eja: change OctonionMatrixAlgebra paramater field -> scalars.

[sage.d.git] / mjo / eja / eja_algebra.py

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index 4d0c802c38c8320a8f10f8faeb50678a85d84e95..d7010bcfd63ff838c0219dc7208a9b8e1988c889 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -1725,6 +1725,21 @@ class ConcreteEJA(RationalBasisEJA):
 
 
 class MatrixEJA:
+    @staticmethod
+    def jordan_product(X,Y):
+        return (X*Y + Y*X)/2
+
+    @staticmethod
+    def trace_inner_product(X,Y):
+        r"""
+        A trace inner-product for matrices that aren't embedded in the
+        reals.
+        """
+        # 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()
+
+class RealEmbeddedMatrixEJA(MatrixEJA):
     @staticmethod
     def dimension_over_reals():
         r"""
@@ -1770,9 +1785,6 @@ class MatrixEJA:
             raise ValueError("the matrix 'M' must be a real embedding")
         return M
 
-    @staticmethod
-    def jordan_product(X,Y):
-        return (X*Y + Y*X)/2
 
     @classmethod
     def trace_inner_product(cls,X,Y):
@@ -1781,29 +1793,11 @@ class MatrixEJA:
 
         SETUP::
 
-            sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
-            ....:                                  ComplexHermitianEJA,
+            sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
             ....:                                  QuaternionHermitianEJA)
 
         EXAMPLES::
 
-        This gives the same answer as it would if we computed the trace
-        from the unembedded (original) matrices::
-
-            sage: set_random_seed()
-            sage: J = RealSymmetricEJA.random_instance()
-            sage: x,y = J.random_elements(2)
-            sage: Xe = x.to_matrix()
-            sage: Ye = y.to_matrix()
-            sage: X = J.real_unembed(Xe)
-            sage: Y = J.real_unembed(Ye)
-            sage: expected = (X*Y).trace()
-            sage: actual = J.trace_inner_product(Xe,Ye)
-            sage: actual == expected
-            True
-
-        ::
-
             sage: set_random_seed()
             sage: J = ComplexHermitianEJA.random_instance()
             sage: x,y = J.random_elements(2)
@@ -1839,14 +1833,7 @@ class 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 RealMatrixEJA(MatrixEJA):
-    @staticmethod
-    def dimension_over_reals():
-        return 1
-
-
-class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
+class RealSymmetricEJA(ConcreteEJA, MatrixEJA):
     """
     The rank-n simple EJA consisting of real symmetric n-by-n
     matrices, the usual symmetric Jordan product, and the trace inner
@@ -1975,12 +1962,12 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
         # because the MatrixEJA is not presently a subclass of the
         # FDEJA class that defines rank() and one().
         self.rank.set_cache(n)
-        idV = matrix.identity(ZZ, self.dimension_over_reals()*n)
+        idV = self.matrix_space().one()
         self.one.set_cache(self(idV))
 
 
 
-class ComplexMatrixEJA(MatrixEJA):
+class ComplexMatrixEJA(RealEmbeddedMatrixEJA):
     # A manual dictionary-cache for the complex_extension() method,
     # since apparently @classmethods can't also be @cached_methods.
     _complex_extension = {}
@@ -2282,7 +2269,7 @@ class ComplexHermitianEJA(ConcreteEJA, ComplexMatrixEJA):
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
-class QuaternionMatrixEJA(MatrixEJA):
+class QuaternionMatrixEJA(RealEmbeddedMatrixEJA):
 
     # A manual dictionary-cache for the quaternion_extension() method,
     # since apparently @classmethods can't also be @cached_methods.
@@ -2598,6 +2585,67 @@ class QuaternionHermitianEJA(ConcreteEJA, QuaternionMatrixEJA):
         n = ZZ.random_element(cls._max_random_instance_size() + 1)
         return cls(n, **kwargs)
 
+class OctonionHermitianEJA(FiniteDimensionalEJA, MatrixEJA):
+
+    def __init__(self, n, field=AA, **kwargs):
+        if n > 3:
+            # Otherwise we don't get an EJA.
+            raise ValueError("n cannot exceed 3")
+
+        # We know this is a valid EJA, but will double-check
+        # if the user passes check_axioms=True.
+        if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+        super().__init__(self._denormalized_basis(n,field),
+                         self.jordan_product,
+                         self.trace_inner_product,
+                         **kwargs)
+
+        # TODO: this could be factored out somehow, but is left here
+        # because the MatrixEJA is not presently a subclass of the
+        # FDEJA class that defines rank() and one().
+        self.rank.set_cache(n)
+        idV = self.matrix_space().one()
+        self.one.set_cache(self(idV))
+
+
+    @classmethod
+    def _denormalized_basis(cls, n, field):
+        """
+        Returns a basis for the space of octonion Hermitian n-by-n
+        matrices.
+
+        SETUP::
+
+            sage: from mjo.eja.eja_algebra import OctonionHermitianEJA
+
+        EXAMPLES::
+
+            sage: B = OctonionHermitianEJA._denormalized_basis(3)
+            sage: all( M.is_hermitian() for M in B )
+            True
+            sage: len(B)
+            27
+
+        """
+        from mjo.octonions import OctonionMatrixAlgebra
+        MS = OctonionMatrixAlgebra(n, scalars=field)
+        es = MS.entry_algebra().gens()
+
+        basis = []
+        for i in range(n):
+            for j in range(i+1):
+                if i == j:
+                    E_ii = MS.monomial( (i,j,es[0]) )
+                    basis.append(E_ii)
+                else:
+                    for e in es:
+                        E_ij  = MS.monomial( (i,j,e)             )
+                        E_ij += MS.monomial( (j,i,e.conjugate()) )
+                        basis.append(E_ij)
+
+        return tuple( basis )
+
 
 class HadamardEJA(ConcreteEJA):
     """