X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=6a9d10f65b7164627394c5ddb32bd682c323c5b2;hb=af79c1d027cf737d125b11fd41bb0bc2150778fb;hp=22fa870fb551624f9b9c47f181cd8aa023944cb6;hpb=c8af8b316ce0f238fea8a994d24776f74dc1e271;p=sage.d.git diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 22fa870..6a9d10f 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -3,8 +3,7 @@ from sage.matrix.constructor import matrix from mjo.eja.eja_algebra import FiniteDimensionalEuclideanJordanAlgebra from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement - -class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): +class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): """ SETUP:: @@ -23,6 +22,17 @@ class FiniteDimensionalEuclideanJordanElementSubalgebraElement(FiniteDimensional 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: set_random_seed() + sage: x = random_eja(AA).random_element() + sage: A = x.subalgebra_generated_by(orthonormalize_basis=True) + sage: y = A.random_element() + sage: y.operator()(A.one()) == y + True + """ def superalgebra_element(self): @@ -46,6 +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: @@ -60,79 +78,134 @@ 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()) ) + # 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) -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, + ....: 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: + + Ensure that our generator names don't conflict with the superalgebra:: + + sage: J = JordanSpinEJA(3) + sage: J.one().subalgebra_generated_by().gens() + (f0,) + sage: J = JordanSpinEJA(3, prefix='f') + sage: J.one().subalgebra_generated_by().gens() + (g0,) + sage: J = JordanSpinEJA(3, prefix='b') + sage: J.one().subalgebra_generated_by().gens() + (c0,) + + 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, 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() - n = len(superalgebra_basis) + 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 + # 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 = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ] + try: + prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1] + except ValueError: + prefix = prefixen[0] + + # 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 ) + + 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] - mult_table[i][j] = W.coordinate_vector(product.to_vector()) - - # TODO: We'll have to redo this and make it unique again... - prefix = 'f' + 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 + # 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() + self._inner_product_matrix = matrix(field, ip_table) + natural_basis = tuple( b.natural_representation() for b in basis ) - category = superalgebra.category().Associative() - natural_basis = tuple( b.natural_representation() - for b in superalgebra_basis ) - self._superalgebra = superalgebra self._vector_space = W - self._superalgebra_basis = superalgebra_basis + fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self) + fdeja.__init__(field, + mult_table, + prefix=prefix, + category=category, + natural_basis=natural_basis, + check_field=False, + check_axioms=check_axioms) - fdeja = super(FiniteDimensionalEuclideanJordanElementSubalgebra, self) - return fdeja.__init__(field, - mult_table, - rank, - prefix=prefix, - category=category, - natural_basis=natural_basis) def _element_constructor_(self, elt): @@ -144,22 +217,55 @@ 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) - 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 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()! 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() + + # 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) + + + + 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() def superalgebra(self): @@ -174,28 +280,33 @@ 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 = sum( i*J.gens()[i] for i in range(6) ) - sage: K = FiniteDimensionalEuclideanJordanElementSubalgebra(x) + 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 Rational Field + Vector space of degree 6 and dimension 2 over... 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) + [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