X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=def2028a6a29a0d64733c8ed5c8a38e226c37f81;hb=cbe4ee91e7a2b24ec573ba623b77dfa8a30a1574;hp=10b8eb338514a0ec611ced0bb0a9ca169dc67411;hpb=16c76a123e1657907d4559518fbb4ea0ef9a8e2f;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index 10b8eb3..def2028 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -57,7 +57,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): prefix='e', category=None, natural_basis=None, - check=True): + check_field=True, + check_axioms=True): """ SETUP:: @@ -78,14 +79,14 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): TESTS: - The ``field`` we're given must be real with ``check=True``:: + The ``field`` we're given must be real with ``check_field=True``:: sage: JordanSpinEJA(2,QQbar) Traceback (most recent call last): ... - ValueError: field is not real + ValueError: scalar field is not real - The multiplication table must be square with ``check=True``:: + The multiplication table must be square with ``check_axioms=True``:: sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),())) Traceback (most recent call last): @@ -93,12 +94,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): ValueError: multiplication table is not square """ - if check: + if check_field: 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') + # field, the rationals, and any quadratic field where + # we've specified a real embedding. + raise ValueError("scalar field is not real") + + # The multiplication table had better be square + n = len(mult_table) + if check_axioms: + if not all( len(l) == n for l in mult_table ): + raise ValueError("multiplication table is not square") self._natural_basis = natural_basis @@ -106,12 +113,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): category = MagmaticAlgebras(field).FiniteDimensional() category = category.WithBasis().Unital() - # The multiplication table had better be square - n = len(mult_table) - if check: - if not all( len(l) == n for l in mult_table ): - raise ValueError("multiplication table is not square") - fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, range(n), @@ -130,7 +131,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): for ls in mult_table ] - if check: + if check_axioms: if not self._is_commutative(): raise ValueError("algebra is not commutative") if not self._is_jordanian(): @@ -263,8 +264,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Whether or not this algebra's multiplication table is commutative. This method should of course always return ``True``, unless - this algebra was constructed with ``check=False`` and passed - an invalid multiplication table. + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. """ return all( self.product_on_basis(i,j) == self.product_on_basis(i,j) for i in range(self.dimension()) @@ -279,7 +280,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): ``True`` result to be truly true, you should also check :meth:`_is_commutative`. This method should of course always return ``True``, unless this algebra was constructed with - ``check=False`` and passed an invalid multiplication table. + ``check_axioms=False`` and passed an invalid multiplication table. """ return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j)) == @@ -293,13 +294,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): associative; that is, whether or not `B(xy,z) = B(x,yz)`. This method should of course always return ``True``, unless - this algebra was constructed with ``check=False`` and passed - an invalid multiplication table. + this algebra was constructed with ``check_axioms=False`` and + passed an invalid multiplication table. """ # Used to check whether or not something is zero in an inexact # ring. This number is sufficient to allow the construction of - # QuaternionHermitianEJA(2, RDF) with check=True. + # QuaternionHermitianEJA(2, RDF) with check_axioms=True. epsilon = 1e-16 for i in range(self.dimension()): @@ -747,7 +748,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): gens = tuple( self.from_vector(b) for b in eigspace.basis() ) subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens, - check=False) + check_axioms=False) if eigval == 0: J0 = subalg elif eigval == 1: @@ -1006,81 +1007,6 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): Element = FiniteDimensionalEuclideanJordanAlgebraElement -class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra): - """ - Return the Euclidean Jordan Algebra corresponding to the set - `R^n` under the Hadamard product. - - Note: this is nothing more than the Cartesian product of ``n`` - copies of the spin algebra. Once Cartesian product algebras - are implemented, this can go. - - SETUP:: - - sage: from mjo.eja.eja_algebra import HadamardEJA - - EXAMPLES: - - This multiplication table can be verified by hand:: - - sage: J = HadamardEJA(3) - sage: e0,e1,e2 = J.gens() - sage: e0*e0 - e0 - sage: e0*e1 - 0 - sage: e0*e2 - 0 - sage: e1*e1 - e1 - sage: e1*e2 - 0 - sage: e2*e2 - e2 - - TESTS: - - We can change the generator prefix:: - - sage: HadamardEJA(3, prefix='r').gens() - (r0, r1, r2) - - """ - def __init__(self, n, field=AA, **kwargs): - V = VectorSpace(field, n) - mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] - for i in range(n) ] - - super(HadamardEJA, self).