SETUP:: @@ -57,6 +56,14 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional f1 sage: A(x).superalgebra_element() e0 + e1 + e2 + e3 + e4 + e5 + sage: y = sum(A.gens()) + sage: y + f0 + f1 + sage: B = y.subalgebra_generated_by() + sage: B(y) + g1 + sage: B(y).superalgebra_element() + f0 + f1 TESTS: @@ -71,22 +78,50 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional sage: y = A.random_element() sage: A(y.superalgebra_element()) == y True + sage: B = y.subalgebra_generated_by() + sage: B(y).superalgebra_element() == y + True """ - return self.parent().superalgebra().linear_combination( - zip(self.parent()._superalgebra_basis, self.to_vector()) ) + W = self.parent().vector_space() + V = self.parent().superalgebra().vector_space() + A = W.basis_matrix().transpose() + W_coords = A*self.to_vector() + V_coords = V.coordinate_vector(W_coords) + return self.parent().superalgebra().from_vector(V_coords) -class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): +class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): """ - 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 FiniteDimensionalEuclideanJordanSubalgebra + + 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 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),)) + sage: K1.one().natural_representation() + [1 0] + [0 0] + sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),)) + sage: K2.one().natural_representation() + [0 0] + [0 1] TESTS: @@ -110,11 +145,12 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 1 """ - def __init__(self, elt, orthonormalize_basis): - self._superalgebra = elt.parent() - category = self._superalgebra.category().Associative() + def __init__(self, superalgebra, basis, category=None, check_axioms=True): + self._superalgebra = superalgebra V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() + if category is None: + category = self._superalgebra.category() # A half-assed attempt to ensure that we don't collide with # the superalgebra's prefix (ignoring the fact that there @@ -128,41 +164,18 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide except ValueError: prefix = prefixen[0] - # This list is guaranteed to contain all independent powers, - # because it's the maximal set of powers that could possibly - # be independent (by a dimension argument). - powers = [ elt**k for k in range(V.dimension()) ] - - if orthonormalize_basis == False: - # In this case, we just need to figure out which elements - # of the "powers" list are redundant... First compute the - # vector subspace spanned by the powers of the given - # element. - 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) - else: - # If we're going to orthonormalize the basis anyway, we - # might as well just do Gram-Schmidt on the whole list of - # powers. The redundant ones will get zero'd out. - superalgebra_basis = gram_schmidt(powers) - basis_vectors = [ b.to_vector() for b in superalgebra_basis ] + basis_vectors = [ b.to_vector() for b in basis ] + # 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. W = V.span_of_basis( V.from_vector(v) for v in basis_vectors ) - n = len(superalgebra_basis) + + n = len(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 = basis[i]*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 @@ -170,53 +183,20 @@ class FiniteDimensionalEuclideanJordanElementSubalgebra(FiniteDimensionalEuclide 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 ) + natural_basis = tuple( b.natural_representation() for b in 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. - - SETUP:: - - sage: from mjo.eja.eja_algebra import random_eja - - TESTS:: - - sage: set_random_seed() - sage: J = random_eja().random_element().subalgebra_generated_by() - sage: J._a_regular_element().is_regular() - True + fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self) + fdeja.__init__(field, + mult_table, + prefix=prefix, + category=category, + natural_basis=natural_basis, + check_field=False, + check_axioms=check_axioms) - """ - if self.dimension() == 0: - return self.zero() - else: - return self.monomial(1) def _element_constructor_(self, elt): @@ -228,109 +208,36 @@ 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 FiniteDimensionalEuclideanJordanSubalgebra EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) - 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 = FiniteDimensionalEuclideanJordanSubalgebra(J, basis) + sage: K(J.one()) + f1 + sage: K(J.one() + x) + f0 + f1 :: """ - 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 elt not in self.superalgebra(): + raise ValueError("not an element of this subalgebra") + # The extra hackery is because foo.to_vector() might not + # live in foo.parent().vector_space()! + coords = sum( a*b for (a,b) + in zip(elt.to_vector(), + self.superalgebra().vector_space().basis()) ) + return self.from_vector(self.vector_space().coordinate_vector(coords)) - 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 superalgebra identity - restricted to this algebra is its identity. Note that we can't - count on the first basis element being the identity -- it migth - have been scaled if we orthonormalized the basis. - - 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 over the rationals:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: A = x.subalgebra_generated_by() - sage: x = A.random_element() - sage: A.one()*x == x and x*A.one() == x - True - - The identity element acts like the identity over the algebraic - reals with an orthonormal basis:: - - sage: set_random_seed() - sage: x = random_eja(AA).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) - sage: x = A.random_element() - sage: A.one()*x == x and x*A.one() == x - True - - The matrix of the unit element's operator is the identity over - the rationals:: - - sage: set_random_seed() - sage: x = random_eja().random_element() - sage: A = x.subalgebra_generated_by() - sage: actual = A.one().operator().matrix() - sage: expected = matrix.identity(A.base_ring(), A.dimension()) - sage: actual == expected - True - - The matrix of the unit element's operator is the identity over - the algebraic reals with an orthonormal basis:: - - sage: set_random_seed() - sage: x = random_eja(AA).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) - sage: actual = A.one().operator().matrix() - sage: expected = matrix.identity(A.base_ring(), A.dimension()) - sage: actual == expected - True - - """ - if self.dimension() == 0: - return self.zero() - else: - sa_one = self.superalgebra().one().to_vector() - sa_coords = self.vector_space().coordinate_vector(sa_one) - return self.from_vector(sa_coords) - def natural_basis_space(self): """ @@ -356,28 +263,33 @@ 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 FiniteDimensionalEuclideanJordanSubalgebra EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x,False) + sage: E11 = matrix(ZZ, [ [1,0,0], + ....: [0,0,0], + ....: [0,0,0] ]) + sage: E22 = matrix(ZZ, [ [0,0,0], + ....: [0,1,0], + ....: [0,0,0] ]) + sage: b1 = J(E11) + sage: b2 = J(E22) + sage: basis = (b1, b2) + sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis) sage: K.vector_space() - Vector space of degree 6 and dimension 3 over... + Vector space of degree 6 and dimension 2 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) + [1 0 0 0 0 0] + [0 0 1 0 0 0] + sage: b1.to_vector() + (1, 0, 0, 0, 0, 0) + sage: b2.to_vector() + (0, 0, 1, 0, 0, 0) """ return self._vector_space - Element = FiniteDimensionalEuclideanJordanElementSubalgebraElement + Element = FiniteDimensionalEuclideanJordanSubalgebraElement