Reorganize the Hurwitz (matrix) algebra stuff.

diff --git a/mjo/eja/TODO b/mjo/eja/TODO
index 17422851f81f5196cd47cc159004c326a848b115..310e3073c667fb42b7591c0b8a989f590455e4fa 100644 (file)
--- a/mjo/eja/TODO
+++ b/mjo/eja/TODO
@@ -27,8 +27,7 @@ sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6
 6. Can we convert the complex/quaternion algebras to avoid real-
    (un)embeddings? Quaternions would need their own
    QuaternionMatrixAlgebra, since Sage matrices have to have entries
-   in a commutative ring. Those and the octonion stuff could be moved
-   to hurwitz.py along with the HurwitzMatrixAlgebra.
+   in a commutative ring.
 
 7. Every once in a long while, the test
 

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index c86cdff0e4ab056292ac248289aee6a876332739..2ecf8cd274bbee0be8c31e25c8a35a6e9561123a 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -2732,7 +2732,7 @@ class OctonionHermitianEJA(RationalBasisEJA, ConcreteEJA, MatrixEJA):
             27
 
         """
-        from mjo.octonions import OctonionMatrixAlgebra
+        from mjo.hurwitz import OctonionMatrixAlgebra
         MS = OctonionMatrixAlgebra(n, scalars=field)
         es = MS.entry_algebra().gens()
 

diff --git a/mjo/octonions.py b/mjo/hurwitz.py
similarity index 78%
rename from mjo/octonions.py
rename to mjo/hurwitz.py
index 78365e298ec7a46fcaff0603039f6410aa768c96..ff1792bd9f3dc26d8b337fe3a6ff1549ab1f68e2 100644 (file)
--- a/mjo/octonions.py
+++ b/mjo/hurwitz.py
@@ -7,14 +7,14 @@ from sage.rings.all import AA, ZZ
 from sage.matrix.matrix_space import MatrixSpace
 from sage.misc.table import table
 
-from mjo.matrix_algebra import HurwitzMatrixAlgebra
+from mjo.matrix_algebra import MatrixAlgebra, MatrixAlgebraElement
 
 class Octonion(IndexedFreeModuleElement):
     def conjugate(self):
         r"""
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -47,7 +47,7 @@ class Octonion(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -80,7 +80,7 @@ class Octonion(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -111,7 +111,7 @@ class Octonion(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -152,7 +152,7 @@ class Octonion(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -198,7 +198,7 @@ class Octonion(IndexedFreeModuleElement):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES::
 
@@ -230,7 +230,7 @@ class Octonions(CombinatorialFreeModule):
     r"""
     SETUP::
 
-        sage: from mjo.octonions import Octonions
+        sage: from mjo.hurwitz import Octonions
 
     EXAMPLES::
 
@@ -280,7 +280,7 @@ class Octonions(CombinatorialFreeModule):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         TESTS:
 
@@ -305,7 +305,7 @@ class Octonions(CombinatorialFreeModule):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
 
         EXAMPLES:
 
@@ -347,6 +347,106 @@ class Octonions(CombinatorialFreeModule):
 
 
 