__init__(field, - mult_table, - check=False, - **kwargs) - self.rank.set_cache(n) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. - - SETUP:: - - sage: from mjo.eja.eja_algebra import HadamardEJA - - TESTS: - - Ensure that this is the usual inner product for the algebras - over `R^n`:: - - sage: set_random_seed() - sage: J = HadamardEJA.random_instance() - sage: x,y = J.random_elements(2) - sage: X = x.natural_representation() - sage: Y = y.natural_representation() - sage: x.inner_product(y) == J.natural_inner_product(X,Y) - True - - """ - return x.to_vector().inner_product(y.to_vector()) - def random_eja(field=AA): """ @@ -1107,6 +1033,65 @@ def random_eja(field=AA): +class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): + r""" + Algebras whose basis consists of vectors with rational + entries. Equivalently, algebras whose multiplication tables + contain only rational coefficients. + + When an EJA has a basis that can be made rational, we can speed up + the computation of its characteristic polynomial by doing it over + ``QQ``. All of the named EJA constructors that we provide fall + into this category. + """ + @cached_method + def _charpoly_coefficients(self): + r""" + Override the parent method with something that tries to compute + over a faster (non-extension) field. + + SETUP:: + + sage: from mjo.eja.eja_algebra import JordanSpinEJA + + EXAMPLES: + + The base ring of the resulting polynomial coefficients is what + it should be, and not the rationals (unless the algebra was + already over the rationals):: + + sage: J = JordanSpinEJA(3) + sage: J._charpoly_coefficients() + (X1^2 - X2^2 - X3^2, -2*X1) + sage: a0 = J._charpoly_coefficients()[0] + sage: J.base_ring() + Algebraic Real Field + sage: a0.base_ring() + Algebraic Real Field + + """ + if self.base_ring() is QQ: + # There's no need to construct *another* algebra over the + # rationals if this one is already over the rationals. + superclass = super(RationalBasisEuclideanJordanAlgebra, self) + return superclass._charpoly_coefficients() + + mult_table = tuple( + map(lambda x: x.to_vector(), ls) + for ls in self._multiplication_table + ) + + # Do the computation over the rationals. The answer will be + # the same, because our basis coordinates are (essentially) + # rational. + J = FiniteDimensionalEuclideanJordanAlgebra(QQ, + mult_table, + check_field=False, + check_axioms=False) + a = J._charpoly_coefficients() + return tuple(map(lambda x: x.change_ring(self.base_ring()), a)) + + class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): @staticmethod def _max_test_case_size(): @@ -1155,43 +1140,44 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Override the parent method with something that tries to compute over a faster (non-extension) field. """ - if self._basis_normalizers is None: - # We didn't normalize, so assume that the basis we started - # with had entries in a nice field. + if self._basis_normalizers is None or self.base_ring() is QQ: + # We didn't normalize, or the basis we started with had + # entries in a nice field already. Just compute the thing. return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients() - else: - basis = ( (b/n) for (b,n) in zip(self.natural_basis(), - self._basis_normalizers) ) - - # Do this over the rationals and convert back at the end. - # Only works because we know the entries of the basis are - # integers. The argument ``check=False`` is required - # because the trace inner-product method for this - # class is a stub and can't actually be checked. - J = MatrixEuclideanJordanAlgebra(QQ, - basis, - normalize_basis=False, - check=False) - a = J._charpoly_coefficients() - - # Unfortunately, changing the basis does change the - # coefficients of the characteristic polynomial, but since - # these are really the coefficients of the "characteristic - # polynomial of" function, everything is still nice and - # unevaluated. It's therefore "obvious" how scaling the - # basis affects