eja: add a test for power-operator-commutativity.

[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index a1102461714c43e318130052669fb898d218914e..1426d5e16be4b6acc7c68a4494e2c6f1c4d61819 100644 (file)
--- 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.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).matrix()
+                sage: Lxn = (x^n).matrix()
+                sage: Lxm*Lxn == Lxn*Lxm
                 True
 
             """
@@ -166,7 +200,7 @@ 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 +367,7 @@ 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 +385,60 @@ 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 +466,7 @@ 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 +511,7 @@ 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 +561,7 @@ 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 +802,7 @@ 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 +837,7 @@ 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 +1014,7 @@ 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 +1038,7 @@ 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