eja: add is_nilpotent() for elements.

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 8f623b4491ca1439d0905b5b91bcfe25cf754ecb..603ef29021d41f53404a84513c2d854f8e4a8c64 100644 (file)
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -263,6 +263,61 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
             return elt.minimal_polynomial()
 
 
+        def is_nilpotent(self):
+            """
+            Return whether or not some power of this element is zero.
+
+            The superclass method won't work unless we're in an
+            associative algebra, and we aren't. However, we generate
+            an assocoative subalgebra and we're nilpotent there if and
+            only if we're nilpotent here (probably).
+
+            TESTS:
+
+            The identity element is never nilpotent::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(2,10).abs()
+                sage: J = eja_rn(n)
+                sage: J.one().is_nilpotent()
+                False
+                sage: J = eja_ln(n)
+                sage: J.one().is_nilpotent()
+                False
+
+            The additive identity is always nilpotent::
+
+                sage: set_random_seed()
+                sage: n = ZZ.random_element(2,10).abs()
+                sage: J = eja_rn(n)
+                sage: J.zero().is_nilpotent()
+                True
+                sage: J = eja_ln(n)
+                sage: J.zero().is_nilpotent()
+                True
+
+            """
+            # The element we're going to call "is_nilpotent()" on.
+            # Either myself, interpreted as an element of a finite-
+            # dimensional algebra, or an element of an associative
+            # subalgebra.
+            elt = None
+
+            if self.parent().is_associative():
+                elt = FiniteDimensionalAlgebraElement(self.parent(), self)
+            else:
+                V = self.span_of_powers()
+                assoc_subalg = self.subalgebra_generated_by()
+                # Mis-design warning: the basis used for span_of_powers()
+                # and subalgebra_generated_by() must be the same, and in
+                # the same order!
+                elt = assoc_subalg(V.coordinates(self.vector()))
+
+            # Recursive call, but should work since elt lives in an
+            # associative algebra.
+            return elt.is_nilpotent()
+
+
         def characteristic_polynomial(self):
             return self.matrix().characteristic_polynomial()