X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_subalgebra.py;h=ca8efa1fd410b8f5f3f6d177b62779b1a24ccaf5;hp=92fd296b3003a06e7bafb79f9d9c37d9cb6b13cb;hb=HEAD;hpb=d395668ab9c439d2ee5ec6224d2061656da5ae04 diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py index 92fd296..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 FiniteDimensionalEJA -from mjo.eja.eja_element import FiniteDimensionalEJAElement +from mjo.eja.eja_algebra import EJA +from mjo.eja.eja_element import (EJAElement, + CartesianProductParentEJAElement) -class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement): +class EJASubalgebraElement(EJAElement): """ SETUP:: @@ -14,7 +16,6 @@ class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement): The matrix representation of an element in the subalgebra is the same as its matrix representation in the superalgebra:: - sage: set_random_seed() sage: x = random_eja(field=QQ,orthonormalize=False).random_element() sage: A = x.subalgebra_generated_by(orthonormalize=False) sage: y = A.random_element() @@ -27,11 +28,10 @@ class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement): works like it does in the superalgebra, even if we orthonormalize our basis:: - sage: set_random_seed() - sage: x = random_eja(field=AA).random_element() - sage: A = x.subalgebra_generated_by(orthonormalize=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 """ @@ -51,26 +51,25 @@ class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement): sage: J = RealSymmetricEJA(3) sage: x = sum(J.gens()) sage: x - e0 + e1 + e2 + e3 + e4 + e5 + 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 + 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(field=QQ, orthonormalize=False) sage: x = J.random_element() sage: A = x.subalgebra_generated_by(orthonormalize=False) @@ -84,12 +83,12 @@ class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement): True """ - return self.parent().superalgebra()(self.to_matrix()) + return self.parent().superalgebra_embedding()(self) -class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): +class EJASubalgebra(EJA): """ A subalgebra of an EJA with a given basis. @@ -98,7 +97,7 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, ....: JordanSpinEJA, ....: RealSymmetricEJA) - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + sage: from mjo.eja.eja_subalgebra import EJASubalgebra EXAMPLES: @@ -110,28 +109,29 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): ....: [0,0] ]) sage: E22 = matrix(AA, [ [0,0], ....: [0,1] ]) - sage: K1 = FiniteDimensionalEJASubalgebra(J, (J(E11),), associative=True) + sage: K1 = EJASubalgebra(J, (J(E11),), associative=True) sage: K1.one().to_matrix() [1 0] [0 0] - sage: K2 = FiniteDimensionalEJASubalgebra(J, (J(E22),), associative=True) + 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:: @@ -139,7 +139,6 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): sage: B = A.one().subalgebra_generated_by() sage: B.dimension() 1 - """ def __init__(self, superalgebra, basis, **kwargs): self._superalgebra = superalgebra @@ -152,7 +151,7 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): # 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: @@ -171,6 +170,7 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): jordan_product, inner_product, field=field, + matrix_space=superalgebra.matrix_space(), prefix=prefix, **kwargs) @@ -185,7 +185,7 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): SETUP:: sage: from mjo.eja.eja_algebra import RealSymmetricEJA - sage: from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra + sage: from mjo.eja.eja_subalgebra import EJASubalgebra EXAMPLES:: @@ -195,37 +195,34 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): ....: [1,0,0] ]) sage: x = J(X) sage: basis = ( x, x^2 ) # x^2 is the identity matrix - sage: K = FiniteDimensionalEJASubalgebra(J, + 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 in self.superalgebra(): - return super()._element_constructor_(elt.to_matrix()) + # 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 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() - - def superalgebra(self): """ Return the superalgebra that this algebra was generated from. @@ -233,4 +230,78 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA): return self._superalgebra - Element = FiniteDimensionalEJASubalgebraElement + @cached_method + def superalgebra_embedding(self): + r""" + Return the embedding from this subalgebra into the superalgebra. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + EXAMPLES:: + + 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 + + """ + 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 + + + +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