X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=ca8efa1fd410b8f5f3f6d177b62779b1a24ccaf5;hp=8c0f9092f040e659b5e550cfa90d4f27f4487ad4;hb=HEAD;hpb=52341fdeb29a68a5cb88e53b9ee42c695e24d9d9 diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 8c0f909..97a7978 100644 --- a/mjo/eja/eja_subalgebra.py +++ b/mjo/eja/eja_subalgebra.py @@ -1,9 +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 FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclideanJordanAlgebraElement): +class EJASubalgebraElement(EJAElement): """ SETUP:: @@ -14,8 +16,8 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide 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.to_matrix() sage: expected = y.superalgebra_element().to_matrix() @@ -26,11 +28,10 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide 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 + 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 """ @@ -50,56 +51,44 @@ class FiniteDimensionalEuclideanJordanSubalgebraElement(FiniteDimensionalEuclide 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 - f0 + f1 - sage: B = y.subalgebra_generated_by() + c0 + c1 + sage: B = y.subalgebra_generated_by(orthonormalize=False) sage: B(y) - g1 + d1 sage: B(y).superalgebra_element() - f0 + f1 + 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() + sage: B = y.subalgebra_generated_by(orthonormalize=False) sage: B(y).superalgebra_element() == y True """ - # 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) + return self.parent().superalgebra_embedding()(self) -class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJordanAlgebra): +class EJASubalgebra(EJA): """ A subalgebra of an EJA with a given basis. @@ -108,7 +97,7 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA) - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + sage: from mjo.eja.eja_subalgebra import EJASubalgebra EXAMPLES: @@ -120,28 +109,29 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda ....: [0,0] ]) sage: E22 = matrix(AA, [ [0,0], ....: [0,1] ]) - sage: K1 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E11),)) + sage: K1 = EJASubalgebra(J, (J(E11),), associative=True) sage: K1.one().to_matrix() [1 0] [0 0] - sage: K2 = FiniteDimensionalEuclideanJordanSubalgebra(J, (J(E22),)) + 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:: @@ -149,14 +139,11 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda sage: B = A.one().subalgebra_generated_by() sage: B.dimension() 1 - """ - def __init__(self, superalgebra, basis, category=None, check_axioms=True): + def __init__(self, superalgebra, basis, **kwargs): 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 @@ -164,58 +151,28 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda # 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 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), - check=check_axioms) - - n = len(basis) - if check_axioms: - # The tables are square if we're verifying that they - # are commutative. - mult_table = [[W.zero() for j in range(n)] for i in range(n)] - ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j]) - for j in range(n) ] - for i in range(n) ] - else: - mult_table = [[W.zero() for j in range(i+1)] for i in range(n)] - ip_table = [ [ self._superalgebra.inner_product(basis[i],basis[j]) - for j in range(i+1) ] - for i in range(n) ] - - for i in range(n): - for j in range(i+1): - 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) - if check_axioms: - mult_table[j][i] = mult_table[i][j] - + # 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)) - self._vector_space = W - - fdeja = super(FiniteDimensionalEuclideanJordanSubalgebra, self) - fdeja.__init__(field, - mult_table, - ip_table, - prefix=prefix, - category=category, - matrix_basis=matrix_basis, - check_field=False, - check_axioms=check_axioms) + super().__init__(matrix_basis, + jordan_product, + inner_product, + field=field, + matrix_space=superalgebra.matrix_space(), + prefix=prefix, + **kwargs) @@ -228,7 +185,7 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra + sage: from mjo.eja.eja_subalgebra import EJASubalgebra EXAMPLES:: @@ -238,45 +195,32 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda ....: [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 = EJASubalgebra(J, + ....: basis, + ....: associative=True, + ....: orthonormalize=False) sage: K(J.one()) - f1 + c1 sage: K(J.one() + x) - f0 + f1 + c0 + c1 :: """ - 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 matrix_space(self): - """ - Return the matrix 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 matrix basis elements to - infer anything from) and the parent is not. - """ - return self.superalgebra().matrix_space() + if elt in self.superalgebra(): + # 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): @@ -286,38 +230,78 @@ class FiniteDimensionalEuclideanJordanSubalgebra(FiniteDimensionalEuclideanJorda 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 FiniteDimensionalEuclideanJordanSubalgebra + sage: from mjo.eja.eja_algebra import HadamardEJA EXAMPLES:: - sage: J = RealSymmetricEJA(3) - 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 2 over... - User basis matrix: - [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) + 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 = EJASubalgebraElement - Element = FiniteDimensionalEuclideanJordanSubalgebraElement + +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