X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=4713ff0859876b5832e6e5e9551f632444c6a138;hb=8ee3c1ac3dd5a2b08f91cfd4c700d87f617196e6;hp=a1102461714c43e318130052669fb898d218914e;hpb=ce3d3c5a22123dd16d074e2909e6d647fd721ce2;p=sage.d.git diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py index a110246..4713ff0 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() + 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: x.matrix()*x.vector() == (x**2).vector() + sage: m = ZZ.random_element(0,10) + sage: n = ZZ.random_element(0,10) + sage: Lxm = (x^m).matrix() + sage: Lxn = (x^n).matrix() + sage: Lxm*Lxn == Lxn*Lxm True """ @@ -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.matrix() + B = other.matrix() + return (A*B == B*A) + + def det(self): """ Return my determinant, the product of my eigenvalues. @@ -166,7 +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(): @@ -333,7 +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 @@ -351,6 +422,60 @@ 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.matrix() + sage: Ly = y.matrix() + sage: Lxx = (x*x).matrix() + sage: Lxy = (x*y).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.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: 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.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: 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() @@ -378,7 +503,7 @@ 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(): @@ -423,7 +548,7 @@ 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() @@ -473,7 +598,7 @@ 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 @@ -714,7 +839,7 @@ 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, @@ -749,7 +874,7 @@ 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 @@ -926,7 +1051,7 @@ 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 @@ -950,7 +1075,7 @@ 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