From: Michael Orlitzky <michael@orlitzky.com>
Date: Tue, 9 Mar 2021 02:21:28 +0000 (-0500)
Subject: hurwitz: add ComplexMatrixAlgebra.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=917a76ab0ae48ddd52e7d24de6f51ca8f91aa3c1;p=sage.d.git

hurwitz: add ComplexMatrixAlgebra.
---

diff --git a/mjo/hurwitz.py b/mjo/hurwitz.py
index ff1792b..75e8568 100644
--- a/mjo/hurwitz.py
+++ b/mjo/hurwitz.py
@@ -1,11 +1,7 @@
 from sage.misc.cachefunc import cached_method
-from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
 from sage.combinat.free_module import CombinatorialFreeModule
 from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
-from sage.categories.magmatic_algebras import MagmaticAlgebras
-from sage.rings.all import AA, ZZ
-from sage.matrix.matrix_space import MatrixSpace
-from sage.misc.table import table
+from sage.rings.all import AA
 
 from mjo.matrix_algebra import MatrixAlgebra, MatrixAlgebraElement
 
@@ -34,6 +30,8 @@ class Octonion(IndexedFreeModuleElement):
             True
 
         """
+        from sage.rings.all import ZZ
+        from sage.matrix.matrix_space import MatrixSpace
         C = MatrixSpace(ZZ,8).diagonal_matrix((1,-1,-1,-1,-1,-1,-1,-1))
         return self.parent().from_vector(C*self.to_vector())
 
@@ -181,50 +179,6 @@ class Octonion(IndexedFreeModuleElement):
         return self.conjugate()/self._norm_squared()
 
 
-    def cayley_dickson(self, Q=None):
-        r"""
-        Return the Cayley-Dickson representation of this element in terms
-        of the quaternion algebra ``Q``.
-
-        The Cayley-Dickson representation is an identification of
-        octionions `x` and `y` with pairs of quaternions `(a,b)` and
-        `(c,d)` respectively such that:
-
-        * `x + y = (a+b, c+d)`
-        * `xy` = (ac - \bar{d}*b, da + b\bar{c})`
-        * `\bar{x} = (a,-b)`
-
-        where `\bar{x}` denotes the conjugate of `x`.
-
-        SETUP::
-
-            sage: from mjo.hurwitz import Octonions
-
-        EXAMPLES::
-
-            sage: O = Octonions()
-            sage: x = sum(O.gens())
-            sage: x.cayley_dickson()
-            (1 + i + j + k, 1 + i + j + k)
-
-        """
-        if Q is None:
-            Q = QuaternionAlgebra(self.base_ring(), -1, -1)
-
-        i,j,k = Q.gens()
-        a = (self.coefficient(0)*Q.one() +
-             self.coefficient(1)*i +
-             self.coefficient(2)*j +
-             self.coefficient(3)*k )
-        b = (self.coefficient(4)*Q.one() +
-             self.coefficient(5)*i +
-             self.coefficient(6)*j +
-             self.coefficient(7)*k )
-
-        from sage.categories.sets_cat import cartesian_product
-        P = cartesian_product([Q,Q])
-        return P((a,b))
-
 
 class Octonions(CombinatorialFreeModule):
     r"""
@@ -245,6 +199,7 @@ class Octonions(CombinatorialFreeModule):
                  prefix="e"):
 
         # Not associative, not commutative
+        from sage.categories.magmatic_algebras import MagmaticAlgebras
         category = MagmaticAlgebras(field).FiniteDimensional()
         category = category.WithBasis().Unital()
 
@@ -343,6 +298,7 @@ class Octonions(CombinatorialFreeModule):
                                     for j in range(n) ]
                for i in range(n) ]
 
+        from sage.misc.table import table
         return table(M, header_row=True, header_column=True, frame=True)
 
 
@@ -398,8 +354,8 @@ class HurwitzMatrixAlgebra(MatrixAlgebra):
 
     def entry_algebra_gens(self):
         r"""
-        Return the generators of (that is, a basis for) the entries of
-        this matrix algebra.
+        Return a tuple of 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.
@@ -565,5 +521,64 @@ class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
     """
     def __init__(self, n, scalars=AA, **kwargs):
         # The -1,-1 gives us the "usual" definition of quaternion
+        from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
         Q = QuaternionAlgebra(scalars,-1,-1)
         super().__init__(Q, scalars, n, **kwargs)
+
+
+class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
+    r"""
+    The algebra of ``n``-by-``n`` matrices with complex entries over
+    (a subfield of) the real numbers.
+
+    These differ from the usual complex matrix spaces in SageMath
+    because the scalar field is real (and not assumed to be the same
+    as the space from which the entries are drawn). The space of
+    `1`-by-`1` complex matrices will have dimension two, for example.
+
+    SETUP::
+
+        sage: from mjo.hurwitz import ComplexMatrixAlgebra
+
+    EXAMPLES::
+
+        sage: ComplexMatrixAlgebra(3)
+        Module of 3 by 3 matrices with entries in Algebraic Field
+        over the scalar ring Algebraic Real Field
+        sage: ComplexMatrixAlgebra(3,QQ)
+        Module of 3 by 3 matrices with entries in Algebraic Field
+        over the scalar ring Rational Field
+
+    ::
+
+        sage: A = ComplexMatrixAlgebra(2)
+        sage: (I,) = A.entry_algebra().gens()
+        sage: A([ [1+I, 1],
+        ....:     [-1, -I] ])
+        +-------+----+
+        | I + 1 | 1  |
+        +-------+----+
+        | -1    | -I |
+        +-------+----+
+
+    ::
+
+        sage: A1 = ComplexMatrixAlgebra(1,QQ)
+        sage: A2 = ComplexMatrixAlgebra(2,QQ)
+        sage: cartesian_product([A1,A2])
+        Module of 1 by 1 matrices with entries in Algebraic Field over
+        the scalar ring Rational Field (+) Module of 2 by 2 matrices with
+        entries in Algebraic Field over the scalar ring Rational Field
+
+    TESTS::
+
+        sage: set_random_seed()
+        sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
+        sage: x = A.random_element()
+        sage: x*A.one() == x and A.one()*x == x
+        True
+
+    """
+    def __init__(self, n, scalars=AA, **kwargs):
+        from sage.rings.all import QQbar
+        super().__init__(QQbar, scalars, n, **kwargs)