eja: fix missing test import.

[sage.d.git] / mjo / eja / DESIGN

diff --git a/mjo/eja/DESIGN b/mjo/eja/DESIGN
index 467e2ae027d2a20af5576b292576edf21a71b10b..75302ca0f9ba4323ac634834f4be816aa8866010 100644 (file)
--- a/mjo/eja/DESIGN
+++ b/mjo/eja/DESIGN
@@ -48,14 +48,46 @@ the matrix algebras?
    subalgebras B and C of A, we can use to_matrix() to go from, say,
    C -> A -> B rather than having to convert from C to B directly.
 
    subalgebras B and C of A, we can use to_matrix() to go from, say,
    C -> A -> B rather than having to convert from C to B directly.
 

-3. When constructing a Cartesian product algebra, we don't know a
-   priori whether or not the result will have matrix-algebra
-   factors. We can figure it out at runtime, but it would be nice if
-   DirectSumEJA always returned the same class. Maybe more
-   importantly, if a Cartesian product has one matrix and one
-   non-matrix factor, then what would its own matrix representation
-   look like? We want to delegate to the factors...

 
 

+Fielding questions
+------------------
+
+All Euclidean Jordan algebras are over the real scalar field. This
+presents a problem: in SageMath, the matrix and vector classes don't
+support scalar fields that are different than their entries. And three
+of the simple families of Euclidean Jordan algebras are matrices with
+non-real entries: the Hermitian comples, quaternion, and octonion
+algebras.
+
+At least in the complex and quaternion case, we can "embed" the
+complex numbers and quaternions into the space of 2-by-2 or 4-by-4
+matrices. But the octonions are not associative, so they can't be
+embedded (via a homomorphism) into any real matrix space. So what
+do we do? Write it ourselves, obviously.
+
+The octonion matrix algebra is implemented separately, as a subclass
+of CombinatorialFreeModule, to work around that issue. The custom
+class supports a scalar field that is different than the entries of
+the matrices. However, this means that we actually have FOUR
+different types of "matrices" to support:
+
+  (1) Sage vectors
+  (2) Sage matrices
+  (3) Our custom matrices
+  (4) Cartesian products of the (1) through (3)
+
+All other Euclidean Jordan algebras could of course be implemented in
+the same way as the octonion algebra, but for the sake of the user
+interface, we must also support at least the usual SageMath vectors
+and matrices.
+
+Note: this has one less-than-obvious consequence: we have to assume
+that the user has supplied an entirely-correct basis (with entries in
+the correct structure). We generally cannot mess witht the entries of
+his basis, or use them to figure out what (for example) the ambient
+scalar ring is. None of these are insurmountable obstacles; we just
+have to be a little careful distinguishing between what's inside the
+algebra elements and what's outside them.

 
 Basis normalization
 -------------------
 
 Basis normalization
 -------------------


@@ -67,3 +99,12 @@ process. Undoing the change-of-coordinates allows us to perform some
 computations in the original basis (like the "characteristic
 polynomial of"), which can be much faster when the original basis
 contains only rational entries.
 computations in the original basis (like the "characteristic
 polynomial of"), which can be much faster when the original basis
 contains only rational entries.

+
+Debugging
+---------
+There are several places in the code where we set check_field=False
+and check_axioms=False because the theory guarantees that they will be
+satisfied. Well, you know what they say about theory and practice. The
+first thing you should do when a problem is discovered it replace all
+of those with check_field=True and check_axioms=True, and then re-run
+the test suite.