X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=287a217e785ff3e44109d3fc8d8fd85ed1ca4771;hb=17aee61574caf7f62a70d181840c2be69879a3e7;hp=00a15a1c56897172f57aeec2d43a391f3b367a45;hpb=8b70663d4c5e51aa5bd0a567c289f67e5ff8c000;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 00a15a1..287a217 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1,6 +1,6 @@ -from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement from sage.matrix.constructor import matrix from sage.modules.free_module import VectorSpace +from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement # TODO: make this unnecessary somehow. from sage.misc.lazy_import import lazy_import @@ -10,7 +10,7 @@ lazy_import('mjo.eja.eja_subalgebra', from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator from mjo.eja.eja_utils import _mat2vec -class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraElement): +class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement): """ An element of a Euclidean Jordan algebra. """ @@ -25,68 +25,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle dir(self.__class__) ) - def __init__(self, A, elt=None): - """ - - SETUP:: - - sage: from mjo.eja.eja_algebra import (RealSymmetricEJA, - ....: random_eja) - - EXAMPLES: - - The identity in `S^n` is converted to the identity in the EJA:: - sage: J = RealSymmetricEJA(3) - sage: I = matrix.identity(QQ,3) - sage: J(I) == J.one() - True - - This skew-symmetric matrix can't be represented in the EJA:: - - sage: J = RealSymmetricEJA(3) - sage: A = matrix(QQ,3, lambda i,j: i-j) - sage: J(A) - Traceback (most recent call last): - ... - ArithmeticError: vector is not in free module - - TESTS: - - Ensure that we can convert any element of the parent's - underlying vector space back into an algebra element whose - vector representation is what we started with:: - - sage: set_random_seed() - sage: J = random_eja() - sage: v = J.vector_space().random_element() - sage: J(v).vector() == v - True - - """ - # Goal: if we're given a matrix, and if it lives in our - # parent algebra's "natural ambient space," convert it - # into an algebra element. - # - # The catch is, we make a recursive call after converting - # the given matrix into a vector that lives in the algebra. - # This we need to try the parent class initializer first, - # to avoid recursing forever if we're given something that - # already fits into the algebra, but also happens to live - # in the parent's "natural ambient space" (this happens with - # vectors in R^n). - try: - FiniteDimensionalAlgebraElement.__init__(self, A, elt) - except ValueError: - natural_basis = A.natural_basis() - if elt in natural_basis[0].matrix_space(): - # Thanks for nothing! Matrix spaces aren't vector - # spaces in Sage, so we have to figure out its - # natural-basis coordinates ourselves. - V = VectorSpace(elt.base_ring(), elt.nrows()**2) - W = V.span( _mat2vec(s) for s in natural_basis ) - coords = W.coordinates(_mat2vec(elt)) - FiniteDimensionalAlgebraElement.__init__(self, A, coords) def __pow__(self, n): """ @@ -228,7 +167,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle """ p = self.parent().characteristic_polynomial() - return p(*self.vector()) + return p(*self.to_vector()) def inner_product(self, other): @@ -255,7 +194,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: y = vector(QQ,[4,5,6]) sage: x.inner_product(y) 32 - sage: J(x).inner_product(J(y)) + sage: J.from_vector(x).inner_product(J.from_vector(y)) 32 The inner product on `S^n` is ` = trace(X*Y)`, where @@ -437,7 +376,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # -1 to ensure that _charpoly_coeff(0) is really what # appears in front of t^{0} in the charpoly. However, # we want (-1)^r times THAT for the determinant. - return ((-1)**r)*p(*self.vector()) + return ((-1)**r)*p(*self.to_vector()) def inverse(self): @@ -465,13 +404,13 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: x = J.random_element() sage: while not x.is_invertible(): ....: x = J.random_element() - sage: x_vec = x.vector() + sage: x_vec = x.to_vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar)) sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list()) sage: x_inverse = coeff*inv_vec - sage: x.inverse() == J(x_inverse) + sage: x.inverse() == J.from_vector(x_inverse) True TESTS: @@ -551,7 +490,9 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle False """ - zero = self.parent().zero() + # In fact, we only need to know if the constant term is non-zero, + # so we can pass in the field's zero element instead. + zero = self.base_ring().zero() p = self.minimal_polynomial() return not (p(zero) == zero) @@ -765,8 +706,8 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): ....: y = J.random_element() - sage: y0 = y.vector()[0] - sage: y_bar = y.vector()[1:] + sage: y0 = y.to_vector()[0] + sage: y_bar = y.to_vector()[1:] sage: actual = y.minimal_polynomial() sage: t = PolynomialRing(J.base_ring(),'t').gen(0) sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2) @@ -806,7 +747,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: J = ComplexHermitianEJA(3) sage: J.one() - e0 + e5 + e8 + e0 + e3 + e8 sage: J.one().natural_representation() [1 0 0 0 0 0] [0 1 0 0 0 0] @@ -819,7 +760,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: J = QuaternionHermitianEJA(3) sage: J.one() - e0 + e9 + e14 + e0 + e5 + e14 sage: J.one().natural_representation() [1 0 0 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0 0 0] @@ -837,7 +778,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle """ B = self.parent().natural_basis() W = B[0].matrix_space() - return W.linear_combination(zip(self.vector(), B)) + return W.linear_combination(zip(B,self.to_vector())) def operator(self): @@ -862,11 +803,12 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle """ P = self.parent() - fda_elt = FiniteDimensionalAlgebraElement(P, self) + left_mult_by_self = lambda y: self*y + L = P.module_morphism(function=left_mult_by_self, codomain=P) return FiniteDimensionalEuclideanJordanAlgebraOperator( P, P, - fda_elt.matrix().transpose() ) + L.matrix() ) def quadratic_representation(self, other=None): @@ -887,7 +829,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: n = ZZ.random_element(1,10) sage: J = JordanSpinEJA(n) sage: x = J.random_element() - sage: x_vec = x.vector() + sage: x_vec = x.to_vector() sage: x0 = x_vec[0] sage: x_bar = x_vec[1:] sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)]) @@ -1000,11 +942,17 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle sage: from mjo.eja.eja_algebra import random_eja - TESTS:: + TESTS: + + This subalgebra, being composed of only powers, is associative:: sage: set_random_seed() - sage: x = random_eja().random_element() - sage: x.subalgebra_generated_by().is_associative() + sage: x0 = random_eja().random_element() + sage: A = x0.subalgebra_generated_by() + sage: x = A.random_element() + sage: y = A.random_element() + sage: z = A.random_element() + sage: (x*y)*z == x*(y*z) True Squaring in the subalgebra should work the same as in @@ -1070,7 +1018,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) A = u_next.operator().matrix() - c = J(A.solve_right(u_next.vector())) + c = J.from_vector(A.solve_right(u_next.to_vector())) # Now c is the idempotent we want, but it still lives in the subalgebra. return c.superalgebra_element() @@ -1116,7 +1064,7 @@ class FiniteDimensionalEuclideanJordanAlgebraElement(FiniteDimensionalAlgebraEle # -1 to ensure that _charpoly_coeff(r-1) is really what # appears in front of t^{r-1} in the charpoly. However, # we want the negative of THAT for the trace. - return -p(*self.vector()) + return -p(*self.to_vector()) def trace_inner_product(self, other):