+
+
+class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+    def is_hermitian(self):
+        r"""
+
+        SETUP::
+
+            sage: from mjo.hurwitz import HurwitzMatrixAlgebra
+
+        EXAMPLES::
+
+            sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2)
+            sage: M = A([ [ 0,I],
+            ....:         [-I,0] ])
+            sage: M.is_hermitian()
+            True
+
+        """
+        return all( self[i,j] == self[j,i].conjugate()
+                    for i in range(self.nrows())
+                    for j in range(self.ncols()) )
+
+
+class HurwitzMatrixAlgebra(MatrixAlgebra):
+    r"""
+    A class of matrix algebras whose entries come from a Hurwitz
+    algebra.
+
+    For our purposes, we consider "a Hurwitz" algebra to be the real
+    or complex numbers, the quaternions, or the octonions. These are
+    typically also referred to as the Euclidean Hurwitz algebras, or
+    the normed division algebras.
+
+    By the Cayley-Dickson construction, each Hurwitz algebra is an
+    algebra over the real numbers, so we restrict the scalar field in
+    this case to be real. This also allows us to more accurately
+    produce the generators of the matrix algebra.
+    """
+    Element = HurwitzMatrixAlgebraElement
+
+    def __init__(self, entry_algebra, scalars, n, **kwargs):
+        from sage.rings.all import RR
+        if not scalars.is_subring(RR):
+            # Not perfect, but it's what we're using.
+            raise ValueError("scalar field is not real")
+
+        super().__init__(entry_algebra, scalars, n, **kwargs)
+
+    def entry_algebra_gens(self):
+        r"""
+        Return the generators of (that is, a basis for) the entries of
+        this matrix algebra.
+
+        This works around the inconsistency in the ``gens()`` methods
+        of the real/complex numbers, quaternions, and octonions.
+
+        SETUP::
+
+            sage: from mjo.hurwitz import Octonions, HurwitzMatrixAlgebra
+
+        EXAMPLES:
+
+        The inclusion of the unit element is inconsistent across
+        (subalgebras of) Hurwitz algebras::
+
+            sage: AA.gens()
+            (1,)
+            sage: QQbar.gens()
+            (I,)
+            sage: QuaternionAlgebra(AA,1,-1).gens()
+            [i, j, k]
+            sage: Octonions().gens()
+            (e0, e1, e2, e3, e4, e5, e6, e7)
+
+        The unit element is always returned by this method, so the
+        sets of generators have cartinality 1,2,4, and 8 as you'd
+        expect::
+
+            sage: HurwitzMatrixAlgebra(AA, AA, 2).entry_algebra_gens()
+            (1,)
+            sage: HurwitzMatrixAlgebra(QQbar, AA, 2).entry_algebra_gens()
+            (1, I)
+            sage: Q = QuaternionAlgebra(AA,-1,-1)
+            sage: HurwitzMatrixAlgebra(Q, AA, 2).entry_algebra_gens()
+            (1, i, j, k)
+            sage: O = Octonions()
+            sage: HurwitzMatrixAlgebra(O, AA, 2).entry_algebra_gens()
+            (e0, e1, e2, e3, e4, e5, e6, e7)
+
+        """
+        gs = self.entry_algebra().gens()
+        one = self.entry_algebra().one()
+        if one in gs:
+            return gs
+        else:
+            return (one,) + tuple(gs)
+
+
+
 class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
     r"""
     The algebra of ``n``-by-``n`` matrices with octonion entries over
@@ -358,7 +458,7 @@ class OctonionMatrixAlgebra(HurwitzMatrixAlgebra):
 
     SETUP::
 
-        sage: from mjo.octonions import OctonionMatrixAlgebra
+        sage: from mjo.hurwitz import OctonionMatrixAlgebra
 
     EXAMPLES::
 
@@ -418,7 +518,7 @@ class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
 
     SETUP::
 
-        sage: from mjo.octonions import QuaternionMatrixAlgebra
+        sage: from mjo.hurwitz import QuaternionMatrixAlgebra
 
     EXAMPLES::
 

diff --git a/mjo/matrix_algebra.py b/mjo/matrix_algebra.py
index a67a9b4a1c0692d31aceffa3097343aa0df99dae..94a8410f9929986a8737f9bbb846c04490751fe5 100644 (file)
--- a/mjo/matrix_algebra.py
+++ b/mjo/matrix_algebra.py
@@ -236,7 +236,7 @@ class MatrixAlgebra(CombinatorialFreeModule):
 
         SETUP::
 
-            sage: from mjo.octonions import Octonions
+            sage: from mjo.hurwitz import Octonions
             sage: from mjo.matrix_algebra import MatrixAlgebra
 
         TESTS::
@@ -322,29 +322,3 @@ class MatrixAlgebra(CombinatorialFreeModule):
             return self
         else:
             return self.from_list(elt)
-
-
-class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
-    def is_hermitian(self):
-        r"""
-
-        SETUP::
-
-            sage: from mjo.matrix_algebra import HurwitzMatrixAlgebra
-
-        EXAMPLES::
-
-            sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2)
-            sage: M = A([ [ 0,I],
-            ....:         [-I,0] ])
-            sage: M.is_hermitian()
-            True
-
-        """
-        return all( self[i,j] == self[j,i].conjugate()
-                    for i in range(self.nrows())
-                    for j in range(self.ncols()) )
-
-
-class HurwitzMatrixAlgebra(MatrixAlgebra):
-    Element = HurwitzMatrixAlgebraElement