From 0fbef149002b9bec70c64eab9170cb22e59afd9f Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 23 Jun 2019 22:05:30 -0400
Subject: [PATCH] eja/euclidean_jordan_algebra.py: begin implementing an
 element class.

---
 mjo/eja/euclidean_jordan_algebra.py | 60 ++++++++++++++++++++++++++++-
 1 file changed, 59 insertions(+), 1 deletion(-)

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 624806f..52e1383 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -5,8 +5,8 @@ are used in optimization, and have some additional nice methods beyond
 what can be supported in a general Jordan Algebra.
 """
 
-from sage.structure.unique_representation import UniqueRepresentation
 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
+from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
 
 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
     @staticmethod
@@ -29,6 +29,64 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         """
         return "Euclidean Jordan algebra of degree {} over {}".format(self.degree(), self.base_ring())
 
+    def rank(self):
+        """
+        Return the rank of this EJA.
+        """
+        raise NotImplementedError
+
+
+    class Element(FiniteDimensionalAlgebraElement):
+        """
+        An element of a Euclidean Jordan algebra.
+        """
+
+        def __pow__(self, n):
+            """
+            Return ``self`` raised to the power ``n``.
+
+            Jordan algebras are always power-associative; see for
+            example Faraut and Koranyi, Proposition II.1.2 (ii).
+            """
+            A = self.parent()
+            if n == 0:
+                return A.one()
+            elif n == 1:
+                return self
+            else:
+                return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
+
+
+        def degree(self):
+            """
+            Compute the degree of this element the straightforward way
+            according to the definition; by appending powers to a list
+            and figuring out its dimension (that is, whether or not
+            they're linearly dependent).
+
+            EXAMPLES::
+
+                sage: J = eja_ln(4)
+                sage: J.one().degree()
+                1
+                sage: e0,e1,e2,e3 = J.gens()
+                sage: (e0 - e1).degree()
+                2
+
+            """
+            d = 0
+            V = self.vector().parent()
+            vectors = [(self**d).vector()]
+            while V.span(vectors).dimension() > d:
+                d += 1
+                vectors.append((self**d).vector())
+            return d
+
+        def minimal_polynomial(self):
+            return self.matrix().minimal_polynomial()
+
+        def characteristic_polynomial(self):
+            return self.matrix().characteristic_polynomial()
 
 
 def eja_rn(dimension, field=QQ):
-- 
2.44.2