the coordinate variables X1, X2, et - # cetera. Scaling the first basis vector up by "n" adds a - # factor of 1/n into every "X1" term, for example. So here - # we simply undo the basis_normalizer scaling that we - # performed earlier. - # - # The a[0] access here is safe because trivial algebras - # won't have any basis normalizers and therefore won't - # make it to this "else" branch. - XS = a[0].parent().gens() - subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i] - for i in range(len(XS)) } - return tuple( a_i.subs(subs_dict) for a_i in a ) + + basis = ( (b/n) for (b,n) in zip(self.natural_basis(), + self._basis_normalizers) ) + + # Do this over the rationals and convert back at the end. + # Only works because we know the entries of the basis are + # integers. The argument ``check_axioms=False`` is required + # because the trace inner-product method for this + # class is a stub and can't actually be checked. + J = MatrixEuclideanJordanAlgebra(QQ, + basis, + normalize_basis=False, + check_field=False, + check_axioms=False) + a = J._charpoly_coefficients() + + # Unfortunately, changing the basis does change the + # coefficients of the characteristic polynomial, but since + # these are really the coefficients of the "characteristic + # polynomial of" function, everything is still nice and + # unevaluated. It's therefore "obvious" how scaling the + # basis affects the coordinate variables X1, X2, et + # cetera. Scaling the first basis vector up by "n" adds a + # factor of 1/n into every "X1" term, for example. So here + # we simply undo the basis_normalizer scaling that we + # performed earlier. + # + # The a[0] access here is safe because trivial algebras + # won't have any basis normalizers and therefore won't + # make it to this "else" branch. + XS = a[0].parent().gens() + subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i] + for i in range(len(XS)) } + return tuple( a_i.subs(subs_dict) for a_i in a ) @staticmethod @@ -1258,16 +1244,11 @@ class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra): Yu = cls.real_unembed(Y) tr = (Xu*Yu).trace() - if tr in RLF: - # It's real already. - return tr - - # Otherwise, try the thing that works for complex numbers; and - # if that doesn't work, the thing that works for quaternions. try: - return tr.vector()[0] # real part, imag part is index 1 + # Works in QQ, AA, RDF, et cetera. + return tr.real() except AttributeError: - # A quaternions doesn't have a vector() method, but does + # A quaternion doesn't have a real() method, but does # have coefficient_tuple() method that returns the # coefficients of 1, i, j, and k -- in that order. return tr.coefficient_tuple()[0] @@ -1416,7 +1397,7 @@ class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra): basis = self._denormalized_basis(n, field) super(RealSymmetricEJA, self).__init__(field, basis, - check=False, + check_axioms=False, **kwargs) self.rank.set_cache(n) @@ -1715,7 +1696,7 @@ class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra): basis = self._denormalized_basis(n,field) super(ComplexHermitianEJA,self).__init__(field, basis, - check=False, + check_axioms=False, **kwargs) self.rank.set_cache(n) @@ -2019,12 +2000,88 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra): basis = self._denormalized_basis(n,field) super(QuaternionHermitianEJA,self).__init__(field, basis, - check=False, + check_axioms=False, **kwargs) self.rank.set_cache(n) -class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): +class HadamardEJA(RationalBasisEuclideanJordanAlgebra): + """ + Return the Euclidean Jordan Algebra corresponding to the set + `R^n` under the Hadamard product. + + Note: this is nothing more than the Cartesian product of ``n`` + copies of the spin algebra. Once Cartesian product algebras + are implemented, this can go. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + EXAMPLES: + + This multiplication table can be verified by hand:: + + sage: J = HadamardEJA(3) + sage: e0,e1,e2 = J.gens() + sage: e0*e0 + e0 + sage: e0*e1 + 0 + sage: e0*e2 + 0 + sage: e1*e1 + e1 + sage: e1*e2 + 0 + sage: e2*e2 + e2 + + TESTS: + + We can change the generator prefix:: + + sage: HadamardEJA(3, prefix='r').gens() + (r0, r1, r2) + + """ + def __init__(self, n, field=AA, **kwargs): + V = VectorSpace(field, n) + mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] + for i in range(n) ] + + super(HadamardEJA, self).