X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=e2d644a903d8eca09a585064f96e31278197e4d6;hb=2a3d7383e72864ac42a532c6c4250cdc45b4c363;hp=636149a92a33a143be9fca84e6a2924ca9377613;hpb=e09be8bb4f245fd948a073f28d52ca9efa372bbb;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 636149a..e2d644a 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -52,6 +52,19 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): assume_associative=False, category=None, rank=None): + """ + EXAMPLES: + + By definition, Jordan multiplication commutes:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: x*y == y*x + True + + """ self._rank = rank fda = super(FiniteDimensionalEuclideanJordanAlgebra, self) fda.__init__(field, @@ -95,11 +108,32 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): instead of column vectors! We, on the other hand, assume column vectors everywhere. - EXAMPLES: + EXAMPLES:: sage: set_random_seed() sage: x = random_eja().random_element() - sage: x.matrix()*x.vector() == (x**2).vector() + sage: x.operator_matrix()*x.vector() == (x^2).vector() + True + + A few examples of power-associativity:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x*(x*x)*(x*x) == x^5 + True + sage: (x*x)*(x*x*x) == x^5 + True + + We also know that powers operator-commute (Koecher, Chapter + III, Corollary 1):: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: m = ZZ.random_element(0,10) + sage: n = ZZ.random_element(0,10) + sage: Lxm = (x^m).operator_matrix() + sage: Lxn = (x^n).operator_matrix() + sage: Lxm*Lxn == Lxn*Lxm True """ @@ -109,7 +143,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): elif n == 1: return self else: - return A.element_class(A, (self.matrix()**(n-1))*self.vector()) + return A( (self.operator_matrix()**(n-1))*self.vector() ) def characteristic_polynomial(self): @@ -127,6 +161,43 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): raise NotImplementedError('irregular element') + def operator_commutes_with(self, other): + """ + Return whether or not this element operator-commutes + with ``other``. + + EXAMPLES: + + The definition of a Jordan algebra says that any element + operator-commutes with its square:: + + sage: set_random_seed() + sage: x = random_eja().random_element() + sage: x.operator_commutes_with(x^2) + True + + TESTS: + + Test Lemma 1 from Chapter III of Koecher:: + + sage: set_random_seed() + sage: J = random_eja() + sage: u = J.random_element() + sage: v = J.random_element() + sage: lhs = u.operator_commutes_with(u*v) + sage: rhs = v.operator_commutes_with(u^2) + sage: lhs == rhs + True + + """ + if not other in self.parent(): + raise ArgumentError("'other' must live in the same algebra") + + A = self.operator_matrix() + B = other.operator_matrix() + return (A*B == B*A) + + def det(self): """ Return my determinant, the product of my eigenvalues. @@ -160,6 +231,26 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): We can't use the superclass method because it relies on the algebra being associative. + EXAMPLES: + + The inverse in the spin factor algebra is given in Alizadeh's + Example 11.11:: + + sage: set_random_seed() + sage: n = ZZ.random_element(1,10) + sage: J = JordanSpinSimpleEJA(n) + sage: x = J.random_element() + sage: while x.is_zero(): + ....: x = J.random_element() + sage: x_vec = x.vector() + sage: x0 = x_vec[0] + sage: x_bar = x_vec[1:] + sage: coeff = 1/(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) + True + TESTS: The identity element is its own inverse:: @@ -313,7 +404,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): aren't multiples of the identity are regular:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() + sage: n = ZZ.random_element(1,10) sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x == x.coefficient(0)*J.one() or x.degree() == 2 @@ -323,7 +414,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return self.span_of_powers().dimension() - def matrix(self): + def operator_matrix(self): """ Return the matrix that represents left- (or right-) multiplication by this element in the parent algebra. @@ -331,11 +422,66 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): We have to override this because the superclass method returns a matrix that acts on row vectors (that is, on the right). + + EXAMPLES: + + Test the first polarization identity from my notes, Koecher Chapter + III, or from Baes (2.3):: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lxx = (x*x).