X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feuclidean_jordan_algebra.py;h=08bce5f7c7b24a96d9fa982bb1ec0c77dc01807a;hb=6db5d3bdea85cdc88c4dcdce1a75b888d1c4f1d8;hp=e0a6f2da12982932838c2dbef2ae0e3d0fe8a1b0;hpb=8b3542a01a689367ae4692bc2658a6864f1be53f;p=sage.d.git

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index e0a6f2d..08bce5f 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -5,7 +5,89 @@ are used in optimization, and have some additional nice methods beyond
 what can be supported in a general Jordan Algebra.
 """
 
-from sage.all import *
+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
+    def __classcall__(cls, field, mult_table, names='e', category=None):
+        fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
+        return fda.__classcall_private__(cls,
+                                         field,
+                                         mult_table,
+                                         names,
+                                         category)
+
+    def __init__(self, field, mult_table, names='e', category=None):
+        fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
+        fda.__init__(field, mult_table, names, category)
+
+
+    def _repr_(self):
+        """
+        Return a string representation of ``self``.
+        """
+        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):
     """
@@ -39,5 +121,55 @@ def eja_rn(dimension, field=QQ):
     # component of x; and likewise for the ith basis element e_i.
     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
            for i in xrange(dimension) ]
-    A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
-    return JordanAlgebra(A)
+
+    return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
+
+
+def eja_ln(dimension, field=QQ):
+    """
+    Return the Jordan algebra corresponding to the Lorentz "ice cream"
+    cone of the given ``dimension``.
+
+    EXAMPLES:
+
+    This multiplication table can be verified by hand::
+
+        sage: J = eja_ln(4)
+        sage: e0,e1,e2,e3 = J.gens()
+        sage: e0*e0
+        e0
+        sage: e0*e1
+        e1
+        sage: e0*e2
+        e2
+        sage: e0*e3
+        e3
+        sage: e1*e2
+        0
+        sage: e1*e3
+        0
+        sage: e2*e3
+        0
+
+    In one dimension, this is the reals under multiplication::
+
+      sage: J1 = eja_ln(1)
+      sage: J2 = eja_rn(1)
+      sage: J1 == J2
+      True
+
+    """
+    Qs = []
+    id_matrix = identity_matrix(field,dimension)
+    for i in xrange(dimension):
+        ei = id_matrix.column(i)
+        Qi = zero_matrix(field,dimension)
+        Qi.set_row(0, ei)
+        Qi.set_column(0, ei)
+        Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
+        # The addition of the diagonal matrix adds an extra ei[0] in the
+        # upper-left corner of the matrix.
+        Qi[0,0] = Qi[0,0] * ~field(2)
+        Qs.append(Qi)
+
+    return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)