__init__(field, + mult_table, + check_axioms=False, + **kwargs) + self.rank.set_cache(n) + + def inner_product(self, x, y): + """ + Faster to reimplement than to use natural representations. + + SETUP:: + + sage: from mjo.eja.eja_algebra import HadamardEJA + + TESTS: + + Ensure that this is the usual inner product for the algebras + over `R^n`:: + + sage: set_random_seed() + sage: J = HadamardEJA.random_instance() + sage: x,y = J.random_elements(2) + sage: X = x.natural_representation() + sage: Y = y.natural_representation() + sage: x.inner_product(y) == J.natural_inner_product(X,Y) + True + + """ + return x.to_vector().inner_product(y.to_vector()) + + +class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra): r""" The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the half-trace inner product and jordan product ``x*y = @@ -2105,7 +2162,7 @@ class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra): # by the ambient dimension). super(BilinearFormEJA, self).__init__(field, mult_table, - check=False, + check_axioms=False, **kwargs) self.rank.set_cache(min(n,2)) @@ -2234,7 +2291,7 @@ class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra): mult_table = [] super(TrivialEJA, self).__init__(field, mult_table, - check=False, + check_axioms=False, **kwargs) # The rank is zero using my definition, namely the dimension of the # largest subalgebra generated by any element. @@ -2267,6 +2324,7 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): """ def __init__(self, J1, J2, field=AA, **kwargs): + self._factors = (J1, J2) n1 = J1.dimension() n2 = J2.dimension() n = n1+n2 @@ -2285,6 +2343,139 @@ class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra): super(DirectSumEJA, self).__init__(field, mult_table, - check=False, + check_axioms=False, **kwargs) self.rank.set_cache(J1.rank() + J2.rank()) + + + def factors(self): + r""" + Return the pair of this algebra's factors. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: JordanSpinEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(2,QQ) + sage: J2 = JordanSpinEJA(3,QQ) + sage: J = DirectSumEJA(J1,J2) + sage: J.factors() + (Euclidean Jordan algebra of dimension 2 over Rational Field, + Euclidean Jordan algebra of dimension 3 over Rational Field) + + """ + return self._factors + + def projections(self): + r""" + Return a pair of projections onto this algebra's factors. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: ComplexHermitianEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = JordanSpinEJA(2) + sage: J2 = ComplexHermitianEJA(2) + sage: J = DirectSumEJA(J1,J2) + sage: (pi_left, pi_right) = J.projections() + sage: J.one().to_vector() + (1, 0, 1, 0, 0, 1) + sage: pi_left(J.one()).to_vector() + (1, 0) + sage: pi_right(J.one()).to_vector() + (1, 0, 0, 1) + + """ + (J1,J2) = self.factors() + n = J1.dimension() + pi_left = lambda x: J1.from_vector(x.to_vector()[:n]) + pi_right = lambda x: J2.from_vector(x.to_vector()[n:]) + return (pi_left, pi_right) + + def inclusions(self): + r""" + Return the pair of inclusion maps from our factors into us. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (JordanSpinEJA, + ....: RealSymmetricEJA, + ....: DirectSumEJA) + + EXAMPLES:: + + sage: J1 = JordanSpinEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = DirectSumEJA(J1,J2) + sage: (iota_left, iota_right) = J.inclusions() + sage: iota_left(J1.zero()) == J.zero() + True + sage: iota_right(J2.zero()) == J.zero() + True + sage: J1.one().to_vector() + (1, 0, 0) + sage: iota_left(J1.one()).to_vector() + (1, 0, 0, 0, 0, 0) + sage: J2.one().to_vector() + (1, 0, 1) + sage: iota_right(J2.one()).to_vector() + (0, 0, 0, 1, 0, 1) + sage: J.one().to_vector() + (1, 0, 0, 1, 0, 1) + + """ + (J1,J2) = self.factors() + n = J1.dimension() + V_basis = self.vector_space().basis() + I1 = matrix.column(self.base_ring(), V_basis[:n]) + I2 = matrix.column(self.base_ring(), V_basis[n:]) + iota_left = lambda x: self.from_vector(I1*x.to_vector()) + iota_right = lambda x: self.from_vector(I2*+x.to_vector()) + return (iota_left, iota_right) + + def inner_product(self, x, y): + r""" + The standard Cartesian inner-product. + + We project ``x`` and ``y`` onto our factors, and add up the + inner-products from the subalgebras. + + SETUP:: + + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: QuaternionHermitianEJA, + ....: DirectSumEJA) + + EXAMPLE:: + + sage: J1 = HadamardEJA(3) + sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False) + sage: J = DirectSumEJA(J1,J2) + sage: x1 = J1.one() + sage: x2 = x1 + sage: y1 = J2.one() + sage: y2 = y1 + sage: x1.inner_product(x2) + 3 + sage: y1.inner_product(y2) + 2 + sage: J.one().inner_product(J.one()) + 5 + + """ + (pi_left, pi_right) = self.projections() + x1 = pi_left(x) + x2 = pi_right(x) + y1 = pi_left(y) + y2 = pi_right(y) + + return (x1.inner_product(y1) + x2.inner_product(y2))