X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=8bee7297fa170c0faf47610dc0ac435a881531fb;hb=b1cae89cc2e8049c43638d7a8bed2256a72f1650;hp=d31b5b71a0df190dee14d86f1544cff547f025ef;hpb=44182089cc3e0f0a6aecb6e652ad2a5c42bbb36a;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index d31b5b7..8bee729 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -26,13 +26,30 @@ lazy_import('mjo.eja.eja_subalgebra', from mjo.eja.eja_utils import _mat2vec class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): - # This is an ugly hack needed to prevent the category framework - # from implementing a coercion from our base ring (e.g. the - # rationals) into the algebra. First of all -- such a coercion is - # nonsense to begin with. But more importantly, it tries to do so - # in the category of rings, and since our algebras aren't - # associative they generally won't be rings. - _no_generic_basering_coercion = True + + def _coerce_map_from_base_ring(self): + """ + Disable the map from the base ring into the algebra. + + Performing a nonsense conversion like this automatically + is counterpedagogical. The fallback is to try the usual + element constructor, which should also fail. + + SETUP:: + + sage: from mjo.eja.eja_algebra import random_eja + + TESTS:: + + sage: set_random_seed() + sage: J = random_eja() + sage: J(1) + Traceback (most recent call last): + ... + ValueError: not a naturally-represented algebra element + + """ + return None def __init__(self, field, @@ -111,7 +128,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES: @@ -139,7 +156,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): vector representations) back and forth faithfully:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() + sage: J = HadamardEJA.random_instance() sage: x = J.random_element() sage: J(x.to_vector().column()) == x True @@ -149,15 +166,22 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): True """ + msg = "not a naturally-represented algebra element" if elt == 0: # The superclass implementation of random_element() # needs to be able to coerce "0" into the algebra. return self.zero() + elif elt in self.base_ring(): + # Ensure that no base ring -> algebra coercion is performed + # by this method. There's some stupidity in sage that would + # otherwise propagate to this method; for example, sage thinks + # that the integer 3 belongs to the space of 2-by-2 matrices. + raise ValueError(msg) natural_basis = self.natural_basis() basis_space = natural_basis[0].matrix_space() if elt not in basis_space: - raise ValueError("not a naturally-represented algebra element") + raise ValueError(msg) # Thanks for nothing! Matrix spaces aren't vector spaces in # Sage, so we have to figure out its natural-basis coordinates @@ -582,12 +606,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA, + sage: from mjo.eja.eja_algebra import (HadamardEJA, ....: random_eja) EXAMPLES:: - sage: J = RealCartesianProductEJA(5) + sage: J = HadamardEJA(5) sage: J.one() e0 + e1 + e2 + e3 + e4 @@ -903,8 +927,7 @@ class KnownRankEJA(object): return cls(n, field, **kwargs) -class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, - KnownRankEJA): +class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ Return the Euclidean Jordan Algebra corresponding to the set `R^n` under the Hadamard product. @@ -915,13 +938,13 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA EXAMPLES: This multiplication table can be verified by hand:: - sage: J = RealCartesianProductEJA(3) + sage: J = HadamardEJA(3) sage: e0,e1,e2 = J.gens() sage: e0*e0 e0 @@ -940,7 +963,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, We can change the generator prefix:: - sage: RealCartesianProductEJA(3, prefix='r').gens() + sage: HadamardEJA(3, prefix='r').gens() (r0, r1, r2) """ @@ -949,7 +972,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ] for i in range(n) ] - fdeja = super(RealCartesianProductEJA, self) + fdeja = super(HadamardEJA, self) return fdeja.__init__(field, mult_table, rank=n, **kwargs) def inner_product(self, x, y): @@ -958,7 +981,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA TESTS: @@ -966,7 +989,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, over `R^n`:: sage: set_random_seed() - sage: J = RealCartesianProductEJA.random_instance() + sage: J = HadamardEJA.random_instance() sage: x,y = J.random_elements(2) sage: X = x.natural_representation() sage: Y = y.natural_representation() @@ -1876,11 +1899,129 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) -class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): +class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): + 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 = + (x0*y0 + , x0*y_bar + y0*x_bar)`` where ``B`` is a + symmetric positive-definite "bilinear form" matrix. It has + dimension `n` over the reals, and