eja: add operator_commutes_with() for elements.

[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 747e198f8922e3d6c195c4c7552829526463d405..4713ff0859876b5832e6e5e9551f632444c6a138 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -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).matrix()
+                sage: Lxn = (x^n).matrix()
+                sage: Lxm*Lxn == Lxn*Lxm
+                True
+
             """
             A = self.parent()
             if n == 0:
@@ -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.matrix()
+            B = other.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
@@ -454,7 +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():
@@ -499,7 +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()
@@ -549,7 +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
@@ -790,7 +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,
@@ -825,7 +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
 
@@ -1002,7 +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
@@ -1026,7 +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