X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=12207b7c5a8e897738ab21a73361883cae03626f;hb=5d646c586de50b571d2983b546a05899bf0c20c2;hp=430f233114b09f36361322e5f2d4b4e408fd2aeb;hpb=42e532c7a3cf00e2d6ce1e70d05435bf0d30010c;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 430f233..12207b7 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -16,12 +16,9 @@ from sage.misc.cachefunc import cached_method from sage.misc.prandom import choice from sage.misc.table import table from sage.modules.free_module import FreeModule, VectorSpace -from sage.rings.integer_ring import ZZ -from sage.rings.number_field.number_field import QuadraticField -from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing -from sage.rings.rational_field import QQ -from sage.rings.real_lazy import CLF, RLF - +from sage.rings.all import (ZZ, QQ, RR, RLF, CLF, + PolynomialRing, + QuadraticField) from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement from mjo.eja.eja_utils import _mat2vec @@ -40,11 +37,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): rank, prefix='e', category=None, - natural_basis=None): + natural_basis=None, + check=True): """ SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja) EXAMPLES: @@ -56,7 +54,23 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: x*y == y*x True + TESTS: + + The ``field`` we're given must be real:: + + sage: JordanSpinEJA(2,QQbar) + Traceback (most recent call last): + ... + ValueError: field is not real + """ + if check: + if not field.is_subring(RR): + # Note: this does return true for the real algebraic + # field, and any quadratic field where we've specified + # a real embedding. + raise ValueError('field is not real') + self._rank = rank self._natural_basis = natural_basis @@ -237,7 +251,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): """ (A_of_x, x, xr, detA) = self._charpoly_matrix_system() R = A_of_x.base_ring() - if i >= self.rank(): + + if i == self.rank(): + return R.one() + if i > self.rank(): # Guaranteed by theory return R.zero() @@ -346,7 +363,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import JordanSpinEJA + sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA EXAMPLES: @@ -360,12 +377,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): sage: p(*xvec) t^2 - 2*t + 1 + By definition, the characteristic polynomial is a monic + degree-zero polynomial in a rank-zero algebra. Note that + Cayley-Hamilton is indeed satisfied since the polynomial + ``1`` evaluates to the identity element of the algebra on + any argument:: + + sage: J = TrivialEJA() + sage: J.characteristic_polynomial() + 1 + """ r = self.rank() n = self.dimension() - # The list of coefficient polynomials a_1, a_2, ..., a_n. - a = [ self._charpoly_coeff(i) for i in range(n) ] + # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n. + a = [ self._charpoly_coeff(i) for i in range(r+1) ] # We go to a bit of trouble here to reorder the # indeterminates, so that it's easier to evaluate the @@ -377,17 +404,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): S = PolynomialRing(S, R.variable_names()) t = S(t) - # Note: all entries past the rth should be zero. The - # coefficient of the highest power (x^r) is 1, but it doesn't - # appear in the solution vector which contains coefficients - # for the other powers (to make them sum to x^r). - if (r < n): - a[r] = 1 # corresponds to x^r - else: - # When the rank is equal to the dimension, trying to - # assign a[r] goes out-of-bounds. - a.append(1) # corresponds to x^r - return sum( a[k]*(t**k) for k in xrange(len(a)) ) @@ -428,15 +444,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import ComplexHermitianEJA + sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, + ....: TrivialEJA) EXAMPLES:: sage: J = ComplexHermitianEJA(3) sage: J.is_trivial() False - sage: A = J.zero().subalgebra_generated_by() - sage: A.is_trivial() + + :: + + sage: J = TrivialEJA() + sage: J.is_trivial() True """ @@ -611,14 +631,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): return self.linear_combination(zip(self.gens(), coeffs)) - def random_element(self): - # Temporary workaround for https://trac.sagemath.org/ticket/28327 - if self.is_trivial(): - return self.zero() - else: - s = super(FiniteDimensionalEuclideanJordanAlgebra, self) - return s.random_element() - def random_elements(self, count): """ Return ``count`` random elements as a tuple. @@ -683,11 +695,15 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): TESTS: Ensure that every EJA that we know how to construct has a - positive integer rank:: + positive integer rank, unless the algebra is trivial in + which case its rank will be zero:: sage: set_random_seed() - sage: r = random_eja().rank() - sage: r in ZZ and r > 0 + sage: J = random_eja() + sage: r = J.rank() + sage: r in ZZ + True + sage: r > 0 or (r == 0 and J.is_trivial()) True """ @@ -760,6 +776,11 @@ class KnownRankEJA(object): Beware, this will crash for "most instances" because the constructor below looks wrong. """ + if cls is TrivialEJA: + # The TrivialEJA class doesn't take an "n" argument because + # there's only one. + return cls(field) + n = ZZ.random_element(cls._max_test_case_size()) + 1 return cls(n, field, **kwargs) @@ -838,32 +859,10 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, return x.to_vector().inner_product(y.to_vector()) -def random_eja(): +def random_eja(field=QQ, nontrivial=False): """ Return a "random" finite-dimensional Euclidean Jordan Algebra. - ALGORITHM: - - For now, we choose a random natural number ``n`` (greater than zero) - and then give you back one of the following: - - * The cartesian product of the rational numbers ``n`` times; this is - ``QQ^n`` with the Hadamard product. - - * The Jordan spin algebra on ``QQ^n``. - - * The ``n``-by-``n`` rational symmetric matrices with the symmetric - product. - - * The ``n``-by-``n`` complex-rational Hermitian matrices embedded - in the space of ``2n``-by-``2n`` real symmetric matrices. - - * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded - in the space of ``4n``-by-``4n`` real symmetric matrices. - - Later this might be extended to return Cartesian products of the - EJAs above. - SETUP:: sage: from mjo.eja.eja_algebra import random_eja @@ -874,8 +873,11 @@ def random_eja(): Euclidean Jordan algebra of dimension... """ - classname = choice(KnownRankEJA.__subclasses__()) - return classname.random_instance() + eja_classes = KnownRankEJA.__subclasses__() + if nontrivial: + eja_classes.remove(TrivialEJA) + classname = choice(eja_classes) + return classname.random_instance(field=field) @@ -1840,3 +1842,40 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ return x.to_vector().inner_product(y.to_vector()) + + +class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + """ + The trivial Euclidean Jordan algebra consisting of only a zero element. + + SETUP:: + + sage: from mjo.eja.eja_algebra import TrivialEJA + + EXAMPLES:: + + sage: J = TrivialEJA() + sage: J.dimension() + 0 + sage: J.zero() + 0 + sage: J.one() + 0 + sage: 7*J.one()*12*J.one() + 0 + sage: J.one().inner_product(J.one()) + 0 + sage: J.one().norm() + 0 + sage: J.one().subalgebra_generated_by() + Euclidean Jordan algebra of dimension 0 over Rational Field + sage: J.rank() + 0 + + """ + def __init__(self, field=QQ, **kwargs): + mult_table = [] + fdeja = super(TrivialEJA, self) + # The rank is zero using my definition, namely the dimension of the + # largest subalgebra generated by any element. + return fdeja.__init__(field, mult_table, rank=0, **kwargs)