reduces to the ``JordanSpinEJA`` + when ``B`` is the identity matrix of order ``n-1``. + + SETUP:: + + sage: from mjo.eja.eja_algebra import (BilinearFormEJA, + ....: JordanSpinEJA) + + EXAMPLES: + + When no bilinear form is specified, the identity matrix is used, + and the resulting algebra is the Jordan spin algebra:: + + sage: J0 = BilinearFormEJA(3) + sage: J1 = JordanSpinEJA(3) + sage: J0.multiplication_table() == J0.multiplication_table() + True + + TESTS: + + We can create a zero-dimensional algebra:: + + sage: J = BilinearFormEJA(0) + sage: J.basis() + Finite family {} + + We can check the multiplication condition given in the Jordan, von + Neumann, and Wigner paper (and also discussed on my "On the + symmetry..." paper). Note that this relies heavily on the standard + choice of basis, as does anything utilizing the bilinear form matrix:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis() + sage: V = J.vector_space() + sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list())) + ....: for ei in eis ] + sage: actual = [ sis[i]*sis[j] + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: expected = [ J.one() if i == j else J.zero() + ....: for i in range(n-1) + ....: for j in range(n-1) ] + sage: actual == expected + True + """ + def __init__(self, n, field=QQ, B=None, **kwargs): + if B is None: + self._B = matrix.identity(field, max(0,n-1)) + else: + self._B = B + + V = VectorSpace(field, n) + mult_table = [[V.zero() for j in range(n)] for i in range(n)] + for i in range(n): + for j in range(n): + x = V.gen(i) + y = V.gen(j) + x0 = x[0] + xbar = x[1:] + y0 = y[0] + ybar = y[1:] + z0 = x0*y0 + (self._B*xbar).inner_product(ybar) + zbar = y0*xbar + x0*ybar + z = V([z0] + zbar.list()) + mult_table[i][j] = z + + # The rank of this algebra is two, unless we're in a + # one-dimensional ambient space (because the rank is bounded + # by the ambient dimension). + fdeja = super(BilinearFormEJA, self) + return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) + + def inner_product(self, x, y): + r""" + Half of the trace inner product. + + This is defined so that the special case of the Jordan spin + algebra gets the usual inner product. + + SETUP:: + + sage: from mjo.eja.eja_algebra import BilinearFormEJA + + TESTS: + + Ensure that this is one-half of the trace inner-product when + the algebra isn't just the reals (when ``n`` isn't one). This + is in Faraut and Koranyi, and also my "On the symmetry..." + paper:: + + sage: set_random_seed() + sage: n = ZZ.random_element(2,5) + sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular') + sage: B = M.transpose()*M + sage: J = BilinearFormEJA(n, B=B) + sage: x = J.random_element() + sage: y = J.random_element() + sage: x.inner_product(y) == (x*y).trace()/2 + True + + """ + xvec = x.to_vector() + xbar = xvec[1:] + yvec = y.to_vector() + ybar = yvec[1:] + return x[0]*y[0] + (self._B*xbar).inner_product(ybar) + + +class JordanSpinEJA(BilinearFormEJA): """ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)`` with the usual inner product and jordan product ``x*y = - (, x0*y_bar + y0*x_bar)``. It has dimension `n` over + (, x0*y_bar + y0*x_bar)``. It has dimension `n` over the reals. SETUP:: @@ -1913,42 +2054,9 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: JordanSpinEJA(2, prefix='B').gens() (B0, B1) - """ - def __init__(self, n, field=QQ, **kwargs): - V = VectorSpace(field, n) - mult_table = [[V.zero() for j in range(n)] for i in range(n)] - for i in range(n): - for j in range(n): - x = V.gen(i) - y = V.gen(j) - x0 = x[0] - xbar = x[1:] - y0 = y[0] - ybar = y[1:] - # z = x*y - z0 = x.inner_product(y) - zbar = y0*xbar + x0*ybar - z = V([z0] + zbar.list()) - mult_table[i][j] = z - - # The rank of the spin algebra is two, unless we're in a - # one-dimensional ambient space (because the rank is bounded by - # the ambient dimension). - fdeja = super(JordanSpinEJA, self) - return fdeja.__init__(field, mult_table, rank=min(n,2), **kwargs) - - def inner_product(self, x, y): - """ - Faster to reimplement than to use natural representations. - - SETUP:: - - sage: from mjo.eja.eja_algebra import JordanSpinEJA - - TESTS: + TESTS: - Ensure that this is the usual inner product for the algebras - over `R^n`:: + Ensure that we have the usual inner product on `R^n`:: sage: set_random_seed() sage: J = JordanSpinEJA.random_instance() @@ -1958,8 +2066,11 @@ class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): sage: x.inner_product(y) == J.natural_inner_product(X,Y) True - """ - return x.to_vector().inner_product(y.to_vector()) + """ + def __init__(self, n, field=QQ, **kwargs): + # This is a special case of the BilinearFormEJA with the identity + # matrix as its bilinear form. + return super(JordanSpinEJA, self).__init__(n, field, **kwargs) class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):