X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=e2d644a903d8eca09a585064f96e31278197e4d6;hb=2a3d7383e72864ac42a532c6c4250cdc45b4c363;hp=747e198f8922e3d6c195c4c7552829526463d405;hpb=ec37d0e9ac40f4345da8e52a1b3dd35599e161fb;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index 747e198..e2d644a 100644 --- a/mjo/eja/euclidean_jordan_algebra.py +++ b/mjo/eja/euclidean_jordan_algebra.py @@ -112,7 +112,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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:: @@ -124,6 +124,18 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): 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 + """ A = self.parent() if n == 0: @@ -131,7 +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): @@ -149,6 +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. @@ -188,7 +237,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): Example 11.11:: 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: while x.is_zero(): @@ -355,7 +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 @@ -365,7 +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. @@ -383,10 +432,10 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: J = random_eja() sage: x = J.random_element() sage: y = J.random_element() - sage: Lx = x.matrix() - sage: Ly = y.matrix() - sage: Lxx = (x*x).matrix() - sage: Lxy = (x*y).matrix() + 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 @@ -398,12 +447,12 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: x = J.random_element() sage: y = J.random_element() sage: z = J.random_element() - sage: Lx = x.matrix() - sage: Ly = y.matrix() - sage: Lz = z.matrix() - sage: Lzy = (z*y).matrix() - sage: Lxy = (x*y).matrix() - sage: Lxz = (x*z).matrix() + 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 @@ -415,13 +464,13 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): sage: u = J.random_element() sage: y = J.random_element() sage: z = J.random_element() - sage: Lu = u.matrix() - sage: Ly = y.matrix() - sage: Lz = z.matrix() - sage: Lzy = (z*y).matrix() - sage: Luy = (u*y).matrix() - sage: Luz = (u*z).matrix() - sage: Luyz = (u*(y*z)).matrix() + 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) @@ -432,6 +481,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra): return fda_elt.matrix().transpose() + def minimal_polynomial(self): """ EXAMPLES:: @@ -454,7 +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(): @@ -499,7 +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() @@ -549,7 +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 @@ -561,9 +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): @@ -596,7 +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 """ @@ -668,7 +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 @@ -685,7 +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 @@ -790,7 +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, @@ -825,7 +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 @@ -1002,7 +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 @@ -1026,7 +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