From: Michael Orlitzky <michael@orlitzky.com>
Date: Thu, 18 Jul 2019 23:44:45 +0000 (-0400)
Subject: eja: finally give Euclidean Jordan algebras an inner product.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=7668f7ca04051e748cc072e6cf3e637e0b4c8861;p=sage.d.git

eja: finally give Euclidean Jordan algebras an inner product.
---

diff --git a/mjo/eja/euclidean_jordan_algebra.py b/mjo/eja/euclidean_jordan_algebra.py
index 8d9b27e..7c1f249 100644
--- a/mjo/eja/euclidean_jordan_algebra.py
+++ b/mjo/eja/euclidean_jordan_algebra.py
@@ -21,7 +21,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                               assume_associative=False,
                               category=None,
                               rank=None,
-                              natural_basis=None):
+                              natural_basis=None,
+                              inner_product=None):
         n = len(mult_table)
         mult_table = [b.base_extend(field) for b in mult_table]
         for b in mult_table:
@@ -45,7 +46,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                                  names=names,
                                  category=cat,
                                  rank=rank,
-                                 natural_basis=natural_basis)
+                                 natural_basis=natural_basis,
+                                 inner_product=inner_product)
 
 
     def __init__(self, field,
@@ -54,7 +56,8 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                  assume_associative=False,
                  category=None,
                  rank=None,
-                 natural_basis=None):
+                 natural_basis=None,
+                 inner_product=None):
         """
         EXAMPLES:
 
@@ -70,6 +73,7 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         """
         self._rank = rank
         self._natural_basis = natural_basis
+        self._inner_product = inner_product
         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
         fda.__init__(field,
                      mult_table,
@@ -85,6 +89,20 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
         return fmt.format(self.degree(), self.base_ring())
 
 
+    def inner_product(self, x, y):
+        """
+        The inner product associated with this Euclidean Jordan algebra.
+
+        Will default to the trace inner product if nothing else.
+        """
+        if (not x in self) or (not y in self):
+            raise ArgumentError("arguments must live in this algebra")
+        if self._inner_product is None:
+            return x.trace_inner_product(y)
+        else:
+            return self._inner_product(x,y)
+
+
     def natural_basis(self):
         """
         Return a more-natural representation of this algebra's basis.
@@ -208,6 +226,44 @@ class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
                 raise NotImplementedError('irregular element')
 
 
+        def inner_product(self, other):
+            """
+            Return the parent algebra's inner product of myself and ``other``.
+
+            EXAMPLES:
+
+            The inner product in the Jordan spin algebra is the usual
+            inner product on `R^n` (this example only works because the
+            basis for the Jordan algebra is the standard basis in `R^n`)::
+
+                sage: J = JordanSpinSimpleEJA(3)
+                sage: x = vector(QQ,[1,2,3])
+                sage: y = vector(QQ,[4,5,6])
+                sage: x.inner_product(y)
+                32
+                sage: J(x).inner_product(J(y))
+                32
+
+            TESTS:
+
+            Ensure that we can always compute an inner product, and that
+            it gives us back a real number::
+
+                sage: set_random_seed()
+                sage: J = random_eja()
+                sage: x = J.random_element()
+                sage: y = J.random_element()
+                sage: x.inner_product(y) in RR
+                True
+
+            """
+            P = self.parent()
+            if not other in P:
+                raise ArgumentError("'other' must live in the same algebra")
+
+            return P.inner_product(self, other)
+
+
         def operator_commutes_with(self, other):
             """
             Return whether or not this element operator-commutes
@@ -885,7 +941,13 @@ def eja_rn(dimension, field=QQ):
     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
            for i in xrange(dimension) ]
 
-    return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
+    # The usual inner product on R^n.
+    ip = lambda x, y: x.vector().inner_product(y.vector())
+
+    return FiniteDimensionalEuclideanJordanAlgebra(field,
+                                                   Qs,
+                                                   rank=dimension,
+                                                   inner_product=ip)
 
 
 
@@ -1215,13 +1277,6 @@ def JordanSpinSimpleEJA(n, field=QQ):
         sage: e2*e3
         0
 
-    In one dimension, this is the reals under multiplication::
-
-      sage: J1 = JordanSpinSimpleEJA(1)
-      sage: J2 = eja_rn(1)
-      sage: J1 == J2
-      True
-
     """
     Qs = []
     id_matrix = identity_matrix(field, n)
@@ -1236,7 +1291,13 @@ def JordanSpinSimpleEJA(n, field=QQ):
         Qi[0,0] = Qi[0,0] * ~field(2)
         Qs.append(Qi)
 
+    # The usual inner product on R^n.
+    ip = lambda x, y: x.vector().inner_product(y.vector())
+
     # The rank of the spin factor algebra is two, UNLESS we're in a
     # one-dimensional ambient space (the rank is bounded by the
     # ambient dimension).
-    return FiniteDimensionalEuclideanJordanAlgebra(field, Qs, rank=min(n,2))
+    return FiniteDimensionalEuclideanJordanAlgebra(field,
+                                                   Qs,
+                                                   rank=min(n,2),
+                                                   inner_product=ip)