X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_algebra.py;h=6b5b090fbefc4a51bfbedaf5a64d997ec7cfad64;hb=d6c744ecba0a22fdd76cb17e663594d323d1bb38;hp=6750f228451b5ec2052efc3974ddee72cd0d0c12;hpb=02cbbda68c8efd79b3c9735d5eca7e7162b24840;p=sage.d.git

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 6750f22..6b5b090 100644
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -36,17 +36,21 @@ for these simple algebras:
 
 In addition to these, we provide two other example constructions,
 
+  * :class:`JordanSpinEJA`
   * :class:`HadamardEJA`
+  * :class:`AlbertEJA`
   * :class:`TrivialEJA`
 
 The Jordan spin algebra is a bilinear form algebra where the bilinear
 form is the identity. The Hadamard EJA is simply a Cartesian product
-of one-dimensional spin algebras. And last but least, the trivial EJA
-is exactly what you think it is; it could also be obtained by
-constructing a dimension-zero instance of any of the other
-algebras. Cartesian products of these are also supported using the
-usual ``cartesian_product()`` function; as a result, we support (up to
-isomorphism) all Euclidean Jordan algebras.
+of one-dimensional spin algebras. The Albert EJA is simply a special
+case of the :class:`OctonionHermitianEJA` where the matrices are
+three-by-three and the resulting space has dimension 27. And
+last/least, the trivial EJA is exactly what you think it is; it could
+also be obtained by constructing a dimension-zero instance of any of
+the other algebras. Cartesian products of these are also supported
+using the usual ``cartesian_product()`` function; as a result, we
+support (up to isomorphism) all Euclidean Jordan algebras.
 
 SETUP::
 
@@ -58,8 +62,6 @@ EXAMPLES::
     Euclidean Jordan algebra of dimension...
 """
 
-from itertools import repeat
-
 from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 from sage.categories.magmatic_algebras import MagmaticAlgebras
 from sage.categories.sets_cat import cartesian_product
@@ -1573,7 +1575,11 @@ class RationalBasisEJA(FiniteDimensionalEJA):
         if check_field:
             # Abuse the check_field parameter to check that the entries of
             # out basis (in ambient coordinates) are in the field QQ.
-            if not all( all(b_i in QQ for b_i in b.list()) for b in basis ):
+            # Use _all2list to get the vector coordinates of octonion
+            # entries and not the octonions themselves (which are not
+            # rational).
+            if not all( all(b_i in QQ for b_i in _all2list(b))
+                        for b in basis ):
                 raise TypeError("basis not rational")
 
         super().__init__(basis,
@@ -1830,7 +1836,7 @@ class RealEmbeddedMatrixEJA(MatrixEJA):
         # as a REAL matrix will be 2*a = 2*Re(z_1). And so forth.
         return (X*Y).trace()/cls.dimension_over_reals()
 
-class RealSymmetricEJA(ConcreteEJA, RationalBasisEJA, MatrixEJA):
+class RealSymmetricEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
     """
     The rank-n simple EJA consisting of real symmetric n-by-n
     matrices, the usual symmetric Jordan product, and the trace inner
@@ -2112,7 +2118,7 @@ class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
         return matrix(F, n/d, elements)
 
 
-class ComplexHermitianEJA(ConcreteEJA, RationalBasisEJA, ComplexMatrixEJA):
+class ComplexHermitianEJA(RationalBasisEJA, ConcreteEJA, ComplexMatrixEJA):
     """
     The rank-n simple EJA consisting of complex Hermitian n-by-n
     matrices over the real numbers, the usual symmetric Jordan product,
@@ -2414,8 +2420,8 @@ class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
         return matrix(Q, n/d, elements)
 
 
-class QuaternionHermitianEJA(ConcreteEJA,
-                             RationalBasisEJA,
+class QuaternionHermitianEJA(RationalBasisEJA,
+                             ConcreteEJA,
                              QuaternionMatrixEJA):
     r"""
     The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
@@ -2584,7 +2590,7 @@ class QuaternionHermitianEJA(ConcreteEJA,
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
-class OctonionHermitianEJA(ConcreteEJA, MatrixEJA):
+class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
     r"""
     SETUP::
 
@@ -2770,7 +2776,29 @@ class OctonionHermitianEJA(ConcreteEJA, MatrixEJA):
         """
         return (X*Y).trace().real().coefficient(0)
 
-class HadamardEJA(ConcreteEJA, RationalBasisEJA):
+
+class AlbertEJA(OctonionHermitianEJA):
+    r"""
+    The Albert algebra is the algebra of three-by-three Hermitian
+    matrices whose entries are octonions.
+
+    SETUP::
+
+        sage: from mjo.eja.eja_algebra import AlbertEJA
+
+    EXAMPLES::
+
+        sage: AlbertEJA(field=QQ, orthonormalize=False)
+        Euclidean Jordan algebra of dimension 27 over Rational Field
+        sage: AlbertEJA() # long time
+        Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
+
+    """
+    def __init__(self, *args, **kwargs):
+        super().__init__(3, *args, **kwargs)
+
+
+class HadamardEJA(RationalBasisEJA, ConcreteEJA):
     """
     Return the Euclidean Jordan Algebra corresponding to the set
     `R^n` under the Hadamard product.
@@ -2862,7 +2890,7 @@ class HadamardEJA(ConcreteEJA, RationalBasisEJA):
         return cls(n, **kwargs)
 
 
-class BilinearFormEJA(ConcreteEJA, RationalBasisEJA):
+class BilinearFormEJA(RationalBasisEJA, ConcreteEJA):
     r"""
     The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
     with the half-trace inner product and jordan product ``x*y =
@@ -3110,7 +3138,7 @@ class JordanSpinEJA(BilinearFormEJA):
         return cls(n, **kwargs)
 
 
-class TrivialEJA(ConcreteEJA, RationalBasisEJA):
+class TrivialEJA(RationalBasisEJA, ConcreteEJA):
     """
     The trivial Euclidean Jordan algebra consisting of only a zero element.