X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=afe0a677aaafd9ddf67355ed979d5ef023beb9c0;hb=0d7746bc8dbe22bd5ce4ece76354e34454eda5d2;hp=e16fd97c4f0e55591728dc8204da3d941529380c;hpb=28d86108d78f1ea5a93d9cc8d69bad9cc8cd74fd;p=sage.d.git

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index e16fd97..afe0a67 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -1737,7 +1737,7 @@ class MatrixEJA:
         """
         # We take the norm (absolute value) because Octonions() isn't
         # smart enough yet to coerce its one() into the base field.
-        return (X*Y).trace().abs()
+        return (X*Y).trace().real().abs()
 
 class RealEmbeddedMatrixEJA(MatrixEJA):
     @staticmethod
@@ -2586,7 +2586,89 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
         return cls(n, **kwargs)
 
 class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
+    r"""
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import (FiniteDimensionalEJA,
+        ....:                                  OctonionHermitianEJA)
 
+    EXAMPLES:
+
+    The 3-by-3 algebra satisfies the axioms of an EJA::
+
+        sage: OctonionHermitianEJA(3,                    # long time
+        ....:                      field=QQ,             # long time
+        ....:                      orthonormalize=False, # long time
+        ....:                      check_axioms=True)    # long time
+        Euclidean Jordan algebra of dimension 27 over Rational Field
+
+    After a change-of-basis, the 2-by-2 algebra has the same
+    multiplication table as the ten-dimensional Jordan spin algebra::
+
+        sage: b = OctonionHermitianEJA._denormalized_basis(2,QQ)
+        sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
+        sage: jp = OctonionHermitianEJA.jordan_product
+        sage: ip = OctonionHermitianEJA.trace_inner_product
+        sage: J = FiniteDimensionalEJA(basis,
+        ....:                          jp,
+        ....:                          ip,
+        ....:                          field=QQ,
+        ....:                          orthonormalize=False)
+        sage: J.multiplication_table()
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | *  || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+        +====++====+====+====+====+====+====+====+====+====+====+
+        | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b1 || b1 | b0 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b2 || b2 | 0  | b0 | 0  | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b3 || b3 | 0  | 0  | b0 | 0  | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b4 || b4 | 0  | 0  | 0  | b0 | 0  | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b5 || b5 | 0  | 0  | 0  | 0  | b0 | 0  | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b6 || b6 | 0  | 0  | 0  | 0  | 0  | b0 | 0  | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b7 || b7 | 0  | 0  | 0  | 0  | 0  | 0  | b0 | 0  | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b8 || b8 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | b0 | 0  |
+        +----++----+----+----+----+----+----+----+----+----+----+
+        | b9 || b9 | 0  | 0  | 0  | 0  | 0  | 0  | 0  | 0  | b0 |
+        +----++----+----+----+----+----+----+----+----+----+----+
+
+    TESTS:
+
+    We can actually construct the 27-dimensional Albert algebra,
+    and we get the right unit element if we recompute it::
+
+        sage: J = OctonionHermitianEJA(3,                    # long time
+        ....:                          field=QQ,             # long time
+        ....:                          orthonormalize=False) # long time
+        sage: J.one.clear_cache()                            # long time
+        sage: J.one()                                        # long time
+        b0 + b9 + b26
+        sage: J.one().to_matrix()                            # long time
+        +----+----+----+
+        | e0 | 0  | 0  |
+        +----+----+----+
+        | 0  | e0 | 0  |
+        +----+----+----+
+        | 0  | 0  | e0 |
+        +----+----+----+
+
+    The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
+    spin algebra, but just to be sure, we recompute its rank::
+
+        sage: J = OctonionHermitianEJA(2,                    # long time
+        ....:                          field=QQ,             # long time
+        ....:                          orthonormalize=False) # long time
+        sage: J.rank.clear_cache()                           # long time
+        sage: J.rank()                                       # long time
+        2
+    """
     def __init__(self, n, field=AA, **kwargs):
         if n > 3:
             # Otherwise we don't get an EJA.
@@ -2599,6 +2681,7 @@ class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
         super().__init__(self._denormalized_basis(n,field),
                          self.jordan_product,
                          self.trace_inner_product,
+                         field=field,
                          **kwargs)
 
         # TODO: this could be factored out somehow, but is left here
@@ -2621,7 +2704,7 @@ class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
 
         EXAMPLES::
 
-            sage: B = OctonionHermitianEJA._denormalized_basis(3)
+            sage: B = OctonionHermitianEJA._denormalized_basis(3,QQ)
             sage: all( M.is_hermitian() for M in B )
             True
             sage: len(B)
@@ -2629,7 +2712,7 @@ class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
 
         """
         from mjo.octonions import OctonionMatrixAlgebra
-        MS = OctonionMatrixAlgebra(n, field=field)
+        MS = OctonionMatrixAlgebra(n, scalars=field)
         es = MS.entry_algebra().gens()
 
         basis = []
@@ -2641,7 +2724,13 @@ class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
                 else:
                     for e in es:
                         E_ij  = MS.monomial( (i,j,e)             )
-                        E_ij += MS.monomial( (j,i,e.conjugate()) )
+                        ec = e.conjugate()
+                        # If the conjugate has a negative sign in front
+                        # of it, (j,i,ec) won't be a monomial!
+                        if (j,i,ec) in MS.indices():
+                            E_ij += MS.monomial( (j,i,ec) )
+                        else:
+                            E_ij -= MS.monomial( (j,i,-ec) )
                         basis.append(E_ij)
 
         return tuple( basis )