X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=6a9d10f65b7164627394c5ddb32bd682c323c5b2;hb=af79c1d027cf737d125b11fd41bb0bc2150778fb;hp=b07f7e25ee599322cad6ec0d94bdd23bf0a964fe;hpb=008446f3a13b4fc117e1adfbc66f86784a6495c9;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index b07f7e2..6a9d10f 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -56,6 +56,14 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide 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: @@ -70,10 +78,23 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide 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()) ) + # As with the _element_constructor_() method on the + # algebra... even in a subspace of a subspace, the basis + # elements belong to the ambient space. As a result, only one + # level of coordinate_vector() is needed, regardless of how + # deeply we're nested. + W = self.parent().vector_space() + V = self.parent().superalgebra().vector_space() + + # Multiply on the left because basis_matrix() is row-wise. + ambient_coords = self.to_vector()*W.basis_matrix() + V_coords = V.coordinate_vector(ambient_coords) + return self.parent().superalgebra().from_vector(V_coords) @@ -85,7 +106,28 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda 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: @@ -109,7 +151,7 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda 1 """ - def __init__(self, superalgebra, basis, rank=None, category=None): + def __init__(self, superalgebra, basis, category=None, check_axioms=True): self._superalgebra = superalgebra V = self._superalgebra.vector_space() field = self._superalgebra.base_ring() @@ -128,16 +170,20 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda except ValueError: prefix = prefixen[0] - basis_vectors = [ b.to_vector() for b in basis ] - superalgebra_basis = [ self._superalgebra.from_vector(b) - for b in basis_vectors ] + # 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(b.to_vector()) for b in basis ) - 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)] + ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j]) + 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 @@ -145,21 +191,20 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda product_vector = V.from_vector(product.to_vector()) mult_table[i][j] = W.coordinate_vector(product_vector) - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) + self._inner_product_matrix = matrix(field, ip_table) + natural_basis = tuple( b.natural_representation() for b in basis ) self._vector_space = W - self._superalgebra_basis = superalgebra_basis - fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) + fdeja.__init__(field, + mult_table, + prefix=prefix, + category=category, + natural_basis=natural_basis, + check_field=False, + check_axioms=check_axioms) @@ -177,11 +222,16 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: x = sum( i*J.gens()[i] for i in range(6) ) - sage: basis = tuple( x^k for k in range(J.rank()) ) - sage: K = FiniteDimensionalEuclideanJordanSubalgebra(J,basis) - 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 :: @@ -189,87 +239,21 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda if elt not in self.superalgebra(): raise ValueError("not an element of this subalgebra") - coords = self.vector_space().coordinate_vector(elt.to_vector()) - return self.from_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. + # The extra hackery is because foo.to_vector() might not live + # in foo.parent().vector_space()! Subspaces of subspaces still + # have user bases in the ambient space, though, so only one + # level of coordinate_vector() is needed. In other words, if V + # is itself a subspace, the basis elements for W will be of + # the same length as the basis elements for V -- namely + # whatever the dimension of the ambient (parent of V?) space is. + V = self.superalgebra().vector_space() + W = self.vector_space() - 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: + # Multiply on the left because basis_matrix() is row-wise. + ambient_coords = elt.to_vector()*V.basis_matrix() + W_coords = W.coordinate_vector(ambient_coords) + return self.from_vector(W_coords) - 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): @@ -301,21 +285,25 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda EXAMPLES:: sage: J = RealSymmetricEJA(3) - sage: x = J.monomial(0) + 2*J.monomial(2) + 5*J.monomial(5) - sage: basis = (x^0, x^1, x^2) + 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