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly) + True + + Test the second polarization identity from my notes or from + Baes (2.4):: + + sage: set_random_seed() + sage: J = random_eja() + sage: x = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: Lx = x.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Lxy = (x*y).operator_matrix() + sage: Lxz = (x*z).operator_matrix() + sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly) + True + + Test the third polarization identity from my notes or from + Baes (2.5):: + + sage: set_random_seed() + sage: J = random_eja() + sage: u = J.random_element() + sage: y = J.random_element() + sage: z = J.random_element() + sage: Lu = u.operator_matrix() + sage: Ly = y.operator_matrix() + sage: Lz = z.operator_matrix() + sage: Lzy = (z*y).operator_matrix() + sage: Luy = (u*y).operator_matrix() + sage: Luz = (u*z).operator_matrix() + sage: Luyz = (u*(y*z)).operator_matrix() + sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz + sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly + sage: bool(lhs == rhs) + True + """ fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self) return fda_elt.matrix().transpose() + def minimal_polynomial(self): """ EXAMPLES:: @@ -358,7 +504,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): identity:: sage: set_random_seed() - sage: n = ZZ.random_element(2,10).abs() + sage: n = ZZ.random_element(2,10) sage: J = JordanSpinSimpleEJA(n) sage: y = J.random_element() sage: while y == y.coefficient(0)*J.one(): @@ -403,7 +549,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Alizadeh's Example 11.12:: sage: set_random_seed() - sage: n = ZZ.random_element(1,10).abs() + sage: n = ZZ.random_element(1,10) sage: J = JordanSpinSimpleEJA(n) sage: x = J.random_element() sage: x_vec = x.vector() @@ -453,7 +599,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Property 6: - sage: k = ZZ.random_element(1,10).abs() + sage: k = ZZ.random_element(1,10) sage: actual = (x^k).quadratic_representation() sage: expected = (x.quadratic_representation())^k sage: actual == expected @@ -465,9 +611,9 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): elif not other in self.parent(): raise ArgumentError("'other' must live in the same algebra") - return ( self.matrix()*other.matrix() - + other.matrix()*self.matrix() - - (self*other).matrix() ) + L = self.operator_matrix() + M = other.operator_matrix() + return ( L*M + M*L - (self*other).operator_matrix() ) def span_of_powers(self): @@ -500,7 +646,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: set_random_seed() sage: x = random_eja().random_element() sage: u = x.subalgebra_generated_by().random_element() - sage: u.matrix()*u.vector() == (u**2).vector() + sage: u.operator_matrix()*u.vector() == (u**2).vector() True """ @@ -572,7 +718,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): s = 0 minimal_dim = V.dimension() for i in xrange(1, V.dimension()): - this_dim = (u**i).matrix().image().dimension() + this_dim = (u**i).operator_matrix().image().dimension() if this_dim < minimal_dim: minimal_dim = this_dim s = i @@ -589,7 +735,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): # Beware, solve_right() means that we're using COLUMN vectors. # Our FiniteDimensionalAlgebraElement superclass uses rows. u_next = u**(s+1) - A = u_next.matrix() + A = u_next.operator_matrix() c_coordinates = A.solve_right(u_next.vector()) # Now c_coordinates is the idempotent we want, but it's in @@ -694,7 +840,7 @@ def random_eja(): Euclidean Jordan algebra of degree... """ - n = ZZ.random_element(1,5).abs() + n = ZZ.random_element(1,5) constructor = choice([eja_rn, JordanSpinSimpleEJA, RealSymmetricSimpleEJA, @@ -729,7 +875,7 @@ def _complex_hermitian_basis(n, field=QQ): TESTS:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5).abs() + sage: n = ZZ.random_element(1,5) sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) ) True @@ -906,7 +1052,7 @@ def RealSymmetricSimpleEJA(n, field=QQ): The degree of this algebra is `(n^2 + n) / 2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5).abs() + sage: n = ZZ.random_element(1,5) sage: J = RealSymmetricSimpleEJA(n) sage: J.degree() == (n^2 + n)/2 True @@ -930,7 +1076,7 @@ def ComplexHermitianSimpleEJA(n, field=QQ): The degree of this algebra is `n^2`:: sage: set_random_seed() - sage: n = ZZ.random_element(1,5).abs() + sage: n = ZZ.random_element(1,5) sage: J = ComplexHermitianSimpleEJA(n) sage: J.degree() == n^2 True