X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=26b72480ffd6d18c1bb8ca43ae3358afda951996;hb=1846049dafeb625173550d00539989e14939544a;hp=d31b5b71a0df190dee14d86f1544cff547f025ef;hpb=44182089cc3e0f0a6aecb6e652ad2a5c42bbb36a;p=sage.d.git diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index d31b5b7..26b7248 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -111,7 +111,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule): SETUP:: sage: from mjo.eja.eja_algebra import (JordanSpinEJA, - ....: RealCartesianProductEJA, + ....: HadamardEJA, ....: RealSymmetricEJA) EXAMPLES: @@ -139,7 +139,7 @@ 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 @@ -582,12 +582,12 @@ 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 +903,7 @@ 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 +914,13 @@ 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 +939,7 @@ 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 +948,7 @@ 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 +957,7 @@ class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra, SETUP:: - sage: from mjo.eja.eja_algebra import RealCartesianProductEJA + sage: from mjo.eja.eja_algebra import HadamardEJA TESTS: @@ -966,7 +965,7 @@ 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 +1875,130 @@ class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra, super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs) +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 + + 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:: + + sage: set_random_seed() + sage: n = ZZ.random_element(5) + sage: M = matrix.random(QQ, 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) ] + + """ + 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(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA): """ 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::